This blog post outlines a philosophy of mathematics centered on representational structures, rejecting both the view that math is merely a tool for physics and the view that it is solely a game of analytic derivations in abstract logic. I draw on Robert Stalnaker’s pragmatics and the solution used by Stalnaker to address the “problem of equivalence”, which is the puzzle that, since all mathematical truths are necessary, they should logically all have the same content. The solution is that mathematics is not just about the truths themselves, but about how we represent them: it is the study of the specific connections between formulas, proof patterns, and axiomatic systems. Therefore, to learn mathematics is to master these symbolic manipulations, and to explore the capabilities of different representational systems.
Under this view, mathematics acts as a formal model of natural language. Mathematical concepts (like numbers or triangles) originate from ordinary language practices (counting or measuring) and are then “reified” into rigorous systems to allow for consistent operations. This perspective explains why mathematics is universally applicable (because these structural patterns are portable across domains) and why “auxiliary” concepts like hyperreals are useful (they serve as temporary representational bridges). It also explains why reducing mathematical objects to set theory often feels wrong (producing “junk theorems”): such reductions might work formally, but they sever the connection to the original representational roles that the concepts play in our natural language.
We cannot deny the axioms of mathematics; for they exhibit nothing more than a consistent use of words, and affirm of some idea that it is itself and not something else. —William Godwin, Political Justice, book 1, chapter 4
Philosophies of mathematics
This blog post is about philosophy of mathematics. As I’ve said before, there are roughly three types of philosophy of mathematics:
- an excuse to do more mathematics: here we have how intuitionism created intuitionist mathematics, or how there are foundation movements that try to recast all of mathematics in terms of type theory or category theory.
- “math is only real if it’s applied”: here we have so-called “mathematical naturalism”, which gives us the famous arguments that mathematical objects must be real because, and only insofar as, they are indispensable to physics.
- you can accept all of current math if you just accept this very unclear idea: the idea here is that you can give an illuminating philosophical interpretation to all of current mathematics—not just the parts that get applied practically—, and without even needing to reinterpret it in any restricted logic (as in intuitionism) or foundational theory (such as category theory or type theory). This is an ambitious goal, and the catch seems to be that every proposal that claims to achieve it seems to do its work by way of providing a very vague and unsatisfying interpretive scheme, such as by saying mathematics is a meaningless game of symbols, or that it consists of tautologies, etc., where it isn’t clear what makes it true that all-and-only mathematics fits the interpretive scheme, and how this fits the various appearances about mathematical practice.
I don’t think any idea about philosophy of mathematics fully escapes being one of these types, and this blog post belongs to type 3, but I hope I have mitigated the vagueness enough that it says something interesting.
The fact that I’m vaguely gesturing at something that isn’t very precise makes it all important to give the reader a feel for the kind of thing that was said in the literature that constitutes the broad kind of background that I’m coming from, without making the reader go read all of those books in full; which I hope explains why I’ve thrown all these quotations in there, including the famous Kant quotation that everyone has seen before (but who knows if someone hasn’t?).
What mathematics is about
Now if mathematical truths are all necessary, then on the possible worlds analysis there is no room for doubt about the truth of the propositions themselves. There are, it seems, only two mathematical propositions, the necessarily true one and the necessarily false one, and we all know that the first is true and the second false. But the functions that determine which of the two propositions is expressed by a given mathematical statement are just the kind that are sufficiently complex to give rise to reasonable doubt about which proposition is expressed by a statement. Hence it seems reasonable to take the objects of belief and doubt in mathematics to be propositions about the relation between statements and what they say. —Robert Stalnaker, Inquiry, p. 73
Stalnaker’s pragmatics, as explained in his books Inquiry (1984; especially chapter 4) and Context and Content (1999; especially chapter 12), begins with what he calls a “coarse-grained” semantic picture. Propositions, at the most basic level, are sets of possible worlds (or functions from worlds to truth-values). With that in hand, you can model assertion as eliminating worlds from the conversational context, presupposition as fixing a common-ground set, and inquiry as progressively narrowing what remains live. As spare as it is, that model earns its keep: Stalnaker can explain a lot of linguistic practice while keeping the semantic core simple and truth-conditional.
But the same simplicity creates what Stalnaker calls the problem of equivalence. If propositions are just sets of worlds, then necessarily equivalent statements—especially in mathematics—collapse to the same content. If every mathematical truth is necessary, it can look as if there are, informationally, only two mathematical propositions: the necessary truth and the necessary falsehood. And yet we plainly distinguish “17 is prime” from “the angles of a Euclidean triangle sum to 180°,” can believe one without believing the other, and can learn one without learning the other.
Stalnaker’s motivating response is to treat this not as a reason to abandon the possible-worlds model of informational content, but as a reason to notice something special about mathematical inquiry: much of what we learn in mathematics is about representational structures and their relations to that coarse-grained content. We do not just “latch onto” a necessary truth; we learn (and can be ignorant of) whether a given formula, proof pattern, or axiom-system representation determines that truth. Mathematics, on this view, is not merely a repository of necessary contents; it is, centrally, a disciplined study of structures of representation—structures that can be instantiated in different languages and notations, and manipulated by calculation and proof.
How mathematics is learned
Suppose there were a community of English speakers that grew up doing its arithmetic in a base eight notation. The words “eight” and “nine” don’t exist in its dialect; the words “ten” and “eleven,” like the numerals “10” and “11” denote the numbers eight and nine. Now suppose that some child in this community has a belief that he would express by saying “twenty-six times one hundred equals twenty-six hundred.” Would it be correct to say that this child believes that twenty-two times sixty-four equals fourteen hundred eight? This does not seem to capture accurately his cognitive state. His belief, like our simple arithmetical beliefs, is not really a belief about the numbers themselves, independently of how they are represented. —Robert Stalnaker, Context and Content, p. 237
The representational-structure picture fits the phenomenology of learning mathematics unusually well. Mathematical competence is inseparable from activities like: transforming inscriptions, applying rules of inference, calculating, constructing proofs, translating between equivalent forms.
Those are not optional “presentational” extras. They are how mathematical information is acquired and used. If learning that 689×43=29627 were merely ruling out worlds where a necessary truth fails, it would be mysterious how computation helps—since computation doesn’t teach us which worlds are actual. But if what you are learning is (roughly) that this representational procedure connects these numerals and operations to this result, then calculation is exactly the right kind of epistemic act: it is an exploration of the representational system’s structure.
So the “object” of mathematical belief is often best seen as involving claims like: a representation with this structure has this content; a derivation of this form preserves truth relative to these axioms; this symbolic construction yields a representation equivalent to that one. Those are propositions we can genuinely be ignorant of and come to know by manipulating the system.
How mathematics is applied
An initial hint that numbers are magnitudes comes from their algebra. The natural numbers, the positive real numbers and the ordinal numbers each have an associative operation of addition which defines a linear order. I call a system with that precise algebraic structure a positive semigroup. We find positive semigroups cropping up not only in mathematics but in the physical sciences as well. The fundamental physical magnitudes have this same algebraic structure: for example, mass is a positive semigroup because addition of masses is associative, and the addition defines a linear order. The same is true for length, area, volume, angle size, time and electric charge: these and other physical magnitudes all have the structure of a positive semigroup. —Keith Hossack, Knowledge and the Philosophy of Number, p. 1
Our picture also helps with a classic puzzle: mathematics is generally applicable—it shows up everywhere—yet it also seems to be about very particular objects (the number 2, the empty set, π, triangles, and so on).
If mathematics is fundamentally about representational structures, its generality is no surprise. Representational structures are general: they are patterns of form and transformation that can be implemented in many domains. The same algebraic structure can organize bookkeeping, physics, logic, probability, and geometry because those practices can be represented in ways that share the relevant structure.
At the same time, it feels like mathematics is about particular objects because our linguistic practice encourages reification. We talk as if “2” names an object; we quantify over numbers; we introduce constants (“0”, “∅”); we build theories that treat these as a stable domain. That practice is extremely useful: it streamlines inference and stabilizes coordination. But on the representational view, the “particularity” of mathematical objects is tightly connected to how our public language fixes and reuses representational roles.
How mathematics is motivated
The trouble with this objection is that it completely ignores history: the theory of real numbers, and the theory of differentiation etc. of functions of real numbers, was developed precisely in order to deal with physical space and physical time and various theories in which space and/or time play an important role, such as Newtonian mechanics. Indeed, the reason that the real number system and the associated theory of differentiation etc. is so important mathematically is precisely that so many of the problems to which we want to apply mathematics involve space and/or time. It is hardly surprising that mathematical theories developed in order to apply to space and time should postulate mathematical structures with some strong structural similarities to the physical structures of space and time. It is a clear case of putting the cart before the horse to conclude from this that what I’ve called the physical structure of space and time is really mathematical structure in disguise. —Hartry Field, Science Without Numbers
Mathematics developed, initially, from ordinary language, and then grew more complex to account for more technical, scientific language. The natural numbers are a model of the counting numbers we deploy in ordinary language: “two apples,” “three steps,” “four chairs.” We naturally used them for measurement of continuous quantities as well, since measurement tools (such as rulers) require counting (of markings on the rulers, for instance) to be used.
Extensions beyond the naturals—negative numbers, rationals, reals, complex numbers—are not forced on us by everyday counting talk, but they can be similarly motivated by ordinary counting contexts themselves, insofar as there is a practical need to perform operations (subtraction, division, solving equations) in a way that preserves and extends successful inferential patterns. We introduce new entities so that the representational system remains closed under operations we already treat as coherent.
On this view, the payoff of “new numbers” is that they yield correct, systematic consequences about the original practice. We extend the representational framework, do the work there, and recover truths that constrain ordinary counting-number claims.
Nonstandard analysis is a good illustration, since it introduces the hyperreal numbers, but only does so in order to formulate theorems of calculus, whose interest lies in what they say about real numbers. Hyperreal numbers function as an auxiliary representational domain: you begin with real-valued problems, temporarily move into the hyperreals to make certain reasoning patterns (infinitesimals, transfer principles) tractable, and then return with a theorem stated purely in terms of the reals. The hyperreals are not forced on us as “the real subject matter”; they are a representational device that systematizes certain operations and delivers results about the original target domain.
More exotic mathematical theories, which are more removed from ordinary practices, must be seen to correspond to more unusual, specialized, or tightly constrained contexts of representation—ways of carving up possibilities that aren’t our everyday default, but that could in principle become useful for unusual particular tasks.
How mathematics is interpreted
Give a philosopher the concept of a triangle, and let him try to find out in his way how the sum of its angles might be related to a right angle. He has nothing but the concept of a figure enclosed by three straight lines, and in it the concept of equally many angles. Now he may reflect on this concept as long as he wants, yet he will never produce anything new. He can analyze and make distinct the concept of a straight line, or of an angle, or of the number three, but he will not come upon any other properties that do not already lie in these concepts. But now let the geometer take up this question. He begins at once to construct a triangle. Since he knows that two right angles together are exactly equal to all of the adjacent angles that can be drawn at one point on a straight line, he extends one side of his triangle, and obtains two adjacent angles that together are equal to two right ones. Now he divides the external one of these angles by drawing a line parallel to the opposite side of the triangle, and sees that here there arises an external adjacent angle which is equal to an internal one, etc. In such a way, through a chain of inferences that is always guided by intuition, he arrives at a fully illuminating and at the same time general solution of the question. —Immanuel Kant, Critique of Pure Reason (Cambridge ed.), A716/B744
Kant famously treated mathematics as synthetic a priori: not merely unpacking meanings, but extending knowledge while remaining non-empirical. Others—especially in the Fregean and later formalist traditions—press the opposite thought: mathematics is analytic, derivable by logic and definitions.
The representational-structure view suggests a reconciliation: Within an axiom system, many results are “analytic” in a formal sense: they follow by rule-governed transformations from the axioms. But choosing the axiom system is not itself analytic. It is a modeling decision about which representational structure we are using to regiment some practice (counting, measuring, spatial reasoning, etc.). So mathematics is “analytic” only conditional on the axioms and rules—on the representational framework you adopt. As a model of natural-language practices (counting, describing space, tracking quantities), it is not analytic in any simple way, because natural-language meanings are not fixed enough to determine a unique axiom system. We can adopt different axioms and get different theorems—precisely because we are building different precise models of an imprecise practice.
Geometries as precisifications of spatial intuitions
In Kant’s time, only Euclidean geometry existed. Kant claimed geometry provided synthetic a priori knowledge about all possible sense experience, because it rested, according to him, on the spatial structure which is inherent to our sensory perception. The later development of non-Euclidean geometries has been thought to undermine Kant’s privileging of Euclidean geometry in this way; but as I see it, the fact that it’s even possible to see Euclidean geometry and alternative geometries as somehow in conflict, rather than simply talking about entirely different things, is a powerful illustration of the point I just made.
In both Euclidean and hyperbolic geometry, despite their different axioms and theorems, we really mean the same thing by the word “triangle”: a triangle is a figure determined by three non-collinear points joined pairwise by straight lines (geodesics). What changes across Euclidean and hyperbolic geometry is not that we suddenly mean something different by “triangle,” but that we add different background modeling assumptions about the ambient space—assumptions that determine which geometrical inferences are licensed. Once those modeling assumptions are in place, predictions diverge: Euclidean triangles have angle-sum 180°, while hyperbolic triangles have angle-sum less than 180°.
The key thought is that these are two precise mathematical models of the flexible, somewhat indeterminate natural-language apparatus of “straight,” “triangle,” and “space.” Different contexts of use—surveying small regions vs. reasoning about curved or non-Euclidean spaces—pull us toward different regimentations.
Reduction and junk theorems
I suggest that both of the above difficulties with the set-theoretical foundational consensus arise from the same source – namely, its strongly reductionistic tendency. Most mathematical objects, as they originally present themselves to us, are not sets. A natural number is not a transitive set linearly ordered by the membership relation. An ordered pair is not a doubleton of a singleton and a doubleton. A function is not a set of ordered pairs. A real number is not an equivalence class of Cauchy sequences of rational numbers. Points in space are not ordered triples of real numbers, and lines and planes are not sets of ordered triples of real numbers. Probability is not a normalized countably additive set function. A sentence is not a natural number. A proof is not a sequence of finite strings of symbols formed in accordance with the rules of some formal system. Each reader can supply his own examples of cases in which mathematical objects have been replaced in our thought and in our teaching by other, purely conceptual, objects. These conceptual objects may form structures which are isomorphic to relevant aspects of the structures formed by the objects we were originally interested in. They are, however, distinct from the objects we were originally interested in. Moreover, they do not fit smoothly into the larger structures to which the original mathematical objects belong. —Nicolas D. Goodman, The Knowing Mathematician
The representational view sheds light on how reductions of one mathematical theory to another always produce “junk theorems”, like “2 ∈ 3” or “1 is the powerset of 0.” (J.D. Hamkins, in his Lectures on the Philosophy of Mathematics, adds the example of John Conway’s account of numbers as games, which produces all familiar theorems about numbers, but where one may ask, “who wins 17?”) Inside a particular coding scheme (say, identifying numbers with particular sets), such statements can come out true. But we recoil because these are artifacts of the encoding, not stable generalizations that track our ordinary linguistic practice with numerals and membership talk.
In other words: reductions can preserve formal structure while scrambling representational roles that matter for interpretation. The resulting theorems may be harmless as bookkeeping inside the reduction, yet they fail as a model of how we actually use “2,” “element of,” or “powerset” in natural language. Our discomfort is pragmatic and semantic at once: the reduction has shifted us into a representational regime where the sentences no longer line up with the inferential and explanatory roles those expressions play in ordinary discourse.
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| Illustration for this blog post, drawn by Nano Banana. |
Postscript: I never watch videos, but the day after I posted this, I watched this video about Euclid by Ben Syversen, which I think is related to this post somehow.
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