Many people will attest to their dislike of math. Yet, it is difficult to navigate the modern world without it. We routinely mentally count and calculate distances, times, speeds, weights, volumes, money, various objects we come across, and other everyday things. When talking about numbers that are associated with units (meters, minutes, grams, liters, dollars, etc.) it seems fairly straightforward what the numbers are referring to out there in the real world**, but what do the numbers themselves actually refer to?
Consider the following proposition:
- The number 3 is a prime number.
Is this a true proposition? In what way is it true? You might think that it is true because “3” is divisible only by 1 and itself. But what do the words “the number 3” and “prime number” actually refer to out there in the real world? Where is the number 3 located? Does it exist in time? Was it in Bangkok four days ago? Does it take up space? How much does the number 3 weigh?
These all sound like absurd questions. Few people, if any, think that the number 3 is some sort of concrete object existing at a certain place at a certain time. Yet, few (though probably more) people would say that the number 3 does not exist – few would say that the sentence “the number 3 is a prime number” is false just because it fails to refer to any concrete objects out there in the real world.
But then we are forced to ask ourselves: what sort of thing is the number 3? Numbers (and other mathematical objects) are thought by many to be prototypical examples of abstract objects: things that exist but are non-spatial, non-temporal, and causally inert (i.e., nobody would ever think that the number 3 caused anything to occur, nor that the number 3 was itself caused by anything). But these all sound like descriptions of (and perhaps even sufficient conditions for) something that does not exist: it is nowhere, no-when, does nothing, and cannot interact with anything. Indeed, many people (variously called anti-realists, anti-Platonists, nominlists, among other names, some of which have their own nuanced views, but for our purposes we can think of them as a single school of thought for now) believe that numbers (and other abstract objects) do not exist. Those who think they do exist are usually called Platonists (or, to differentiate their views from Plato’s canonical theory of forms, platonists with a lowercase “p”).
One of the main arguments used in favor of the existence of numbers is called the indispensability argument. This says, in essence, that many of our best and most successful scientific theories, which make great predictions with astonishing accuracy, require math (and thus numbers). If it were not the case that numbers exist – if all of the mathematics used were false due to any of the signifiers failing to refer to anything – then there would be something very wrong with our scientific theories. More formally, the argument goes like this:
(P1) We ought to have ontological commitment to all [holism] and only [naturalism] 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.
There are issues with this argument. If you are like me, the one that probably stood out to you most is holism from premise 1. Why ought we have an ontological commitment to all of the indispensable entities (and what makes an entity indispensable)? For instance, in quantum field theory, the use of virtual particles is indispensable (or, at the very least, most useful). Yet, most (though not all) physicists would probably say that virtual particles don’t really exist (that, instead, they are only mathematical tools).
But more than this, why should things like numbers get brought along for the ontological ride, just because the theory they are used to describe is so successful? In other words, when experiments confirm the predictions of a theory, the experiment is not simultaneously confirming the existence of the numbers used to perform the calculations involved in making those predictions.
The thinking, as far as holism is concerned, is probably something like this: if, say, some new piece of genetic evidence is discovered that further supports the evolutionary model that has whales evolving from land animals, that also lends further credence to the model of humans evolving from a common ancestor with chimpanzees, by virtue of the fact that it further confirms the entirety of the theory of evolution. By the same token, the thinking seems to be, it must also confirm the existence of the numbers used in by the computer algorithm utilized to perform bioinformatics on the genetic data serving as evidence. But why must it be the case that the mathematics used in the algorithm are referring to some ontologically real, but abstract, object out in the real world? And could the theory work even without numbers?
As to the latter question, Hartry Field has attempted to show that scientific theories, such as Newtonian gravity, can be formulated without using numbers (or other mathematical objects). This is not meant as a way to replace scientific theories using mathematics with “nominalist” theories, but instead is meant as an undercutting defeater for the indispensability argument.
What is also of note when it comes to scientific theories is that scientists will be quick to adopt new mathematics for their theory. What I mean by that is, if some new mathematical formalism is constructed (discovered?) or some new theorem is proven, a fully fleshed-out scientific theory can adopt it and still function in the same way. Our theories are overdetermined by the mathematical objects we use within them. This itself is somewhat of an undercutting defeater for at least some mathematical objects (though perhaps not for numbers in general).
One issue I see with Field’s project is that our universe is one that is related by quantities, such as sizes, masses, and amount. When we say, for instance, that Jupiter is larger than Saturn, we are, in a broad sense, ontologically committing ourselves to something quantitative. Putting actual numbers to it is just translating something less precise (“is larger than”) to something more precise (142,800 kilometer diameter > 116,500 kilometer diameter). Furthermore, regardless of what units we use, the size of these objects has a true zero to which they can be compared, namely the non-existence of that object. For instance, we could say that Jupiter is “one Jupiter larger than the absence of Jupiter” (where “the absence of Jupiter” would be an infinitely small point in space). Thus, with a true zero, it makes sense to discuss objects that are “half a Jupiter in size” or “two Jupiters in size” as being understood in terms of ratios of the unit “Jupiter”. This attests to some intrinsic “quantity” inherent in the existence of objects that is independent of what metric is being used.
There is also the fact that the number of something determines other properties, i.e., there is a grounding relationship between the number of something and other concrete facts about it. This is most easily observed in chemistry, where the number of subatomic particles (protons, neutrons, electrons) grounds the physical properties of the elements. In other words, that physical stuff behaves the way it does is grounded in the number of the physical objects composing the stuff. That, say, oxygen has the chemical properties that it has is physically dependent on the number of protons, neutrons, and electrons that it has; any description of oxygen that does not account for the number of these objects will fail to describe oxygen. Put differently, the word “oxygen” is a paraphrase for something like “the object composed of 8 protons, 8 neutrons, and 8 electrons” where the numbers are essential to the object.
My point here being that, it seems to be that it would be impossible to construct a physical theory that does not use something like magnitude or amount, even if only using these things in very imprecise terms (with terminology such as “larger”, “further away”, “more”, and so on).
Aside from the indispensability argument, though, there is also the (modal) necessity of mathematical relationships. In other words, that 2 + 2 = 4 is something that carries the force of (modal) necessity. It could not have been otherwise than that 2 + 2 = 4.
Consider the following three propositions:
- The number 3 is a prime number.
- Frodo Baggins is a hobbit.
- Frodo Baggins is an elf.
I discussed above how the first one seems to fail to refer to anything out there in the real world. There is no “number 3” that exists anywhere at any time – numbers are not things or events that can be located in space and time, engaged in any causal chains. But the same is true of the second and third sentences: neither “Frodo Baggins” nor “hobbit” nor “elf” refer to anything that exists in the world (barring the fact that I’m sure, somewhere, there is at least one person named after the character; for now, we are talking only about the fictional character and classes of people from Tolkien’s Lord of the Rings; I do not want to get into a whole discussion about reference-fixing here). Yet, anyone familiar with Lord of the Rings will say that the second sentence is true while the third one is false (the various problems with this could go down a rabbit (hobbit?) hole on the ontology of fictional entities, but I will not discuss that further here).
What is importantly different, though, is that Tolkien could have made it otherwise (Frodo could have been an elf, for instance) or even have not made up neither Frodo nor hobbits. There are possible worlds where Frodo is an elf, or where the character of Frodo (or the species of hobbits) was never invented. That the number 3 is not able to be factored into non-trivial whole numbers, on the other hand, was not decreed by some author. If anything, this was a discovered fact, a property that the number 3 possessed prior to anyone having words or symbols for these things (or, indeed, prior to anyone existing at all). The seeming necessity of mathematics even motivates some religious apologists to be number-nihilists, since the numbers’ existence would be independent of God and thus violate God’s aseity (and undercut some of the justification for the Kalam cosmological argument, namely because of the existence of an “actual infinity” in the form of actually existing infinite series of numbers).
Likewise, simple arithmetic seems to carry the weight of (modal) necessity: in no possible world will anyone with 2 apples, upon acquiring another single apple (and losing none), thereby have 4 apples. In no possible world, if I have 3 chairs and I absolutely insist on splitting them exactly evenly between two people, will both people end up with a whole number of chairs. And so on.
This (modal) necessity also seems to go against any sort of intuitionist view of math, that it is all something that exists only in the minds of conscious beings. Besides, if numbers exist only in people’s minds, then we run into the same problem we do for colors: is your “17” the same as my “17”? For instances, does 17 ≠ 17 when the two instances of “17” are held in different people’s minds? While, just like for colors (or any other qualia) there is a sense in which skepticism could be defended here, the case does not seem to be as strong as it is for qualia. While the qualia of, say, the color red depends on the existence of an experiencer of red, it is conceivable that there can be 17 of something even if there are no minds to observe or comprehend it. Any sort of psychologism for numbers would also commit someone to the belief that, when not being thought about by anyone, a number does not exist. For instance, the following 100-digit number (most likely) did not exist until I thought of it and wrote it down, and will cease to exist until you or someone else is thinking about it:
4,785,200,136,053,235,775,700,788,354,448,552,172,604,541,730,847,786,156,147,610,144,748,058,640,208,748,000,422,154,060,816,083,483
This hints at another way in which our words/symbols (our signifiers) can refer to numbers: they are not mental, but are, in fact, collections of concrete objects. This works if the written/typed symbols or mouth sounds for “the number 3” refer to all instances of three things out there in the world – to the set of all possible sets of three things. An issue with this, as you can probably see, is that we are now ontologically committed to the existence of sets, which are just another mathematical object. Not only that, but then it opens up issues of mereology and what counts as a member of these sets – while the mereological realist / mathematical realist would say that three small stones count as a member of the set of all sets of three things, the mereological nihilist / mathematical realist would say that each stone is some 1023 objects, and thus belong to the set of all 1023 member sets. Then the mereological universalist / mathematical realist will say that there are some 210²³ objects in each stone (the power set of all the particles). There is also then the issue that (at least on mereological nihilism) any number above 1082 would not exist, since that is the (high end) estimate for the number of particles in the universe. It would also mean that only positive whole numbers exist since there are no sets with negative or fractional cardinality.
We might then be tempted to adopt a formalist theory of mathematics. In this way of thinking about it, math is a set of rules like in a game. We can think of chess, where each piece has possible movements, which we could think of as the operations within math. The starting positions are sort of the axioms. From the axioms (starting positions) and operations (moves) more theorems (arrangements of pieces on the board) can be derived. Yet we would not think that the rules of chess refer to some abstract object out there in the real world.
This position also suffers some of the same problems already encountered. Why, for instance, does math seem to carry the weight of (modal) necessity? Just like with fictional characters, the moves (and starting positions) in chess could have been different (there are possible worlds where they are different), and are still subject to change or modification. But, again, there is no possible world where 2 + 2 = 4 is not true. The formalist theory might undercut some mathematical objects, but few would say that simple arithmetic will cease to hold if someone “changed the rules” about what it means for a person with two apples to be given another one now having three apples.
Really any non-platonist view will suffer one glaring problem: how do we know that theorems (even simple ones like 2 + 2 = 4) in mathematics are true? It would need to be true either by convention (e.g., if someone writes “2 + 2 = ” then, by convention, you ought to write “4” to the right of “=” sign) or by induction (every instance in which you had 2 of something and added 2 more of that thing, it just so happened that it worked out that you now had 4 of that thing). As such, that math works is a sort of blackbox or brute fact. I might liken it to John Searle’s Chinese Room Argument: all you are doing is manipulating symbols, but you don’t actually understand what you are doing, because what you are doing does not actually refer to anything real.
The formalist will likely argue that it is true by virtue of internal consistency; the intuitionist might say that the symbols refer to thoughts in people’s heads. But these just run into the problems I’ve brought up before about these positions. What we want is a theory of epistemology. We want to know how math can be true in-itself and true of things in the real world, not just whether it is internally consistent or universally agreed upon. But this gets to another argument against platonism, namely epistemological access.
This argument essentially asks, if numbers are not anywhere, at any time, nor within any causal chain, then how can we come to know about them? Put differently, if abstract objects cannot cause anything, then how than they cause us to have knowledge of them? The SEP article “Platonism in the Philosophy of Mathematics” states this in the way Harty Field does:
Premise 1. Mathematicians are reliable, in the sense that for almost every mathematical sentence S, if mathematicians accept S, then S is true.
Premise 2. For belief in mathematics to be justified, it must at least in principle be possible to explain the reliability described in Premise 1.
Premise 3. If mathematical platonism is true, then this reliability cannot be explained even in principle.…
… The first two premises are relatively uncontroversial. Most platonists are already committed to Premise 1. And Premise 2 seems fairly secure.
…
Premise 3 is far more controversial. Field defends this premise by observing that “the truth-values of our mathematical assertions depend on facts involving platonic entities that reside in a realm outside of space-time” (Field 1989, p. 68) and thus are causally isolated from us even in principle. However, this defense assumes that any adequate explanation of the reliability in question must involve some causal correlation. This has been challenged by a variety of philosophers who have proposed more minimal explanations of the reliability claim. (See Burgess & Rosen 1997, pp. 41–49 and Lewis 1991, pp. 111–112; cf. also Clarke-Doane 2016. See Linnebo 2006 for a critique.)[13]
To me, this brings up the issue that a lot of philosophy on this topic seems to dance around, which is what it even means to be abstract, or what kind of a thing is an abstract object? In the above quote, Field says that platonists accept some “…realm outside of space-time” where numbers “reside”. But what is the nature (transcendental? immanent? substrative? something else?) of this so-called realm? In what way does it affect (causality? composition? grounding? emanation?) the world of concrete objects? The reason these questions are important here is because of what premise 3 of the epistemological access argument says. If there is not some accounting for what exactly an abstract object, such as a number, even is, then premise 3 is smuggling in all sorts of assumptions.
And really, even beyond just the epistemological access argument, I think the nature of these abstract objects is a question not examined as much as it should be. If you go down the rabbit hole on this topic, where you often end up is at discussions of the ontological commitment inherent to quantifiers: does the existential quantifier “∃” (which can be read as “there is/are a(n) _” or “there exists a(n) _”) commit us to the actual existence of the variable it binds? Certainly the wording here appears to support this, where we are saying “there is” or “there exists” before whatever we are going to discuss. And so, the argument goes, if we are going to use something like “∃x such that x is a number” then we are ontologically committed to the existence of numbers. But then we would not be able to use it in the following way: “∃x such that x is a hobbit” because there does not exist anything that is a hobbit. Yet, many people would say that yes, there is at least one hobbit: Frodo Baggins. They are just fictional rather than existing in the real, actual world. We thus get deflationary senses of the existential quantifier, which says we can use it in an “ontologically heavy” way that does commit us to the real, actual existence of the object, and an “ontologically lightweight” way that does not so commit us to some real, actual existence of the object.
The platonist wants to say that numbers have an “ontologically heavy” way of existing. It seems to me, then, that the onus is on them to define the nature of this “ontologically heavy” way of existing. The minimal definition seems to be, according to the SEP article:
Mathematical platonism can be defined as the conjunction of the following three theses:
Existence.
There are mathematical objects.
Abstractness.
Mathematical objects are abstract.
Independence.
Mathematical objects are independent of intelligent agents and their language, thought, and practices.
But this seems unsatisfactory, as it does not tell us, for instance, what relationship an abstract number has to an instantiation of that abstract number in a collection of concrete objects, or how anyone can come to be acquainted with abstract numbers, or why multiple mathematical objects can be used to describe a single concrete, physical phenomenon, amongst other questions.
Concluding Remarks
This post is somewhat meandering, but, as with most topics in philosophy, there has been a lot said on the subject. This post is really only a brief overview, with some of my own commentary, on the topic. As for my own stance on the question of whether numbers exist, I am thoroughly agnostic. I understand and agree with many of the arguments for both sides, and with many of the objections and counterarguments from both sides. Like on many topics, I find myself sympathetic to all points of view.
When I first started digging into this topic, I thought I was going to find anti-platonism the most convincing. Or, put another way, before knowing anything about this topic, I was a naive anti-platonist. This comes from my bias towards physicalism (though I am by no means committed to this position, my first-approximation judgement of most ideas is that if it must postulate non-physical things, it is likely untrue; in other words, I am biased against non-physical explanations of things).
To me, though, I think the greatest argument in favor of platonism is just the seeming (modal) necessity of math. That it always works, and that it must always work in all possible worlds, is the biggest reason I cannot reject platonism. When it comes to other mathematical objects (sets, abstract spaces, and the like) I lean more toward anti-platonism, but something about numbers themselves, and simple arithmetic, just seem too “real” in some way to be dismissed so easily as useful fictions.
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**While numbers with units may seem fairly straightforward, they can sometimes be a little more slippery. For instance, if I ask you to count out the number of soda cans in a pack, you could mark them and arrange them in some order. But if I ask you to arrange in order the number of grams of soda in one can, this does not make any sense. And while it would make sense to say that one can is the same as itself and different from the other cans, it would not make sense to say that the grams in one can are different from the grams in another (at least not without grouping up the molecules within the can and designating some of them to the “first gram” and some to the “second gram” and so on, but then this is just a way of labeling things that were counted rather than comparing grams as such).