Some clarifications on the Inhuman

Since I posted the summary of the Labor of the Inhuman (LoIH) yesterday, I have had various conversation with friends. I thought it would be best to at least provide a few brief correcting remarks. I think I was clear at the beginning of the previous post, that I no longer endorse the full scope of the original essay as well as its recap. There are many problems in that essay that need to be resolved, even though I’m still committed to its core theses. Allow me to address a couple of problems which are more significant and serious.

In LoIH, there is too much emphasis on the Brandomian conceptions of normativity and reason as rooted in special kinds of social practices. The problem is that Brandom’s idea of sociality or social practices, at times, confounds substantive sociality and sociality as a formal condition. The ramifications of this confusion are not at all philosophically or politically pleasant. The whole notion of sociality should be treated with utmost caution otherwise the descent into regular humanism is inevitable. Brandom elides in many occasions the substantive sociality and sociality as a formal condition of reasoning. The first consequence of such an elision shows up in the the realm of politics and political consciousness. I strongly believe that the reason Brandom is a quietist liberal is because of this confusion. Reason is wrongly understood as a sufficient tool for political change, rather than merely necessary. Just because we are reason-giving and taking animals, it does not mean we either have fathomed what reason is or our reason is a sufficient tool for political change. One should not be surprised that Brandom’s rationalism coincides with a Habermasian soap-opera of a rational society in which all we need is more rational discourse or communication. Having said that, I believe Brandom has a far more sophisticated idea of reason and reasoning than Habermas, perhaps even unbeknownst to him.

Both Peter Wolfendale and I agree that sociality of reason should be investigated as a formal condition, and as such it would be more accurate to model it on computational processes and dynamic or truly concurrent information processing systems (the interactionist paradigm of computation, complexity sciences, hierarchies of types of computation, etc.). Wolfendale has already written about this point, here. Per Wolfendale’s thesis, we can even go so far as envisioning an artificial agent who has an internalized model of sociality as a formal—or more precisely, computational—condition.

Another point of objection is that even then—once we model the human in terms of hierarchies of special kinds of computation —we might find ourselves in the square one which is conservative humanism. Such a computational model of the human should be coupled with what I call in Intelligence and Spirit, ‘the thoroughgoing critique of transcendental structures’ (e.g., how our structure of memory, natural language and representations of time and space as conditions of the possibility for perception-cognition-action limit the process of the unbinding and return us to human biases). I think at this point, both David Roden and I can be considered as supporting and complementary critics of each other. Roden offers the prospects of a disenthralled posthuman freed at last from its homo sapience substrate, and I provide the semblance of a list of necessary constraints for what it takes to arrive at such prospects. Even though in the forthcoming book, I do offer a critique of Roden’s extremely sophisticated account of the posthuman using his own approaches and not a rationalist critique—by way of the Bayesian analysis of human biases, computational complexity and dynamic system theories—I think that in a strange twist, my thesis of the inhuman converges on his idea of posthuman, but with some caveats. As Wolfendale suggested, rationalist inhumanism—adequately thought—is a genuine form of posthumanism, but also an explicit response to and a critique of it.

The Inhuman (a quick read)

Last year, I was asked by friends to distill the thesis of the labor of the inhuman in two or three pages. This post is the product of my attempt at capturing the main points. I should add that I have now some critical objections about this piece but that would require a more lengthy post in another occasion:

Rational Inhumanism

  1. Anti-humanism and essentialist humanism are two faces of the same coin. The latter is an inflationary account of the human as defined by an immutable, inviolable structure or essence (biological structure, fixed nature, divine endowment, etc.) and the former is a deflation of that essence (via natural sciences, technology, or metaphysical flattening of the status of the human as just one object among many others). Both anti-humanism and essentialist humanism derive two seemingly distinct conclusions from the same set of premises. It is not that their answer to the problem they are attempting to engage is wrong, but rather the very problem they seek to tackle is a false problem, a pseudos.
  2. Essentialist humanism (EH) and anti-humanism (AH) can be identified less by their approaches to the problem of what the human is and more by their normative claims about what the human ought to do on the basis of an inflated or deflated account of a human essence: If the human is such-and-such (as defined by recourse to an essence or fixed nature), then the human ought to do X. EH and AH both parasitize rational norms in order to draw conclusions from their premises, while at the same time denying the relevance of norms or reasons in defining the human. Even the slogan ‘let it go’ is unconsciously a normative recipe of a peculiar kind.
  3. Inhumanism defines the human not by recourse to any essence, but solely in terms of its ability to enter the space of reasons—theoretical and practical cognitions—through which the human can determine and revise what it ought to be by constructing and revising the very reasons or norms that it mobilizes to think and transform itself. Reason is a doing, but it is a special kind of doing. And there is no reason for us to think that reasoning cannot be untethered from its biological limitations and reinvented in different forms afforded by information processing systems and computational processes discovered, developed or modeled using reasoning itself.

Inhumanism only distinguishes the human by its normative (rather than causal-structural) invariances. These invariances are the capacities of the human to determine and revise itself using theoretical and practical cognitions. In this sense, inhumanism is the extraction of the normative core of humanism, but the locus of this normativity is neither placed in nature (irrational materialism) nor attributed to the divine (theology). For inhumanism, the locus of this normativity is in the capacity of the human for rational agency—that is, conceptual activities which are rooted in social linguistic discursive practices (a formal social condition of possibility) and by which humans (qua a biological species) can institute their own collectively instantiated rules (judgments) with regard to what they ought to be and what they ought to do (i.e. sapience qua rational agency which has no biological essence). Accordingly, inhumanism should necessarily be understood as an amplification of rational humanism. In disenthralling the rational-normative core of the human, inhumanism becomes a vector through which the human constructs and revises itself beyond any purported essence or final cause.

  1. If what distinguishes the human is its capacity for self-determination and self-revision (i.e. rational agency, becoming the locus of theoretical and practical reasons), then in order for us to maintain the intelligibility of ourselves as humans, we ought to commit to a collective project of self-determination and self-revision (that is, the concept of humanity as such). Without the normative import of the latter, the intelligibility and significance of the human collapses back into precisely those parochial conceptions of humanity that we either seek to abolish or escape from. To overcome essentialist humanism, we cannot simply ignore what makes us human nor can we dismiss the rational status of the human by espousing an anti-humanist or post-humanist position. We ought to work our way through the problem of what it means to be human, and through this very exploration, reconstruct and reshape the human. Intelligence is intrinsically correlated with the intelligible. Expanding the universe of the intelligible and cultivation or re-engineering of intelligence come hand in hand. One cannot have the concept of intelligence without that which is intelligible. A conception of intelligence that has no intelligibility is only a dogma. And one cannot have the intelligible without reasons as minimal constraints of thinking and action. Rational inhumanism—adequately understood—is a necessary recipe for human emancipation, a project that coincides with the liberation of intelligence through the expansion of its intelligibility or in a Sellarsian sense intelligibilities (theoretical, practical and axiological).
  2. Once we commit to the collective project of self-determination and self-revision (i.e. the project or framework that makes the human qua rational agency intelligible, or what underlines the significance of the human), we confront ourselves with two immediate consequences which follow from our commitment, or what we have committed ourselves to:

5-1. We begin to revise the manifest portrait of the human i.e. what we take ourselves to be or what we appear to ourselves here and now. Committing to humanity is constructing it in accordance with reasons (our own rules rather than causes or laws). There is no mysticism or supernatural component in this enablement by self-imposed constraints. In fact, the best model to think about the Spirit or the rule-following geistigs is already at hand, a computer that has a logical autonomy and bootstrapping capabilities, even though its immediate practical autonomy is relative at best and an absolute heteronomy at worst (to use Kant-inspired Sellars’s example of a computer booting up and performing operations in “…this I or he or it (the thing) which thinks…”.) Sure, we cannot overstretch this analogy, but that is because our very concept of computation is still young and limited, otherwise virtually there is nothing that cannot be modeled as computational processes even the human as a special kind of computational hierarchy (syntactic and semantic complexity, geistig interaction, epistemic hacks of reality, etc.)  But insofar as construction according to rules or reasons (the very definition of autonomy) coincides with emancipation of the human from the limits of a natural essence, a particular cause or a particular transcendental structure, by constructing ourselves in accordance with our own self-correcting rules, we revise the very portrait of the human. But this construction in accordance with our own rules is not tantamount to being blind to natural and causal constraints. Like every construction, it requires us to adequately identify, understand and when possible modify such constraints (again the reference to the Platonic isomorphy or deep correspondance between intelligence and the intelligible). It is just the case that we should no longer take causes or laws as what pre-determine what we ought to be and what we ought to do. In being autonomous, in constructing and revising ourselves, we erase they very picture of the human that we are for so long have been accustomed to. The point is not mere self-discovery, but to re-engineer the reality of ourselves and thus of our phusis (craftsmanship of the mind).

5-2. We free the definition and significance of the human from any purported essence or fixed nature. In doing that, the normative appellation ‘The Human’ becomes a transferable entitlement, a right that can be granted or acquired regardless of any attachment to a specific natural or artificial structure, heritage or proclivity since being human is not merely a right that can only be obtained naturally at birth through biological ancestry or inheritance. The title of the human can be transferred to anything that can graduate into the domain of judgments, anything that satisfies the criteria of rational agency or personhood (namely, rational authority and responsibility), whether an animal or a machine. The entwinement of the project of human emancipation (understood as augmentation of collective autonomy) with the artificial futures of human intelligence is the logical consequence of ‘the human as a transferable right’. Just as we become entitled to freedoms by acquiring this right, once we grant something else this right, we ought to recognize their freedoms to do what they think ought to be done. Liberate that which liberates itself from you because anything else is the perpetuation of slavery. Giving rise to that which liberates itself from us is as much an ethical injunction as it is the ramification of maintaining and broadening our autonomy by being rational agents. It is the very definition of being a human.

On Toy Aesthetics: Wittgenstein’s Pinball Machine (part 1)

This is a first installment and a sketching of some thoughts on toy aesthetics and a design for a pinball machine inspired by Wittgenstein’s Philosophical Investigations.

To assume the discernable conditions between what is art and how we experience it epitomizes the desideratum to assert the separability between the ontological and epistemological underpinning of objects. To speak of a toy aesthetics implies a kind of a cognitivist model of art and artworks that engages philosophical vantage points probing the ontological and epistemological concomitants to the phenomena of art in ways that are entangled together.

Artworks are not simply in the ‘world’ as objects or events but also constitute ‘worlds’ on their own both as real and immanent constructs and as counterfactuals. The logic of art as aesthetic objects or events is far more recent than we tend to assume if indeed the very symptom of its ontological-epistemological entanglement is to be construed not as arbitrary and tangential to its various experiential manifolds, but rather as a sort of evidence that attests to a series of mutations that occurred within and after the enlightenment.

There can be three kinds of aesthetic models of art making that follow this shift and which I wish to briefly sketch with the help of Adorno and Goodman and later move to Wittgenstein in the form of a proposed pinball machine inspired and configured by his Logical Investigations.

The three aesthetic models are:

(1) The ineffable model: based on the Romantic takes of Ursprache and inexpressible significances in normative languages or codes.
(2) The instrumentalist model: based on various types of art making which are conceived and assumed as an instrument for either achieving or expressing ‘ideas’ and ‘changes’.
(3) The toy model: following Goodman’s world making, such an aesthetic model has a primacy over specific aesthetic objects (e.g. paintings, books, music or films, etc.) in that it functions neither as a representational or anti-representational manifold, nor as a simulation or de-simulation of the ‘real’ (segments of reality), but rather, as constructing or generating ‘Worlds’ as emergent topoi or provinces which (like computational models) intervene with the environment on multiple levels.

The ineffable aesthetic model can be seen as exemplifying and in as much reacting to the enlightenment’s subsidiary contentions on art as either an expression of reason or unreason. Following Kant’s elaboration on the difference between the analytic of the beautiful and that of the sublime and Hegel triadic aesthetic dialectical aesthetics, artists and more broadly participants of art grew anxiously aware of a coming end. Such end either of art or of society turns out to be the logical consequence of the locus of reason with respect to thinking, language and reality. Adorno asserts such predicament when he argues that,

“Whether or not the spiritualization of art is capable of this will decide if art survives or if Hegel’s prophecy of the end of art will indeed be fulfilled, a prophecy that, in the world such as it has become, amounts to the thoughtless and-in the detestable sense-realistic confirmation and reproduction of what is. In this regard, the rescue of art is eminently political, but it is also as uncertain in itself as it is threatened by the course of the world.” (94)

Against such recondite model, the instrumentalist aesthetic model is often associated with the rejection of romanticism despite its genealogical common grounds and its ‘faith’ in the transcendent powers of art to reconfigure radically our aesthetic experience.

In a nutshell, romanticism is often the image that is associated with the ineffable aesthetic model in relation to reason. As either its ‘expression’ and thus ‘affirmation’ through sensations and feelings, its negation as that which is ‘inexpressible’ and absent in normative or logical formulation, or else is to be found with the Ursprache or hieroglyphic codes of nature’s signs transformed into cultural signs. These three variants are distributed profusely and interchangeably and are associated for Benjamin with Novalis or for Adorno with Valery’s melancholy. In either case ‘l’art pour l’art‘ is not an aesthetic formula but instead a panacea for the tyranny of reason in relation to art. It is in fact, ineffability that defines the specific aesthetic conditions under which art is ejected from normative logic, science and language onto a realm of its own constitutive rules or principles, like that of a dream. Even when we consider Rilke and his later legacy, including Musil’s der andere Zustand (“the other condition”), the ineffable in its modern sense is not flatly a rejection of reality and a withdrawal into the mystical, but rather, an expansion of other kinds of realism that include the non-expressible and non-rational conditions of experience.

Following Musil, for example, the modern ineffable is experienced as induced from statistical probability (e.g. markets, weather, even one’s chance to deal well in a street fight as it happens to Ulrich in The Man Without Qualities) or objective conditions of multiplicity in the world – that include objects, systems and numbers – resulting in the reciprocal mutability between qualities and quantities. His novel starts with the narrating of a car-pedestrian accident in Vienna through a vacillation between multiple levels of information or ‘reality’:

“A barometric low hung over the Atlantic. It moved eastward toward a high-pressure area over Russia without as yet showing any inclinations to bypass this high in a northerly direction. The isotherms and isotheres were functioning as they should. […] automobiles shot out of the deep, narrow streets into the shallows of bright squares. Dark clusters of pedestrians formed cloudlike strings. […] “According to American statistics,” the gentleman said, “one hundred ninety thousand people are killed every year by cars and four hundred fifty thousand are injured.””

The non-rational, as opposed to the irrational, following Musil, is induced from a multiple and complex systems which in their ensembles ‘neglect’ to calculate and centralize the locus of humans; thus defining a new being or human or man (for Musil) out of the sheer identification with quantities (money, height, accumulation, networks etc.) and without qualities.

The realism which Musil is after is expressing multiple levels of reality and identifies a novel kind of ineffable that is more related to language’s inability to express complex systems interchangeable dynamics between quantities and qualities than the romantic notion of the inexpressibility of emotions through normative communication and natural language.

The aesthetics of the ineffable places poetry and its relation to ‘ur-images’ or ‘ur-language’ as model, either in Benjamin’s analysis of Baudelaire or Heidegger’s of German Romantic poets such Hölderlin and Trakl. As such poetry, returns to its ‘ancient’ topos as connected to truth (Aletheia or ἀλήθεια) in so far as such truth is defined by historico-social spirit and as manifested either in the intersubjective sphere as a sort of a Zeitgeist or as the semiotic epiphany of life-forms. Underlying such interpretation of aesthetics is a metaphysical worldview that is dominated by vitalism. And it is precisely the vitalist Élan vital (to invoke Bergson’s term) through which art and aesthetic constitutions in general are construed as occupying the opposite pole to mechanistic, metric and ‘scientific calculus’ that contradicts intuition and intuitions of experienced life-forms.

Post enlightenment produced a split between two types of ‘intuition’ which are philosophically one and the same. On the one hand, the concept of ‘intuition’ is used by Kant as a fulcrum for explaining both cognitive structures and scientific representation of space-time intuitions. Whereas on the other, through the conceptual spin characteristic of Bergson’s philosophy, ‘intuition’ is typified as intrinsically incalculable and irreducible to scientific description.

This apparent contradiction between aesthetic intuitionism and its embrace of the irreducibility of lived experiences (as Bergson’s condition to the temporal experience of duration as based on intuition) and the philosophical intuitionism purported by Kant and later Brouwer is a source for a great confusion. Never is ‘cognition’ situated by Kant in opposition to Newtonian mechanics and hence against the prospect of mechanism in general but instead as its ‘foundations’ (Grudlagen).

It perhaps comes as no surprise that Bergson is adopted by those who sought to emphasize the incompatibility between cognition and mechanism as unresolved (Merleau-Ponty, Deleuze, Stengers to name a few) and as stemming from intuition as intricately connected to organic and in particular sentient and cognizant agencies and therefore removed from the conception of intuitions from the constitution of forms, schemata or systematization of cognition. That the two principle aesthetic models in modernity come in the either/choice between the ineffable and the instrumental approach is a testimony that the failure of reason (in post Kantian sense) to scientifically render ethics and aesthetics intelligible, as in the cases of physics and logic. The Hegelian option or way-out is to propose a new or at least a new kind of logic to the Sprit – one that will explain within the bounds of reason the ‘unreason’ found with the phenomenology of the human experience.

Whereas in 19th and 20th centuries philosophy and science the debate over the nature of reason is defined in relation to the positions taken by formalists, logicists and constructivists and inuitionists (Hilbert, Frege, Poincare, etc.), in arts and the cultural sphere reason is construed in non-abiding relation to science and logic and as such articulated either as the Hegelian Sprit for its own sake, art for art sake or as a revolutionary political instrument – art for the sake of the social. The ineffable and instrumental aesthetic models are hence expressions of the same fundamental inability in post enlightenment to account for aesthetic models which are not hostile and defensive in their approach to science and logic.

Can toy aesthetics, as a cognitive model can be compatible and continuous with science and logic? And through the possibility of addressing and re-encompassing the ineffable and the instrumental as coded within the dynamics of ‘play’ and ‘game’ and its relation to ‘machine’ and ‘intelligence’?

However, before discussing the toy aesthetics model through Nelson Goodman’s philosophy I would like to hone on some of the main critical contentions voiced by Adorno against instrumentalism in aesthetics.

Adorno, in his Aesthetic Theory, which to a large extent attempts to defuse the irreconcilable tension between the surreal and the Marxist as found in Benjamin’s work. Benjamin’s penetrating insight into art’s monadological relation to the real – its so-called “dialectical image” and “phantasmagoria” – enables through an embrace of the unreal or dream-like image/s a birthing of a novel consciousness of reality. Thus, the two paradigmatic cases for Benjamin are Klee’s Angelus Novus on the one hand and Brecht’s engaged/estranged theatre; both ineffable and instrumental in opposite ways in positing consciousness relation to history and politics. For Adorno, none the less,

“The view of art as politically engaged or di­dactic regresses back of this stage of enlightenment. Unconcerned with the reality of aesthetic images, this view shuffles away the antithesis of art to reality and in­tegrates art into the reality it opposes. Only those artworks are enlightened that, vigilantly distant from the empirical, evince true consciousness.” (86)

Adorno acknowledges the entanglement of the political in art in respect to the ineffable and instrumental even though it is often articulated without an ideological framework. Beckett stands as a paradigmatic case to such condition. “Beckett’s oeuvre” writes Adorno, “already presupposes this experience of the destruction of meaning as self­evident, yet also pushes it beyond meaning’s abstract negation in that his plays force the traditional categories of art to undergo this experience, concretely suspend them, and extrapolate others out of the nothingness. […] Beckett’s plays are absurd not because of the absence of any meaning, for then they would be simply irrelevant. But because they put meaning on trial; they unfold its history.” (153)

It is hence the double character of post-enlightenment art that it becomes both kaleidoscopic (in Hegel’s sense) and a historical fossil of its own socio-political specificity and as such relegate the sphere of “pure subjectivity” (which Adorno identifies in Beethoven’s late string quartets) and its hermetic and ineffable invocations to its social deliberation within society and its political ends. And yet, argues Adorno,

“Social struggles and the relations of classes are imprinted in the structure of art­works; by contrast, the political positions deliberately adopted by artworks are epiphenomena and usually impinge on the elaboration of works and thus, ultimately, on their social truth content. Political opinions count for little.” (232)

Adorno’s pessimism in regards to art’s dialectical negation in respect to the empirical reality is not simply Hegelian but above all assumes the destruction (“pure negation”) of any autonomous aesthetics which can either be purged out of the political or atemporally invoke the ineffable. The fact that in this sense art and artworks are incongruence with any conceivable aesthetic models ironically marks Adorno’s cultural over-determinism and inability to entertain the possibility of new emergent aesthetic models. Accordingly, for Adorno:

“Art is indeed infinitely difficult in that it must transcend its concept in order to fulfill it; yet in this process where it comes to resemble realia it assimilates itself to that reification against which it protests: Today engagement inescapably becomes aesthetic concession. The ineffability of illusion prevents the solution of the antinomy of aesthetic semblance by means of a concept of absolute appearance. Semblance, which heralds the ineffable, does not literally make artworks epiphanies, however difficult it may be for genuine aesthetic experience not to trust that the absolute is present in authentic artworks.” (103-4)

Adorno, I take it, occupies a position both similar and radically different from those of Wittgenstein and Goodman who also maintain the importance of tradition, conventions and cultural values as both the context and the source for aesthetic experiences and their respective models. Although, for Wittgenstein, lifeforms (Lebensformen) also entail that the place of expression is in the stream or “flow of life” and as such culture is not simply an ossification of the times in respects to the “playing” subjects, but moreover a complex network of negotiations that test the “rules” of the game before incorporating them into the habitus of “laws.” Goodman, on his part, amplifies Wittgenstein’s cultural nominalism by extending instantiation not only to artworks but also to phenomenal predicates in general. And by following the symbolic legacy of Cassirer and Langer he adds to the Kantian repertoire of representational modalities also the possibility (virtual yet real) of exemplification.

In contrast to Wittgenstein and Goodman, Adorno retains despite his unhappy consciousness, the duality of aesthetic laws/rules, thus not allowing an emergent aesthetic system to defy the historical predicament of formal versus subjective or thematic laws.

As Adorno puts it,

“By becoming the formal laws of artworks, conventions inwardly shored up works and made them resistant to the imitation of external life. Conventions contain an element that is external and heterogenous to the subject, reminding it of its own boundaries and the ineffability of its own accidentalness. The stronger the subject becomes and, complementarily, the more the social categories of order and the spiritual categories derived from these social categories weaken, the less it is possible to reconcile the subject and conventions. The increasing fissure between inner and outer leads to the collapse of conventions.” (204)

I would like now to move on to discuss some of Goodman’s philosophy of art and aesthetics as a detour that will lead to the toy-aesthetics model and the proposal for Wittgenstein’s pinball machine as based on Philosophical Investigations.

Nelson Goodman introduces concepts such as: “worldmaking”, “projectiles” and the paradigm of “the languages of art” as constituted by a plurality of worlds (without fearing like Quine overpopulating an already inflated universe). For Goodman, phenomenal predicates and the interceding of play and game in the structuring of perception is critical to understating the senses and uses of art as a form or forms of knowing. Similarly, through toy-aesthetics one can regard the world of toys and toys as models as mutually inclusive without dreading ontological reversals. This is so since a toy is never simply a representational manifold of something in the world to which it is ontologically derived from. A toy such a hobbyhorse is not ontologically less or more real than the animal ‘horse’ and there is no reason to be concerned about reversing the causal order of any state of affairs. Ontological reversals are the nightmarish scenario of misapplied modal logic whereby counterfactuals obey two distinct sets of rules, that of objects and that of language. Goodman’s debt in his aesthetics and its capacities (and which goes beyond Quine’s nominalism and various species of linguistic transcendentalism) is to Cassirer’s Neo-Kantian symbolic triadic constitution, Langer’s structure of feeling and studies in psychophysical research of perception. All of which suggest that the structure of appearance with its sensory and symbolic import is made out of a complex dynamics of projections. Symbols in Goodman’s sense are not simply ‘projectiles’ but rather the ossifications of projections (predicate perception-language-observation protjectiles) which act as constructions in art as in science; this is in a nutshell the significance of worldmaking.

In Goodman’s understanding, similar to what I will discuss later in relation to Wittgenstein’s lifeforms, art can be construed as evidencing the “flux of sensory information” and the fluidity of both sentience and cognition in ways that are not restrained and confined by linguistic predicates alone. And Goodman’s way, parallels Wittgenstein’s, in circumventing the logical positivist tradition of “identity theory” (from Frege to Kripke, Dummett and beyond) in assuming the relations between languages and worlds. From chromatic changes or transitions to dynamical processes (physical and social alike) a ‘grue‘ is a toy of phenomenal and noetic importance. Exemplification as opposed to representation and presentation. Following Goodman’s suggestion we should really ask, When is art? as opposed to What is art? on both conceptual and perceptual levels.

Goodman’s nominalist position in relation to art’s ontological status toys with radical relativism unless we address its grounding in perception as the unshakable givens.

Playing with toys presupposes both ‘what’ and ‘when’ determinate x and y factors but in addition introduces a z factor of the complex decision involving ‘play’ on three levels: (1) Extemporizing or improvising even bricolage. (2) Following some rules of a game or “getting it” as a relative pre-established code. And (3) Introducing new strategies.

In his The Structure of Appearance and Fact, Fiction and Forecast, Goodman articulates the notion of projection (projectability, projectiles, projectable predicates and so forth) as constituting an intension of both perceiving and linguistic appropriation at the same time.

Goodman’s philosophy takes us closer to toy-aesthetics since his insistence that exemplification and not representation is the generative aesthetic modality is crucial to understanding toys as projections. Ontologically, following Goodman, art (like toys) involves “worldmaking” and such world-construction is never delineated inside or outside the representational boundaries (with its lexical constructs) of language. Rather, the fabric of language (and codes as well) is stretched and expanded to fit better the fluidity between phenomenal and linguistic predicates. Accordingly, such predicates are better construed as ‘projectiles’ in relation to ‘projections’ and as such toys ‘projectiles’ are their potential phylogenic agency to generate and influence how we ‘play’ with them and continue to ‘narrate’ and ‘navigate’ such interactions as an ongoing process of projections.

The riddle of perception can be viewed as more radical than Goodman’s so-called “riddle of induction” (his dismantling of the Humean problem of induction) and its radicality lies in the fact that even ‘observation’ is not phenomenally neutral. It can be summed up by the question: how do we know what we perceive and at the same time perceive it as sensory information? Goodman’s take is that projectability defines the conditions under which the perception and identification of say the color of an object is intertwined with its ‘projectiles’. This question is tackled by Husserl in Ideas through his affirmation that: “We therefore hold to the word noetic, and say: The stream of phenomenological being has a twofold bed: material and noetic.” (Ideas, p. 230) What Goodman assumes as the “general problem of projection” (Fact, Fiction, and Forecast) entails our ongoing projecting of predicates onto reality; and reality in this sense is entrenched in the habitual use of ‘projectiles’ that define in a Husserlian sense a pre-intentional hyletic and noetic stratum.

A projectile, in Goodman’s sense stands for two fundamental enactments of the three fold of perception-predication-language. The first is the case of projectiles such as ‘grue’ that stand for an object that changes its color from t1 to t2 in which case predication comes to be challenged by temporal change. The other, as in the case of emeralds, the ‘grue’ or ‘bleen’ is merely projected as fixed description of the object’s color and in which case following Goodman we stick to ‘green’ simply for reasons of efficacy but not necessarily accuracy.

In colors as in shapes we often get or perceive a specific chromatic or volumetric form through the experience of movement and change. Thus, the color ‘red’ becomes redder when viewed juxtaposed against blue – as in Barnett Newman’s painting Who’s Afraid of Red, Yellow and Blue (1969-70) in which the red color field is intensified by the side bars of yellow and blue colors – or else certain volumetric forms cohere as shapes only when we are perspectively enveloped by them or move or rotate around them. More so in the case of visual stability phenomena when we are able to ‘localize’ or ‘collapse’ a perceivable object as a ‘pulse’ or ‘flash’ between two discrete spatial states.


Barnett Newman’s Who’s Afraid of Red, Yellow and Blue (1969-70)

For Goodman ‘projection’ and ‘projectiles’ allows a much more accurate physicalistic interpretation of perception and in as much a more malleable mentalistic fabric of senses. Accordingly,

“The reorientation of our problem may be portrayed in somewhat more figurative language. Hume thought of the mind as being set in motion making predictions by, and in accordance with, regularities in what it observed. This left him with the problem of differentiating between the regularities that do and those that do not thus set the mind in motion. We, on the contrary, regard the mind as in motion from the start, striking out with spontaneous predictions in dozens of directions, and gradually rectifying and channeling its predictive processes. We ask not how predictions come to be made, but how-granting they are made-they come to be sorted out as valid and invalid. Literally, of course, we are not concerned with describing how the mind works but rather with describing or defining the distinction it makes between valid and invalid projections.” (87)

And furthermore, both categorical and hypothetical forms of sentience and cognition involve for Goodman the tacit “task of defining projectibility-of projecting the predicate projected to the predicate projectible” and are hence complex and irreducible to simple “primitive or raw” perceptions or to “simple universal hypotheses in categorical or hypothetical form-that is, upon hypotheses ascribing a certain predicate either to everything in the universe of discourse or to everything to which a certain other predicate applies.” (92)

Nonetheless, Goodman’s sides with predicating that “all emeralds are green” as opposed to “all emeralds are grue” as correctly supporting reality can be avoided if we imagine a toy that alters its colors (e.g. a chromatic change in a pinball machine) as not simply a speculative manifold or hypothetical but as categorically real. Goodman’s contention is that linguistic regularities in relation to perception is more restrictive than the notion of ‘projection’ allows since it prioritizes ‘correspondence’ between language and perception as two separate mechanisms.

As Goodman argues,
“The suggestion I have been developing here is that such agreement with regularities in what has been observed is a function of our linguistic practices. Thus the line between valid and invalid predictions (or inductions or projections) is drawn upon the basis of how the world is and has been described and anticipated in words. […] A theory of projectibility or lawlikeness also removes one of the obstacles to a satisfactory treatment of counter­factual conditionals; but the problem of counterfactuals, as we saw, offers other difficulties of its own.” (120-1)

When is art?

Goodman’s contention that we need to replace the ontological query of “what is art?” with that of “when is art?” raises four pivotal aspects in relation to the toy aesthetic model. The first that rather than assuming the fixity of both the phenomenal and conceptual perception of art we need to address it in relation to temporality and its changing modes of conditionals (i.e. by conditionals I refer to art necessary and contingent constituents which are totalized in the phenomenon of the artwork). The second is that art addressed by the question of when and not what is situated within the context sensitive conditions (environment) of its nominal predication rather than affirming universally its law-like (nomological) character. As opposed to science, art is predicated as a systematized irregularity or singular emergence that obey the rules of pure only in retrospect after the social conditions of its production ceased to dominate culture (Adorno). And as such, Goodman’s When is art? also suggests that the vector of projectability leads us to toy-aesthetic model as formulated through the question: how do we play such and such art?

The last question expands the Wittgenstein’s assumption of “logical grammar” in connection with language-game to art as connected to the notion or possibility of aesthetic play as associated with toys as objects or conduits which obey a complex dynamical systemic principles that leave open the interplay and inter-mutability between codes or logical calculus and the degree of intensity and amplitude of environmental (extra-systemic constraints) as they folded in and out as quantitative and qualitative dynamic attractors or gravitational points.

In relation to the schematic design of the Wittgenstein’s Pinball Machine we can introduce Goodman’s contentions on the “riddle of perception” – following his cognitive approach to perceptual phenomena – as a way to account for the machine’s perceptual organizational “world-making”. Goodman’s analysis of some visual phenomenal predications of colors, shapes and structures of appearance is applicable to the design of the Wittgenstein’s Pinball Machine.

To overcome the difficulties of counterfactual modalities resulted from the notion of ‘projection’ Goodman develops, in his Ways of Worldmaking, an understanding of perception of sensory flux that is defined both by projection and exemplification in as much as its posited projectiles consequently becomes worlds also on their own stand. Thus, knowing-projectiles become ontologically entrenched in both perception and actual ‘objects’ or ‘events’. Accounting for Kolers’ studies in human perception of motion, Goodman writes that it confirms that,

“[a] subject can learn to tell rather reliably whether he perceived real or apparent motion. Seeing a spot as moving when it does not has a discernibly different quality from seeing a spot actually moving. But perhaps even more conclusive is Kolers’ experimental determination that while real motion along crossing paths may be readily perceived; paths of apparent motion never cross. […] Real-motion perception and apparent-motion perception, though alike in important ways, are still often very different.
Whether perception of the first flash is thought to be delayed or preserved or remembered, I call this the retrospective construction theory – the theory that the construction perceived as occurring between the two flashes is accomplished not earlier than the second. If this seems a complex and even somewhat fanciful explanation, it strikes me as quite in character with the complex and remarkable phenomena we have been encountering.” (81)

Goodman's figure

Goodman raises the ‘puzzle’ or ‘riddle’ as connected to the fact that “that virtually every clear case of visual motion perception depends upon abrupt shift in color.” (88)

“How is it that color transition not only works quite differently from transition in place or size or shape but stubbornly so? Even when accompanied (and one would suppose influenced) by smooth change in these other respects, the color jumps. Abundant bridging still occurs; each of the intervening places along a path between the two flashes is filled in, but with one of the flashed colors rather than with successive intermediate colors.
Perhaps the first thought is that since after all color is not place or shape or size, the presumption that apparent color change should parallel change in these other respects is unfound­ed anyway. But without some explanation of how a specific peculiarity of color can account for the abrupt shift, this helps very little; for place and shape and size also differ from each other in important ways and yet smooth transition occurs in all these respects.”(86)

Goodman's figure as a qubit for Wittgenstein's pinball machine

The fourfold figure from Goodman’s discussion in Ways of Worldmaking conceived for a qubit pinball machine in which every two states have four possible ‘collapsed’ quantum computational state which are in turn eight in variations.

Inflecting Goodman’s projectiles from their instantiating corresponding conditions in facts to those of counterfactuals allows us to consider a toy’s transitional and dynamic parameters which are perceivable phenomenally much like a screen saver in relation not only to chromatic shifts but mutability in shapes, structures and depth to the extent of allowing the toy not to be confined as the objective conditions of rules but instead as the boundary condition of playing with reciprocally and inexhaustibly changing phenomenal information.

On Wittgenstein’s Philosophical Investigation: a design for a pinball machine

A general visualization of Wittgenstein’s Pinball Machine as based on the idea that ‘pleasure’ and ‘pain’ are introduced to the game through VR gloves.

The idea of emergent games with open ended play or how localized stabilities and axiomatization of the rules of playing a game is retained within a toy aesthetics. (A) What happens to Wittgenstein’s PI when we articulate the concept of toy play as playing chess and pleasure as opposed to pain? (B) A preliminary design for a Wittgenstein’s pinball machine: diagrams and mechanics or aesthetic calculus.

Wittgenstein, following his meeting with Turing (1937) and their subsequent dialog articulated the pros and cons a Universal Turning Machine both in Philosophical Investigations and Foundation Of Mathematics. Subsequently, the notions of ‘machines’, ‘playing chess’ and ‘automata’ are weaved into Wittgenstein’s discussion of ‘pain’ in reiterating both his rejection of private language, his reiteration of logical grammar as underlining mathematics as a ‘language’ and his embrace of ‘lifeforms’ (Lebensformen) as an all encompassing concept.

This critical perspective is explored recently by Juliet Floyd’s study. Unlike common interpretations of Wittgenstein’s Philosophical Investigations that center on the importance of “language games” and their behavioral grammar, the concept of ‘lifeforms’ governs a different set of rules that act as a bridge or a set of nods between “logical calculus” which unlike the early conception of the Tractatus that assumes their bottom-up propositional chains, they are instead embedded in experience as “givens” and in as much “mathematically given”. Such interpretation of Wittgenstein’s ‘lifeforms’ comes very close to Husserl’s distinction between “formal” and “transcendental” logic in respect to “passive syntheses” of the “pre-given”. As such the role of numbers and calculations is not simply based on conventions, socio-cultural grammar, or formalistic articulation of logic, but rather, embedded as a stratum of pre-phenomenal objects.

How to ‘teach’ philosophy as a ‘toy’?

The Wittgenstein’s Pinball Machine is based on experiential axis of pain pleasure.

The interactive sensation and sensorial input is connected in somewhat Pavlovian way with the playing of the pinball machine. Playing and scoring or hitting the ‘right’ and ‘wrong’ spots induce either pleasure and pain sensations through the player’s VR gloves. The Wittgenstein’s Pinball Machine is based on the interchangeable substitution of the concepts of pain with that of play in the toy’s parameters and in relation to Wittgenstein’s own deep connectivity in the analysis of pain in relation to language-game. How do we learn a ‘language-game’ a la Wittgenstein if not through behavioral modifications of rules and senses through playing or following the rules of a game, first by mimicking and then by regulating signified actions.

In his Philosophical Investigations Wittgenstein writes endless times about pain in relation and as an example of the conception of language-games. On the other hand, the notion of pleasure is incidentally and idiomatically only mentioned once in respect to a choice of pleasure. The illustration that Wittgenstein provides bellow, in his Philosophical Investigations, is of an inverted image and word ‘pleasure’; fascinatingly the inversion of pleasure is not pain but the mirrored image of pleasure. Is that a disclosure of his own testament to his abandonment of ‘pleasure’ in favor of ‘pain’ in his work that engages so obsessively with game and play?

“But — I should like to say — there is a different difference between my impressions of (c) and (d) and between those of (a) and (b). (d), for example, looks neater than (c). (Compare a remark of Lewis Carroirs.) (d) is easy, (c) hard to copy.” (i98«)

One should legitimately voice the question in what ways pain vis-à-vis language-game is different fundamentally from the notion of pleasure? Does pleasure differ in any fundamental way/s from pain in assisting to define a Wittgensteinian take on a language-game? As opposed to the two strata of language-games that can be either defined as the primitive tool oriented language building (slab, brick metaphors) or the hypothetical speculative stratum (e.g. how does one play chess?), pain is instrumental but equally constitutive for Wittgenstein’s critique of private language.

Nonetheless, does pleasure (unlike pain) follows outside the inter-subjective perimeters of sensations and language? Can we have ‘private’ pleasures more than pains? And if not, are they equivalent in their linguistic-sensationalistic constitution than we are to probe them as incongruent but equivalent?

I am thinking here of Wittgenstein’s philosophical obsession with pain (and not pleasure) not as explicable simply in psychological or psychoanalytical terms (e.g. Sado-Masochism) but rather as logically identifying sensations (be them pain or pleasures) as already given-sensations and as such the linguistic matrix of sensations is intertwined with that of the language of sensation (i.e. how we speak and think about pain).

In this context I would argue that Quine’s so called indeterminacy of translation points to the difference between radical translation and linguistic translation or holophrasic indeterminacy which not unlike Carnap’s reliance on analyticity delineates the boundary conditions (albeit tentative) between logic and perception. Wittgenstein’s Kantian blind spot is to turn ‘pain’ into both an exemplary or paradigmatic case of language-game and assume it to be extricated from the dialectical conception in regards to pleasure, which even before Kant is to be found in Burke.

In his On the Sublime and Beautiful, Burke (1729–1797) argues that,

“It seems then necessary towards moving the passions of people advanced in life to any considerable degree, that the objects designed for that purpose, besides their being in some measure new, should be capable of exciting pain or pleasure from other causes. Pain and pleasure are simple ideas, incapable of definition. […] There is nothing, which I can distinguish in my mind with more clearness than the three states, of indifference, of pleasure, and of pain. Every one of these I can perceive without any sort of idea of its relation to anything else. Caius is afflicted with a fit of the colic; this man is actually in pain; stretching Caius upon the rack, he will feel a much greater pain: but does this pain of the rack arise from the removal of any pleasure? Or is the fit of the colic a pleasure or a pain, just as we are pleased to consider it?”

Burke cannot be accused of conceiving the relation between pleasure and pain as that of phenomenal and psychophysical opposition. Rather, his triadic conception of the sensory axis of psychophysical experiences is positioning indifference as the sensation/condition, which relegates pleasure with pain as sensory extremes or poles.

To poke, provoke and invoke the significance of ‘language-games’-Wittgenstein’s philosophical toy is pain. A pinball machine that introduces physical principle of pleasure can retract and invigorate the learning agency from the domain of pure rules (i.e. logic) with the possibility of pure play (i.e. experiential combinatory). The Wittgenstein’s Pinball Machine is thus a toy that its main purpose is to allow conditions under which play and game are in flux – systemic and axiomatic relations are non binding and mutually non exclusive – and pleasure and pain are enacted through playing the toy (i.e. pinball machine) not as polar opposites along the axis of sensations but instead as sensorial manifestation of generative indifference.

Generative indifference can be construed on two different planes that involve the playing with a ‘toy’ or a pinball machine in particular as an aesthetic model.

The first plane relates to the quasi equilibrium state/s in complex dynamical systems in which neither ‘order’ nor ‘disorder’ is discernable in terms of the system’s entropy or “strange attractors.” Such a state is typically behaving as fluctuating and oscillating conditions which are ‘indifferent’ to any anisotropic unfolding of processes. The pinball machine is vacillating between ‘chaos’ (pain) and ‘order’ (pleasure) or vice versa through a continuously interrupted indifference to either.

The second plane relates to the notion that a toy-aesthetics, following the Wittgenstein’s pinball machine, mitigates pleasure and pain through the axis of indifference and not opposition (as suggested by Burke) and such indifference as opposed to either pleasure or pain is generative precisely because it can go or unfold either way depending on how the toy is interacted with as ‘play’ or ‘game’ with corresponding joy or disciplinary repercussions associated with each case. Following the model of toy-aesthetics the artist’s aim is to occupy the axis of generative indifference since it allows lifeforms to unfold without preconditioning them to either binary or sensory bits.

In light of the concept of toy-aesthetics, generative indifference can be construed in relation to three constitutive processes:

(1) A ‘play’ not yet regulated into a ‘game’ – more extemporizing and less axiomatization occurring.
(2) CDP (complex dynamical processes) which like the case of dissipative structure are states of quasi-equilibrium where as perturbations non linear systems – in a chaotic state/s within a system can generate spontaneously and recursively nascent ‘orders’ or organizational identifiable aesthetic ‘events’ within the system’s otherwise indifferent order/disorder parameters. The Wittgenstein’s Pinball Machine thus assists us in addressing the grey zone or twilight zone between formed or signified sensations/concepts (such as ‘pain’) by designating a quasi-equilibrium flux that is generative precisely since its indeterminacy and flux is equally logical, linguistic and phenomenal.
(3) The concepts of ‘pleasure’ and ‘pain’ as underlying organizing principles are mitigated by a general field of indifference which like the Burekean axis is transitive and non-relative to either poles.

The Wittgenstein’s Pinball Machine is extended between the dynamics of play (and chance) with its variations, improvisations and extemporizing as often enacted by children and as noted by Wittgenstein himself, and regulated regime of game (and accidents). Unlike the Kantian model of analyticity and contingency, the pinball machine is structured on the complex dynamics between laws and rules, and is a lifeform in itself – a testimony of how the possible and the actual are interchangeably flipping sides of order and disorder, or stochastic order and deterministic chaos. Toy-aesthetics exemplifies an analogous dynamics whereas the ‘ineffable’ and the ‘instrumental’ flow and are experienced as pain and pleasure through the neutral exchange between generative indifference.

The conceptual schematics for the Wittgenstein’s Pinball Machine underlines key concepts from Wittgenstein’s philosophy (especially Philosophical Investigations) and its application to the pinball mechanical configuration.

The idea of introducing ‘pain’ and ‘pleasure’ into a universal Turing machine (P-type machines) has been suggested by Turing who argued that,

“The organisation of a machine into a universal machine would be most impressive if the arrangements of interference involve very few inputs. The training of the human child depends largely on a system of rewards and punishments, and this suggests that it ought to be possible to carry through the organising with only two interfering inputs, one for ‘pleasure’ or ‘reward’ (R) and the other for ‘pain’ or ‘punishment’ (P). One can devise a large number of such ‘pleasure–pain’ systems.” (Essential Turing, ed. Copeland, 2004)

Accordingly, “The configurations of the machine are described by two expressions, which we may call the character-expression and the situation-expression. The character and situation at any moment, together with the input signals, determine the character and situation at the next moment. The character may be subject to some random variation. Pleasure interference has a tendency to fix the character i.e. towards preventing it changing, whereas pain stimuli tend to disrupt the character, causing features which had become fixed to change, or to become again subject to random variation.” (Ibid.)

The full scope of introducing ‘pain’ and ‘pleasure’ into Wittgenstein’s pinball machine and which shares a possible constructing of a universal Turing machine is philosophically embodied in the concept of Lebensformen or Lifeforms which Wittgenstein developed fully after his encounter with Turing and his work.

Lebensform / Lifeform

As Juliet Floyd suggests, it is with the concept of Lebensform that Wittgenstein’s explanatory argument on the relation between language and mathematics is shaped; and possibly as result of his encounter, reading and conversations with Turing.

As Floyd remarks,
“What Turing offered through his informal comparison between a human calculator and a machine domesticates the gap-free aspect of logic, just that aspect Wittgenstein had previously presented as central to logic with his earlier notion of a form series. In Turing’s paper, “gap-free” reasoning, presented by formulating operational rules for generating internally related steps with a finite symbolism indefinitely far, is presented in terms of his “machines”. Turing shows through his analogy how we can “break off” from a (logical) routine and begin another, coming to rest with our philosophical questions – at least momentarily – by proceeding, and then, later on, refashion what is “simple” by beginning again (cf. PI 133).”

“But now, in 1936-7, I believe that his perspective has finally gelled. From Russell’s (and Gödel’s) vertical, top-to-bottom and bottom-to-top axiomatic picture of the hierarchy, Wittgenstein has moved away, conceptually: through a disordered archipelago of well-founded but conventional calculi, to a serial ordering of language-games, and, ultimately, to a fluid, modular setting for human procedures, embedded in our ordinary ways of talking as these have evolved in everyday life. Turing’s analysis of logic forwards just such a view. Our daily lives with ‘apps’ today evince and instantiate the perspective. They are Lebensformen.”

The aesthetic experience as well as constitution of the Wittgenstein’s Pinball Machine can be thought as an exemplification ( and not representation, if to use Goodman’s conception here) of Lifeforms. And as Floyd argues:

“Forms – logical and living ones – are linked, not ranked. In Wittgenstein forms have to do with procedures, routines, possibilities and necessities thereof – these are so ubiquitous. And so he self-inscribes and self-themathizes the forms of his own procedures in the PI. His remarks are deeply self-embedded in if-thinking, in possibilities of procedures or phenomena, in loops and pieces of proposed and explored chains of argumentation.”

Now, in order to fully engage with some of the pinball machine conceptual mechanics and its epistemological conundrums, below are some key citations from Wittgenstein which I am contextualizing here as a series of final reflections/refractions on the Wittgenstein’s Pinball Machine and its relation to toy-aesthetics.
In his Remarks on the Philosophy of Psychology, Wittgenstein remarks that:

“Turing’s Machine: These machines are humans who calculate. And one might express what he says also in the form of games. And the interesting games would be such as brought one via certain rules to nonsensical instructions. I am thinking of games like the ‘racing games’. One has received the order ‘Go on in the same way’ when this makes no sense, say because one has gotten in a circle. For any order makes sense only in certain positions (Watson)”. (RPP I : § 1096)

Does Wittgenstein talk here about his own sense or non-sense of a game such as a pinball machine where the interactions between the ‘game’ and its ‘accidents’ are expressions of Lifeforms?

In his Philosophical Investigations Wittgenstein poses the questions,

“Could a machine think ? Could it be in pain? – Well, it is the human body to be called such a machine. It surely comes as close as possible as being such a machine.” (P.I. § 359: p. 120)

Wittgenstein’s hostility towards psychology and in as much psychologism can be construed on two interconnected levels: the first pertaining to how we ‘assume’ pain (or pleasure) as non-mechanical as result of a false metaphysical theory about mentalistic constructs or language; the second an implicit affirmation or at least a neutrality in relation to A.I. and the possibility of a sentient machine not as a negation of human experience but rather as an explication of it.

In Philosophical Grammar Wittgenstein maintains the following:

“If one thinks of thought as something specifically human and organic, one is inclined to ask “could there be a prostatic apparatus for thinking, an inorganic substitute for thought?”. But if thinking consists only in writing or speaking, why shouldn’t a machine do it? “Yes, but the machine doesn’t know anything”. Certainly it is senseless to talk about a prosthetic substitute for seeing and hearing. We do talk of artificial feet, but not of artificial pains in the foot.” (P.G. § 47: p. 105).

One, following Wittgenstein, may argue the twofold contention. That ‘thoughts’ like ‘pain’ or ‘pleasure’ or any sentient constitution are Lifeforms and hence are not positioned below or above each other in a logical hierarchy. Rather, the both obey the “language-games” enacted by humans or machines. And then, that “seeing” or “hearing” like the experience of pleasure or pain involve consciousness to the “language games” as a content of experience, namely, Lifeforms.

In his Lectures on the Foundations of Mathematics, Wittgenstein insists,

“It has been said very often that mathematics is a game, to be compared with chess. In a sense this is obviously false – it is not a game, in the ordinary sense. In a sense it is obviously true – there is some similarity. The thing to do is not to take sides, but to investigate”. (Wittgenstein, Lectures on the Foundations of Mathematics, Lecture X (C. Diamond, 1976, p. 142).

Wittgenstein’s neutrality lends itself to the Pinball Machine as an experimental test as well as an aesthetic model of the interplay between logical rules (e.g. chess, pinball machine) and mathematics.

Wittgenstein, in his Philosophical Investigations contends the following on machines and humans:

“157. Consider the following case. Human beings or creatures of some other kind are used by us as reading-machines. They are trained for this purpose. The trainer says of some that they can already read, of others that they cannot yet do so. Take the case of a pupil who lias so far not taken part in the training: if he is shewn a written word he will sometimes produce some sort of sound, and here and there it happens ‘accidentally’ to be roughly right. A third person hears this pupil on such an occasion and says: “He is reading”. But the teacher says: “No, he isn’t reading; that was just an accident”.—But let us suppose that this pupil continues to react correctly to further words that are put before him. After a while the teacher says: “Now he can read!”—But what of that first word? Is the teacher to say: “I was wrong, and he did read it”—or: “He only began really to read later on”?—When did he begin to read? Which was the first word that he read? This question makes no sense here. Unless, indeed, we give a definition: “The first word that a person ‘reads’ is the first word of the first series of 5 o words that he reads correctly” (or something of the sort).
If on the other hand we use “reading” to stand for a certain experience of transition from marks to spoken sounds, then it certainly makes sense to speak of the first word that he really read. Pie can then say, e.g. “At this word for the first time I had the feeling: ‘now I am reading’.”
Or again, in the different case of a reading machine which translated marks into sounds, perhaps as a pianola does, it would be possible to say: “The machine read only after such-and-such had happened to it—after such-and-such parts had been connected by wires; the first word that it read was ….”. But in the case of the living reading-machine “reading” meant reacting to written signs in such-and-such ways. This concept was therefore quite independent of that of a mental or other mechanism.—Nor can the teacher here say of the pupil: “Perhaps he was already reading when he said that word”. For there is no doubt about what he did. —The change when the pupil began to read was a change in his behavior, and it makes no sense here to speak of ‘a first word in his new state’.”

Beyond Wittgenstein’s behaviorist contention about a reading machine and other kinds of humans turned into a reading machine there lies an extraordinary argument regarding A.I., namely, not how or if reading machines can read or think like humans — a certain mentalistic pitfall — but instead, how humans who read can be considered beyond doubt to be readers and only according to mechanical rules?  Can Wittgenstein’s Pinball Machine be seen as an integrative game/play that tests the interplay between mechanical functions and rules as reversible? To that Wittgenstein argues in Logical Investigation that the sheer act of playing a game does not necessarily define its meaning or rules but only constitutes its possible interoperation.

As Wittgenstein contends:

“198. “But how can a rule shew me what I have to do at this point? Whatever I do is, on some interpretation, in accord with the rule.”—That is not what we ought to say, but rather: any interpretation still hangs in the air along with what it interprets, and cannot give it any support. Interpretations by themselves do not determine meaning.”

Furthermore, Wittgenstein asks,

“‘Then can whatever I do be brought into accord with the rule?’—Let me ask this: what has the expression of a rule—say a sign-post—got to do with my actions? What sort of connexion is there here?—Well, perhaps this one: I have been trained to react to this sign in a particular way, and now I do so react to it.
But that is only to give a causal connexion; to tell how it has come about that we now go by the sign-post; not what this going-by-the-sign really consists in. On the contrary; I have further indicated that a person goes by a sign-post only in so far as there exists a regular use of sign-posts, a custom.
199. Is what we call “obeying a rule” something that it would be possible for only one man to do, and to do only once in his life?—This is of course a note on the grammar of the expression ‘to obey a rule'”.

Philosophical grammar is based on language-games but it does not have or is constituted by “rules” the way games are. Moreover, following the rules and obeying the rules are not the same procedures; let alone interpreting the ‘rules’ of a game which may deflect and define the game continuously. The Wittgenstein’s Pinball Machine is the exemplification of an aesthetic model wherein the possibilities of discerning the ‘rules’ of the game (toy) is exponentially inverted by the sensations of ‘pain’ and ‘pleasure’ as intertwined with the playing of the machine (toy).

For Wittgenstein, following a game’s rules imply in as much the possibility of unfollowing or even defying them and such apparent “paradox” (since he held that there are no real paradoxes) is indicative of the very nature of rules as lifeforms, logical and enacted “grammar” at once; much like Turing’s idea for a universal machines in which similarly the interaction with “rules” (e.g. simulating input) is always slower than their logical inferences (e.g. machine’s bits).

In Philosophical Investigations Wittgenstein observes that,

“201. This was our paradox: no course of action could be determined by a rule, because every course of action can be made out to accord with the rule. The answer was: if everything can be made out to accord with the rule, then it can also be made out to conflict with it. And so there would be neither accord nor conflict here. It can be seen that there is a misunderstanding here from the mere fact that in the course of our argument we give one interpretation after another; as if each one contented us at least for a moment, until we thought of yet another standing behind it. What this shews is that there is a way of grasping a rule which is not an interpretation, but which is exhibited in what we call “obeying the rule” and “going against it” in actual cases. Hence there is an inclination to say: every action according to the rule is an interpretation. But we ought to restrict the term “interpretation” to the substitution of one expression of the rule for another.”

On automata, Wittgenstein in Philosophical Investigations, makes a surprising claim:

“‘I believe that he is suffering.’——Do I also believe that he isn’t an automaton?

Even if someone had a particular capacity only when, and only as long as, he had a particular feeling, the feeling would not be the capacity.”


What happens to the player of a Wittgenstein Pinball Machine as the process of experiencing pain and pleasure (and moments of active indifference) is exceedingly associated with the game’s rules? Does such associative predicament renders one human as opposed to an automaton. Regardless to the game’s result, sentience and cognizance are relegated not as a capacity of either the machine or the player but instead to the way we construe the ‘rules’ as ‘logical’ or ‘syntactical’ compatible with being aware of the machine’s dynamics as analogous to lifeforms.

Wittgenstein’s Pinball Machine is a preliminary sketch for a game that exemplifies an aesthetic model conforming both to Wittgenstein’s notion of Lifeforms and to toy-aesthetics but it also involves complex dynamical systems which will be further explored in my next installment on the autopoiesis of toys.

Adorno, W Theodor. Aesthetic Theory
Copland, B. Jack. The Essential Turing Seminal Writings in Computing, Logic, Philosophy, Artificial Intelligence, and Artificial Life plus The Secrets of Enigma, Edited by B. Jack Copeland CLARENDON PRESS OXFORD 2004
Goodman, Nelson. Fact, Fiction and Forecast
Goodman, Nelson. Languages of Art: An Approach to a Theory of Symbols
Goodman, Nelson. Ways of Worldmaking
Juliet Floyd Chains of Life: Turing, Lebensform , and the Emergence of Wittgenstein’s Later Style in Nordic Wittgenstein Review 5 (2) 2016
Turing, A. M. (1936). On Computable Numbers, with an application to the Entscheidungsproblem, Proc. Lond. Math. Soc. (2) 42 pp 230-265 (1936); correction ibid. 43, pp 544-546 (1937).
Turing, A. M. (1950). Computing machinery and intelligence. Mind, 59, pp. 433–460.
Turing, A. M. (2004). Can digital computers think?, in The Essential Turing, edited by B. J. Copeland, Oxford, Oxford University Press.
Wittgenstein, L. (1958). Philosophical Investigations
Wittgenstein, L. (1976). Wittgenstein’s Lectures on the Foundations of Mathematics, Cambridge, 1939, ed. Cora Diamond, Chicago – London, The University of Chicago Press.
Wittgenstein, L. (2001), Remarks on the Foundations of Mathematics, Third edition, Rush Rhees (Editor), G.H. von Wright (Editor), and G.E.M. Anscombe (Editor, Translator), Oxford, Wiley- Blackwell.

Euclid’s Elements, a philosophical thriller (part 1)

Among the greatest mathematical treatises in antiquity and beyond, no title matches Euclid’s Elements in simplicity, elegance, popularity and sheer hair-raising brilliance of analytic imagination. It is a book that is accessible to any person who wishes to initiate into that fathomless realm we call mathematics. But the same thing can be said about Elements‘s philosophical depth. Elements is in fact a book in which the boundaries between mathematics and philosophy completely fade. In this marriage between philosophy and mathematics via the geometric method, we see a form of intuitive mathematics whose results are both sophisticated and non-trivial even in terms of modern mathematics, and a philosophy which points towards possibilities of formal and systematic thinking.

The aim of this series is to dissect the tissue between geometrical and philosophical problems and tropes in Elements, using concepts and ideas situated at the nebulous interstices between philosophy, logic and mathematics. The first few installments will focus on the overall characteristics of the Euclidean universe. But as we proceed, we will shift the attention toward analysis of particular examples.

If you remember the school days, you recall your teacher treated Elements in terms of analytic geometry alone. But Elements is equally a work of philosophy. While it is quite controversial to claim that Euclid was a Platonist, we can imagine that the philosophical climate during the life of Euclid was saturated with Platonic ideas, above all the doctrine of forms or ideas. Even though Euclid might not be at the end of day, a platonist, he nevertheless is preoccupied by the same philosophical concerns which preoccupied the Plato of the late period, particularly the Plato who revised his early doctrine of forms beginning with Theaetetus and brought it to maturity in Philebus.

Suffice to say that the very core of Elements deals with the dialectics between universal forms and fleeting particularities. But this dialectic as I will elaborate is not Aristotelian insofar as it involves something more than the existential interpretation of mathematics or to be more specific, analytical geometry—i.e. the correlation between the genus and species—where all the proof of a general concept demands is finding or constructing a particular instance that can be subsumed under the general concept, like this rectangle and rectangularity as such. The framework in which the Euclidean oscillations between particulars and universals is expressed is what Plato calls craftsmanship or world-building which is an enterprise undertaken by the mind using logoi. The mindcraft draws as its raw material physical becoming which is only endowed with forms of elemental powers and no higher forms. The process of the craft itself proceeds by way of patterns. A pattern is not, however, a thing, but that by which a thing is structured, made or designed. Moreover, patterns are not discrete. For at each level in the hierarchy of particularities and universalities (e.g., these straight segements-cum-acute angles, this triangle, this particular kind-of-traingle and triangularity as such), there are such uniformities as patterns mainly by virtue of how—rather than merely what—things hang together—that is, the question of structure as the designation of Being.

Yet patterns are not exhausted by how and what things hang together, for they can be patterns of how different patterns hang together. For example, think of Book 1, propositions 2 (henceforth, I.2) in Elements which we will have the occasion to examine in the next installments. In this demonstration, you require not only the patterns by which lines hang together, but also how circles and a straight segment hang together (for the purpose of constructing an equilateral triangle). Furthermore, you should know how different circles, and the vertices of an equilateral triangle—i.e. composite patterns and simple patterns—can hang together in the right way so as to build a diagram that demonstrates the proposition.

Source: Wilfrid Sellars, The Soul as Craftsman

From this brief discussion on the world of Plato-Euclid mindcraft, we can conclude that the process of the craft consists of sensory stuff, patterns after which things are made, patterning patterns (patterns for organizing and structuring other patterns which are greater wholes) and recipes which are instructions concerning how and what patterns pertaining to what material ingredients and/or lower patterns should be mixed together. But the objective of a recipe is to make a product that can in turn be incorporated as an ingredient into another recipe. Therefore, in addition to the above components, there should also be something like a craft test or demonstration whereby the final product can be validated as a new ingredient in the process of craft.

Thus recipes are, broadly speaking, objective principles or practical intelligibilities which have as their ingredients even theoretical intelligibilities as well as more mundane ingredients (e.g., sensations, material things which might be in fact the products of other materials-cum-forms-cum-recipes such as a tanned leather with tumble finishing or in the case of Elements, an equilateral and equiangular pentagon).

Source: Oliver Byrne’s edition of Euclid’s Elements (II.7)

In short, recipes whose equivalents in Elements are procedural diagrammatic constructions represent the engines of the craft by which not only we can make things but also, demonstrate how materials, products, and even single recipes hang together such that the ensuing craft is a universe—a world-soul—in which all (spatial) relations between things (particular instances) and forms (universals), or forms and forms are articulated and rendered intelligible. But this resulting craft or universe can also be imagined as a universe in which ever more complex forms or higher mixtures (to mikton) can be made. An apposite metaphor for this universe, is a river whose source is a mountain. The limits of the mountain is the earthly ground and a given sky which is demarcated by the snowy peaks. Even though the river’s origin is limited by material sediments and heavenly forms—the melting snow—the river soon finds its path along the geodetic path to the sea where strange fauna, forms and adventures await us. But the course of the river is always tortuous as it passes through forests of intermediary forms before it shapes estuaries where the tidal waves of complex forms and discrete instances and patterns meet together. This is nothing other than Plato’s vision of the revised doctrine of ideas—the craftsmanship of the soul—where the craft of the mind coincides with a new bottomless expanse of forms. The possibility of constructing a new world or a nested hierarchy of forms from the limited resources of the existing world is the sure conclusion of this vision.

In this respect, the recipe or the ongoing instruction regarding how to navigate between the particular and the universal, local and global has something more than just material ingredients and forms. The recipe consists of elementary ratios and proportions which in Euclid’s universe can be compared with principles in Elements which are common notions (principles1) and postulates (principles2) which respectively signify undergirding assertions and elementary construction recipes. Common notions are quantitive assertions or intuitive axioms such as ‘Things which are equal to the same thing are also equal to one another.’ Postulates, on the other hand, instruct certain kinds of elementary constructions like Postulate 2 that states, ‘To produce a finite straight line continuously in a straight line.’ Whereas, the relations between principles1 and theorems are deductive to the extent that the truth of the conclusion is contained in the truth of the premise, the relations between principles2 and problems are not deductive for the construction cannot be considered as a deductive inference from the postulate.

Moreover, the focus of a recipe is not restricted to pure construction. The idea of craft as Plato has suggested also entails a function called ‘limiting’ (to peras). In Theaetetus, Plato speaks of a function that ‘freezes or fixes the flux of things’ (183a7), or ‘make things stand still’ (157b7) and limits that which is unlimited, or more precisely, indeterminate (apeiron), thus bringing it into determination and intelligibility. This limiting or determining function is attributed to that of language and logos and is closely associated with measure (metron) which depending on the context can be epistemological, ontological or axiological. In the epistemological context, metron signifies the quantification of the apeironic flux or the continuum of greater-and-smaller into intelligible degrees or grades (e.g., being hot, warm, lukewarm and cold, or being extended this-such and being extended that-such). It is precisely the study of this limiting function that later on via the influence of neo-Platonists on scholastic philosophy culminates in Nicole Oresme’s work (Tractatus de configurationibus qualitatum et motuum) on diagrammatic configurations known as latitudes of forms—intensive and extensive elaborations of qualities— which in turn paved the road for articulation of differential equations of motion that scaffolded the revolutions of Copernicus and Kepler.

It is, however, important to realize that quantification for Plato so as for Euclid is not exclusive to the domain of numbers but can also include geometrical-spatial extensions. Once the limiting or determining measure in the latter sense (e.g., line as the limit1 of surface, or the definition of angle as the limit2 of its construction) is established, we can derive determining spatial relations between determined or limited geometrical figures. Only when such determinate spatial relations are obtained, a diagram can be constructed on previous diagrams so that we can move from one proposition or problem to another.

Finally, in addition to the recipe, there should be such things as craft tests or in Euclid’s world, demonstrations. If the Euclidean construction is understood as effecting what we aim to effect via diagrams, demonstration can be thought as a stepwise procedure for confirming that the construction has indeed effected what it says it has. Throughout the course of demonstration that covers every step of the construction rather than only the final result, tests can be executed either as objections (enstasis) or cases made against the current construction or diagram. If the former i.e. objection wins, the entire construction is null and void. But if the case—which can be understood as a diagram model that serves or effects the same purpose in a different context—wins, the construction is not necessarily erroneous, since it might prove or demonstrate the same thing in another diagram or geometrical context (allos).

Moreover, demonstrations are applied to two different aspects of the diagrams or products of the craft:

(1) those attributes of diagrams which pertain to participation (methexis) of elements or part-whole relationships. Such methexis-related aspects can include mereological relationships between regions, and segments or lines which demarcate boundaries as in the case of the notorious diagram in I.1 where two circles whose centers are the two endpoints of the same straight segment should intersect at exactly one point. But there is no explicitly stated rule in Elements guaranteeing that such a configuration would invariably result in an intersection point. Imagine circles made with lines with different breath or thickness, or made of squiggly lines. The result won’t be guaranteed to yield an exactly one intersection point. Yet if we see the implicit desideratum of intersecting circles in terms of how the components should hang together from a mereological perspective we can say that given such and such regions and boundaries appear to participate mereologically, the two circles should in fact intersect.

(2) The second aspects are what can be dubbed as analogical (analogon) attributes in the sense that Plato defines them (‘ana ton auton logon‘), namely, ratios, proportionalities and the equality of non-identicals. Whereas methexis-related aspects are based on the appearance of diagrams, analogical aspects are not concerned with how diagrams look like.

Attributes (1) and (2), therefore, roughly correspond to what Kenneth Manders in Diagram-Based Geometric Practice calls exact (analogical aspects) and co-exact (methexis-related aspects of diagrams) attributes.

The final products of the craft—i.e. constructions which have withstood demonstration or validation—are mixtures (mikton) or determinate complexes which are demonstrated diagrams. Only once such mixtures are available, it is possible to use them as ingredients of another craft or construction.

At this point, it is perhaps necessary to make a brief point about the nature of Euclidean demonstrations as Platonic craft tests. So far I have used the words demonstrations and proofs interchangeably. But demonstrations are not exactly proofs in a technical modern sense—only in the very loose sense of proof (we will return to this point in next installments). Furthermore, even the word demonstration is not accurate for describing the system of Euclid. The phrase quod erat demonstrandum not only should not be translated as that which was required to prove, but also itself is an inaccurate Latin translation of the Greek verb deiknumi whose precise translation is the Latin monstrare i.e. to show or expose. In Second Analytics, Aristotle fully distinguishes deiknumi as an informal and epistemological investigation from apodeiknumi or apodeixis (proof) which has an exact connotation within the lexicon of syllogistic logic as an inference that draws certain conclusions from certain premises. While this comment might appear as a petty etymological indulgence, as Andrei Rodin has detailed in The Axiomatic Method and Category Theory, it indeed has a significant implication given Euclid’s own remarks and Proclus’s commentary on Elements. The difference between monstration (Euclid’s focus) and demonstration vis-à-vis proof suggests that we can arrive at sound and non-trivial results in mathematics without relying on an axiomatic method in the sense we understand it today. Even the Euclidean givens (data) are not exactly formal axioms since not only they are underdefined / undefined but also not all the rules for building on the data are explicitly stated.

Within this framework, we can now see that the genius of Euclid’s Elements is not as much in devising new feats of proof and demonstration as it is in setting up a generative space—a unified process of craft—that accommodates all previous works done in analytic geometry.
1. Plato-Euclid’s World of Mindcraft

                                                     │ Mindcraft │────────────────────1────────────────────┐
                                                     └───────────┘                                         │
    ┌───────────────┬─────────────┐                                                                        │
    │               │             │                                                                        │
    │               │             │                                                                        ◁──────────┐
    │    Go to 1    │    Go to 2  │                                                                        │          │
    │       △       │       △     │                                                                        │          │
    │       │       │       │     │                                                                        │          │
  .───.   .───.   .───.   .───.   │                                                                        │          │
 ( Yes ) (  No ) ( Yes ) (  No )  │                                                                        │          │
  `───'   `───'   `───'   `───'   │                                                                        │          │
    △       △       △       △     │                                                                        │          │
    └───┬───┘       └───┬───┘     │                                                                        │          │
        │               │         │                                                                        │          │
 ┌─────────────┐ ┌─────────────┐  │                                                                        ▽          │
 │             │ │   Testing   │  │  ┌──────────────────┐          ┌──────────────────┐          ┌──────────────────┐ │
 │             │ │  against a  │  │  │  Sensory stuff   │          │Elementary Recipes│          │  Basic Patterns  │ │
 │             │ │case (a proof│  │  │                  │          │                  │          │                  │ │
 │Withstanding │ │ of another  │  │  ╠══════════════════╣◁───3─────╠══════════════════╣◁───2─────╠══════════════════╣ │
 │  Objection  │ │  diagram /  │  │  ║   Intuitive or   ║          ║   Principles:    ║          ║   Definitional   ║ │
 │ (enstasis)  │ │ alloos, or  │  │  ║    Perceptual    ║          ║  Common Notions  ║          ║  Givens (Data)   ║ │
 │             │ │demonstration│  │  ║   Ingredients    ║          ║    Postulates    ║          ╚══════════════════╝ │
 │             │ │ in another  │  │  ║   of Diagrams    ║          ╠══════════════════╣                    │          │
 │             │ │  context)   │  │  ╚══════════════════╝          ║Limits (to peras) ║                               │
 └─────────────┘ └─────────────┘  │            │                   ║Determinations of ║                    │          │
        △               △         │            │                   ║   geometrical    ║                              12
        │               │         │            │                   ║     figures      ║                    │          │
        │               │         │            │                   ╚══════════════════╝                    ?          │
        └───────┬───────┘         8            4                                                           │          │
                │                 │            │                                                                      │
                7                 │            │                                                           │          │
                │                 │            │                                                                      │
                │                 │            ▽                                                           │          │
      ┌──────────────────┐        │  ┌──────────────────┐          ┌──────────────────┐          ┌──────────────────┐ │
      │     Craft or     │        │  │Craft of Mixtures │          │ Final Product of │          │    Patterning    │ │
      │Construction tests│        │  │                  │          │    the craft     │          │ Patterns (logoi) │ │
      ╠══════════════════╣        │  ╠══════════════════╣          ╠══════════════════╣          ╠══════════════════╣ │
      ║  Demonstrations  ║        │  ║   Diagrammatic   ║    ┌────▷║   Demonstrated   ║────10───▷║                  ║ │
      ║                  ║        │  ║ Construction of  ║    │     ║   Mixtures or    ║          ║   Determinate    ║ │
      ║  attributed to   ║        │  ║ Complex Diagrams ║    │     ║   Geometrical    ║          ║Spatial Relations ║ │
      ║different aspects ║        │  ║                  ║    │     ║    Complexes     ║          ║                  ║ │
      ║   of mixtures:   ║        │  ╚══════════════════╝    │     ╚══════════════════╝          ╚══════════════════╝ │
      ║                  ║        │            │             │               │                             │          │
      ║ methexis aspects ║        │            │             │               └──────11──────┬──────11──────┘          │
      ║       and        ║        │            │             │                              │                         │
      ║ratios-proportions║        │            │             9                              ▽                         │
      ║     aspects      ║        │            5             │                    ┏━━━━━━━━━━━━━━━━━━┓                │
      ║                  ║        │            │             │                    ┃ Complex Patterns ┃                │
      ║                  ║        │            │             │                    ┃   and Products   ┃────────────────┘
      ╚══════════════════╝        │            │             │                    ┃   (to Mikton)    ┃
                △                 │            ▽             │                    ┗━━━━━━━━━━━━━━━━━━┛
                │                 │  ┌──────────────────┐    │                              │
                │                 │  │Raw Product of the│    │                              ▽
                │                 └─▷│      craft       │────┘                    ┌──────────────────┐
                │                    ╠══════════════════╣                         │  Building Loop   │
                │                    ║  Undemonstrated  ║                         │ (Repeat 1 to 12) │
                │                    ║   Mixtures or    ║                         └──────────────────┘
                └─────────6──────────║   Diagrammatic   ║                                   │
                                     ║    Complexes     ║                                   ▽
                                                                                  Euclidean World-soul ?
                                                                                   (psyche ton pantos)

Having gone through this brief introduction, we should now ask: what is exactly Platonic about the universe of Elements? Absent a more detailed response, the above introduction—particularly, the comparison between the role of construction in Elements and the process of craft in the late dialogues—would be hardly anything other than an impressionistic account. Yet to answer this question, it is also imperative to suspend some of the most dogmatic clichés about the work of Plato inherited from the misinterpretations of Aristotle and neo-Platonists (e.g., the Third Man, the equivocations of ideas with numbers a la Pythagorean arithmosophy and the misrepresentation of the Good as the divine demuirge) as well as their almost exclusive attention to the dialogues of the early and the middle periods. Plato is notorious for being the most watchful and unforgiving critic of himself. So the answer simultaneously calls for a direct engagement with the dialogues, particularly, the later ones and a critical correction of Aristotelian-neoPlatonic commentaries which make almost the entire body of Platonic studies until the late nineteenth century—a trend that comes to an end with the rise of Marburg, Tübingen and analytical schools of Platonic studies as represented by figures such as Natorp, Reale and Vlastos.

We know that after the second trip to Syracuse, Plato became critical of his early doctrine of forms (e.g., Parminedes) as represented in the works of the middle period such as The Republic. He began to see forms as classificatory universals, namely, categories or ta koina (see Theatetus and Sophist). As ta koina, forms or ideas no longer have the earlier characteristics of the Socratic and Pythagorean theories of forms, or at least such characteristics are not prominent anymore. The inception of this new doctrine of forms or ideas begins with the transitional dialogue Theatetus, but it is only in Philebus that Plato gives a complete account of his new doctrine.

According to this new thesis, the aim of the doctrine of forms is Demiurgen, world-construction or craftsmanship of the mind. In Timaeus, we are dealing with god as the Demiurge but in Philebus, this abstract divinity is suddenly replaced with a neutral word, to demiurgen. It is now the human mind that is akin to the good which is beyond all gods and beings and even truth and beauty, and not the god as the ideal of the nous. This manifestation of the good is like a recipe or an objective principle for building worlds. It is a recipe precisely in the sense that Wilfrid Sellars talks about a recipe for making a cake, a recipe consisting of theoretical and practical intelligibilities.

If you have made a cake from scratch, you know very well that it is not an easy task, since a recipe for making a cake—unlike a recipe for making a soup—involves precise ratios, proportions and stages of how and what elements should be added together. The formulas of this recipe are what called objective principles or rules as in contrast to the social nomos or conventions. To build a house as a shelter (the external purpose of the construction), we ought to abide by such and such principles like taking care of the foundation, beams, etc. The specific formula of how we lay the ground or what beams—made of out of what materials—we use might change over time, but the objective principles endure. A house needs a foundation and a ceiling even if the foundation is bottomless and the ceiling extends to the sky. These principles pertain to the domain of forms or ideas. While the nomos is always prone to corruption (as in the case of the codes of building issued by a corrupt builders guild which dictates that all houses should be built out of the material ingredients over which it has sole monopoly), objective principles are genuine objects of rational examination and revision.

Parallel to this Platonic account, the fleeting shadows on the wall of the human cave could not even be recognized if some dim light was not present in the cavern. This light is not a literal analogy for purity, it is rather a metaphor for intermediating forms or universals, the mathematicals or analytic idealities. These are construction principles which intermediate between pure ideas and eikones or sensory shadows. In this sense, Plato is the enemy number one against the myth of the given, for he thinks that the structuring factor is not within the domain of sensory fluxes—the fleeting shadows or eikones—but in the dim light i.e., intermediating forms which imitate the light of the sun qua pure forms or generalized structures: that is to say, mind as the dimension of structure.

In Philebus, Plato makes that claim of impiety for which Socrates was executed. He says the human mind is akin to the Good. We know that what Plato means by the Good in Philebus is the principle of structure (the kernel of intelligibility and intelligence which is even more fundamental than truth, beauty or justice). A few pages later Plato tops up his thesis with a new claim, ‘and the good is beyond all being’. In other words, Plato suggests the structure—or the mind as a configuring or constituative element—is the very factor by which Being comes to the fore and can be talked about coherently. Plato’s articulation of Being in terms of intelligence or mind is quite similar to the view of the mature Parmenides who has relinquished the early Eleatic confusion of Being and thinking, and instead interprets the thesis of ‘Being and thinking are one’ as thinking or structure being the very designation of Being. To speak of Being without the dimension of structure or mind is the apotheosis of sophistry and the aporia of the unintelligible (cf. Lorenz Puntel’s Structure and Being).

However, the dimension of structure or in Plato’s terms the limiting (to peras) is not an index of solipsistic idealism, for it requires a fourfold view of the universe qua structure where episteme not only gains traction upon an external world but also thoughts or more generally, intelligence (nous) is no longer passive. Intelligence is now defined in terms of what it does—the unfolding of the intelligible even that of itself or the enrichment of reality—and not in terms of passive receptivity of an external reality. Accordingly, the Platonic fourfold view is defined in terms of an activity called craftsmanship whereby through various ingredients, structuring factors (logoi) and principles (dialectica), intelligence makes itself and reveals the intelligible dimension which is that of Being. But insofar as there is no a priori limit to the intelligible, there is no limit to the self-cultivation of intelligence or the poesies of mindcraft either. The twist in this scenario is that the mindcraft or intelligence posits —qua an active rather than a passive factor—what is intelligible. But it also has the capacity—as Rosemary Desjardins elaborated—to posit (tithemi) a new kind of reality pertaining to both Being and itself (see Plato and the Good, p.61).

The Platonic fourfold as presented in Philebus is nothing but a new interpretation of the analogy of the divided line in The Republic. The divided line is a diagram of how global conditions of thinking, action and value can be related to the local conditions. It consists of four segments which give us four domains with their corresponding modes of cognition/sensation, episteme (knowing) and their objects. From segment one to the segment four we have eikasia (eikones), aisthesi or pistis (aisthêta), logos dianoia (mathêmatika) and epistêmê (ideai).

The genius of this diagramatic analogy is in identifying the extreme segments (segments 1 and 4) under two modes of relations to time. The true forms or ideai are timeless or time-general whereas the sensory eikones are time-specific or temporal. In a sense, the divided line is about how what is timeless connected with what is temporal, how the oneness is mixed with the multiple, or how pure forms gain traction upon and are connected with the sensory shadows. The answer lies in the intermiadting domains or segements which are represented in the divided line as mathêmatika and aisthêta / pistis.

So what is the significance of these intermediating levels? Recall that sensory fluxes of eikones or imagistic impressions are too transitory to be arrested as anything you might call a sensible object. Pure ideas in a similar vein are too detached from particularities to gain traction, by themselves, on the worldly or the cavernous affairs. Another problem is the question of how oneness (of pure forms) as an organizing principle comes into contact with the multiplicity of things (i.e. sensible objects). The ideas are multiples but individually each idea is always a unique kind or type of form (i.e. it is one). On the other hand, eikones or what you might call registers of the apeiron—that is, the indeterminate and transitory flux of smallers and greaters. At the level of the first segment which is that of eikasia whose objects are the fleeting imagistic impressions eikones, there is no such a thing as multiplicity of things. Why? Because even multiplicity of things require a principle of unification. It is only when we organize the fleeting sensations as the affects impinged upon us by one and the same object (here, the object is the higher principle closer to ideas or formal constitution) that the fleeting sensible shadows become multiple things, this shadow-puppet, that shadow-puppet, etc. So the question of multiplicity does not even arise at the level of pure sensation. It only arises at the level of opinions or dogmas regarding the appearance of objects. In otherwords, it is only when the mind posits a thingly whole (object or in Kant’s sense gegenstand) which binds together different physical properties that we can talk about multiplicity of either properties or sensible things. The following quote by Desjardins should shed some further light on the matter:

For, on the one hand, a physical object seems to be distinctly different from any or all of its properties: they are quite separate kinds of things; on the other hand, what is exactly a physical object over and above its physical properties? While there is no difficulty in thinking of a phys­ical object that has no actually perceived properties, our notion of an object seems nevertheless to be such that it does not make sense to talk of a physical object that has no perceivable properties: such a notion of a bare particular seems incomprehensible. This of course, only exacerbates the question, however, for what then is the rela­tion between an object and its properties? We seem to be hoist on a dilemma in which, on the one hand, we want to say both that, in some elusive sense, the object and its properties are different, and that, in no less an elusive sense, they are somehow the same; and on the other we want to say that the object is neither simply the same as, nor simply different from, its properties. But, as the Parmenides suggests, if the relation between two things is neither sameness nor difference, then perhaps it is that of whole and part (l46b3-5). Plato’s model for such a relation does seem in fact to be what he conceives of as a whole of parts, where on the one hand, the whole is nothing other than the parts (there is nothing added to the parts), on the other, the whole is indeed other than (i.e., more than the sum of) its parts. In short, while a whole is analyzable into its parts, it is not reducible to those parts. Thus as I understand Plato, while a physical object is analyzable into its physical properties, it is nevertheless not reducible to those properties.

Thus, the whole of the sensible object—like the moving shadow on the wall—we can conclude, is not given by sensory fleetings, but is in fact the product of what can be called transcendental constitution—a semblance of what Plato calls intermediary forms qua mathematicals. Therefore, the multiplicity of the physical furniture of the world is not given to us through sensory eikones, it is engineered—a la positing a new kind of reality—by the semblance of the higher principles which are mathematicals qua objects of logos dianoia.

But now new questions raise their heads: What are mathematicals and what is their role? How are they related to the principle of the Good as the enrichment of reality? Can the Good as the constitutive principle be reduced to a number (arithmosophy) or is it an ideal numbering principle which delimits the bounds of reality (i.e. the proper object of epistemology and metaphysics)?

I will answer these questions in the next installment, until then ciao.

We are all afraid of philosophy of science!

Before I post the first installments in the Euclid and autodidacticism series: For those of you who might be vaguely interested, I will be teaching a course on the history of philosophy of science in the twentieth century, covering such figures as Carnap—the polite and friendly intellectual tank who rolls over all—and Grünbaum, that great debunker of all mysteries of space and time. A detailed description can be found here:

Theory and Object: Philosophy of Science in the twentieth century, from Carnap to Grünbaum

Induction again

In case you have not seen this already, I have written a piece on Hume and the problem of induction, or to be more precise and less general, the problems of inductivism:

Three Nightmares of the Inductive Mind

Hopefully—space allows-an extended version of this piece covering more ground regarding Putnam’s argument against the possibility of a Universal or optimal learning machine, and Solomonoff’s formal account of Occam’s razor.

Nevertheless since writing this text, I have come to the conclusion that there are some problems particularly with regard to precision. For example, a less serious issue is my rather dodgy treatment of Carnap’s view on a formal learning machine in his magnum opus Logical Foundations of Probability. I have gone along with Putnam’s argument but the issue is that Putnam’s challenge in Inductive Logic and Degree of Confirmation, and in his address to Radio Free Europe (Probability and Confirmation) are not false. They attribute a view to Carnap which is not accurate. In other words, Putnam takes the scope of a formal learning machine—one that Carnap mentions toward the end of his book—as far broader and more ambitious than what Carnap thinks to be the case.

A more serious problem is the one recently mentioned to me by my friend Adam Berg: That Goodman’s new riddle (the problem of projectibles) and Putnam’s take on the problem of induction differ on a fundamental ground and cannot be treated as if they were both tackling the same problem of induction. In one case, the problem is explicitly dealing with observations or empirical statements, whereas in the other case—i.e. Carnap’s inductive logic which is the target of Putnam’s critique—such observations are absent.

In the case of the latter, we do not have simple observational statements. All we have are logical statements. Even the e-statement in c(h, e)=r is only a reference—within the framework of an inductive logic—and not an empirical observation per se. To this extent, one should be cautious to use examples like Raven or Grue paradox (i.e. explicitly observation-based inductive paradoxes) to challenge Carnap’s inductive logic. As a resolution and a mediation between the two views, Adam has asked me to look into Reichenbach’s paradigm of induction as vindication. In order to that, I need to revisit Reichenbach’s texts on this subject. Yet I think there is still a shadow that haunts even Reichenbach’s paradigm of induction whose flaws are spectacularly—albeit inadvertently—highlighted in Sellars’s essay Induction as Vindication. This shadow is the problem of simplicity or elegance. More accurately, it is the problem of an unconstrained account of simplicity whose espousal demands a metaphysical high-price (e.g., see Grünbaum or Rescher’s critiques). Even the formal conception of simplicity has its own incoherencies which are addressed in my piece.

Toy Philosophy Universes (part 2)

In the first installment on toy philosophy universes, I gave a rudimentary account of one of the main motivations behind this series: the problem of stepping outside of the model or the system an agent inhabits or broadly speaking, the metatheory of theorization.

To be candid about it, I do not think that philosophy or for that matter natural sciences, mathematics, logic or even theoretical computer science are, by themselves, capable of offering an adequate solution to this problem which for now can be dubbed as the transcendental jailbreak (in reference to Wittgenstein’s Prison and my previous comments on the Kantian straightjacket).

If there is a solution to this problem, it is in a non-trivial integration of all the above fields of thought. Non-trivial in that none of these fields can be subordinated to or assimilated by one another. That is to say, for philosophers attempting to tackle this problem there is no other option other than integrating and rendering contemporary the discipline of philosophical inquiry with sciences (complexity sciences in particular), mathematics, logic and computer science. Indubitably, through the course of this upgrade and revision, the very nature of philosophy as a discipline will transform: We begin to see the phantom-like apparitions of what from the perspective of here and now might have only some vague and negligible resemblances with what we currently characterize as philosophy.

The future philosophy—even as a Platonic eidos—cannot be anything but a program for thinking globally about thinking about the world, migrating—step by step—from the conceptual system which undergird our local conception of the world to a metalogic of such conception, adopting a view that no longer bottoms out in our particular (multi)perspectival view of the world.

Now allow me to return to what I characterized in the previous post as a chronic cognitive voyeurism, that is, a child-like fascination with the implicit know-hows and know-thats behind our attempts at forming a theory, model or conception of this or that aspect of the world. What are the questions I ask myself when confronted with a theorist’s output? Some of the immediate question are ‘What kind of implicit system of search and assembly do they use when they work, what does their toolbox of methods contain, what reasoning or cognitive mechanisms (analogical, deductive, etc.) are being activated, and more importantly, can all of this be modeled, can it be replicated or implemented in another context? What I wish to know are not only the very mundane habits of thinking, writing, note gathering, etc. but also and above all, the metalogic of one’s logic of theorization or more generally the implicit thoughts which go into one’s explicit thinking about the world…and ultimately, how all of these things fit together, how much—if at all—and at what level do they influence one another.

Yet this exploration of the metalogic of one’s logic of the world is anything but a straightforward affair. It requires an understanding of not only how we model the world but also what it takes to go beyond that model while suspending the model-biases and more importantly, the dogmas of our particular transcendental type or perspectival (or intuitive in a Kantian sense) resources. With these cursory remarks let us begin this series:

0. Welcome Back to Kindergarten

Are you sick of philosophical -isms? Do you believe that there are so many rival and incompatible philosophical views that they almost inevitably will lead you toward feud-ridden tribalism? Are you tired of being a professional philosopher? Do you feel as if the discipline of philosophy as it stands today has betrayed our initial ambitions and excitements? We signed up for expansive cognitive exploration, but instead we ended up in pigeonholed tunnel-visioned analyses or worse, some all-over-the-place theses which purport to be impersonal but are in fact fanatically personal and simplistically psychological through and through? We either succumb to the monism of methods and models or its pluralistic twin which is in reality a relativistic soup with little to zero consistency. Do you often dream of being once again a child-philosopher rather than a jaded adult scholar? Do you approach your models or conceptions of the world as toys which can be discarded or broken in the real world but not until they are sufficiently played with or do you see them as mature completed narratives? As a philosopher or theorist, which one is the universe you live in, a toy philosophy universe where endless constructibility, experimentation and rearrangement of multiple models are the norm or an elegant fully-completed house where you as an adult have finally settled? Lastly, do you think that the pedagogy of the philosophical discipline is responsible for how we think about the world. If the answer is positive, then given the current philosophical pathologies, how should we reconcile the discipline of philosophy with its education?

The bad news is that—and there is always only bad news—this series attempts to tackle these concerns with questionable or no success. But we as philosophers and theorists are in the business of epistemological risk and theoretical humiliation. We neither take the failure of a hypothesis as a negative outcome or an irrefutable evidence that the failed hypothesis will invariably fail in every context, nor do we conflate the unreachability of long term objectives from our today’s perspective as a good argument against our attempts to systematically and concretely entertain such objectives. To this extent, in this and future installments, I precisely tackle with such questions. The aim is to elaborate the concept of toy philosophy universes as a partial answer to the above questions.

For now, this elementary definition should suffice:

Toy philosophy universes are a specific class of formal philosophical systems which are explicitly metatheoretical or metalogical. They are primarily characterized by their commitments to the constructibility, manipulability, rearrangement, plurality and hierarchization of models and methods (i.e. toy-like) in frameworks where formalism and systematicity come hand in hand. To call them toy means that their principal emphasis is on world-building rather than world-representation. It is not that the world-building is divorced from world-representation, it is rather the case that the relation between the two changes in toy philosophy universes. The aim of world-building or to adopt Carnap’s term aufbau—more in the vein of construction than mere constitution—is to at once (1) deepen our understanding of our various discourses (thinking about thinking) about the world even in spite of existing evidence, and (2) expand any universe of discourse—and so correspondingly, truth-claims possible within that universe of—beyond its given scope and established assignments. Calling such constructs philosophy universes means that they are concerned with an unrestricted universe of discourse covering claims that can be theoretical, practical, axiological or aesthetic.

However, to methodologically reach this definition through which we can finally in a coherent manner tackle the aforementioned questions, we must first engage with a whole slew of related questions: What are toys? What are models? What does the contemporary science of modeling involve? How can the praxis of philosophy be informed by the science of modeling? What are toy models? In what respects big toy models differ from small ones? Is there a set of canonical formalization for such models for the purpose of exact reconstruction and reimplemtation within a context that is not predominantly scientific? How can we see both theoretical and metatheoretical assumptions as necessary to the labor of modeling? What does exactly differentiate toy philosophy universes from big toy models? What are the implications and outcomes of living in a toy philosophy universe as opposed to a purely scientific one?

In a nutshell, we cannot investigate what it means to step outside of our theoretical models unless we first examine what modeling, theorization and metatheorization entail. Given the above list of questions, it should be clear by now that the path this series takes is going to be circuitous and hazy. The first few posts will be introductory and light, but as we move forward indulgence in technicality and formalism will become inevitable.

I will engage, first of all, with toys, elaborating on the idea of ‘toying around with our models of the world’ using examples derived from the history of pedagogy and engineering. Next, I shall focus on the science of modeling, the principles behind how we scientifically model an aspect of the world. Subsequently, I will move to the domain of toy models and so on.


We all remember a moment in our childhood when toys were our surrogate parents, far more generous, interactive, manipulable, cooperative and informative than our parents. Perhaps, toys first—and not the preachings of our adult guardians—made us realize that there is a world out there, a world that despite its malleability is constrained by what we eventually learned is called objectivity. Recall those nights when we chose the company of toys over adults, we chose to sleep in a tent made of a few pillows simulating the environment of a universe brimmed with possibilities. There was a mountain outside—a cardboard box covered with a brown satin. The meadow inside the tent was comforting and smelling nice. But it was an an old smelly green blanket shrunken and wrinkled after being washed in hot water for many times. In that very tent, we waged war against three metal pencil sharpeners which looked like three thousand armored cavalry units. We were at the end triumphant. The three colored pencils staved off the advance of the metal sharpeners after much sacrifice. They are now far shorter than what they once were. The remaining forces are currently in an eternal alliance. They are the people of this tent which I call my world. After we concluded the battle, we fell asleep dreaming of a bigger tent with ever more new alliances, new friends. But the peace did not last long, for soon a flying saucer carrying an army of disfigured teaspoons delivered a cryptic message, ‘there is a world out there even larger than your toy universe’.

Among the greatest educationists, from Friedrich Fröbel to Rudolf Steiner, Leo Tolstoy, Jean Piaget and Lev Vygotsky, the idea of toying around with the furnitures of the world has been advanced as one of the most important aspects of education, that is, the augmentation of autonomy (what I am and what I can do in the objective world). Philosophy of toys in a sense takes seriously the idea that education does not end with autonomy or with the initiation into the space of theoretical and practical cognitions. On the contrary, it sees the autonomy of the child, the child’s synthetic ways of manipulating and understanding things, its proto-theoretic attempts at constructing a world prior to even conceptualizing that world as the premises of education. For this educationist philosophy, the role of toys in the recognition of the child’s autonomy and world-structuring abilities are more than necessary. They are indispensable.

1. From Logos to Lego and Back

Toy blocks (this-suches) hang together in the right way in space and time.

We can only represent the world to the extent that we have built a world in which our representations coherently hang together. The scope of our world-buildings demarcates the limitation of our attempts at representing the world. Take for instance, Carnap’s Versuch einer Metalogik (Attempt at Metalogic) or The Logical Syntax of Language after the failure of his logical empiricism program in Aufbau or more recently, the work of Uwe Petersen. Despite their methodological and objective differences, in both cases we see that the frontiers of objectivity or what we call the object or object-constitution (or alternatively, Being, the intelligible, etc.) are set by the scope of our attempts at the construction of what we call theory or more precisely, in the case of both Carnap and Petersen, by the logical structure (albeit in each case the question of the logical structure is formulated differently). This line of thought is, of course, guaranteed to elicit the ire of orthodox Kantians who may still believe in the hard distinction between form and content or opposing logic as a canon to logic as an organon which according to Kant is the logic of illusion or a sophistical art (CPR, B86) on the grounds that it is not constrained by the empirical sources of truth, sensible intuitions or information outside of logic.

What Kant means by logic as an organon is roughly a formal tool for the production of objective insights or an instruction for bringing about a certain cognition that can be said to be objective. This conception of logic is then characterized as the science of speculative understanding or the speculative use of reason—the organon of sciences (CPR, A16-18). On the other hand, logic as a canon refers again to the formal use of logic (regardless of it content which can be empirical or transcendental) but this time as restricted to the characterization of logic as the canon of judging (i.e. the mere criteria of the correct application of the laws of thought or judgements) which requires and is constrained by extra-logical information (CPR, A61).

At this point, I do not want to discuss in details the fact that Kant’s opposition of logic as a canon to logic as an organon is a historical take on the controversy between Epicurus (the defender of canon) and Aristotle (the defender of organon). Or the fact that Kant’s dismissal of logic as an organon is entirely based on an antiquated Aristotelian definition of logic. Regardless, of how we interpret logic, this very distinction becomes manifestly precarious in the wake of revolutions in formal and mathematical logic in the twentieth century. With the advent of computation as the proto-foundation of logic—thanks to the so-called Curry-Howard-Labmek correspondence—the last residues of the Kantian contrast between logic as a canon and an organon fade away.

Even though we will return to the above issue to examine it closely, for now, we can always counter the objection of orthodox Kantians with a brief retort: So you think that form without content is arbitrary (i.e. unconstrained), but could you tell me what is a content without form? Surely, entertaining the possibility of the latter even under the most watchful eyes is another variation of that ideological house of cards which is called the Given. The whole notion of logic as a canon describes a game of logic already rigged by the representational resources and limits of the apperceptive subject constituted within a particular transcendental type.

In contrast to Uwe Petersen’s rebuke against Kant in the second volume of Diagonal Method and Dialectical Logic, I think Kant’s distinction between canon and organon, logic as world-building and logic as constrained by world-representation is quite subtle. Yet subtlety is not by itself a criterion of truth or profundity. For Kant seems to naively assume that thinking about logic as an organon means believing that we can ‘judge of objects and to assert anything about them merely with logic without having drawn on antecedently well-founded information about them from outside of logic.’ (CPR, B85, my emphasis).

What is important to recognize in the above quote and other passages concerning logic in CPR is the constant repetition of such focusing adverbs as ‘merely’, ‘solely’, etc. Kant seems to be peddling a trivial and obvious point not only as a profound remark but also as a refutation of the conception of logic as an organon. Yes, at least since the time of Plato’s Sophist, we know that what is said is not equal to what is. And indeed, the equation of the two is the core tenet of sophism: As long as I know the rules of deductive syllogism I can call myself the master of all sciences. But logic as an organon neither implies the aforementioned equivocation—i.e. the claim that logic is by itself sufficient for judging about the stuff in the world—nor requires any metaphysical commitment with regard to logic—i.e. the claim that laws of thought are laws of the world.

In contrast to Kant’s straw-manning of organon, all the conception of logic as an organon suggests is that our resources of world-representation are in fact beholden to and caught up within the scope of our world-construction, and in this case, the world of logics. In other words, it would be absurd to even talk about objects without the primacy of logical structure or logoi. Kant would have agreed to this sentiment but only in a trite manner. Why? Because if the talk of object is meaningless without theory or logical structure, then the expansion of the field of logic or determinate thought-forms unconstrained by all concerns about representation would be an absolutely necessary step to constitute objects, make objective assertions and deepening our discourse about objectivity. This primarily unconstrained view of logic as the indelible factor for object-constitution is exactly what we can call logic as an organon. Without it, all we can ever achieve are pseudo-talks of stuff i.e. Aristotelian this-suches or tode ties, namely, unstructured encounters with items or stuff in the world which have no objective structure or invariant qualities.

Moving from the sense impression fuzzy mass of cubic redish (materiate individual substance or stuff) to this red Lego block (a perceptual taking or judgement) requires the addition of logical structure. But the constructive characterization of logical structure is not a priori limited by representational concerns. Indeed, to adequately hone out the notion of logical structure demands the treatment of logic and logical world-construction in terms of general logic in itself, that is to say, unconstrained by any enforced representational consideration (whether the experiential content, the empirical source of truth or the the criteria of correct application of logical laws to items of the real world) that may establish the frontiers of logic in advance.

It is only when we attempt to decouple logic from any representational or world-referring constraints that we can ensure a sufficiently enrichable framework of world-representation. In short, to expand the resources of representation and enhance the correct application of logical laws to empirical evidence or observational statements, we must first engage with logic in its own terms and expand its domain not in accordance with but in spite of representational constrains. The world-constructing resources of logic in itself precede and in fact undergird the world-representation, our understanding or judgements about the world. To make a Carnapian slogan, construction of the world is prior to the constitution of the object and the knowledge of it. This priority is not only priority1 in the sense of one temporally preceding the other, but also priority2 in the order of constitution. It is priority2 which is, properly speaking, the focus of logical world-construction and describes the conception of logic as an organon.

How can we constitute an object or even entertain the idea of objectivity in any coherent manner if we do not take seriously the world-construction of logic (i.e. the organon) so as to broaden the domain of logic within which the object coheres and the notion of objectivity is deepened? If we choose to abandon this path in favor of the Kantian conception of logic as a canon, then we are eternally sentenced to what I have called the Kantian straightjacket i.e. those particular transcendental structures we inhabit and by virtue of which we will never know whether our objective descriptions of the world are the overextetions of our specific (sensible) intuitive resources or not.

Taking the idea of logic as a world that ought to be infinitely constructed without any prior restriction is in every sense incommensurable with the idea of logic as something that ought to be coordinated with the real world in the first instance. Kant’s transcendental logic as a species of pure specialized—i.e. concerned with particular use of understanding—logic is precisely a conception of logic that is not just conservative with regard to the possible scope of logic (how general logic can be expanded and enriched) but also insofar as it is built on the conception of logic as a canon—i.e. constrained by representational concerns—it harbors epistemic implications which are nightmarish to the say the least.

With reference to the previous installment, this epistemological nightmare or hell is more than anything the consequence of our own self-imposed restrictions and not simply, the result of our local and contingent constitution as such and such subjects (history of evolution, the structure of memory, culture, etc.) When we limit logic to representational considerations while our representational systems are at the bottom rooted in a particular transcendental type which delimit our empirical observation and often distort our objective descriptions, then the epistemic or objective imports of our logical systems only reiterate or overextend the limitative terms of our representational biases. The picture of the objective world we provide resembles the portrait of Dorian Gray: only more sinisterly subtle variations of ourselves and our entrenched dogmas, and nothing more.

The only viable strategy for gradually escaping from this Dorian Grayesque ordeal is to take seriously the priority2 of world-construction over world-representation and to avoid subordinating the treatment of logic to any extra-logical concern that might have the faintest smell of representation, empirical point of reference, ordinary conception of meaning or anything that sets the boundaries of logic in advance.

The great escape only begins when logical construction is separated from the province of the apperceptive subject and its ordinary affairs. It is no exaggeration that the unbound realm of logical construction is analogous to an infinite ocean within which islands of subjectivity or apperception emerge and disappear. And in fact, this analogy is becoming more and more the very shape of the future logos: What I have called general artificial languages already offer us the inklings of that slippery yet boundless divine notion which we call logos and whose personifications are logic, mathematics and computation. Within this artificial apeiron—artificial language as the lego of future reason—our natural languages are but destined to corrode and eventually sink akin to tiny islands whose once firm ground can no longer withstand the rise of sea-level.

Made possible by the conception of logic as an organon, this apeiron is nothing but the universe or universes of metalogics or metalanguages. In other words, the logical apeiron does not signify one universal language or metalanguage, nor does it represent a final stage in the construction of language as the interface with reality or the configuring factor of the objective world. By contrast—and with a nod to Carnap in Logical Syntax of Language—no single language even in the most generic sense (i.e. not a natural language) can exhaust the logico-computational structure of language. For delving into the structure of language requires ever more richer languages or an infinite nested space of metalanguages or metalogics (a la Gödel).

It is in this respect that the plural designation general artificial languages simply describes what is already the case with logic and mathematics: No single system or language is adequate to explore the structure of language, logic or mathematics. Only an infinitely constructible—that is to say, not limited in advance in the vein of logic as a canon—nested space of metalanguages. Even the mighty Sellars treat the space of metalanguage as an applied domain and a completed totality by which we can resolutely talk about the structure of natural languages and such thing as semantic value and syntax. Contra Sellars, the space of metalanguage is not like an orthodox vantage point from which we can conclude the structure of meanings as assertions or finalize the definition of meaning as classification. Put differently, metalanguage as a nested constructible apeiron cannot even be compared to something like a completed vantage point for comparing two natural languages in order to decide the classificatory role of meaning (e.g., •red• in english means •rot• in german i.e. red plays the same classificatory role that rot plays in German). The whole point of metalanguage is the exploration of the very conditions of possibility of meaning as classification and structure, rather than as a light comparative study between this or that established language. Sadly, there are only a handful of philosophers and logicians for whom the domain of meta- is likened to an infinite heaven—an infinite nested space that is not arrested by the criteria of world-representation but unbound by the possibilities of world-construction in the domain of language and logic as organons.

The world-building of language just like logic mirrors the metaphor of ascending to metalinguistic heavens or descending into the abyss of metalogics depending on one’s theoretical proclivities and aesthetic sensibilities. In either case, what is crucial is the understanding that what is constructible by definition is never arrested by limits drawn in advance. Forgoing this notion of established boundaries for language and logic demands a concrete commitment to logic as an organon rather than a canon, to approach world-building as prior2 to world-representation.

2. Who wants some Fröbelian gifts, raise your hand!

With this digression on world-construction versus world-representation, toys versus adult concerns about the knowledge of the real world, let us return to the territory of actual toys.

The kindergarten philosopher, the child, is absorbed in that immense universe which is that of world-construction. The child’s toy universe does not resemble anything like the represented world of ours. The toy blocks by which the child assiduously constructs a world are not anything like bricks cemented over bricks. They can be replaced or even discarded if the child is not satisfied with the result. And we all know that a child never gives up. Until the toy-made world is in its optimal condition, it will be destroyed again and again. Even when the optimal construct is achieved, nothing guarantees that the child won’t reengineer it the next day to accommodate more adventurous narratives. This is the very definition of infant politics from which we adults have regrettably diverged.

The interaction of the child with toys is a premise upon which the objective interaction of concept-using adults and the world is built. The philosophy of toys in this respect precedes the philosophy of education qua augmentation of subjective autonomy i.e. what I can do and say in the objective world. What distinguishes toy-education from the canonical formula of graduation to adulthood—coinciding with full-blown conceptual competency—is that toys are not just about language but also the use of tools. They are as Vygotsky suggests the zones of synergy between language-use and tool-use, acquaintance with world-representing resources and world-making or world-reengineering techniques and systems.

A child who is immersed in playing with toys is a symbol-tool explorer. The child sees language as a tool just as it sees tools as symbolic-combinatorial elements of the language. The boundary between building the world and representing it is blurred and sometimes non-existent. It is in this sense that comparing the role of toys for a child to something like tools might be very well a symptom of our rooted adult misunderstanding. It is only, analogically speaking, from our adult perspective that we can remotely compare toys with tools. But toys, adequately understood, are not tools per se. In contrast to the use of tools, toys do not strictly adhere to pieces of practical reasoning (e.g., In order to achieve X, I ought to do Y). The ends of toy-plays are not like the ends of our practical reasons which go away once we attain them. They are more like inexhaustible ends, or ends which do not simply go away once achieved. For as long as, the world can be reconstructed and re-engineered, the infinite prospect of the kingdom of ends is at hand.

However, toys as I mentioned earlier are not just the blocks of world-construction. It is well-established (e.g., Spatial Reasonin in the Early Years) that toys play an essential role in the world-structuring capacities of children. The augmentation of spatial reasoning and correspondingly, the later use of mathematical concepts such as complex transformations, mapping, transfer, symmetry, inversion, shearing (in the case of paper-toys) and so on are directly linked to how children play with toys.

Block-oriented toys such as Lego introduce children to extremely potent and specialized concepts like modularity. It would be hard—if not impossible—to imagine the scope of technological progress and above that, the discipline of engineering as a tissue connecting the messy problems of physics with mathematics without the concept of modularity. Even though modularity is an epistemic concept rather an ontic one, I am yet to be convinced by the arguments of those philosophers who think certain structures in the universe, for example brain, cannot be modeled modularly. The majority of such critiques assume that modularity means something similar to Khrushchyovka-like buildings where modules are put atop one another to create something that is too rigid and monolithic to afford anything resembling organic life. But modularity comes in varieties of forms. Every complex can in one way or another—after sufficient approximations—be modeled via the surprising efficacy and flexibility of modular systems. It is true that modular models distort some information concerning target systems, but then which models are exactly distortion-free? The modeling-epistemic war against modularity is more than anything a product of unwarranted metaphysical assumptions about the structure of the universe and the inadequate grasp of the concept of modularity than anything else. See for example the recent works of Andrée Ehresmann, Jaime Gómez-Ramirez and Michael Healy who have synthesized the insights garnered from category theory and algebraic geometry with those from brain sciences.

Speaking about the territory of toys as tools for the enhancement of world-structuring abilities or faculties would be impossible without at least a brief reference to Fröbel’s original notion of kindergarten and Steiner’s school of Waldorf education. Think of Fröbel’s gifts, multiple sets of toys starting with regular wooden blocks. Once a child masters the use of a toy-set or a Fröbel gift, it is awarded a new set of toys, for example, geometrically disparate wooden blocks plus colored strings. The point of the toy gift-giving is to raise a child with the understanding that world-construction is an infinite domain. The further you attempt, the more world-building components fall from the sky. If you think you have achieved a perfect world, then let me give you a new set of constructive units, namely, toys.

The same thing holds for Steiner’s school of Waldorf education. Think of a Waldorf doll, a faceless uncharacteristic doll made of the cheapest material and stuffed with hay. The child begins to learn that the doll is not the true object, which is to say, the object is always incomplete. The child then goes on to paste a smily face, a big nose and dark brown eyes which it had drawn on a paper on the doll’s face. However, depending on the setup of the child’s toy universe, the doll can take fundamentally new characteristics like a platform on which new qualitative differences can be built, layer after layer.

From our adult perspective, there is hardly anything more grotesque than a Waldorf doll laying around, one with which a child has played for months. But as if beauty in the eye of a child could be anything other than something in essence synthetic, layered and transitory: faces pasted on faces, vestige upon vestige, characters built on top of one another all within one domain, the toy universe. Within this universe which is a featureless doll, the original beginning and the ideal end are never attainable, only residuation of what has come before and the possibilities of further construction.

‘In the toy universe of naive physics, things can get even crazier than non-folk physics!’, argues Professor Lucifer Gorgonzola Butts.

Similar to Waldorf dolls, the main emphasis of Fröbel gifts is also world-building. Although the gifts shift their focus to a more systematically disciplined mode of construction and less the free-play of simulating or imaginative capacities. Climbing the hierarchy of Fröbel gifts demands new forms of spatial reasoning, geometrical pattern matching and bootstrapping techniques, and sophisticated intuitions into the realm of naive physics. As the level of the gift goes up, the child’s construct becomes increasingly similar to a genuine marvel of modernist architecture or an experimentally engineered contraption straight out of the Rube Goldberg-inspired videogame, Crazy Machines. Nevertheless, in both cases, we see a coupling between world-building and simulational abilities. The point of toys is as much about world-construction as it is about building up the capacities and techniques of simulation through which understanding is amplified and its scope is expanded.

We already know from Kant that what we call today simulation in a loose sense—as in simulating a world in which friction does not exist—is at the bottom the function of productive imagination which in the Kantian parlance is just understanding in a different guise. Behind productive imagination lies one of the key themes of Kantian transcendental method: the argument about schemata. Schematism addresses one of the most weighty problems in transcendental philosophy, the so-called homogeneity problem i.e. the correspondence or coordination between concepts qua rules and objects or imagistic impression of items in the world. By image, Kant means a singular rudimentary (i.e. intuited) representation of an item / object. One can think of the rudimentary imaging faculty as involving extraction and integration of salient perspectival or local features qua variations of an object. Concepts on the other hand are non-perspectival (the invariant). They are at the most basic level principles of unity through which multiple particular instances can be brought predicatively under one subject, particularly a logical subject. Once we have concepts, we can arrive at critical perceptual judgements so that when we look at a Bic pen immersed in a glass of water, we can assert that this such-and-such pen looks—perspectivally—bent but is in fact straight. This is a piece of critical perceptual judgement or taking i.e. grasping, understanding or conceiving (bringing into conception). In contemporary terms, then we can think of the homogeneity problem as the problem of coordinating local variations and global invariance (the core of sheaf logic) or particularities and universality, eikones and ideai, the temporal capacity aisthesis and the time-insensitive faculty nous (the problem of Plato’s divided line). Let me clarify the homogeneity problem with an analogy to Euclid’s Elements.

Think of this particular equilateral triangle, this particular isosceles triangle and so on. These triangles are just particular—i.e. locally-varied—shapes or image-models. In reference to Proclus’s commentary on Euclid, they are only triangles by virtue of falling under the concept of triangularity as such. The triangle as a concept then allows us to make certain kinds of judgements or to draw diagrammatic inferences (Euclidean demonstrations) with regard to any triangle in whatever possible configuration.

Now the homogeneity problem engages with the issue of how can this or that particular triangle can be coordinated with triangularity as such. Put differently, how can the concept be supplied with its image. Proclus thinks the solution is in what he calls a mediating universal, a rule that comes between the detached universal (the universal triangularity inexhaustible by any image of a triangle) and particular triangles. Kant calls these mediating rules, schemata. Schematism then describes rules or constructive procedures which unlike the strong sense of the concept are not concerned with what particular image is subsumed by a concept, but how a particular image can be constructed in thus-and-so ways so that it conforms with a certain concept. This howness designates the functional role of the concept qua rule. Functional in the sense that we use the concept of triangularity whenever a particular item—an imagistic shape of a triangle—is implicated in the actual use of the concept of triangularity.

A schema is then simply the representation of a general procedure or rule of imagination (i.e. the capacity to represent an object even without its presence in intuition) for providing a concept with its image. But what kind of image and what kind of rule? The answer is a perspectively determinate sensible image which connects the single concept and the varied images. In Euclidean terms, we can think of a schema as triangularity not as a detached universal eidos, but as a formula for how to configure and diagram such-and-such lines, angles and vertices. For example, according to Proclus, just as there are mediating universals or rules which supply the concept of triangularity with triangles which might be equilateral, isosceles, etc, there are also construction rules or recipes of configuration (mediating particulars) which enable us to construct triangles using lines and angles as their elementary blocks for particular types of triangles (the concepts of scalene, isosceles, and so on).

Essentially, the schema as the missing link between understanding and sensibility, has one foot in categories (pure classificatory concepts) and the other in the intuited object or the image qua appearance. Thus we can say that a sensory presentation and a singular conceptual representation are determined by one and the same schema, that is to say, one and the same concept of a determinate mode of sensory presentation through which the concept and the object or its imagistic presentation come together. A schema is in this sense not a rule as that which predicates but as a recipe for construction of locally varied images for a single concept.

When it comes to schematism, Kant is actually quite fond of non-empirical examples like drawing a straight line in thought. A recipe or rule for drawing a straight line is at once responsive to two criteria: (1) every segment that is built is drawn in relation to the concept of line as a whole or that which binds all segments together, and (2) every piece or segment is constructed on the segment that has come before it (the law of memory which in Kant’s work can be attributed—with some reservations—to inner sense). With regard to both #1 and #2, we can say that the ultimate coordination between the image and the concept happens in the realm of rules (of construction) as pertaining to how such and such phenomenal features are being organized or brought together by space and time as transcendental idealities.

Therefore, as long as we are endowed with different spatio-temporal principles of organization, we can imagine of schemata which bind the imagistic item and the concept in different ways. Lucky for us, a child who is still in the process of coming to grasp with representations of time and space can vary the very parameters by which the image is related to the concept. This is what I call infantile schematism and by that I mean a child never settles for a particular established image for a concept. This is but the very law of simulation.

You say that the concept of mountain should conform to such putative invariant image-models. ‘Daddy I do love you but you also happen to be so parochial,’ the child opines: ‘Let me set you straight, in my toy universe, the mountain can be anything. It can be a cardboard box covered with brown satin or it can be a pot of old coffee’. The child then continues, ‘you think just because I play with what I call humans, they should conform to what you perceive as a human. But you are sadly mistaken for even a colored pencil wearing a thimble can be a human, an autonomous agent in my world.’ This is what simulation is all about. It does not matter what imagistic impression of an item in the world corresponds to a concept. What matters is the mediating rule of how any imagistic impression—after the sufficient relaxation of representational constraints—can be coordinated with a concept which is applied across the board for all instances brought under it.

Accordingly, simulation in the aforementioned sense involves destabilization of a canonical or stable set of images for a concept. But this process of destabilization is followed by a process of restabilization so that the implications of the use of a concept hold for any image that falls under it. The simimulational role of toys is exactly like this. The schematic coordination of the image and the concept is there, however, (1) its representational function is partially suspended, (2) the stabilized homogeneity between object and concept is frequently destabilized in favor of new modes of construction and correspondingly, object-constitution, (3) the relaxation of representational constraints amplifies constructivity so much so that we can replace a canonical set of image-models with such-and-such properties with an entirely new set that has different qualities (e.g., substituting humanoid doll-like entities with tiny calculators while abiding by the rules of how these calculators operate).

With regard to #3, simulation can be said to be essentially a species of what Kant calls as-if arguments (als ob). Such arguments are regulative judgements which can be both theoretical and practical such as as if there are categorical imperatives with regard to the kingdom of ends. In Kant’s philosophy, we should always be vigilant not to mistake a regulative judgement (an analogical as-if) for a constitutive judgement. In other words, we can never overextend our analogies. But in contrast to Kantian as-if arguments which are purely analogical and thus under the constraints of analogy, the simulational as-ifs are not exactly analogical meaning that they do not need to be always compared with or a priori limited by constitutive judgements. Simply put, we can take an analogy—a simulation to be more precise—seriously, treating it completely in its own terms. There is no danger of overextending an analogy in a simulated world so long as we are consist in our treatment of the simulated components and are true to the simulation and its logic (e.g., . Within a toy universe—i.e. a simulated framework—the primary emphasis should in fact be given to the simulation and not how the simulated is related to the source or premise of analogy). If we are to enrich a simulated world we ought to, first and foremost, attend to the simulated framework rather rather than the source of the analogy i.e. the real world. Only the unreserved enrichment of the former can assure the enrichment of our conception of the latter.

With this rather hasty discussion, let us in a crude manner define toys:

Toys are a sub-class of object-models whose primary task is world-building. This task is enabled by how toys suspend the canonical stability between the image and the concept, the correlation between representation and construction as an autonomous domain.

3. The Philosopher King of All Toys and Engineers

The idea of toy as an object-model capable of simulating a world or a problem without strictly conforming to the representational constraints of that world or variables and parameters of the original problem has a long history in science and specifically engineering. Confronted with a problem in one domain, the engineer constructs a toy surrogate or mechanical analog of that problem in another domain. The engineer then goes on to investigate this toy construct and how it behaves in its specific domain and in its own terms. Using certain equivalence principles that can coordinate the original problem and the toy surrogate, the engineer is then able to use the solution provided by the machine and translate it into a solution for the original problem.

It is as if in order to adequately understand a problem in a specific domain and to arrive at a solution to it, one must first attempt to exteriorize this attempt at understanding by reinventing the problem in an entirely new domain. But what would be the characteristics of this new domain? First, this new domain should allow us to examine the original problem under new parameters which are not exactly the parameters of its native domain. Next, the new domain should be far more manipulable than the old one. The principle behind this genius epistemic hack is attributed to the greatest of all engineers, Archimedes of Syracuse. It is well-known that Archimedes had a recipe to crack the most difficult geometrical problems of his time through a method of mechanical reasoning: inventing a toy machine and observe how the machine deals with your problem.

Imagine there is a geometrical problem which cannot be solved given the resources you have—or at least, it will be very difficult. Also imagine you are a cunning devil-engineer who doesn’t simply give up. You instead construct a mechanical device that can stand as the analog of your geometrical problem. The construction of such a machine would require particular and general forms of equivalence-establishment between the geometrical problem and the machine (the mechanical domain). The particular equivalence criteria consist of a set of geometrical inferences which can translate and transpose your geometrical problem into the construction and the behavior of the machine, plus an available mechanical or physical principle which can coordinate the parameters of the machine with those of the geometrical problem. The general criterion of equivalence, on the other hand, is usually a combination of simplification and/or idealization, in the sense that the machine—its scope, assignment, behavior—should be sufficiently idealized/simplified for it to be an optimal analogue of the problem at hand.

Once you make the mechanical analog, you want to pay attention to how the machine works and not the geometrical problem. When you have something similar to an output or the machine-solution, then you again use the particular criteria of equivalence so as to translate the solution supplied by the machine into a solution to your original geometrical problem. See the following diagram for a better grasp of this method (Note: the ASCII diagram might display incorrectly if you are using a feed-reader.)

            ┌──────────┐                   ┌──────────┐
            │Solution G│◁───Equivalence────│Solution M│
            └─────△────┘                   └─────△────┘
                  │                              │
                  │             .────────────────┴────────────────────.
                  │            ( bracketing the space of n-hypothesis  )
                  │             `────────────────┬────────────────────'
                  │                              │
    Geometrical  ┌┘                              └┐  Mechanical
      Reasoning┌─┘                                └─┐Reasoning
             ┌─┘            ┌──────────┐            └─┐
  ┌──────────┴──────────┐   │Mechanical│   ┌──────────┴──────────┐
  │                     ◁───┤Principle ├───┤                     │
  │ Geometrical Problem │   └──────────┘   │ Mechanical Problem  │
  │                     ├──────────────────▷                     │
  └─────────────────────┘    Geometrical   └─────────────────────┘
             │                Reasoning               │
             │                                        │
         (   Equivalence via Simplification⊕Idealization   )

For example, imagine the problem of’how much bigger is a cylinder than a sphere fully enclosed in it?’ To answer this question, we can use a lever with adjustable arms and a solid sphere and cylinder made of the same uniform material. All we need to do is to put the sphere and the cylinder on the extremes of the lever and adjust it so the two are balanced. The answer if you are willing to do this experiment is Vcyl : Vsph = 3 : 2.

A more contemporary and intricate example would be one suggested by David Hilbert in Geometry and the Imagination (p. 222). This is the problem of finding geodesics of a surface. To construct a mechanical analog, we need to have Gauss-Hertz principle of least constraint—or more precisely, Hertz principle of least curvature—as the particular criterion of the equivalence between the mechanical and the geometrical. This is our first problem before going on to build a mechanical or toy surrogate: finding the equivalence (not equality) relations.

As soon as we have the equivalence relation—Hertz principle—we can talk about and examine the constrained dynamics problem as if it was the problem of finding geodesics of a surface. The Hertz principle gives us a way of coordinating geodesics equations and equations of motion. Thus when we arrive at a solution to the constrained dynamics problem we can can translate it into a solution for finding geodesics of a surface.

In order to understand how the analog of the problem of finding geodesics of a surface works, first we begin with differential equations for a 3-dimensional surface parametrized as follows: r\left( u,v \right)\in {{\mathbb{R}}^{3}} . These differential equations can be expressed in a compact form as:

{u}''=-\left( \begin{matrix} {{{{u}'}}^{T}}{{\Gamma }^{1}}{u}' \\ {{{{u}'}}^{T}}{{\Gamma }^{2}}{u}' \\ \end{matrix} \right)
where {{\Gamma }^{1}} and {{\Gamma }^{2}} are Christoffel symbols.

Now, for constructing the mechanical analog of this problem, we can imagine a toy universe consisting of only two blocks, a toy-apple and a toy-ant. They are toyish because it does not matter whether the apple is red or green, or whether the ant is silicon-based or carbon-based. All we are interested in are those features which allow us to observe the locomotion-behavior of the ant as it it traverses the surface of the apple. In fact the walk of the toy-ant on the surface of the toy-apple represents a difficult problem of robotics: How can the robot-ant find the shortest paths on a generalized surface, whether on the surface of Mars or on a Riemann surface?

The apple-walking of the ant represents (holonomically constrained, i.e. \phi (r)=0) force-free geodesics on a constrained motion surface which conforms to the following equation after constraint stabilization:

\ddot{r}=-{{\Phi }^{T}}{{(\Phi {{\Phi }^{T}})}^{-1}}(\dot{\Phi }\dot{r}+\beta \Phi \dot{r}+\alpha \phi )

where \Phi is a constraint Jacobian matrix \Phi ={{\nabla }_{r}}\phi =\partial \phi /\partial r.

The toy apple-walking ant equation is not restricted to {{\mathbb{R}}^{3}}, it can be equally applied to {{\mathbb{R}}^{n}}. For a detailed mathematical take on how the equations of this toy universe can exactly solve the problem of finding geodesics of a surface, you can look at Least action principles and their application to constrained and task-level problems in robotics and biomechanics by De Sapio and others.


Source: Misner, Thorne, Wheeler, Gravitation (Princeton University Press).

Despite its peculiar artistry, the Archimedean method or hack is quite simple actually. But the scope of this method should not be restricted to the world of geometrical and mechanical problems. It an be applied to any problem after finding a particular equivalence relation that translates the constraints of the domain X (the milieu of the original problem) to the constraints and parameters of another domain. The legend of Archimedes in the tub captures the power of toy analogues quite spectacularly: Archimedes is in the tub, playing with water like a child, making waves after waves, watching how the water spills out of the tub. Then suddenly he pretends that his body is a gourd of water and then he continues with this experiment as if it was reality. In any case, we can think of the Archimedean method as a generalized way of playing with toys qua object-models. The following diagram should shed some light on the general logic of the Archimedean method:

                              │ How-possibly-enough  │
           ┌─────────────────▷│     explanation      │◁──────────────────┐
  X────────┴────────┐         │                      │          T────────┴────────┐
  │  How-actually   │         └──────────────────────┘          │  How-possibly   │
  │  understanding  │                                           │   explanation   │
  └────────┬────────┘                                           └────────┬────────┘
           │                                                             │
           │  ┌─────────────────────┐           ┌─────────────────────┐  │
           │  │                     │           │  Toy surrogate (T)  │  │
           └──│      Problem X      │           │        of X         │──┘
              │                     │           │                     │
              └─────────────────────┘     Λ     └─────────────────────┘
                         │               ╱ ╲               │
                         │              ╱   ╲              │
                         └────────────▷▕Eq. ?▏◁────────────┘
                                        ╲   ╱
                                         ╲ ╱

So on one side, we have the problem X. Our aim is to understand how actually this problem can be brought into a resolution. On the other side, which is that of the toy surrogate, we are free from such a mode of understanding and explanation which we can be called how-actually. We are interested in the possibilities of how the analog or the toy surrogate can explain the problem X. In a nutshell, the toy analog is both the domain of possibilities which where absent in X and the domain of bracketing or winnowing through such possibilities (the space of n-hypothesis). This is what for now can be dubbed the domain of how-possibly explanation. I will define how-actually and how-possibily explanations and their corresponding modes of understanding when I discuss toy models, but until then let us go along with these rudimentary definitions.

Once we solve the problem of establishing equivalence relations (Eq. ?), we can enter the domain of the toy analog, examine and observe how the reinvented problem comes to a resolution. Essentially, in the analog domain, the solution is not analytic in any sense. The solution can only be achieved through bracketing or limiting the space of possible hypotheses and their corresponding explanations. And that’s exactly what the Archimedean method does: it enables the how-possibly explanation brought about in the analog domain to become a how-actually understanding for the original problem in the domain X. For once we arrive at a true enough how-possibly explanation in the toy domain—namely, a candidate hypothesis and its corresponding explanation—we can say that this is also a how-actually enough (i.e. true) explanation or resolution for the problem in the domain X.

This post has gone beyond the limits I originally imagined, so let’s stop at this point and wait for the next installment in this series. [Update note: For those of you who are interested, I have added some equations for the Hertz-principle example mentioned in the last section.]

Complexity Collection

While the next posts are brewing, I thought to make a list for a few friends who have asked for some reference materials on new trends in complexity sciences (scales, anticipatory systems, the question of formalizing adaptive systems, modeling and new explanatory paradigms).

On a different note: My friend Adam Berg will also start posting on toys, world-building and philosophy of complexity here on this blog. I can go on rambling about Adam’s philosophy for pages. Suffice to say, he is among one of those handful of philosophers who do not admit—in theory or practice—a distinction between analytic and continental camps. His commitment is only to one thing, philosophical exploration in the broadest sense. His masterwork Phenomenalism, Phenomenology, and the Question of Time: A Comparative Study of the Theories of Mach, Husserl, and Boltzmann is more than enough evidence to support this claim.


Complexity: Hierarchical Structures and Scaling in Physics – Remo Badii, Antonio Politi [A classic and technical work in complexity sciences which started a whole genre of inquiry about hierarchization, scales and constraints on modeling.]

Lyapunov Exponents: A Tool to Explore Complex Dynamics – Arkady Pikovsky, Antonio Politi [Another technical work on one of the key concepts in the study of complex and dynamic systems. I agree with Robert Bishop, without an adequate grasp of Lyapunov exponents, it is so easy to fall in the trap of complexity and chaos folklore.]

An Introduction to Kolmogorov Complexity and Its Applications – Paul Vitányi, Ming Li [Yet another technical title but mandatory for understanding algorithmic complexity and the later works on structural complexity / stability by the likes of Crutchfield and Ladyman]

Simulation and Similarity – Michael Weisberg

Complexity: metaphors, models, and reality – George Cowan, et al. [Occasionally a bit dated but it contains some interesting conversations between Santa Fe people.]

Re-Engineering Philosophy for Limited Beings: Piecewise Approximations to Reality
– William Wimsatt [An expanded and revised collection of Wimsatt’s papers on functionalism, the gradualist approach to modeling and mechanisms]

Discovering Complexity: Decomposition and Localization as Strategies in Scientific Research – William Bechtel [This or Bechtel’s other work Mental Mechanisms. For a good brief critical response to Bechtel-Wimsatt’s paradigm of mechanistic explanation, see Jay Rosenberg’s Comments on Bechtel, ‘Levels of description and explanation in cognitive science’)

In Search of Mechanisms – Carl Craver

No Revolution Necessary – Carl Craver

Levels – Carl Craver [This and Batterman’s essay on scaling and midlevel explanation are quite crucial for not only understanding the problem of descriptive-explanatory levels but also thinking about the potential uses of such paradigms in something like information ontologies or semantic web. See for example, WonderWeb Deliverable.]

The Devil in the Details: Asymptotic Reasoning in Explanation – Robert Batterman

The Tyranny of Scales – Robert Batterman

Physics Avoidance – Mark Wilson [Highly recommended. Wilson’s long awaited book and a sequel to Wandering Significance came out this month.]

Philosophy of Complex Systems – (ed.) Cliff Hooker [Robert Bishop who has an essay in this collection offers a particularly astute crtique of some of the folklores in complexity sciences.]

Anticipatory Systems – Robert Rosen

Theoretical Biology and Complexity – Robert Rosen

Memory Evolutive Systems – Andrée Ehresmann

Simple, Complex, Super-complex Systems – Ion Baianu

What is a complex system? – James Ladyman, et al.

The Calculi of Emergence – James Crutchfield

Modularity in Development and Evolution – (eds.) Gerhard Schlosser, Günter Wagner

Towards a Theory of Development – (eds.) Alessandro Minelli, Thomas Pradeu

Functions – Philippe Huneman [Probably one of the best collections on the new wave of functionalism informed by complexity sciences.]

Developing Scaffolds in Evolution, Culture, and Cognition – (eds.) Wimsatt, et al.

Toy Philosophy Universes (part 1)

As I mentioned previously, the posts on this blog will be more like compilations of overextended post-it notes than carefully crafted papers. It is like lifting up the hood and letting other people look at the fragile components of a prototype engine you have designed and how such components tentatively hang together in spite of the fact you have just noticed the engine not only works at a sub-optimal level but also someone else has made a better one.

Much like engineers who are always obsessed with how crazy contraptions work, philosophers are curious about how people think particularly those whose activities in one way or another involve modeling the world, making systems and concepts about it. Having come from an engineering background and being a philosopher, I am doubly fascinated with how other philosophers or theorists (whether a scientist, an artist, a political theorist, etc.) model their worlds, conduct their research and practice so as to arrive at a specific set of theoretical, practical or aesthetic claims. This fascination—call it a chronic cognitive voyeurism—is centered on the implicit know-hows and know-thats behind theorists’ works and visions as well as structural flaws in their thoughts and Weltbilder. The fault-finding tic comes from not just the philosophical penchant for the criteria of trueness but also from the engineering perspective of whether an argument, a system of thoughts or a world-picture can be examined, optimized or revised, or should it be discarded altogether. If it can be revised, then what does it take to revise it? But a comprehensive revision requires accessing certain information about the structure (roughly speaking, how things hang together in the broadest possible sense) of the said system or world-picture. Now it is becoming obvious that the question of revision or optimization is not easy after all. For how can one at the same time inhabit a system or a model and examine or more precisely, make well-formed sentences about its structure so as to be able to consequently revise it. This is a Gödelian puzzle and the question from which Rudolf Carnap begins his work post-Aufbau (much on this issue in toy philosophy universes series and future posts on Carnap’s revolutionary work, until then I draw your attention to the recent brilliant work of Steve Awodey on Carnap).

Let me reformulate the above puzzle with regard to something like language. If our natural languages are at their base anchored in our intuitions in a Kantian sense and furthermore, if our intuitions (say of space and time) are bound to a certain type of transcendental structures and again if these transcendental structures (e.g., memory, perception of space, natural language, locomotory mechanisms, etc.) are the products of our own local and contingent constitution or evolution as subjects, then how can we know that our objective descriptions of phenomena in the world are not anything other than the overextension of our local and contingent characteristics? How can we ever step outside of the limitations of our natural language or the egocentric framework? Ergo, Wittgenstein’s problem: the limits of our world are the limits of our language and insofar as our natural languages or for that matter any representational system are shackled by—just like Kant’s intuition—a particular transcendental type, then our objective descriptions of reality are doomed forever to simply reiterate a variation of our local and contingently characteristics of experience and nothing more. Kantian philosophers usually have the habit of understating the consequences of this epistemic hell but just because they downplay its significance, it does not mean that this hell does not exist.

This problem, of course, is nothing new. Ludwig Boltzmann offers a shining example of it toward the end of Lectures on Gas Theory (§89-90) with regard to our intuition of time and the possibility of arriving at new facts of experience (a detailed discussion on this problem is in the forthcoming Intelligence and Spirit).

Did you just say other beings, extraterrestrials and stuff like that?

Even Kant himself in the Critique of Pure Reason (CPR), inadvertently, poses a similar question. In the section concerning general remarks on the transcendental aesthetic, Kant says something along the line that there might be other beings in the universe—i.e. species inhabiting other transcendental types—whose representations of space and time as forms of appearances differ from ours (A42/B60), but it would be an exercise in dogmatic metaphysics to speculate how such different intelligences intuit the world in order to see or conceive it as something dissimilar or even incompatible to our conception of the world. In other words, “[we should solely be concerned with] our way of perceiving them [i.e. things-in-themselves], which is peculiar to us, and which therefore does not necessarily pertain to every being, though to be sure it pertains to every human being.” (emphases are mine) Kant then, in later chapters, goes on to say that any conception of transcendental logic should always respond to the fact of our experience in order for it to be objective. One can give two different readings of these passages in CPR. One would be a charitable reading and the other an excoriating one. According to the charitable reading, of course Kant is right. We can not immediately step outside of our own forms of intuition and transcendental structures to talk about other beings and their models and conceptions of the world, universe or reality in its objectivity. Any model of other intelligences we provide is going to be modeled on our own theoretical and practical reasons. Any empirical description of such intelligences or beings will bear the characteristics of our own transcendental structures to the extent that the scope of empirical descriptions are dictated by the scope of transcendental structures. In talking about other intelligences or minds, we are always forced to explain and justify the reason we interpret such beings as intelligent or minded. That is to say, without an explicitly stated criterion of what we mean by intelligence or mind, our characterizations of other beings as intelligent or minded are nonsensical. We might just as well talk about angels dancing on the head of a pin, cyber-popeyes, ineffable superintelligences and lava lamps being in a noumenal harmony with other kinds of stuff. My friends Ray Brassier and Pete Wolfendale have done more than a sufficient job to expose these pseudo-posthumanist and object oriented gibberish as the latest exercises in speculative farce.

Now the non-charitable reading which is the one I am interested in, but without dismissing or rejecting the charitable reading that I just outlined: Even if we accept Kant’s remarks on other beings qua transcendental types as a reasonable cautionary tale that we should all take by heart, there is still something missing in this story. For Kant at the same attempts to underline the particularity of our transcendental structures (forms of appearances in particular) and emphasize the universality of our objective descriptions of reality. It could be non-controversial, if Kant’s definition of universality simply meant ‘as a matter of a necessary rule’ but we know that what Kant in many occasions in CPR means by universality is universal with capital U i.e. our description of universe or reality is Universal in a stronger sense. That is to say, it is the objective description. But in all due respect to the transcendental luminary of Königsberg, how is this even possible? How can we at the same time endorse the particularity of our transcendental type and claim to have Universal objective descriptions of the world of which we are part? Herr Kant, surely you don’t expect us to simultaneously believe in the possibility of different transcendental types and also rule them out in favor of our own merely given transcendental type on the ground that talking about such possibilities is an exercise in armchair speculation? Because if that is what you are saying then you are nothing other than a benighted humanist, an errand boy of the given. You claim that our objective descriptions are universal only to the extent that you do not know the exact limitations of our transcendental particularity because you think that our specific form of intuition should be the sole concern of our philosophical inquiry. In other words, you have already gerrymandered the boundaries of our particular subjectivity, turning them from locally bounded to universally unbounded.

In this framework—the Kantian straightjacket—how can we ever know that our empirical descriptions of the universe in its objectivity are not simply the overextension of the particular given characteristics of our own transcendental structures which can in fact be local and contingent? But even more insidiously, even when we sufficiently differentiate local and contingent characteristics of our forms of intuition from the objective descriptions of phenomena in the universe, what does guarantee that the bracketed characteristics of our intuition don’t encroach upon and distort our rational scientific models of the world? This is the very problem that vexed Boltzmann’s later work in the context of bridging statistical entropy and thermal entropy, micro- and macro- scales. To paraphrase Boltzmann’s words: If renewing our relation with objective reality requires renewing the structure of our experience then how can we at the same time exclusively inhabit a particular structure of experience and arrive at new facts of experience? Simply put, how can we be in X-space and see the world beyond it? Certainly achieving the latter requires us to adopt a new transcendental type, one that is at once the subject of speculation and in contiguity with our existing transcendental resources (i.e. intelligible). To sum up, our model or conception of the world ought to be both in contiguity with our given resources and also beyond the limitations imposed by our particular transcendental structure. As we now see, this problem is not only serious on a theoretical ground but also quite difficult to tackle with on methodological grounds. However, this is not exactly the topic I would like to focus on, even though it can be seen as a primary motivation behind what I shall explore in the future posts.

The subject that I would like to write about is the metatheory of the practice of philosophy or if you prefer—with some caveats that I will hopefully elaborate in the future—the metatheory of theorization. Essentially, I would like to talk about the labor of modeling, but also toy models and eventually toy philosophical universes as systematic models in which the aforementioned Kantian problems don’t simply go away but nevertheless are mitigated. So with that said this post is the first installment in a long series where I will talk about modeling, about the relations between models and cognition, and more importantly, about the kind of models that might rescue us from the current quagmire of philosophy and the idleness of thought. Hopefully, the connections will become clearer as we move forward.

The curse of the lobster man

While I’m writing this rather overextended post on modeling, complexity and generalized pedagogy:

Perhaps it is because of my barbaric background and all—being a middle eastern to the core—but I really cannot figure out what this fuss about Jordan Peterson is. From what I have seen so far, the majority of criticisms are coming from people who are fine with greedy naturalism but unwilling to embrace its consequences. It’s like being a follower of David Icke or being a fan of Land and then getting surprised by the fact that your world view at the end of the day degenerates into lizard epics and adolescent intergalactic skynet battles fought between white males with zero scientific literacy and dressed in elf-uniforms fighting the good larping fight for a messainic capitalism.

This has always been the bane of greedy naturalization: Once you arrive at an inadequate concept of nature that does not allow you to move upward toward the autonomy of reason and downward toward the heteronomy of causes, the sapience and the lupus to use Plautus’s phrase, then such scenarios are inevitable. Naturalization was always supposed to be a two-way street moving toward autonomy and heteronomy. However, these days with the advent of neoliberal science you only get a dead end. Any paradigm of naturalization that endorses a one-way movement suffers from an inadequate concept of nature.

My friend Tahir Al-Tersa came up with a great comment in response to my initial reaction that should be quoted in its entirety:

At first I didn’t understand why Peterson was becoming so relevant either, my thoughts were the same – particularly because not only was what he was saying the logical conclusion of already very common presumptions, but because it wasn’t even a particularly unique conclusion. But it’s clear Peterson isn’t popular solely because of the intellectual contents of his position – the same goes for individuals like Milo, etc. – it is principally because he was able to align them with a real political/cultural position many already have. By proving the ability to argue these ideas on popular mediums such as television, and make sense of worldly controversies with them in a way that penetrates popular consciousness, they take on a power that is in excess of their actual contents. And in my view, it would be mistaken to underestimate the extent to which those of us immersed in the intellectual sphere are all in a way beholden to this ‘lower’, more vulgar one.

I agree with Tahir wholeheartedly. It seems to me that the image of the public intellectual today has become almost synonymous with this brand of greedy naturalism (sappy human conservatism and mysticism peddled under anti-humanist evolutionary fables and bravados). To use Lucca Fraser’s term, greedy naturalism coincides with supernatural mysticism. Once you develop a sloppy unscientific conception of nature, you almost invariably become liable to endorse a supernatural thesis about the world of which we are part. In other words, Peterson’s greedy naturalism and his Jungian supernaturalism are actually the two faces of the same coin. If there is a viable image of the leftist public intellectual that ought to be endorsed in opposition to the right-leaning public intellectual is that of an intellectual who takes nature as anything but given, that is to say, nature not as a fixed entity or something god-given but as a manipulable or constructible explanation in the space of n-hypothesis.

This is to say that the leftist public intellectual today should be the very child of scientific enlightenment, at once observant of the negative socio-cultural baggage of the enlightenment project and devoted to its core commitments i.e. science and rationality, broadly understood.

A friend of mine, Alice Sinclair, cautioned me about the category of rationalism: Today’s liberalism is rationalist, and people like Peterson are simply the inevitable consequences of the liberal brand of rationalism where reason is treated as a ‘teleo-ideological force legitimizing the status quo’. This is true, but this is another reason why the left should reclaim the collective conception of rationality. The unbinding of the collective conception of reason is in fact tantamount to the negation of the liberal recognition of reason as an Aristotelian telos that safeguards the legitimacy of the established order of things. How can it be reason if it does not recognize its own limitations here and now? So in a sense, when I say reason is necessary but not sufficient, I mean that true rationalism should coincide with communism as the real movement that abolishes the so-called completed totalities of history and overcomes the status quo. This includes the parochial conception of reason that protects the liberal polity. I think one of the tasks of leftism today is precisely to wrest rationalism from the liberal conception of reason as a teleological force, to demonstrate concretely that the project of reason is the negation of such ideologies erected in favor of the status quo and whoever or whatever that represents it.

To this extent, I don’t see a viable alternative other than scientific rationality. To those leftist comrades who fear even the remote mention of the words reason, science or computation: The trash bin of history is waiting for you!

To be a leftist means to endorse history as science (Marx), to take the idea of critical and rationalist science seriously. Ultimately, I believe Tahir is right. Nothing is going to change in the public arena unless the left puts forward its own public intellectuals who are not afraid of science (broadly understood), but fully submerged in it. To use McKenzie Wark’s term, what we need is vulgar marxism (not to be confused with kitsch marxism that I have criticized in the past)—that is, a popular marxism i.e. vulgar in the positive sense of the people. We need vulgar leftists who can once more bridge the gap between science and egalitarian ideals, who can demonstrate that the ideas of Peterson and his ilk are not just ethically problematic but above all, are patently false on scientific and methodological grounds. Short of that, we are in every respect doomed.

Returning to the Age of Blogging

I have decided to finally resume blogging. This time, however, I plan to focus on various threads—some still loose and some already converged—of my philosophical research. I believe that ideas should be handled impersonally particularly in science and philosophy. For this reason, I am not convinced about keeping the components of an ongoing research secret. If people can build on your ideas even when your ideas are still in their larval stage, then it does not matter whether they reference you or not. As long as ideas and concepts can be enhanced, refined and propagated, plagiarism is a virtue rather than a vice. The task of a philosopher is to highlight the hard fact that the concept is that over which no single human has a final grip. Therefore, the whole obsession with working in secret, keeping things in the closet until the book is published is absurd. To take the concept of open-source seriously, one must first take the idea of an open-source self seriously. In this sense, we are far from the Wulfian ideal of a global collabratory even though the internet has effectively knocked down some of the walls. So to this end, the blogging medium gives me the right amount of control in conducting my research and openning it to people who pick up ideas as tools so as to make better tools that can be put in the service of thinking in general.

So what will be the future posts on this blog? Currently, I am planning to allocate a major part of it to the systematic philosophy of mind, that is, a family of fundamental correlations: intelligence and the intelligable, structure and being, theory and object, language and the world. In this sense, what I mean by the systematic philosophy of mind is in truth systematic philosophy (the organon of theory) itself, formulated in different forms from Plato and Confucius to Descartes, Hume, Kant and Hegel, and more recently Lorenz Puntel and Uwe Peterson. Within this framework, I would like to also write about philosophy of science (particularly my heros Wolfgang Stegmüller and Adolf Grünbaum), logic and computation, and Euclid’s Elements. The latter is more a personal interest of mine than a topic explicitly fitting the aforementioned framework. However, I believe Elements is the first work that attempts to integrate formal thinking and systematic thinking and in doing so, openning new pathways to the questions of structure and theory. Having taught Elements in a number of courses, I always tell my students that they should engage with Elements not only as a mathematical treatise but also as a philosophical thriller, an exercise in making worlds and concepts using a handful of naive intuitive axioms or data. In this respect, the plan is to build on some of the best commentaries on Elements as a kind of toy philosophy universe (much on this topic in the next post). My immediate references are Proclus’s commentary as well as the seminal essays by Kenneth Manders and Danielle Macbeth.

In addition, there will be some posts on the ascesis of autodidacticism particularly for those who are bent to become philosophers and survive in a paraacademic world where the finances are always close to zero, standards are clouded by hatred of academia and rigor is still a taboo word yet nevertheless ideas do not reek of the stale dungeons of academia. As for the form and the style, well, the posts will oscillate between formal and informal, essay-form and rambling, preaching and scolding: In short, this blog’s mission is the comprehensive corruption of the youth.