Introduction: Zero Existence
In a fun episode on one of my favorite podcasts, Elucidations (Episode 42, 12/12/12), philosopher Agustín Rayo argues that the number zero exists independently of human thought and behavior.
I like his argument. And it’s simple enough, I think, to get the gist of it across in a blog post.
To do that, I’ll draw from his 2014 paper, “Nominalism, Trivialism, Logicism” (the argument also appears, with more detail, in his 2013 book The Construction of Logical Space, which I haven’t read). I found the paper to be playful and challenging. His writing is clear and his enthusiasm for these difficult topics is contagious; no surprise after hearing him on Elucidations.
This is not to say that I’ve been converted by Rayo’s argument, which can be extended to account for all natural numbers. I remain a mathematical nominalist—that is, someone who does not believe in the independent existence of numbers (or any other mathematical objects). This is in contrast to a platonist, who does believe in the independent existence of numbers (and perhaps other mathematical objects).
Actually, Rayo’s argument suggests what he calls a “subtle” platonism. But before getting into that, I’d like to say a little more about the nominalist–platonist divide—or, more precisely, about where I stand with respect to it.
It’s common, by the way, to see the word “Platonism” capitalized. I follow the standard of not capitalizing in this context, as contemporary mathematical platonism is usually significantly removed from Plato’s theory of eternal forms—e.g., Plato’s forms have causal powers, while the mathematical platonist’s abstract objects usually don’t.
That said, people use the terms in these discussions in sometimes very different ways. I’ll do my best to navigate the discursive terrain without becoming misled, bogged down, or lost in its many rabbit holes (though I’ll links some nice entryways into those rabbit holes).
Nominalism versus Platonism about Mathematics (Or: Why I’m not a Platonist)
In his paper, Rayo describes a common challenge posed to mathematical nominalists:
Mathematical Nominalism is the view that there are no mathematical objets. A standard problem for nominalists is that it is not obvious that they can explain what the point of a mathematical assertion would be. For it is natural to think that mathematical sentences like ‘the number of the dinosaurs is zero’ or ‘1 + 1 = 2’ can only be true if mathematical objects exist. But if this is right, the nominalist is committed to the view that such sentences are untrue. And if the sentences are untrue, it not immediately obvious why they would be worth asserting.
A nominalist could try to address the problem by suggesting nominalistic paraphrases for mathematical sentences. She might claim, for example, that when one asserts ‘the number of the dinosaurs is zero’ one is best understood as making the (nominalistically kosher) claim that there are no dinosaurs, and that when one asserts ‘1 + 1 = 2’ one is best understood as making the (nominalistically kosher) claim that any individual and any other individual will, taken together, make two individuals.1
1More carefully: ∀x∀y(x≠ y → ∃!2z(z = x ∨ z = y)).
[The symbolic notation can be read quasi-formally as “for all x and for all y, if x doesn’t equal y, then there exist exactly two z‘s such that z equals x or z equals y.“]
The paraphrasing solution is roughly what I’m committed to as a nominalist. Actually, “committed” is too strong. It’s simpler than it. It just seems obvious to me that some form of nominalism is true, and that some aspect of that will involve paraphrasing.
That is, it’s easiest for me to accept an origin of numbers that involves something like “put as many apples in my bag as I have children” becoming shortened to, say, “put two apples in my bag.” And from there “two apples” becomes “two whatever” and then “two” becomes reified into a concept in its own right that can be invoked or signified or referred to explicitly with symbols such as the numeral “2” and expressions like “a quarter of eight” and “1/4 times 8” and “the square root of four” and “
There are infinitely many ways to represent the quantity two.
It’s plausible to me that, from something more or less along those lines, we developed over many centuries the beautiful system(s) of pure and applied mathematics we continue to develop today. So we now have the square root of two and of other prime numbers, negative numbers, 
What do I mean by “particular domains”? Calculus students will lose points for not treating 00 as an indeterminate form (just like 0/0), even though mathematicians maintain that the answer is 1, “because mathematicians said so” (as it’s put in this Ask a Mathematician/Ask a Physicist post; does this constitute a Carnapian formalist move?). Depending on which calculator you input 00 into, you’ll get different answers. I put it into Google calculator and Desmos and got 1; Wolfram|Alpha yields “undefined” and lists it as an indeterminate form.
In that same calculus course, students must treat 1 + 2 + 3 + 4 + … onto infinity as getting bigger and bigger without converging to a specific value. Common sense and mathematicians agree with this. But a physicist might tell you the sum of those numbers—or, at least, the proper value to put on the right side of the equal sign—is −1/12.
To see physicist Tony Padilla excited about this result—while assuring us that, “I know this looks like a bit of mathematical hocus-pocus sort of thing, but actually in truth it’s not… we know it’s not… because these kind of sums appear in physics all the time”—see this 2014 video from the popular Numberphile YouTube channel:
The video garnered pushback, such as from another popular YouTube channel, Matheloger:
And I recall seeing comments from math teachers saying things like “math is hard enough to teach without you telling our students wrong things.”
Padilla wrote a defense of his answer that I highly recommend anyone (fake-) outraged by the Numperphile video read, in which he compellingly, and I’d say passionately, maintains that −1/12 “is the only sensible value one can attach to this divergent sum. Infinity is not a sensible value. In my opinion, as a physicist, infinity has no place in physical observables, and therefore no place in Nature”: “What do we get if we sum all the natural numbers?”
Or for a sober, yet still aptly wide-eyed-at-the-weirdness-of-it-all, assessment of the situation, see Numperphile’s follow-up video called “Why −1/12 Is a Gold Nugget”:
It features mathematician Edward Frenkel, who points out that the −1/12 result—which goes at least as far back as Leonhard Euler (born in 1707) and has a place in pure mathematics—arises in applications from an effort to make the sum meaningful or useful (very useful, in fact) for certain contexts. In other words, with sufficient mathematical rigor and self-awareness, the manipulations that enable us to jettison infinity while leaving behind a “golden nugget” of −1/12 are legitimate.
And yet, says Frenkel:
I think, actually, we don’t really fully understand what’s happening here… it certainly is not the result of summation of these numbers; it is something else, but what is it? … it is the best kept secret maybe in physics, in quantum physics, that most of the calculations that physicists do today are like this. … I would say, it’s not exactly the sum… it blows up, it is infinite… does it mean that in some sense, there is a context in which this infinite sum is mysteriously −1/12? I’m not sure.
Returning to the present point, it seems obvious to me that these developments and results and fascinating discussions can occur whether or not numbers exist independently of human activities. I’m unconvinced that the story requires the additional fact that there exists an eternal and immaterial mathematical realm against which our mathematical statements must hold up in order to be true; in order, for example, for −1/12 to be, in some sense, true. Same goes for 1 + 1 = 2.
I believe one and one really can make two without the existence of a realm of independently existing, immaterial, inert abstract objects we call “numbers.”
This is a naive intuition, I admit. And I’m not prepared to spend the next five to 20 years trying (and almost certainly failing) to bear it out to the satisfaction of everyone interested in this problem (myself included). So instead I’ll commit to a weaker claim. Namely, that my naive intuition has yet to be swayed by any platonist account I’ve seen. This includes Rayo’s, which I’m almost ready to get back to.
First, I think I need to say a little more about what sort of objects platonists take numbers to be. A standard view, as I’ve already noted, is that they are abstract, causally inert objects that do not exist in the material world.
This raises many problems that I will only barely touch on here. A great place to start a deeper dive into those problems, along with the core problems nominalists face, is in the Stanford Encyclopedia of Philosophy (SEP) entries “Abstract Objects” and “Platonism in the Philosophy of Mathematics.” In that order. (And I’m sure to link more sources as I go along here.)
I will say that, despite knowing that many brilliant mathematically inclined people believe in a platonic realm of mathematical objects, I have not been swayed. This includes when the above-mentioned Frenkel writes in Love and Math (2014), a book I adore, that he believes in a “Platonic world of mathematics” that “exists independently of physical reality” (p 234).
Same goes for Max Tegmark’s descriptions in his excellent book Our Mathematical Universe (2014) of a “radical Platonism” (p 321) in which “our external physical reality is a mathematical structure” (p 254). These are informal expressions of Tegmark’s “Mathematical Universe Hypothesis” (MUH), the eureka moment for which he recounts in his book:
…a lightbulb went off in my head and I realized that there’s a way out of this philosophical conundrum. I argued to Bill that complete mathematical democracy holds: that mathematical existence and physical existence are equivalent, so that all structures that exist mathematically exist physically as well. (p 321)
Where this takes Tegmark is fascinating and controversial (I haven’t encountered anyone who agrees with MUH). At any rate, this sort of platonism takes numbers to be something more than inert abstract objects. Read the book to get into the deeper implications, of which I’ll share two of the more fascinating examples:
Mathematical structures are eternal and unchanging: they don’t exist in space and time—rather, space and time exist in (some of) them. …
The MUH implies that you’re a self-aware substructure that is part of the mathematical structure. (p 318)
I share Tegmark’s view not only to present perhaps the most radical contemporary form of platonism I can find from a credible source. But rather to show the variety of views on offer, and thus underscore my point that, were I to be swayed, I wouldn’t know whom to agree with.
Though I suppose the easiest to agree with would be Leopold Kronecker, who once reportedly said, “God made the integers, all else is the work of man.” Sometimes that’s translated as “natural numbers” (from “ganzen Zahlen”), perhaps for reasons to do with philology and to do with his overall views on math. Accepting only the natural numbers as real is even easier to do than accepting only the integers as real.
I’ll offer an even easier-to-accept version of this: God gave us the primes, the genetic materials from which we built the naturals and developed the fundamental theorem of arithmetic, and so on—and any number of stories from there could explain the invention of the integers.
But, again, I also feel that any number of similar stories can account for the invention of the natural numbers. Despite agreeing with Haim Gaifman on another Elucidations episode (#66, 12/19/14) I enjoyed, when he observes that the number of whatever objects we currently call “planets” would have that number were humans to cease to exist. I also agree with him that there is some fact of the matter about whether there is a discoverable proof for the list of twin primes being infinite. However, I disagree with him—the man who wrote the symbolic logic textbook I used in college—when he follows such observations to the final statement he makes in the discussion:
You think that rocks are more objective [than the things referred to with mathematical language], you are wrong. Rocks are material. That means you can hit them, you can kick them, and so on. So you get this impression that there is something very realistic about it. But try to say anything meaningful about it, even imagine without humans, you’ll come up with some description in which the whole thing is organized in a particular way and there’s no escaping from that. So to say that the whole thing is a social construct is—excuse me—it’s nonsense.
Gaifman’s arguments are so reasonable, that I almost wonder if we do share the most important beliefs relevant to the present discussion, but we differ on how to label that set of beliefs. I certainly believe that a cultural norm claiming 2 + 3 = 6 (in a decimal base and without changing the meaning of those symbols, etc.) would simply be wrong.
If that’s all some platonists mean, then what’s the fuss? Is it possible that I and Giafman would check all the same boxes with respect to propositions about numbers except for the one that says, “numbers have independent existence”? I wonder.
On the other hand, I think 2 + 3 = 6 would be wrong for reasons that do not require the literal existence of mathematical objects. That seems like a belief relevant to the present discussion—or a proposition that should be included among the boxes to check. Or is it just a proposition implied by the “numbers have independent existence” conclusion/intuition?
I don’t know. Nor do I know what combination of semantic paraphrasing or formalism might fill in the gaps from the premises to my conclusion (or expose the gaps in the platonist’s). But I do know that Gaifman’s discussion of these things hasn’t swayed me, even if taken to be strictly about the natural numbers. Honestly, if this is all we posit existing, I again have to wonder what the point is, and why we can’t imagine a world precisely like ours in which we invented them.
Now, I don’t want to hold Gaifman to things he’s uttered on a podcast as being the most careful expression of his view. That said, statements like this keep me confused about what I’d be expected to believe in were I to proclaim conversion to platonism:
(@ 11:28) “You can prove it” means you can find a proof. A proof is also an abstract entity. And essentially to say that there exists a proof is of the same kind of question like “there exists a natural number.” Because you can actually code the proofs into natural numbers. This was a big, ingenious trick found by Gödel, that you can code proofs into natural numbers. So to ask whether there exists a proof, is to ask whether there exists a natural number. If you think that this question makes sense, then you subscribe to the view that certain mathematical statements have an objective status.
He goes on to say some things I mostly agree with, namely regarding mathematical truths (if not about how the natural numbers are “abstract entities”). But then later, when expounding on the idea that mathematics is an cumulative activity:
(@ 33:57) …if it’s a correct theorem, what will happen later, it will be incorporated within a wider perspective, it won’t be lost. But you’ll have a wider perspective on it. So mathematics grows all the time. What is shifting is the point of view, conceptual innovation, and so on.
For example, in some period, philosophers and mathematicians thought that there are no negative quantities because it’s contradictory, it’s by the very nature of quantity that there cannot be negatives. But negatives were invented as an auxiliary for computations. But nowadays there’s no problem with negatives because we have structures, we have ways of arranging things in which computing with negatives makes perfect sense.
So, there’s a structure of all the real line, positive and negative, imaginary—I mean, “imaginary” gives you the thought [Note: or “source”?] that they thought they are not real numbers, they are imaginary; but, in a certain sense they are as real as the real numbers. So, this is a conceptual innovation, and mathematics invents a lot of abstract ways to organize things, but nothing is lost…
I have no problem with the notion that numbers are real while our conceptual tools for making use of or engaging them are inventions; clearly symbols like “2” and “+” are inventions. But what about, say, logarithms in themselves, a famously useful invention making log tables possible before the availability of calculating machines? Are these just notational devices, or are they abstract entities in their own right?
[Here’s a 2016 video by 3Blue1Brown that recommends replacing that notation with new one more suited to the current age, that unifies logarithms, exponents, and radicals: “Triangle of Power.”]
Maybe that’s a bad example. But the point is that I’m confused about Gaifman’s use of “invention” for things that are supposed to exist independently of human activities. I might ask the same of functions, which Rayo seems to imply also have independent existence as “mathematical objects” (Rayo, p 20, footnote 16), and as such are off-limits to the most zealous nominalists.
To be clear, I’m not at all put off by his point about imaginary numbers having the same status as any sort of number, whatever that status may be. It’s often noted that “imaginary” is an unfortunate name. As Frenkel puts it in his aforementioned Love and Math:
Note that it is customary to denote
That said, let me restate what I said above in stronger terms. I firmly believe that 1 + 1 = 2 (in a decimal system, etc.). Even more fundamentally: a = a. This is a necessary truth. For an apple and another apple to be in a bag is for two apples to be in the bag—this is simply a basic, trivial truth. It’s not arbitrary or made up, so long as the words mean what they usually mean. I just don’t think that the consistency of this system requires that numbers exist in a platonic realm. If they do, and we can’t access them, we’d still have to invent them. And the system would still have to be consistent.
I also believe that the Pythagorean theorem is true whether or not humans find a proof for it, and whether or not it ever occurs to them to do Euclidean geometry with the conceptual objects we call “Euclidean triangles,” and so on.
This might make me some kind of realist about mathematics, but not a platonist, as I see no need for any mathematical entities to have literal, independent existence. There’s also, it seems to me, some non-convoluted story that accounts just as well for it being possible or not to prove that, say, the list of twin primes is infinite.
The plausibility of such stories existing strikes me as entailing the plausibility of mathematical assertions being mappable to paraphrases in non-mathematical human language, particularly for the basic mathematical entities whose independent existence most platonists seem to agree on—e.g., numbers.
An example of “paraphrasing” is, I take it, “put in my bag as many apples as I have children.” Or replace “children” with “fingers” or with “as many times as I open and close my hand,” and so on. The point is that I’m avoiding reference to numbers or mathy language.
Though, on the face of it, I find it silly that we non-platonists should be expected to provide paraphrases for “how many?”-type questions that are most efficiently answered with phrases like “what you gave me yesterday but subtract a half a pound” or “add seventeen to the four I already have,” given that “how many?” is precisely the sort of question such words were developed to answer. I’ll say a little more about paraphrasing when I consider Rayo’s argument (getting there).
Truth be told, though, I haven’t found myself all that taken with developed nominalist views I’ve encountered. I won’t go much into those, but one that seems to merit a mention is “formalism,” by which I here mean any class of non-platonist views holding mathematics to be about the manipulation of (probably) meaningless symbols. Formalism “can be held simultaneously with Platonism or various versions of anti-Platonism, but it is usually conjoined with nominalism,” to cite the informative Encyclopædia Britannica article “Logicism, Intuitionism, and Formalism” (part of their larger “Philosophy of Mathematics” series).
Another great reference is SEP‘s entry on the topic—”Formalism in the Philosophy of Mathematics“—where you’ll encounter nominalist (or at least non-platonist) formalists including Ludwig Wittgenstein, Rudolf Carnap, W. V. O. Quine, David Hilbert, and Haskell Curry. You’ll also encounter the opposition, namely Gottlob Frege and Kurt Gödel. The latter’s incompleteness theorems were in part an anti-platonist response to formalist views, providing indirect support for the literal existence of an independent realm of mathematical objects.*
⇒ START OF RABIT-HOLE–DAPPLED ASIDE …
*For more about that, see this instructive 2005 Edge interview with Rebecca Goldstein: “Gödel and the Nature of Mathematical Truth“; and you might want to read one of the books she mentions: Nagel & Newman’s Gödel’s Proof, first published in 1958, but get the 2001 edition edited by Douglas Hofstadter; though before reading that, you might want to read something like Bertrand Russell’s Introduction to Mathematical Philosophy (1919), which I finally read after seeing Jorge Luis Borges’s assessment of it, along with Russell’s Our Knowledge of the External World (1929) as “unsatisfactory, intense books, inhumanly lucid.”
[The quote is from Borges’s essay “The Perpetual Race of Achilles and the Tortoise” (1929), page 47 of the astonishing collection Selected Non-Fictions (1999)]
Before Gödel, Frege, as we’ll see below, had challenged anti-platonism more directly (note that I’m attempting to be careful in my use of “non-platonisim” versus “anti-platonism). My impression is that anti-platonism was more popular before Gödel, at least as an explicitly stated metaphysical commitment. But I’m unsure of that.
In a quick search, the earliest instance I can find of the term “mathematical platonism” with Google’s Ngram Viewer comes from a 1910 edition of the The Encyclopædia Britannica (11th Edition, Volume 2, page 503) in the entry on Aristotle; the term does in fact seem to be meant in the sense we’d relate to Plato’s metaphysics. By 1980, the term seems to mean essentially what it does for philosopher’s today.
However, the term “mathematical realism” seems to go back further, with, for instance, a fascinating 1920 reference by Paul Carus to Isaac Newton (The Monist, Volume 30, page 340):
His system is impregnated with what may be called mathematical realism. Space as a category of our knowledge of the physical world is confused with the contents of mathematics.
This sort of criticism is in recent times used to push against indespensibility arguments for platonism. I’ll say more about such arguments below, but for now will say that the pushback involves noting that many of our useful mathematical models don’t literally correspond to how the world is (e.g., models that represent matter as continuous).
At any rate, I’m not prepared to try to pinpoint exactly what these early uses of “mathematical realism” mean. Even more difficult may be cataloging what’s meant by the many various instances “mathematical idealism,” whose usage goes even further back in the Ngram search, to the 1850s. I won’t touch on that but will share with you the search link.
Whatever the distribution or fashion with respect to these beliefs in the late-19th and early-20th centuries, G.H. Hardy’s had this to say in his 1940 booklet A Mathematician’s Apology (or read it for free here):
For me, and I suppose for most mathematicians, there is another reality, which I will call ‘mathematical reality’; and there is no sort of agreement about the nature of mathematical reality among either mathematicians or philosophers. Some hold that it is ‘mental’ and that in some sense we construct it, others that it is outside and independent of us. A man who could give a convincing account of mathematical reality would have solved very many of the most difficult problems of metaphysics. If he could include physical reality in his account, he would have solved them all.
… I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our ‘creations’, are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards… (pp 34–35)
Contrast this with Goldstein’s comments in the above-linked interview about Gödel’s experience at the University of Vienna in the 1920s:
Platonism has always had a great appeal for mathematicians, because it grounds their sense that they’re discovering rather than inventing truths. When Gödel fell in love with Platonism, it became, I think, the core of his life. He happened to have been married, but the real love of his life was Platonism, and he fell in love, like so many of us, when he was an undergraduate.
Platonism was an unpopular position in his day. Most mathematicians, such as David Hilbert, the towering figure of the previous generation of mathematicians, and still alive when Gödel was a young man, were formalists. To say that something is mathematically true is to say that it’s provable in a formal system.
And though platonism isn’t technically implied by Gödel’s incompleteness results, notes Goldstein, “Gödel made it harder not to be a Platonist. He proved that there are true but unprovable propositions of arithmetic. That sounds at least close to Platonism.”
I don’t see why it should, in itself, make it harder to be a platonist. Though if its popularity as evidence of platonism bolsters the fashionability of that belief, well—I can understand the difficulty in turning against fashion.
In any event, in their “Concluding Reflections” in Nagel & Newman’s aforementioned 1958 book Gödel’s Proof, they note:
The import of Gödel’s conclusions is far-reaching, though it has not yet been fully fathomed. … whether an all-inclusive definition of mathematical or logical truth can be devised, and whether, as Gödel himself appears to believe, only a thoroughgoing philosophical “realism” of the ancient Platonic type can supply an adequate definition, are problems still under debate and too difficult for further consideration here. (p 110)
They then give, by the way, in footnote 40, a definition of mathematical Platonism that does indeed make use of Plato’s notion of eternal forms, which again, is misaligned with typical contemporary understandings of mathematical platonism.
Interestingly, they end on an optimistic note, despite Gödel’s finding that “there are innumerable problems in elementary number theory that fall outside the scope of a fixed axiomatic method”:
The human brain may, to be sure, have built-in limitations of its own, and there may be mathematical problems it is incapable of solving. But, even so, the brain appears to embody a structure of rules of operation which is far more powerful than the structure of currently conceived artificial machines. There is no immediate prospect of replacing the human mind by robots. (pp 111–112)
… The theorem does indicate that the structure and power of the human mind are far more complex and subtle than any non-living machine yet envisaged. Gödel’s own work is a remarkable example of such complexity and subtlety. It is an occasion, not for dejection, but for a renewed appreciation of the powers of creative reason. (p 113)
I’ve avoided saying anything technical about Gödel’s Incompleteness Theorem as it is far too involved to get into. I’ll only say here that, if you’re not familiar with Gödel’s incompleteness result and have been led to believe that it somehow shows all math to be arbitrary or socially constructed or “relative” (and, bizarrely, that morality or even “reality” itself is therefore “socially constructed” or “relative”), it should be obvious by now that Gödel, a platonist, took from his result the opposite view about math: i.e., there are objective mathematical truths. In fact, his results rely on the formulation of mathematical sentences that are obviously intuitively true, yet cannot be proved from the axioms of the consistent arithmetic system in which those sentences are formed.
See Nagel & Newman’s book for a brief technical introduction.
(I have not yet read Goldstein’s well-received 2005 book Incompleteness: The Proof and Paradox of Kurt Gödel, where she also lays out the proof, but I see that Nagel & Newman’s is first on the list in her book’s “Suggested Readings” section.)
… END OF ASIDE⇒
And Quine himself soon left behind his nominalist formalism to join the opposition with a form of platonism known as the “Quine-Putnam indispensability argument for mathematical realism.” Read about that, too, at a SEP entry: “Indispensability Arguments in the Philosophy of Mathematics.” The article notes that “Quine and [Hilary] Putnam have argued that the indispensability of mathematics to empirical science gives us good reason to believe in the existence of mathematical entities,” and where the argument is syllogistically summarized:
(P1) We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.
(P2) Mathematical entities are indispensable to our best scientific theories.
(C) We ought to have ontological commitment to mathematical entities.
In the article, you’ll find objections to the two premises. For example,
[Penelope] Maddy presents some serious objections to the first premise of the indispensability argument. In particular, she suggests that we ought not have ontological commitment to all the entities indispensable to our best scientific theories. Her objections draw attention to problems of reconciling naturalism with confirmational holism.
I won’t enumerate her specific objections here, but would direct readers looking for a deeper look at how this plays out in philosophy of science to Maddy’s 1997 book, Naturalism in Mathematics. I haven’t read it, but find the appeal to “naturalism” enticing (as is the context to application; I tend to focus my readings more on pure mathematics).
And Hatry Field, to whom the article ascribes a “nominalisation program,” objects to the second premise in a two-part argument claiming that (1) “mathematical theories don’t have to be true to be useful in applications, they need merely to be conservative” and (2) “our best scientific theories can be suitably nominalised … [such that] we could do without quantification over mathematical entities and … what we would be left with would be reasonably attractive theories.”(Referenced there is Field’s intriguing looking rebuttal to indispensability, the 1980 book Science Without Numbers: Defence of Nominalism.)
There are other objections at the SEP article, but you get the idea. I find the objections plausible, and generally agree with those who say mathematics can be indispensable to science even when numbers don’t exist.
Finally, the platonist argument I most often encounter involves, not ontological commitment, but paraphrasing. From the afore-linked SEP article, “Platonism in the Philosophy of Mathematics“:
The most important argument for the existence of abstract mathematical objects derives from Gottlob Frege and goes as follows (Frege 1953). The language of mathematics purports to refer to and quantify over abstract mathematical objects. And a great number of mathematical theorems are true. But a sentence cannot be true unless its sub-expressions succeed in doing what they purport to do. So there exist abstract mathematical objects that these expressions refer to and quantify over. …
Mathematical platonism has been among the most hotly debated topics in the philosophy of mathematics over the past few decades.
The book referenced is Frege’s 1884 The Foundations of Arithmetic, or, as it’s referred to by Rayo in his paper, “Grundlagen” (from the German title, Die Grundlagen der Arithmetik). I’ve read the book. Enjoyed it. Was very challenged by it. Did not come away believing in the literal existence of numbers. Same goes for all of the other books and articles I’ve linked here, that I’ve read, which is many of them, but not all. I haven’t even read all of the relevant SEP entries, such as the one on fictionalism and the one on “Brouwerian intuitionism,” which is probability my favorite anti-platonist account, but is not a view I actually hold (though I admit I’m far from expert in its details).
I remain committed to my naive nominalist intuition that paraphrasing has some role to play in any accurate account of why we have math. This may be true even if there is a mathematical realm of numbers because: How do we access it? Why should our discussions bear any relation to it?
At any rate, I think that even if the platonic realm of mathematical objects does exist, in a reality in which that realm doesn’t exist, we’d have to invent it. Particularly if their causal inertness cuts us off from that realm. And it seems intuitive to me that this would begin as some sort of paraphrasing.
And is it actually reasonable to posit mathematical objects as causally inert or inefficacious, given that they are meant to exist independently of human minds? If thinking “207 + 1” causes me to think “208” [or erroneously “209,” for that matter], what “caused” that thought? And does the object 32 cause me to think of another object, 9, or are they the same object by virtue of being the same quantity?
At any rate, causality is itself a monstrously difficult concept that. It is noted in the above-linked SEP article “Abstract Objects”:
Suppose John is thinking about the Pythagorean Theorem and you ask him to say what’s on his mind. His response is an event—the utterance of a sentence; and one of its causes is the event of John’s thinking about the theorem. Does the Pythagorean Theorem ‘participate’ in this event? There is surely some sense in which it does. The event consists in John’s coming to stand in a certain relation to the theorem, just as the rock’s hitting the window consists in the rock’s coming to stand in a certain relation to the glass. But we do not credit the Pythagorean Theorem with causal efficacy simply because it participates in this sense in an event which is a cause. The challenge is therefore to characterize the distinctive manner of ‘participation in the causal order’ that distinguishes the concrete entities. This problem has received relatively little attention. There is no reason to believe that it cannot be solved. But in the absence of a solution, this standard version of the Way of Negation must be reckoned a work in progress.
But before addressing causality in the realm of abstract objects, it’d be nice to understand the problem more broadly. And, you guessed it, there’s an SEP article for that. Actually, there are many entries for that one. Here’s an survey entry: “The Metaphysics of Causation.”
But let’s set causation aside. Again, I ask: are 32 and 9 the same object? It would seem they aren’t, as one implies an operation (multiplication) and the other is simply the quantity picked out by “9”. But what about this: is 5 the same object as −(−5), since negative numbers also have independent existence? Or is this still implying multiplication? Does it have to imply multiplication? There are other ways to pick conceptualize such entities.
Similarly, the quantity picked out by our numeral “1” can also be picked out by the symbol strings π0 and as “the limit as x goes to 2 of 
To be fair, one mathematical object existing is enough for platonism to be true. Perhaps that object is the number zero.
Which finally brings me back to Rayo’s paper.
Rayo’s Argument for the Existence of Zero (and Numbers in General)
The specifics of Rayo’s argument get rather technical and rely on some assumptions, developed against some reasonable constraints, that are ultimately shown to be unsatisfiable due to Gödel’s incompleteness results. I always worry about trying to summarize such arguments, particularly one that is already relatively clearly and economically expressed in a freely available paper: “Nominalism, Trivialism, Logicism.”
(For the record, I have read the paper very closely several times. I have a few lingering technical questions that I won’t bring up here, but that I think Rayo could easily clear up were I to ask him about them.)
Also, Rayo assumes the reader’s familiarity with specialized terms like “intensional operator” (to get some background on that, here are some 2011 MIT lecture notes by linguists Kai von Fintel and Irene Heim: “Intensional Semantics“; helpfully, they provide background info on modal logic, i.e., the logic of “possibility and necessity”). But that shouldn’t be a problem here.
At any rate, I will try to say no more or less than I need to in order to home in on the aspect of Rayo’s argument I’d like to discuss, which is the formal result he also presents on the Elucidations podcast—i.e., the proof I’ll get to in a moment. Actually, I think the proof can be discussed without worrying too much about the nominalist’s motivation for following Rayo’s lead to its formal conclusion, but I think the motivation is worth reviewing. Plus it’ll provide context.
Here goes.
As I noted at the start of this post, the nominalist is often challenged to justify why on Earth they’d make mathematical assertions if those assertions are in essence meaningless, given the belief that numbers don’t exist in their own right. A tempting answer for the nominalist is that their mathematical assertions can be paraphrased as nominalistic content.
One way to formalize or model this response is with a nominalist paraphrase-function, which Rayo nicely defines as
an effectively specifiable procedure that assigns to each mathematical sentence a non-mathematical paraphrase in such a way that the nominalistic content of the mathematical sentence matches the literal content of its paraphrase (p 26)
Early in his paper, Rayo endorses a nominalistic paraphrase-function inspired by Gottlob Frege, involving a metaphysical thesis he calls trivialist infinitarianism, but he concedes that this “Fregean Paraphrase-Function is not uncontroversially a trivialist paraphrase-function” (p 7).
Actually, this isn’t the only paraphrase view Rayo’s optimistic about. But, ultimately, he concludes that “it is impossible to specify a paraphrase-function for the language of arithmetic that is uncontroversially trivialist” (p 11). The news is not all bad, however, as he offers a way forward that removes the “middle man” (i.e., parphrase-function).
This, by the way, seems to vindicate the aforementioned trivialist infinitarianism thesis, though it of course doesn’t vindicate any sort of paraphrase-function, as it explicitly excludes any such function, which, again, is suggested by Rayo’s results to have been a hopeless and misguided question to begin with, “since it tied the metaphysical thesis that there are no numbers to potentially controversial linguistic theses concerning the legitimacy of particular expressive resources” (p 27).
Bypassing paraphrasing means just what it sounds like:
On such a view, all that is required of the world for ‘the number of the dinosaurs is zero’ to be literally true is that there be no dinosaurs, and nothing is required of the world for ‘1 + 1 = 2’ to be literally true. (p 23)
Rayo’s way forward for the nominalist turns out, however, to also suggest a kind of “trivialist” or “subtle” platonism.
It’s a pretty sneaky, but of course perfectly honest, argument that sets out to help the nominalist fix their paraphrase-function problem with a clever tool called outscoping. Outscoping characterizes an operator by which all arithmetical content may be separated from specified nominalistic content, where “nominalistic content of a sentence is the requirement that the world would have to satisfy in order for a given sentence to be true ‘as far as the non-mathematical facts are concerned’” (p 14). In other words, outscoping as an alternative to paraphrasing for “specifying nominalistic contents” (p 27).
Depending on the scope of the outscoping operator in application to a given mathematical assertion, it may result in platonist content (i.e., when “wide”) or, rather, may remain neutral on the question of whether numbers literally exist (i.e., when “narrow”). For specifics, see the paper. What’s important here is that the nominalist wants their nominalist content to actually be true, or as Rayo puts it, I think rightly:
the real reason to be interested in nominalistic paraphrases is that one can use them to claim that the nominalistic content of a mathematical sentence is the literal content of the sentence’s nominalist paraphrase (pp 13–14)
The move that accomplishes this is, as I understand it, to view what we’re doing here not just as “specifying the nominalistic contents of arithmetical sentences,” but as specifying an “accurate statement of literal truth-conditions” (p 23). And:
If this is right, then the notion of a nominalistic content—which we first introduced in an effort to help nominalists answer a challenge—can also be used to cause trouble for nominalism, by allowing for rigorous development of a rival view. (p 28)
The “rival view” is, of course, the above-noted “subtle” platonism. And this is due to a reliance on a general claim abbreviated by Rayo as “[NUMBERS].” What “[NUMBERS]” means is explained in the following passage, which leads up to a proof. That proof is what he presents on the Elucidations podcast, and is what I ultimately plan to zero in on:
I would like to consider a view whereby it is both the case that ‘the number of the dinosaurs is zero’ is committed to the number zero, and that all that is required of the world for ‘the number of the dinosaurs is zero’ to be literally true is that there be no dinosaurs.
The proposal escapes incoherence by endorsing the following claim:
For the number of the dinosaurs to be zero just is for there to be no dinosaurs.
and, more generally,
[NUMBERS]
For the number of the Fs to be n just is for it to be the case that ∃!nx(Fx).[Note: You can read “∃!nx(Fx)” as “there exist exactly n number of x‘s that have the property of being F‘s.” Or, for short, “there are n F‘s.”]
A friend of [NUMBERS] thinks that there is no difference between there being no dinosaurs and their number’s being zero, in the same sort of way that there is no difference between drinking a glass of water and drinking a glass of H2O. More colorfully: when God crated the world and made it the case that there was water to be drank, there was nothing extra she needed to do or refrain from doing to make it the case that there was H2O to be drank. She was already done. Similarly, a friend of [NUMBERS] thinks that when God created the world and made it the case that there would be no dinosaurs in 2013, there was nothing extra she needed to do or refrain from doing to make it the case that the number of the dinosaurs in 2013 would be zero. She was already done.
An immediate consequence of [NUMBERS] is that a world without numbers would be inconsistent:
Proof: Assume, for reductio, that there are no numbers. By [NUMBERS], for the number of numbers to be zero just is for there to be no numbers. So the number of numbers is zero. So zero exists. So a number exists. Contradiction.
One might therefore think of [NUMBERS] as delivering a trivialist form of mathematical Platonism—the number zero exists, but its existence is a trivial affair. ….
The trivialist semantic theory we set forth in the preceding section [Note: involving the aforementioned “outscoping”] can be used to generalize this idea to every sentence in the language of arithmetic. One can claim that the literal truth-conditions of an arithmetical sentence are accurately stated both by a standard (homophonic) compositional semantics [Note: Rayo defines compositional semantics as “an assignment of semantic values to basic lexical items, together with a set of rules for assigning semantic values to a complex expression on the basis of the semantic values of its constituent parts.”] and by our trivialist semantics with outscoped semantic clauses. But the two semantic theories do not contradict one another because the truth-conditions they associate with a given sentence are, in fact, one and the same: there is no difference between what would be required of the world to satisfy the truth-conditions delivered by one semantic theory and what would be required of the world to satisfy the truth-conditions delivered by the other. (pp 23–25) …
… our outscoped semantics … can be used to reassess nominalism, by allowing one to give a rigorous characterization of a subtle variety of Platonism: a view according to which there is no difference between what would be required of the world to satisfy the nominalistic content of a given arithmetical sentence and what would be required of the world to satisfy the truth-conditions that would be assigned to that sentence by a homophonic semantic theory. (p 27)
Rayo’s work here may have chipped away significantly at my naive assumption that paraphrasing can account for all of mathematics. More specifically, I’m ready to agree with Rayo that a paraphrase-function that maps every possible mathematical, or even just arithmetic, assertion to some “nominalistically kosher” paraphrase. This is not the same, however, as being convinced that math didn’t start by turning phrases like “give me as many apples as I have left-hand fingers” into symbols, and that those took on a complicated life (so to speak) of their own.
And, again, I’ll ask: Just why are we nominalists expected to come up with a “nominalistically kosher” paraphrasing of, say, Newtonian gravitational theory (as I understand the above-mentioned Hartry Field to have done), not to mention the surely much harder job of doing this for quantum mechanics. Again, that’s why we invented the multitudinous and powerful conceptual tools of mathematics.
That said, what I’m most interested in in Rayo’s argument is the proof, which, again, is:
Proof: Assume, for reductio, that there are no numbers. By [NUMBERS], for the number of numbers to be zero just is for there to be no numbers. So the number of numbers is zero. So zero exists. So a number exists. Contradiction.
I have questions. I’ll enumerate them.
(1) Does this claim make whatever it is that’s usually invoked by the word “nothing” nonsensical or paradoxical? For example, suppose I wish to ask “Why is there something rather than nothing?” (or “Why are there some numbers rather than no numbers”?) There is a lot to debate about what is meant by this use of the word “nothing,” which I’ll in fact do elsewhere (namely in contemplation of a certain book by a certain physicist). But the ordinary understanding of that usage of the word “nothing” is that there is literally no thing that exists.
But, on Rayo’s view, to say that “no thing” exists is to say that at least one thing exists: the number zero. Which is, in fact, to say that at least two things exist: {0, 1}. Which is now to say that at least three things exist: {0, 1, 2}. (And that might be undercounting, if we count the sets and subsets and so on formed by the elements in those sets.)
Another way to say this is that it seems to rule the possibility of even positing a counterfactual world without numbers. In fact, when we contemplate “something rather than nothing,” the “nothing” should involve a complete absence of numbers. It doesn’t imply, “Oh yes, there actually is one thing that exists: nothing.”
To be clear, I’m not conflating the terms “zero” and “nothing” here (more on this below), but I am resting on the case that to “zero things” is synonymous with “no things,” or, for short, “nothing.”
This in mind, I agree with Rayo that to say there “there are no dinosaurs” just means that “the number of the dinosaurs is zero.” And I’m happy to put this as: What makes the claim that the number of dinosaurs is zero literally true is for there to be no dinosaurs. (I.e., that there are no dinosaurs is the truth-condition that must be satisfied to make the mathematical assertion literally true.)
And, furthermore, I see that what makes the claim that the number of numbers is zero literally true is for there to be no numbers. And what a perplexing result! Particularly if you want to avoid paraphrasing or merely specifying nominalistic content, and, rather, see that content (“there are no numbers”) as specifying truth-conditions for the mathematical assertion that the number of numbers is zero.
And yet, I struggle to develop the intuition that this is just a matter of vocabulary, and that the phrases are just different utterances for saying the same thing, as in “I have zero shits left to give” and “I have no shits left to give.”
Furthermore, such phrases aren’t always synonymous. Perhaps Rayo’s results can be restricted only to careful mathematical assertions and their corresponding truth-condition specifications; however, I think it’s at least worth considering whether the restriction in fact makes sense, particularly for the benefit of the perspective of what we might call the philosophically unsophisticated “casual, everyday nominalist” (such as myself) who simply takes their ordinary, everyday mathematical assertions to be paraphrasable and so on; even if the consideration turns out to be a nit-picky distraction.
(2) The quantity represented by the numeral “0” and the (English) words we take to be synonymous with the that numeral, such as “no” and “zero,” don’t have a consistent meaning.
For example, people say things like, “no one has ever done such and such” or “you never call” and “I have nothing to say to you,” when they don’t literally mean, for instance, “the number of things I have to say to you is zero” (in fact, this in itself might be saying a whole lot).
Even worse is something like, “I like nothing more than singing,” which could mean “I like singing more than I like anything else” or it could mean “rather than sing, I would prefer to do nothing at all.” Things get vague fast.
Maybe the phrase “there are no dinosaurs in 2020” is clearer; I do agree that this is clearly synonymous with “the number of dinosaurs is zero.” But even clearer would be something like “there are no square circles,” though we don’t need the number zero in order to say that a world consisting only of impossible things doesn’t exist.
At any rate, were Rayo’s proof to rely on a sentence like “I like no thing more than I like singing,” it might not land so cleanly. Though is it asking too much for such a phrase to land cleanly? Maybe so: Rayo may only be committed to clearly stated nominalistic phrases.
But this brings me back to a question I’ve already asked twice. How is not the careful formulating of English-language utterances about quantities—i.e., the answer to the question “How many?”—not essentially the same project as carefully developing a system of manipulable shorthand symbols for those utterances, that can then take on a life (so to speak) of its own?
This worry doesn’t only apply to zero. It is intelligible to say “give me the bigger half of the sandwich” and “I’ll be with you in two nanoseconds.”
But I’d like to say a little more about zero, as that is the particular quantity Rayo uses to motivate our intuitions.
(3) Zero is a strange and powerful number. We could spend all day exploring its properties. I’ll try to restrain myself.
Depending on which philosopher of mathematics you talk to, you might get different answers about sort of thing zero is.
This makes me wonder whether a statement like “there are zero apples in my bag” captures the entity that we take the mathematical entity referred to by the word (in a mathematical context) “zero” and numeral “0” to be.
If this means that entity defies being capturing by such sentences, does this challenge Rayo’s proof? On the other hand, perhaps all we need to capture is the most basic version of zero, and then we can debate whether its more elaborate features, or applications, point to literally existing properties of the thing.
Ontologically speaking, I’m not sure what the essential property of zero is such that, if Rayo has captured it here, it amounts to what we mean in our mathematical statements like “111 Choose 0” = 1 and 120 = 1 and 0! = 1 and “zero is much bigger than negative five quintillion” and 8/0 is undefined and 00 is either indeterminate or 1, depending on context (and, as noted above, what mathematicians happen to decide).
In a 1967 paper called “The Importance of Definitions in Mathematics: Zero,”1 Claire M. Newman points out that:
We can explain n × 0 = 0 easily enough by referring to n sets, each containing 0 elements. But, why 0 × n = 0? Here is an example of the importance of basic agreements in mathematics. We agree that multiplication is commutative. (a × b = b × a.) Namely, the order in which we write the factors does not affect the product. If zero is going to fit into the picture, we must make it behave in the same manner. Hence, we agree that 0 × n = 0. (p 380)
It may be surprising that this would arise as an example in the Newman’s paper (in which there also appear many of the examples I point out here; though it’s taken for granted there that for a0, a ≠ 0). But depending on your interpretation of mathematical foundations, some surprising question or another is bound to arise.
Again, depending on which philosopher of mathematics you talk to, you might get different descriptions of what zero is (rather than only its cardinality being zero). Some will tell you that the empty set just is zero. Because it happened to cruise into my field of attention recently, here’s another relevant statement about zero, from mathematician Eric Weinstein in conversation with his teenage son, Zev, about 70.5 minutes in:
[I remember] the famous story about you distinguishing between the empty set and zero, which many people don’t distinguish because they have a sense of “nothing,” and your point was that you knew that zero wasn’t nothing, it was itself a thing.
[Source, The Portal, Ep 34: Zev Weinstein – On Parenting, Boys & Generation Z, 5/13/20.]
Now, I know that it’s common for people who aren’t mathematically inclined to think of “zero” as being synonymous with something like, “nothing at all,” or something close to this, like “the absence of all magnitude or quantity,” which I’ve lifted from the Merriam-Webster online’s first definition of “zero. And at Google’s default dictionary (when Googling “zero definition” on 6/19/20), it’s defined as “no quantity or number; naught; the figure 0.”
We don’t rely on dictionaries to define such things, but it does reflect popular usage. And that’s the usage of the people uttering the day-to-day mathematical assertions of the sort nominalists are asked to paraphrase, and so on.
That said, back to discussions among the mathematically inclined.
I’m not sure who, if anyone, in the history of the discussion of set theory proper, has said that the empty set is identical to zero. Perhaps Weinstein isn’t referring to set theorists or mathematically accomplished people, but I have seen the view ascribed to such people. For example, in this Quora thread it’s ascribed to John von Neumann, in this Mathematics StackExchange thread, where it’s noted that “Funny enough, some set theorists may say that {0} actually is 1 (and that 0=∅).”
It is common to run into the whole numbers modeled or defined such that {} = 0 and {0} = 1 and {0, 1} = 2, etc., (where “empty set” is represented with “{}”) as discussed in the Wikipedia entry, “Set-Theoretic Definition of Natural Numbers.” But I’m not sure whether the equal sign is taken by any such theorist as a literal identity relation.
Including by von Neumann. In the above Quora thread, the commenter Anton Fahlgren uses phrases like “zero should be represented by the empty set” and “it is natural since the number of element in this set is zero (by our intuitive definition of zero as nothing).” I haven’t attempted to verify Fahlgren’s account of Neumann’s views, but I am equally as interested in Fahlgren’s own views as someone with an M.S. in mathematics—i.e., as a mathematically inclined person who makes mathematical assertions. (And perhaps gives some teeth to Weinstein’s characterization.)
What I’m more familiar with are the likes of Frege, who makes use of such a definition, as I understand it, as an equivalence rather than identity relation. He puts a lot of thought, in his aforementioned Foundations of Arithmetic, into the concept of identity (see page 73, in a section called “To obtain a concept of Number, we must fix the sense of a numerical identity”).
That said, the set-theoretic conversation has gone through many changes over the decades and faced many theoretical or technical challenges.
In other words, these foundations are not perfected, and the discussion is far from over. In the aforementioned book Love and Math, Frenkel writes:
Unlike natural numbers, which form a set, vector spaces form a more sophisticated structure, which mathematicians call a category. A given category has “objects,” such as vector spaces, but in addition, there are “morphisms” from any object to any other object. … It seems inevitable that the next generations of computers will be based more on category theory than on set theory, and categories will enter our daily lives, whether we realize it or not.
The paradigm shift from sets to categories is also one of the driving forces of modern math. It is referred to as categorification. We are, in essence, creating a new world, in which the familiar concepts are elevated to a higher level. For example, numbers get replaced by vector spaces. The next question is: what should become of functions in this new world? (Frenkel, pp 159–160)
And, later in the book, Frenkel refers to
…[mathematicians whose] idea was to rewrite the foundations of mathematics using a new standard of rigor based on the set theory initiated by Georg Cantor in the late nineteenth century. They succeeded only partially, but their influence on mathematics has been enormous. (p 226)
I’ll say more about some of these discussions another day (e.g., surrounding the question of whether we should be ontologically committed to literally differing sizes of infinity due results from set theory).
But now back to zero and nothing.
It strikes me personally as intuitive that zero is not nothing. For a nice discussion of this, see this 1996 Ask Dr. Math response: “Difference Between Zero and Nothing.” The critical point made there for illuminating the difference to point out that the value of x such that (5 + x = 3 + 2) is clearly 0, while the value of x such that (x + 5 = 1) and (x + 1 = 1) has no answer, which is to say that x = {}.
An example I often give for why zero isn’t literally nothing goes something like: it’s better to have zero dollars in your bank account than negative a million. For some of us, zero is a big number that does indeed answer the question, “What is the quantity?”
[As a side-note, I’m reminded here of what strikes me as a similar notion that’s often casually distorted: possibility. Namely, you’ll hear people criticize the statement “It’s possible” with, “Well, anything is possible,” as though a statement of possibility is an effort to avoid commitment. Sometimes it might be an avoidance. But it often isn’t. If you ask me, “Will this standard, 6-sided die I’m about to toss land 4?,” I’d say, “It’s possible.” If you ask me if that die will turn mid-air into a square circle, I’d say, “That’s not possible.” Possibility is its own fascinating topic one could spend a lifetime studying. I’m exploring it elsewhere, namely with respect to free will; actually, in that context I’m interested in impossibility. More on this later.]
As for zero, itself, I’ve already noted that it has many fascinating properties. Multiply as large a finite set of numbers as you like by zero, and it’s the number left standing; it’s as if it can eat any other number whole. This also makes zero the multiple of every number. But zero can also be gentle. If you’re adding, it gives you back exactly what you put in, making it the additive identity for the integers.
(But don’t get from this the idea that you can multiply all the numbers on the number line to get 0, or add them all together to get 0. It is easily shown that such claims can’t be borne out, though they are appealing and one occasional runs into them. I won’t prove it here, unless someone asks for it.)
Zero can also function as a placeholder, as in the numeral 307, where it holds the tens place; but, as is noted in the above-mentioned Newman article, “3” and the “7” are also placeholders, really, so this is just zero being like any other digit, as it is when used as the first character of a password.
Zero can be a “significant digit.” Or it can represent something like nothing, as in there being “no difference” between 5 and 5, or in the sense of “no thing” or “no quantity.” We can quibble with that, as I do at length above, but there’s a reason why this is how so many, perhaps all, standard-usage dictionaries define it.
Absence is another of those surprisingly perplexing topics. There are many, many things you have an absence, or zero of, such that the quantity you have of those things is zero. Though you might not notice the absence until it is pointed out to you, as with your absence of a 316.12 pound pickle). From a certain perspective, perhaps what you have the most of is the quantity zero: you have infinitely many absences.
From another perspective, zero makes it possible for us to map arbitrary, nth-dimensional coordinate systems onto our models of reality. That is, wherever you have a radius of length zero, you have no radius, which is to say you have no circle, though you do have a “degenerate” circle, also known as a point. Is this an abstract mathematical entity that has literal existence or just a. conceptual tool? If it does exist, is there just one, while our numbering of them mounts to tokens?
And let’s not get into to the difficulties wrought by trying to divide by zero, or whether it should be seen as a natural number, or its arbitrary usage in measurements, such as when we refer to “zero degrees Fahrenheit.” The point is, zero is a lot of things. But is it real?
That is, does it actually exist independently of human minds, language, and behavior? Or, rather, is it the formalization of vague, informal concepts developed by humans for integration into a larger conceptual system we call “mathematics”? By “vague” and “informal” I mean, in addition to my point above about colloquial usage, those concepts represented in English by words like “no” and “nothing” and “never” and so on. As noted above, such language or concepts quickly becomes vague or ambiguous, as in the phrase “I like nothing more than singing.”
Ok, I’ve made my point as well as I can. Simply put, there’s a lot going here, even with this one number, and I’m not clear on how much of it is captured by formally well-assembled arguments implying that mathematical assertions have specifiable literal truth conditions such that “there are no numbers” becomes an absurd statement.
(4) Allow me to reiterate my weak commitment to nominalism, or non-platonism, as a state of not having yet been swayed by a platonist account. Platonists present significant disagreements among themselves, and I’m not sure which I would want to anchor myself to as the one with the least amount of challenges to my existing intuitions. On the other hand, my intuitions also aren’t particularly excited by the robust anti-platonist accounts I’ve seen, either. So, I declare affiliation here only with non-platonism about mathematics.
To put my intuition another way, I don’t see why we need numbers to exist in their own right any more than we require the independent existence of relations like “to the left of” or “smellier than,” though I acknowledge that the latter depend on some sort of point of view, or even subjectivity, to be made sense of. It seems to me that mathematics gives us a tool for making such relations less dependent on point of view (e.g., by operationalize “smelly” as, “the presence of <x number of such and such sort of molecules”), but this will take us into some very tricky topics (e.g., relation of entropy to the “arrow of time”), so I will leave it here.
Finally, any theory of math that implies Euclid, Archimedes, G.W. Descartes, Leibniz, Newton, Russell, Hilbert, Frege, Gödel, Turing, and pick your genius, didn’t “understand” math would be ludicrous.
There are scarily brilliant people on both sides of this debate. It strikes me that the present discussion isn’t about an understanding of math, but rather of metaphysics.
Concluding Thoughts: Who Cares?
Whether mathematical objects have objective existence might seem like a non-problem. But, sneakily, it isn’t. (If you actually made it this far, I bet you agree.)
We’re asked to believe in the existence of a lot of invisible things, by some person or another. Often by smart people whose opinions we (are supposed to) respect. (How I define “invisible” is yet another question for another day.) How we determine which things on that list are real, or, rather, how we should deploy the word “real” in our talk of those things, is a larger project. The tools for that project must be honed for anything that can show up on the list. That engaging the numbers question can help with that seems good enough reason to spend a minute or two in contemplation of it. Though I’m also moved by the fact that many mathematicians and physicists are also platonists.
As for the list of “invisible” things whose existence, not to mention nature, it takes effort to contemplate, much less understand: it is long.
Also on the list is God; living in a computer simulation; other minds; strange subatomic particles, branching worlds, the cat being in a dead–alive superposition; there being trillions of cats-on-the-mat; causality; quantum randomness; the fused object composed of Barack Obama’s left nostril and Queen Victoria’s right pinky toe; dark matter; Boltzmann brains; infinitely many different sizes of infinity; black holes; the Big Bang; and on and on and on.
Some of those, I believe in (other minds seems worth mentioning). But not all. I could list plenty more, but will add only one more.
One of the weightiest, and perhaps the most urgent, is race. What sort of thing race is is a fiercely difficult question. Even knowing what the question “What is race?” is actually asking is, in itself, difficult. For an book that does a great job of detailing the extent to which that discussion is at the ground level about whether race is an actually existing abstract entry or merely a (vague) category sustained by our language and behavior, see What Is Race?: Four Philosophical Views (2019), by Joshua Glasgow, Sally Haslanger, Chike Jeffers, and Quayshawn Spencer.
I recently posted some thoughts on that book, while also sharing some of my own points of contemplation and confusion: “What Is Race?: Four Philosophical Views by Glasgow, Haslanger, Jeffers, Spencer.”
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