Mathematical Platonism and its Critics

Logic & Philosophy of Mathematics
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Mathematical Platonism and its Critics

I’ve long been persuaded by the basic intuition of mathematical Platonism regarding the reality of number and other mathematical entities. It came upon me as a kind of epiphany - the realisation that the ancients esteemed mathematical objects because they were non-temporal — they didn’t come into or go out of existence — and they were not composed of parts (although later I came to realise that this strictly speaking only applies to prime numbers).

This indicated that mathematical objects are real in a different manner to the objects of sense which are temporally delimited and composed of parts, and are thus subject to change and decay. And also that the objects of mathematics are in some profound sense nearer to the eternal or the unchanging than are ordinary particulars.

I found a passage in a well-regarded textbook which validated this intution:

Neoplatonic mathematics is governed by a fundamental distinction which is indeed inherent in Greek science in general, but is here most strongly formulated. According to this distinction, one branch of mathematics participates in the contemplation of that which is in no way subject to change, or to becoming and passing away. This branch contemplates that which is always such as it is and which alone is capable of being known: for that which is known in the act of knowing, being a communicable and teachable possession, must be something that is once and for all fixed ~ Jacob Klein, Greek Mathematical Thought and the Origin of Algebra.

When I started posting on philosophy forums I posed my first question about this topic. I was soon to discover that most were not receptive, preferring various forms of constructivism or nominalism, saying that mathematics is created by the mind or at any rate is a human construction. I was to learn that ‘discovered or invented’ was the short-hand description of this division which remains a fault-line in contemporary philosophy.

However to this day there are still mathematical Platonists, a famous example being Kurt Gödel, about whom Rebecca Goldstein said:

Gödel was a mathematical realist, a Platonist. He believed that what makes mathematics true is that it’s descriptive—not of empirical reality, of course, but of an abstract reality. Mathematical intuition is something analogous to a kind of sense perception. In his essay “What Is Cantor’s Continuum Hypothesis?”, Gödel wrote that we’re not seeing things that just happen to be true, we’re seeing things that must be true. The world of abstract entities is a necessary world—that’s why we can deduce our descriptions of it through pure reason.

Another is Sir Roger Penrose:

Mathematics itself indeed seems to have a robustness that goes far beyond what any individual mathematician is capable of perceiving. Those who work in this subject, whether they are actively engaged in mathematical research or just using results that have been obtained by others, usually feel that they are merely explorers in a world that lies far beyond themselves–a world which possesses an objectivity that transcends mere opinion, be that opinion their own or the surmise of others, no matter how expert those others might be.

To those who say that mathematics is created by the mind Penrose will reply that this is a circular argument, because for consensus to be reached requires an external standard to which reasoning must conform. In other words, we refer to mathematical objects and truths to judge whether or not and to what extent our minds understand them.

The Backlash against Platonism

Further research informed me that most modern philosophy of maths is anti-Platonist in orientation. The SEP entry Philosophy of Mathematics notes that:

The general philosophical and scientific outlook in the nineteenth century tended toward the empirical: platonistic aspects of rationalistic theories of mathematics were rapidly losing support. Especially the once highly praised faculty of rational intuition of ideas was regarded with suspicion.

Why the suspicion of Platonism? I say it’s because of a deeper, underlying conflict between Platonism and Naturalism. Consider this passage from another SEP article, this one on the Implications of Platonism in Philosophy of Mathematics:

Mathematical platonism has considerable philosophical significance. If the view is true, it will put great pressure on the physicalist idea that reality is exhausted by the physical. For platonism entails that reality extends far beyond the physical world and includes objects that aren’t part of the causal and spatiotemporal order studied by the physical sciences. Mathematical platonism, if true, will also put great pressure on many naturalistic theories of knowledge. For there is little doubt that we possess mathematical knowledge. The truth of mathematical platonism would therefore establish that we have knowledge of abstract (and thus causally inefficacious) objects. This would be an important discovery, which many naturalistic theories of knowledge would struggle to accommodate.

An explicit argument against Platonism was the subject of a very influential 1973 paper called Mathematical Truth by Paul Beneceraff.

Briefly, the salient point of Benacerraf’s epistemological argument against Platonism is based on what he describes as a causal constraint on knowledge — that to know something is true, you must be situated in an appropriate causal relationship with the facts that make it true. We know there’s a cup on the table because the cup causally affects our perceptual apparatus. Knowledge requires causal contact with its object. Notice the explicitly empiricist assumption behind this, ‘all knowledge through (sensory) experience.’

Mathematical objects, on the Platonist account, are abstract — outside space, time, and the causal order entirely. They have no causal powers and so cannot affect anything. Therefore, on the causal constraint, mathematical knowledge becomes unjustifiable - there is no identifiable causal relationship between them, as objects, and the observing mind.

A similar point is made in a magazine article about philosophy of maths published in the Smithsonian Magazine:

Scholars—especially those working in other branches of science—view Platonism with skepticism. Scientists tend to be empiricists; they imagine the universe to be made up of things we can touch and taste and so on; things we can learn about through observation and experiment. The idea of something existing “outside of space and time” makes empiricists nervous: It sounds embarrassingly like the way religious believers talk about God, and God was banished from respectable scientific discourse a long time ago.

There is a series of arguments called the Indispensability Arguments for Mathematics, associated with analytical philosophers Quine and Putnam, which attempt to address Beneceraff’s dilemma. Notice the article on this topic also calls into question the very idea that the mind possesses a rational faculty which can intuitively grasp mathematical objects:

(Rationalist) philosophers…claim that we have a special, non-sensory capacity for understanding mathematical truths, a rational insight arising from pure thought. But, the rationalist’s claims appear incompatible with an understanding of human beings as physical creatures whose capacities for learning are exhausted by our physical bodies.

One would think that this might give pause to the idea that we are, in fact, ‘physical creatures’ - but for naturalism that possibility is not on the table. As I will show, this is the underlying ground of the argument against Platonism.

A Caveat regarding ‘Abstract Objects’

One point to notice throughout these debates is the underlying ambiguity of the expression ‘mathematical objects’ or ‘intelligible objects’. This expression implies the existence of an ‘abstract object’ perhaps in a so-called ‘ethereal realm’. I’m sure this is the source of the reifications that plague this topic. But the term ‘object’ in this context really means something more like ‘the object of an exercise’ or ‘the object of the argument’ — something to be understood or grasped by reason. My view is that mathematical objects are essentially acts of intellect — not objects per se. (I believe this is a point elaborated by Husserl.)

The Eclipse of Reason

I think it becomes clear where the conflict really lies in these arguments. They are variations of the argument against the idea that reason is able to understand truths on grounds other than the empirical. Recall from the initial SEP article, the deprecation of the ‘once highly praised faculty of rational intuition’. I suggest that the very faculty of reason is called into question in these arguments. And this is because contemporary naturalism seems animated by the empiricist conviction that nature is ‘blind’ or devoid of reason, which must be super-imposed on it by the observing mind. But this fails to account for the ‘unreasonable effectiveness of mathematics in the natural sciences’ (per Wigner and Hamming)

Mind and World

Mathematics is the discipline where mind and world most explicitly meet. At its foundations, mathematical facts are not constructed but encountered - the world has rational structure and the mind is capable of apprehending it through the faculty of reason. Accordingly, mathematics is neither purely subjective, a free construction of the human mind, nor is it purely objective in the sense of existing independently of the observer. It is rather the discipline in which the deep consonance between mind and world are entwined. That this was traditionally regarded as an indication of a higher intelligence is perhaps an implicit reason why it is resisted so arduously by naturalism. But to then demand justification of mathematical insight by reference to sensory knowledge alone seems to call into question the very faculty which underwrote much of ‘natural philosophy’ in the first place.

Does this mean I believe that mathematics is ‘for once and for all fixed’? Not at all. As knowledge expands, so too mathematics develops, but then advances in mathematics itself may open horizons that would otherwise remain unknown. As a well-known saying has it, ‘God made the integers, all else is the work of Man.’ I grant the truth of that, but also observe that were we not able to grasp the integers then those works of Man would remain forever undone. As Wigner concludes in his now-famous essay on the unreasonable effectiveness of mathematics in the natural sciences:

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.

Philosophy of Mathematics SEP
Platonism in Philosophy of MathematicsSEP
The Unreasonable Effectiveness of Mathematics in the Natural Sciences Eugene Wigner
Unreasonable Effectiveness of MathematicsR W Hamming
Penrose On Whether A Platonic Objectivity Can Exist Independent of Human Minds.
The Benacerraf Problem of Mathematical Truth and Knowledge IEP
The Indispensability Argument in the Philosophy of MathematicsIEP
What is Math? Smithsonian Magazine

Gödel and the Nature of Mathematical Truth, Rebecca Goldstein

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I happen to have been looking at some philosophy of mathematics over the past couple of days and I wonder if the field is divided in the way you say, with Platonism on one side and reductive naturalism on the other. What I’m seeing is a lot of structuralism and naturalized Platonism. I was just reading “Naturalized Platonism vs. Platonized Naturalism” by Linsky & Zalta (PDF available online), which defends the latter position. Penelope Maddy is very prominent too, and her position (or positions, since it seems they’ve changed a lot) are not easily classified as one or the other.

Let’s say we can literally perceive abstract objects like sets, in the same way we perceive ordinary things, i.e., immanent to the causal order. Wouldn’t that effectively naturalize Platonism and go some way to dissolving the great battle you’re describing?

And this reminds me of something I was reading in connection with Adorno: Husserl’s categorial intuition, in which our access to abstract objects is rooted in perception but not reducible to the sensory.

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I think it’s more a matter that ‘staunch naturalism’ has had to give ground on account of the difficulties it presents to science. That IEP article on The Indispensability Argument has many passages which throw the distinction between mathematical intuiton and empirical cognition in to sharp relief:

Mathematical objects are in many ways unlike ordinary physical objects such as trees and cars. We learn about ordinary objects, at least in part, by using our senses. It is not obvious that we learn about mathematical objects this way. Indeed, it is difficult to see how we could use our senses to learn about mathematical objects. We do not see integers, or hold sets. …Sets are abstract objects, lacking any spatio-temporal location. Their existence is not contingent on our existence. They lack causal efficacy. Our question, then, given that we lack sense experience of sets, is how we can justify our beliefs about sets and set theory.

and most tellingly:

The indispensability argument in the philosophy of mathematics is an attempt to justify our mathematical beliefs about abstract objects, while avoiding any appeal to rational insight. Its most significant proponent was Willard van Orman Quine.

Quine was a proponent of ‘naturalised epistemology’ i.e. that epistemology should conform to the discoveries of science, and that there was no ‘first philosophy’ or metaphysics. But the question of the reality of mathematical objects is undeniably a metaphysical question, because it revolves around what kind of reality such objects have. I maintain there is no conceptual space for ‘kinds of reality’ in much of empiricism (save perhaps in modal metaphysics, but this is anchored in semantics not in ontology.)

So in answer to that specific question, what do you mean by ‘perceive’? If you grant that they are perceived ‘by reason’, there is no contest! But whether there is such a kind of non-empirical cognition is the point raised in the article I’m referencing.

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Great topic !

For context, my degrees are in math. So I am passionate about math. But not all math is invented equal !

That joke is my attempt to point at the leap from the positive integers to something like the power set of the real numbers. “God made the integers, the rest is the work of man.” This same Kronecker passionately opposed Cantor’s set theory.

The positive integers, many will grant, are “imposed on us.” Hard to think around. “Intuitively secure.” A few radicals, like ultra-finitists, problematize the assumption of an “infinity” of such numbers. But most aren’t troubled by statements about “all” natural numbers. They implicitly accept this particular infinity as vaguely somehow completed, so that it can function as a truthmaker for any sufficiently definite statement.

As you may know, the power set of a set is the set of all its subsets. The problem is that by “taking power sets” we quickly get beyond computationally secure meaning, to a hazy linguistic meaning, which is hard to tell from what some would call poetry.

In a formal system, we might start with an axiom of infinity. Or the axiom that every set “just automatically has” a power set —even if it’s not intuitively secure. The game has strict rules like Chess, which are computationally secure, which lets the question of meaning fall to the side.

Given our previous conversations, I take you as a “Platonist” about integers, at least. But are you a Platonist about the continuum hypothesis ?

I ask this because I find the positive integers to be “imposed on us” ( in a way that deserves explication) while finding the continuum hypothesis too fuzzy to take seriously, given the fuzziness of the real numbers as often conceived/described.

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Though it’s true that naturalized platonism is still committed to ontological naturalism, the point of it is precisely to do justice to the functional independence and objectivity of mathematical objects that Platonism captures, without invoking a separate realm. What I object to in your position, among other things, is that you presuppose that this isn’t possible. You presume that only full-fat Platonism can account for the objectivity and independence of mathematical objects. But this has been contested now for a long time.

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Beyond my scope! Unlike you, I didn’t graduate in maths. (Odd that later in life I’ve discovered this topic). But I’m not going to pretend to be able to answer that question — as I understand it, it puzzled Gödel himself.

I think that’s a bit unfair. I’ve ilustrated a conflict between various schools of philosophy and platonism, and those schools were said by the reference article to have been developed in opposition to Platonism and out of distrust of the ‘once highly-praised faculty of rational intuition’. If there is a ‘naturalised platonism’ I’d be interested to hear it, although I note Lloyd Gerson’s claim that all attempts to reconcile naturalism and platonism outside mathematics have not succeeded, at least in his view. But I’m not foreclosing on anything.

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I agree with the first sentence but not with the second, and I think this is in line with a lot of naturalism. So the big question is whether the features that make mathematical objects real in a different manner than empirical objects require any proximity to an eternal order. But your dedication to the transcendent makes this connection inevitable.

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I think it’s justified, because it’s not like you’re just describing the field. For example:

But naturalism in mathematics is not committed to opposing “the idea that reason is able to understand truths on grounds other than the empirical”.

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But consider the sense in which mathematical theories in natural science are of a higher order of understanding than isolated ad hoc observations of individual phenomena. Didn’t this synthetic ability originate from the platonist background of science in the first place? The capacity to see ratios, relationships and structural patterns across phenomena isn’t itself an empirical observation. It’s an act of rational comprehension of a higher order.

I think here you’re implying that such arguments are based on religious convictions and therefore not amenable to persuasion. But that’s an illustration of the passage from the Smithsonian essay: that the assertion of the reality of non-empirical objects seems like religious conviction.

Notice when I say ‘these arguments’, I gave specific quotes which illustrate the point I’m making. That’s why I picked them. If there are other examples in those articles which present an alternative construction by all means point them out.

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I’m happy to go along with this epistemology, but it doesn’t really support a Platonic metaphysics.

Fair enough.

I’m not denying that your sources support your position. I’m denying that they properly represent the field.

I apologize for my brevity. I’m on the fence as to whether to commit to this topic, given the research it will require :smiley:

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that’s totally OK, they’re all very good questions. I’ve bought one of the key texts in the field, Platonism, Naturalism and Mathematical Knowledge, James Robert Brown (Routledge - review here). He’s the representative Platonist in the Smithsonian article I quoted, which is why I sourced his book. It’s difficult material, especially for those without much background in actual mathematics, but these conversations may help to navigate it. (For instance there’s a whole chapter on Maddy.)

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Again, great OP !

First, we look at Gödel’s paper.

Cantor’s continuum problem is simply the question: How many points are there on a straight line in Euclidean space? In other terms, the question is: How many different sets of integers do there exist?

This is what he’s platonistic about, which is well beyond the positive integers that most intuit as intensely transpersonal or intersubjective. He goes on to justify, in his own eyes, that this question is sufficiently meaningful and has a definite answer. The brief, informal justification fails, in my view — but he understands this already.

This negative attitude towards Cantor’s set theory, however, is by no means a necessary outcome of a closer examination of its foundations, but only the result of certain philosophical conceptions of the nature of mathematics, which admit mathematical objects only to the extent in which they are (or are believed to be) interpretable as acts and constructions of our own mind, or at least completely penetrable by our intuition.

He’s well aware of intuitionism, and that Brouwer rejects “the interesting” infinite sets as non-constructive. Brouwer declared that there are no unexperienced truths.

Platonism, as described in the paper, projects something beyond what can be perfectly intuited.

And this is confirmed here:

Mathematical platonism is any metaphysical account of mathematics that implies mathematical entities exist, that they are abstract, and that they are independent of all our rational activities. For example, a platonist might assert that the number pi exists outside of space and time and has the characteristics it does regardless of any mental or physical activities of human beings.

I would place math rather at the basis of our rational activities.

We might think of pi as an idea that implies various equivalent programs for generating its decimal expansion. Can we understand platonism as the belief that “somewhere” this “entire” decimal expansion exists as a completed infinity ?

We might even find the issue closer to home. The ur-sequence is

1,2,3,4,…

What is the status of this sequence ? A platonist would presumably believe that it is a completed infinity, that “all” of the positive integers “already exist.” This is how statements about “all” positive integers can already —even eternally — be “true” or “false,” whether or not we happen to know which.

For me, this is going too far. Yet I believe that the positive integers ( as a flexible, extensible, and enacted system ) are intensely intersubjective and intercultural.

Elsewhere I suggest that numbers are what we do ( in time ) with generalized numerals. Numbers are pragmatic equivalence classes of “generalized” numerals.

I suggested that a jar of beans could be a generalized numeral for the number that also has the numeral 43. But this is what it means to count, is it not ?

Was this impious of me ? I think not, because physics depends on generalized numerals. The effectiveness of mathematics in the real world, I suggest, is precisely through such generalized numerals. Notches on sticks were an early form of record keeping. One notch for each goat, etc. But it didn’t have to be notches in sticks. It could have been knots in ropes. The “immaterially” of numbers is our genius for switching between their numerals. This “our” is important, because it’s what makes them “eternal” relative to us. We can’t imagine past or future humans doing it differently in a therefore “essential” way.

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Perhaps the ‘eternality’ of number mis-states what is really at issue - which is not eternality, but timelessness. The concept of ‘eternal’ means ‘of endless duration’ or ‘lasting forever’ while ‘timeless’ means ‘outside of time’. That seems somehow less portentous than ‘eternal’.

Regarding Cantor’s continuum hypothesis - is the point that we can posit unimaginable magnitudes of numbers — infinite numbers of infinities - beyond anything we could rationally encounter or understand? And that therefore there is no meaningful way to say that such concepts are real?

In any case, I feel that the basic contention of platonism - that numbers and other primitive constituents of mathematics - are real independently of any particular mind still holds, regardless of such conundrums. Given that we can grasp mathematical concepts, we’re able to generate them also.

You quote Gödel saying 'Mathematical platonism is any metaphysical account of mathematics that implies mathematical entities exist, that they are abstract, and that they are independent of all our rational activities’. But consider this - while they’re independent of our rational activities, they can nevertheless only be grasped by them.

Isn’t it rather that any symbol or counting method can represent the same numerical value? Whether you represent 7 by notches on a stick, knots in a rope, or marbles in a jar, what is being represented is separate from both the media and the symbolic form.

There’s an interesting snippet in the Hamming article on effectiveness of mathematics (attached to OP. It was that essay that prompted this OP in the first place).

Hamming says:

The earliest history of mathematics must, of course, be all speculation, since there is not now, nor does there ever seem likely to be, any actual, convincing evidence. It seems, however, that in the very foundations of primitive life there was built in, for survival purposes if for nothing else, an understanding of cause and effect. Once this trait is built up beyond a single observation to a sequence of, “If this, then that, and then it follows still further that . . . ,” we are on the path of the first feature of mathematics I mentioned, long chains of close reasoning. But it is hard for me to see how simple Darwinian survival of the fittest would select for the ability to do the long chains that mathematics and science seem to require.

I think Hamming gets very close to the core of a larger question - how does evolutionary biology account for or explain the faculty of reason?

Later in the essay he says ‘Some people have further claimed that Darwinian evolution would naturally select for survival those competing forms of life which had the best models of reality in their minds - “best” meaning best for surviving and propagating.’ However he says this without, I think, much conviction. Because, after all, every other creature, many of which have lived far longer than h.sapiens has, have survived and propogated quite successfully without mathematics or language. And reducing mathematics and reason to the level of evolutionary adaptation is to call into question the ‘soveriegnty of reason’ i.e. to try to account for reason in other terms (Thomas Nagel has an essay on this.)

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Basically, yes.

If you look at the first part of Cantor’s set theory, you get beautiful results. We can “enumerate” the positive fractions. That’s a delightful surprise. We can enumerate all finite subsets of the positive integers. Also awesome. This is where Cantor connects to computability.

But the power set of the natural numbers cannot be enumerated. A easy diagonal argument shows this. Every sequence of sequences “casts a shadow” that isn’t on the list. So we can construct, from the list, a sequence not on the list. So Cantor’s work can be read as taking us to the limits of sense. The power set of the natural numbers is a shaky concept. Don’t get me wrong. This is only mildly shaky. It’s get way way stranger.

For others, however, this is just the beginning. It’s what’s on the other side of sense that’s juicy and wonderful. Yet the formal system that (pretends to? ) talk about this stuff can be as decidable as whether a Chess move is legal. So there’s a tension between the computable formalism and what it is taken, by some, to discuss.

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Reading around on Beneceraff - only died in Jan 2025! - I found this morsel in an essay of his on Gödel:

"Paul Rosenbloom attributes to André Weil the saying that “God exists, since mathematics is consistent, and the Devil exists, since we cannot prove it.”

Love it!

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I didn’t stress it or include the link, but that was from a second source. My fault.

Sure. Depending on your vision of reality, maybe that claim can’t be ruled out. For me it’s unwarranted. But also inoffensive. And it’s not a bad initial explication. But can we zero in ?

I’m suggesting that the “immateriality” is ( or can be understood as ) the interchangeability of the numerals, so the number is a fundamentally a temporal phenomenon. This is how I make sense of Plato’s unwritten doctrine. The numeral is only a numeral through its interchangeability with other numerals. So the number has “no true name.” In this sense, it is irreducible to any particular instantiation/numeral. This is one way to read its timelessness. It is not caught up in a single moment, and numeral doesn’t make sense as a frozen thing, apart from relation to other numerals.

The numeral has its “being” as a numeral through its connection to other times and places. The number is an “open possibility” of more numerals. Any kind of thing we might count one day. Pulses of undiscovered radiation. Who knows ?

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This is complex issue, definitely. Because we make a case at the normative level. If we make a case at this normative level against the reality or legitimacy of case-making, there’s a tension. On the other hand, we need not assume a truth-making “static” or “finished” external/objective reality in the first place.

Another point is that explicating statements about the past is not so trivial from my POV. Are statements about the past best understood as statements about the future ? In terms of their implications ? Counter-intuitive, because most assume a kind of symmetry of past and future. But for some the future has a fundamental priority. And this connects to the continued “open-ness” of the number in terms of the possibility of further numerals. Sense itself is broken open toward the future. But perhaps that’s for another thread…

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Surely. Again number, logical principles, scientific laws, can only be grasped by rational thought. They don’t exist as do sensory objects. I can point to a rock, but I can only explain a principle.

Interchangeable with respect to what? The numeral is a symbol - you can have different symbols represented in different media, but what they denote is invariant.

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Remember that an evolutionary approach isn’t committed to saying that mathematics and the faculty of reason are direct adaptations. They may be spandrels that have exceeded their origin (in some sense).

The key term here is reducing. What does it mean, and is naturalism necessarily committed to such reductionism?

If it means that mathematics and reason are nothing but adaptation, and therefore that their results have no more authority than any other survival-enhancing adaptation, then I doubt there are any mathematical or metaphysical naturalists who actually hold this view.

But if it means only that they arose from evolution, it doesn’t call into question reason’s sovereignty. At least, an argument would be required here, even though I know it might seem intuitive.

I am not especially convinced by a dialectics of nature in scientific practice as recommended by Engels and Lewontin, but at the philosophical level, a dialectical approach would seem to be a good way between Scylla and Charybdis: neither reductive naturalism nor Platonic realm, but an immanent transcendence—which might not be as crazy as it sounds. It has to do with this “exceeding their origin” business I mentioned above.

Features or structures arising from material, evolutionary, and historical processes can nevertheless attain an autonomy and normative irresistibility that is not fully explicable in terms of the causal order they emerged from, i.e., this isn’t a “nothing but” reductionism, though it’s not platonic realism either.

It’s like Hilary Putnam said in his televised discussion with Bryan Magee:

higher-level facts about organization have a kind of autonomy

— Putnam on Philosophy of Science

EDIT: In Adorno’s terms, this transcendence is a consequence of non-identity. That mathematics cannot be reduced to its origin in the material, and that it can maintain its sovereignty, is the moment of transcendence—a failure of reductive explanation to exhaust its object, not a transcendence beyond nature.

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This is pretty much the “naturalism vs. normativity” debate, I think. And it’s the question of what you’re calling “authority” that is often unclear. Is it possible to give a completely naturalistic account – perhaps one that speaks in terms of evolutionary adaptation – of math and reason, while still granting them normative weight?

Yes, an argument is certainly required. I don’t know about “sovereignty,” but Nagel’s argument is that reason has “the last word” because we can’t formulate and justify the scientific, evolutionary picture without it.

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