Infinity, Finitude

Here is another area where I find myself with mixed sympathies.

Plato seems to have regarded infinity — or what he called the Unlimited — as something bad. Aristotle argued that infinity exists only in potentiality and not in actuality, a view I find highly attractive. I think I encounter a world of seemingly infinite structure but only finite actualization.

Some time in the later Hellenistic period, notions of a radical spiritual infinity seem to have appeared in the West for the first time, associated with the rise of monotheism and the various trends now commonly called Gnostic. This kind of intensive rather than extensive infinity sometimes seems to be folded back on itself, evoking infinities of infinities and more. The most sophisticated development of a positive theological infinite in later Western antiquity occurred in the more religious rethinking of Greek philosophy by neoplatonists like Plotinus, Proclus, and Damascius.

In 14th century CE Latin Europe, Duns Scotus developed an influential theology that made infinity the principal attribute of God, in contrast to the pure Being favored by Aquinas. Giordano Bruno, burned at the stake in 1600, was a bombastic early defender of Copernican astronomy and notorious critic of established religion who espoused a curious hybrid of Lucretian atomistic materialism, neoplatonism, and magic. He proclaimed the physical existence of an infinity of worlds like our Earth.

Mathematical applications of infinity are a later development, mainly associated with Newton and Leibniz. Leibniz in particular enthusiastically endorsed a speculative reversal of Aristotle’s negative verdict on “actual infinity”. Nineteenth century mathematicians were embarrassed by this, and developed more rigorous reformulations of the calculus based on limits rather than actual infinity. The limit-based formulation is what is generally taught today. Cantor seemingly went in the opposite direction, developing infinities of infinities in pure mathematics. I believe there has been another reformulation of analysis using category theory that claims to equal the rigor of 19th century analysis while recovering an approach closer to that of Leibniz, which might be taken to refute an argument against infinity based solely on lack of rigor according to the standards of contemporary professional mathematicians. One might accept this and still prefer an Aristotelian interpretation of infinity as not applicable to actual things, though it is important to recall that for Aristotle, the actual is not all there is.

The philosophy of Spinoza and even more so Leibniz is permeated with a positive view of the infinite — both mathematical and theological — that in a more measured way was later also taken up by Hegel, who distinguished between a “bad” infinite that seems to have been an “actual” mathematical infinite having the form of an infinite regress, and a “good” infinite that I would gloss as having to do with the interpretation of life and all within it. Nietzsche’s Eternal Return seems to involve an infinite folding back on itself of a world of finite beings. (See also Bounty of Nature; Reason, Nature; Echoes of the Deed; Poetry and Mathematics.)

On the side of the finite, I am tremendously impressed with Aristotle’s affirmative development of what also in a more Kantian style might be termed a multi-faceted “dignity” of finite beings. While infinity may be inspiring or even intoxicating, I think we should be wary of the possibility that immoderate embrace of infinity may lead — even if unwittingly — to a devaluation of finite being, and ultimately of life. I also believe notions of infinite or unconditional power (see Strong Omnipotence; Occasionalism; Arbitrariness, Inflation) are prone to abuse. In any case, ethics is mainly concerned with finite things.

Poetry and Mathematics

Philosophy is neither poetry nor mathematics, but a discursive development.  Poetry may give us visionary symbolism or language-on-language texturings that deautomate perception.  Mathematics offers a paradigm of exactitude, and develops many beautiful structures.  But philosophy is the home of ethics, dialogue, and interpretation.  It is — dare I say it — the home of the human.

Poetry and mathematics each in their own way show us an other-than-human beauty that we as humans can be inspired by.  Ethics on the other hand is the specifically human beauty, the beauty of creatures that can talk and share meaning with one another.

Totality

The last post suggests another nuance, having to do with how “total” and “totality” are said in many ways. This is particularly sensitive, because these terms have both genuinely innocent senses and other apparently innocent senses that turn out to implicitly induce evil in the form of a metaphorically “totalitarian” attitude.

Aiming for completeness as a goal is often a good thing.

There is a spectrum of relatively benign errors of over-optimism with respect to where we are in achieving such goals, which at one end begins to shade into a less innocent over-reach, and eventually into claims that are obviously arrogant, or even “totalitarian”.

Actual achievements of completeness are always limited in scope. They are also often somewhat fragile.

I’ll mention the following case mainly for its metaphorical value. Mathematical concepts of completeness are always in some sense domain-specific, and precisely defined. In particular, it is possible to design systems of domain-specific classification that are complete with respect to current “knowledge” or some definite body of such “knowledge”, where knowledge is taken not in a strong philosophical sense, but in some practical sense adequate for certain “real world” operations. The key to using this kind of mathematically complete classification in the real world is including a fallback case for anything that does not fit within the current scheme. Then optionally, the scheme itself can be updated. In less formal contexts, similar strategies can be applied.

There are also limited-scope, somewhat fragile practical achievements of completeness that are neither mathematical nor particularly ethical.

When it comes to ethics, completeness or totality is only something for which we should strive in certain contexts. About this we should be modest and careful.

Different yet again is the arguably trivial “totality” of preconceived wholes like individuals and societies. This is in a way opposite to the mathematical case, which worked by precise definition; here, any definition is implicitly suspended in favor of an assumed reference.

Another kind of implicit whole is a judgment resulting from deliberation. At some point, response to the world dictates that we cut short our in principle indefinitely extensible deliberations, and make a practical judgment call.

Form as a Unique Thing

Ever since Plato talked about Forms, philosophers have debated the status of so-called abstract entities. To my mind, referring to them as “entities” is already prejudicial. I like to read Plato himself in a way that minimizes existence claims, and instead focuses on what I think of as claims about importance. Importance as a criterion is practical in a Kantian sense — i.e., ultimately concerned with what we should do. As Aristotle might remind us, what really matters is getting the specific content of our abstractions right for each case, not the generic ontological status of those abstractions.

One of Plato’s main messages, still very relevant today, is that what he called Form is important. A big part of what makes Form important is that it is good to think with, and a key aspect of what makes Plato’s version good to think with is what logically follows from its characterization as something unique in a given case. (Aristotle’s version of form has different, more mixed strengths, including both a place for uniqueness and a place for polyvocality or multiple perspectives, making it simultaneously more supple and more difficult to formalize.) In principle, such uniqueness of things that nonetheless also have generality makes it possible to reason to conditionally necessary outcomes in a constructive way, i.e., without extra assumptions, as a geometer might. Necessity here just means that in the context of some given construction, only one result of a given type is possible. (This is actually already stronger than the sense Aristotle gave to “necessity”. Aristotle pragmatically allowed for defeasible empirical judgments that something “necessarily” follows from something else, whenever there is no known counter-example.)

In the early 20th century, Bertrand Russell developed a very influential theory of definite descriptions, which sparked another century-long debate. Among other things (here embracing an old principle of interpretation common in Latin scholastic logic), he analyzed definite descriptions as always implying existence claims.

British philosopher David Corfield argues for a new approach to formalizing definite descriptions that does not require existence claims or other assumptions, but only a kind of logical uniqueness of the types of the identity criteria of things. His book Modal Homotopy Type Theory: The Prospect of a New Logic for Philosophy, to which I recently devoted a very preliminary article, has significant new things to say about this sort of issue. Corfield argues inter alia that many and perhaps even all perceived limits of formalization are actually due to limits of the particular formalisms of first-order classical logic and set theory, which dominated in the 20th century. He thinks homotopy type theory (HoTT) has much to offer for a more adequate formal analysis of natural language, as well as in many other areas. Corfield also notes that most linguists already use some variant of lambda calculus (closer to HoTT), rather than first-order logic.

Using first-order logic to formalize natural language requires adding many explicit assumptions — including assumptions that various things “exist”. Corfield notes that ordinary language philosophers have questioned whether it is reasonable to suppose that so many extra assumptions are routinely involved in natural language use, and from there reached pessimistic conclusions about formalization. The vastly more expressive HoTT, on the other hand, allows formal representations to be built without additional assumptions in the representation. All context relevant to an inference can be expressed in terms of types. (This does not mean no assumptions are involved in the use of a representation, but rather only that the formal representation does not contain any explicit assumptions, as by contrast it necessarily would with first-order logic.)

A main reason for the major difference between first-order logic and HoTT with respect to assumptions is that first-order logic applies universal quantifications unconditionally (i.e., for all x, with x free or completely undefined), and then has to explicitly add assumptions to recover specificity and context. By contrast, type theories like HoTT apply quantifications only to delimited types, and thus build in specificity and context from the ground up. Using HoTT requires closer attention to criteria for identities of things and kinds of things.

Frege already had the idea that logical predicates are a kind of mathematical function. Mathematical functions are distinguished by invariantly returning a unique value for each given input. The truth functions used in classical logic are also a kind of mathematical function, but provide only minimal distinction into “true” and “false”. From a purely truth-functional point of view, all true propositions are equivalent, because we are only concerned with reference, and their only reference (as distinguished from Fregean sense) is to “true” as distinct from “false”. By contrast, contemporary type theories are grounded in inference rules, which are kinds of primitive function-like things that preserve many more distinctions.

In one section, Corfield discusses an HoTT-based inference rule for introduction of the definite article “the” in ordinary language, based on a property of many types called “contractibility” in HoTT. A contractible type is one that can be optionally taken as referring to a formally unique object that can be constructed in HoTT, and whose existence therefore does not need to be assumed. This should also apply at least to Platonic Forms, since for Plato one should always try to pick out the Form of something.

In HoTT, every variable has a type, and every type carries with it definite identity criteria, but the identity criteria for a given type may themselves have a type from anywhere in the HoTT hierarchy of type levels. In a given case, the type of the identity criteria for another type may be above the level of truth-functional propositions, like a set, groupoid, or higher groupoid; or below it, i.e., contractible to a unique object. This sort of contractibility into a single object might be taken as a contemporary formal criterion for a specification to behave like a Platonic Form, which seems to be an especially simple, bottom-level case, even simpler than a truth-valued “mere” proposition.

The HoTT hierarchy of type levels is synthetic and top-down rather than analytic and bottom-up, so everything that can be expressed on a lower level is also expressible on a higher level, but not necessarily vice versa. The lower levels represent technically “degenerate” — i.e., less general — cases, to which one cannot “compile down” in some instances. This might also be taken to anachronistically explain why Aristotle and others were ultimately not satisfied with Platonic Forms as a general basis for explanation. Importantly, this bottom, “object identity” level does seem to be adequate to account for the identity criteria of mathematical objects as instances of mathematical structures, but not everything is explainable in terms of object identities, which are even less expressive than mere truth values.

Traditionally, mathematicians have used the definite article “the” to refer to things that have multiple characterizations that are invariantly equivalent, such as “the” structure of something, when the structure can be equivalently characterized in different ways. From a first-order point of view, this has been traditionally apologized for as an “abuse of language” that is not formally justified. HoTT provides formal justification for the implicit mathematical intuition underpinning this generally accepted practice, by providing the capability to construct a unique object that is the contractible type of the equivalent characterizations.

With this in hand, it seems we won’t need to make any claims about the existence of structures, because from this point of view — unlike, e.g., that of set theory — mathematical talk is always already about structures.

This has important consequences for talk about structuralism, at least in the mathematical case, and perhaps by analogy beyond that. Corfield argues that anything that has contractible identity criteria (including all mathematical objects) just is some structure. He quotes major HoTT contributor Steve Awodey as concluding “mathematical objects simply are structures. Could there be a stronger formulation of structuralism?”

Thus no ontology or theory of being in the traditional (historically Scotist and Wolffian) sense is required in order to support talk about structures (or, I would argue, Forms in Plato’s sense). (In computer science, “ontology” has been redefined as an articulation of some world or domain into particular kinds, sorts, or types, where what is important is the particular classification scheme practically employed, rather than theoretical claims of real existence that go beyond experience. At least at a very high level, this actually comes closer than traditional “metaphysical” ontology did to Aristotle’s original practice of higher-order interpretation of experience.)

Corfield does not discuss Brandom at length, but his book’s index has more references to Brandom than to any other named individual, including the leaders in the HoTT field. All references in the text are positive. Corfield strongly identifies with the inferentialist aspect of Brandom’s thought. He expresses optimism about HoTT representation of Brandomian material inferences, and about the richness of Brandom’s work for type-theoretic development.

Corfield is manifestly more formally oriented than Brandom, and his work thus takes a different direction that does not include Brandom’s strong emphasis on normativity, or on the fundamental role of what I would call reasonable value judgments within material inference. From what I take to be an Aristotelian point of view, I greatly value both the inferentialist part of Brandom that Corfield wants to build on, and the normative pragmatic part that he passes by. I think Brandom’s idea about the priority of normative pragmatics is extremely important; but with that proviso, I still find Corfield’s work on the formal side very exciting.

In a footnote, Corfield also directs attention to Paul Redding’s recommendation that analytic readers of Hegel take seriously Hegel’s use of Aristotelian “term logic”. This is not incompatible with a Kantian and Brandomian emphasis on the priority of integral judgments. As I have pointed out before, the individual terms combined or separated in canonical Aristotelian propositions are themselves interpretable as judgments.

Kantian Discipline

The Discipline of Pure Reason chapter in Kant’s Critique of Pure Reason makes a number of important points, using the relation between reason and intuition introduced in the Transcendental Analytic. It ends up effectively advocating a form of discursive reasoning as essential to a Critical approach.

If we take a simple empirical concept like gold, no amount of analysis will tell us anything new about it, but he says we can take the matter of the corresponding perceptual intuition and initiate new perceptions of it that may tell us something new.

If we take a mathematical concept like a triangle, we can use it to rigorously construct an object in pure intuition, so that the object is nothing but our construction, with no other aspect.

However, he says, if we take a “transcendental” concept of a reality, substance, force, etc., it refers neither to an empirical nor to a pure intuition, but rather to a synthesis of empirical intuitions that is not itself an empirical intuition, and cannot be used to generate a pure intuition. This is related to Kant’s rejection of “intellectual” intuition. We are constantly tempted to act as if our preconscious syntheses of such abstractions referred to objects in the way that empirical and mathematical concepts do, each in their own way, but according to Kant’s analysis, they do not, because they are neither perceptual nor rigorously constructive.

All questions of what are in effect higher-order expressive classifications of syntheses of empirical intuitions belong to “rational cognition from concepts, which is called philosophical” (Cambridge edition, p.636, emphasis in original). This is again related to his rejection of the apparent simplicity and actual arbitrariness of intellectual intuition and its analogues like supposedly self-evident truth. It opens into the territory I have been calling semantic, and associating with a work of open-ended interpretation. (See also Discursive; Copernican; Dogmatism and Strife; Things In Themselves.)

I am more optimistic than Kant that something valuable — indeed priceless — can come from this sort of open-ended work of interpretation. Its open-endedness means no achieved result is ever beyond question, but I think we implicitly engage in this sort of “philosophical” interpretation every day of our lives, and have no choice in the matter. I also think serious ethical deliberation necessarily makes use of such interpretation, and again we have no choice in the matter. So, pragmatically speaking, defeasible interpretation is indispensable.

Kant goes on to polemicize against attempts to import a mathematical style of reasoning into philosophy, like Spinoza tried to do. Spinoza’s large-scale experiment with this in the Ethics I find fascinating, but ultimately artificial. It does make the inferential structure of his argument more explicit, and Pierre Macherey used this to great advantage in his five-volume French commentary on the Ethics. But there is a big difference between a pure mathematical construction — which can be interpreted without remainder by something like formal structural-operational semantics in the theory of programming languages, and so requires no defeasible interpretation of the sort mentioned above, on the one hand — and work involving concepts that can only be fully explicated by that sort of interpretation, on the other. Big parts of life — and all philosophy — are of the latter sort. So it seems Kant is ultimately right on this.

Kant points out that definition only has precise meaning in mathematics, and prefers to use a different word in other contexts. I make similar well-intentioned but admittedly opinionated recommendations about vocabulary, but what is most important is the conceptual difference. As long as we are clear about that, we can use the same word in more than one sense. As Aristotle would remind us, multiple senses of words are an inescapable feature of natural language.

Kant says that unlike the case of mathematics, in philosophy we should not put definitions first, except perhaps as a mere experiment. Again, he probably has Spinoza in mind, and again — personal fondness for Spinoza notwithstanding — I have to agree. (Macherey in his reading of Spinoza actually often goes in the reverse direction, interpreting the meaning of each part in terms of what it is used to “prove”, but the order of Spinoza’s own presentation most obviously suggests the kind of thing to which Kant is properly objecting.) More than anything else, meanings are what we seek in philosophical inquiry, so they cannot be just given at the start. We can certainly discuss or dialectically analyze stipulated meanings, but that is strictly secondary and subordinate to a larger interpretive work.

Following conventional practice, Kant allows for axioms in mathematics, but says they have no place in philosophy. He has in mind the older notion of axioms as supposedly self-evident truths. Contemporary mathematics has vastly multiplied alternative systems, and effectively treats axioms like stipulative definitions instead. If we have in mind axioms as self-evident truths, Kant’s point holds. If we have in mind axioms as stipulative definitions, then his point about stipulative definitions in philosophy applies to axioms as well.

A similar pattern holds for demonstration or proof. Mathematics for Kant always has to do with strict constructions, which do not apply in philosophy, where there is always matter for interpretation. (From the later 19th century, mathematicians began increasingly to invent theories that seemed to require nonconstructive assumptions — transfinite numbers, standard set theories, and so on. This is currently in flux again. Contrary to what was thought at an earlier time, it now appears that all valid “classical” mathematics, including transfinite numbers, can be expressed in a higher-order constructive formalism. Arguments are still raging about which style is better, but I am sympathetic to the constructive side.) Philosophical arguments are informally reasoned interpretations, not proofs.

Kant says that speculative thought in general, because it does not abide by these guidelines, unfortunately ends up full of what he does not hesitate to call dishonesty and hypocrisy. (When I occasionally ascribe honesty or dishonesty to a philosopher, it is with similar criteria in mind — especially the presence or absence of frank identification of speculation as such when it occurs. See also Likely Stories.)

The kind of philosophy I am recommending is concerned with explication of meanings, not a supposed generation of truths, so it is not speculative in Kant’s sense. What may not be obvious is just how large and vital the field of this sort of interpretation really is in life. The most common and compact form by which such interpretations are expressed in the small looks syntactically like ordinary assertion, and in ordinary social interaction, mistaking one for the other has little effect on communication. When the focus is not on practical communication but on improving our understanding, we have to step back and look at the larger context, in order to tell what is a speculative assertion and what is an interpretation expressed in the form of assertion. (See also Pure Reason, Metaphysics?; Three Logical Moments.)

(In the present endeavor, the great majority of what look like simple assertions are actually compact expressions of interpretations!)

Definition

The deeper Hegelian truth of a conceptual content can only be approached diachronically, via a historical recollective expressive genealogy. But in passing in the course of his world-historically groundbreaking interpretation, Brandom says Hegel rejects the very possibility of conveying a conceptual content by defining it, without saying what definition is or elaborating on what this denial means for the status of definition (Spirit of Trust, p.7). I find this to be ambiguous, and potentially a little misleading. At least within any given synchronic context and to some extent even more broadly, I believe definition in the sense of an Aristotelian “what it is” still has a positive role to play. It would not be reasonable to suppose that Brandom really means to ban the philosophical use of definitions; otherwise, we would have an extreme nominalism incompatible with his stated goals, which include what he calls conceptual realism. (See also Abstract and Concrete.)

The ambiguity in the passage has to do with how strong a sense we give to “conveying”. We should not expect a run-of-the-mill definitional representation to literally convey conceptual (inferential) content in its explicit form. But such a representation absolutely does address or concern conceptual content, and therefore can still “convey” that content in the weaker sense of referring to it or reliably picking it out. (We could also atypically construct definitions in terms of explicit material incompatibilities and consequences. These would presumably in a stronger sense convey the conceptual content isomorphic to them. We could even atypically construct definitions in terms of the current best expressive genealogy, so I don’t really see these as counterposed.)

I do not think Hegel would go so far as to deny the high pragmatic value of definition in synchronic contexts. This is part of the necessary moment(s) of determinacy (and Understanding) in the larger process of the development of Spirit. He just wants to make the larger point that diachronically, any realized ground-level definition is ultimately just a stopping point along the way. That does not mean we should not attempt to sum up the best understanding we have achieved at each moment. I think we are deontically obligated to do just that. Every ground-level definition is contextualized by its historical situation and therefore subject to change, but at every moment we should still strive to speak and act in accordance with the best definitions we can achieve. Representational clarity is imperfect and always dependent on other considerations in the background, but it is still a moment to be preserved.

We should distinguish the conceptual-content-related doing associated with developing a definition from the representation produced. Further, I find it difficult to separate a concern for definition from a methodological concern for problems of definition, as evinced by Plato and Aristotle for instance. From this perspective, definition has more to do with a line of questioning than a putative answer. The question of the “what” or conceptual content of things is actually far more substantial and interesting than those of mere fact or abstract existence. Even if it aims at a representation, definition as a practical task is all about inquiry into that whatness of things. The norm to which synchronic representation of whatness is responsible comes down to the best achievable view of the relevant difference and mediation, or material incompatibility and material consequence (as Brandom would put it) in the circumstances of that logical moment. This I think is actually independent of the diachronic moves of expressive genealogy.

Hegel’s “Substance that is also Subject” is explicitly presented as an extension of Aristotle’s (expressive meta) concept of ousia, and I think Aristotle anticipates even more than Hegel recognizes. (Expressive genealogy is distinctively Hegelian, but Substance certainly not, and Hegel himself notes in the History of Philosophy lectures that the concerns he groups under “Subject” were significantly addressed by Socrates, Plato, and Aristotle.)

If Brandom is right that Hegel intended to exclude such expressive metaconcepts from the general prognosis that all (ground-level) concepts eventually elicit their own negation, then it is at least logically possible that Aristotle’s metaconcept had already achieved the requisite stability to be incorporated by Hegel without negating the subordinate aspect of ousia that for Aristotle corresponds to a definition.

Without prejudice to claims about what Hegel added, I would argue that Hegel did in this way intend to incorporate all the multiple nuances of Aristotelian ousia, including the definitional one. With due respect for Brandom’s distinction between determination as Hegelian process and determinateness as Kantian/Fregean property (and the importance of the process as a superior point of view), I also think we need to forgivingly recollect all best attempts at determinateness. (See also Classification.)

I wonder what Brandom would say about the role of definitions in the articulation of mathematical conceptual content. The doing of mathematics seems to join the doing of history as problematic for simple subsumption under a genealogical approach as Brandom has described it. Mathematics needs definitions, and history needs to evaluate data without Whiggish filtering. (But Brandom does not exactly disallow either, and I can’t imagine that he would want to. The meaning of mathematical theorems can certainly be expressed in terms of material incompatibility and consequence, and the concepts used in non-Whiggish historiography could themselves be Whiggishly genealogically grounded.)

We should think about the functional inferential role of stipulative definitions, as well as the definitions of empirical concepts that I expect Brandom has foremost in mind. We could say that in both cases, the meaning sought by definition — as distinct from the definiens — is actually constituted through material incompatibility and material consequence. But a stipulative definition is a making rather than a taking. It in a sense starts a whole course of reasoning, whereas empirical concepts implicitly summarize results of reasoning.

Also, mathematical definition is mostly concerned with structures and structural properties. I believe a case could be made that in general, such structures and structural properties are expressive metaconcepts in much the same sense that logical concepts are.

I don’t think it’s historically right that expressive metaconcepts are a “discovery or invention” of German Idealism (p.5). Aristotle already had quite a few expressive metaconcepts, as at least partially exhibited in this blog. I believe Hegel himself recognized this.

Higher Order

Before and after early modern mechanism and in contrast to it, Plato, Aristotle, Kant, and Hegel all broadly agreed on the normative importance of higher-order things.

In modern terms, Plato’s forms are higher-order things, as distinct from first-order things. Plato trusts higher-order things more than first-order ones, because he considers only higher-order things to be knowable in the sense of episteme, because only higher-order things contain an element of universality, and episteme applies only to universals, not particulars.

Aristotle agrees that higher-order things are ultimately more knowable, but believes it is possible to say more about first-order things, by relating them to each other and to higher-order things; that our initial rough, practical grasp of first-order things can help us to begin to grasp higher-order things by example; and that going up and down the ladder of abstraction successively can help us progressively enrich our understanding of both.

(Incidentally, I have always read the Platonic dialogues as emphasizing the normative importance of acquiring a practical grasp of forms more than specific existence claims about “the forms”. Aristotle’s criticisms make it clear that at least some in the Platonic Academy did understand Plato as making such existence claims, but particularly in what are regarded as later dialogues like Parmenides, Sophist, and Theaetetus, what is said about form seems relatively close to an Aristotelian view. It is even possible that these dialogues were influenced by the master’s even greater student.)

Early modern mechanism completely discarded Plato and Aristotle’s higher-order orientation. Descartes famously recommends that we start by analyzing everything into its simplest components. This temporarily played a role in many great scientific and technological advances, but was bad for philosophy and for people. Hegel calls this bottom-up approach Understanding, as distinct from Reason.

Early and mid-20th century logical foundationalism still adhered to this resolutely bottom-up view, but since the later 20th century, there has been an explosion in the use of higher-order formal concepts in mathematics, logic, and computer science. It turns out that even from an engineering point of view, higher-order representations are often more efficient, due to their much greater compactness.

Leibniz already tried to reconcile mechanistic science with a higher-order normative view. He also contributed to the early development of higher-order concepts in mathematics.

Kant and Hegel decisively revived an approach that is simultaneously higher-order and normative.

Aristotle and Mathematics

Aristotle wrote near the very beginning of the golden age of Greek mathematics. He criticized the mathematics of his day (arithmetic and geometry) as being useful but insufficiently abstract, which was a very valid point at the time. In particular, it did not offer much support for showing the intelligibility of becoming, which was his main goal in the Physics. He also took a strong stand against Pythagorean superstition, which at the time was hard to separate from enthusiasm for mathematics.

We do not know how Aristotle would have responded to category theory or homotopy type theory, or even algebra or calculus. But given the nature of his criticism, it seems extremely questionable to simply assume he would not have welcomed such advances. (See also The Animal’s Leg Joint.)