Argument against Platonism

Question

Platonic view of mathematics states that numbers have abstract reality. One way to test what this really means is to do a thought experiment of extinction of humanity. Also suppose after all evidence of previous civilizations have been eliminated, humanity comes back. It is reasonable to say that, if equally intelligent, they will find utility of counting, and therefore it is highly likely they will be again talking about numbers, prime numbers, etc. To say that numbers will be there, like stars, I think this thought experiment is the only way to understand whatever the assertion (of abstract reality) means.

The issue I am trying to point out is the following:

WE formulated numbers. It is WE who began the process of counting : A process which generates numbers. (We defined 1 as: single instance of counting, 2 as: an instance of counting and again an instance of counting, and then formalized them). If these abstract numbers cannot exert influence on world on their own (which I am sure they don't), does it even makes sense to argue of their independent existence? If it is me who is going to use them, or study them, aren't they a result of my own mental faculties (where mental faculties mean reasoning or logic )? That mathematics is a construct of rigorous reasoning or logic. Why would someone bring in an altogether different notion of their abstract reality, when the argument about numbers being a projection of reasoning (mental faculty/intelligence) is the most logical explanation? I am looking for answers which directly answer the question in bold, and any flaw in my argument. (I am not looking to solve the perennial debate on foundation of mathematics; I am raising argument against the debate itself -that debate is unnecessary for the solution lies right in front of us).

Answer 1

There's a lot to unpack here, so first I'm going to summarize my understanding of your question:

• You disagree with Platonism.
• Based on what you have written, you sound like a nominalist (i.e. you think mathematical entities are representations of "real" or physical entities).
• Some people who support Platonism make an argument of the form "If aliens (or humanity's successors) invent or discover math, it will look similar to our math. Therefore, mathematical concepts must exist independently of humans."
• You agree with the premises of this argument, but disagree with the conclusion, because you believe that mathematics results from a deterministic process of logical deduction anchored in real-world experiences such as counting. So aliens will necessarily arrive at the same mathematics, given the same ancestral environment.
• You are asking for reasons that this argument should be interpreted to support Platonism given that there is an alternative explanation.

I am going to offer a frame challenge: This is not a good argument, and you should not concern yourself with refuting it. It's an intuition pump which presupposes that the aliens will think very similarly to us. There is no reason to assume that their mathematics will actually look similar to ours in the first place.

Humans are a heavily visual species. We (or at least, most of us) define our experiences in terms of the things we see. When humans first developed mathematics, it was heavily focused on geometry. Positive integers were a neat tool for counting, and the Pythagoreans got surprisingly far using integral ratios to describe geometric figures, but the idea of numbers existing "between the integers" took a very long time to catch on in the west. The same was true of zero, negatives, complex numbers, and many more abstract concepts. Even after these things were discovered, time and again western mathematicians stubbornly insisted that they were only meaningful insofar as they could be tied back to geometry. Western mathematics only began to shake this habit in the past couple of centuries or so (around the time of Cantor and Hilbert), which is extraordinarily recent compared to the history of mathematics as a whole.

Now I want to do a thought experiment. Imagine you have an intelligent alien species that hardly uses vision at all. Perhaps they spend most of their time underground, or perhaps they have a heavier reliance on other senses like touch or smell. Either way, they will not have the same relentless focus on geometry that early human mathematicians did.

In fact, let's go further. Imagine these aliens not only don't put geometry first, they take a long time to even invent it. Instead, their early mathematics focuses on algebra, number theory, and eventually modular arithmetic, because these fields are useful for describing (say) the complex network of tunnels that this species likes to build underground. By the time they get around to inventing geometry, they're pretty happy with the rational numbers, and perhaps the Gaussian integers, but they have not really encountered any irrational numbers yet. Geometry is seen as an esoteric, "weird" field, far away from what most of these aliens think of as "math." And, right out of the gate, geometry produces a number (sqrt 2) that does not exist. Obviously, the field must be nonsense. The species rejects geometry as a form of mathematics.

As time goes on, they eventually get around to inventing the p-adic numbers, and proceed to develop calculus in that setting. When talking about "a number," they intuitively think of a p-adic number for some suitable value of p, whereas most humans think of "a number" as meaning a real number. Perhaps, one day, the aliens return to geometry, and investigate it further, eventually developing a theory of the real numbers. But they still don't think of them as "the real numbers." They think of them as "those weird things that occasionally pop up when we play with the zeta function," in much the same way as we think of p-adics as "those weird things that occasionally pop up when we play with modular arithmetic."

If a human mathematician and one of these alien mathematicians were able to converse, they would both learn a lot of things from each other, which seems very counterintuitive if you insist that they're both working in the same overarching Platonic framework which provides the same basic set of truths and falsehoods to each side. It is almost inevitable that our chosen mathematical axioms would be incompatible with theirs, given the profoundly different starting points each of us had. So we would not be able to directly translate our theorems to their mathematics and vice-versa, without extensively tracing each theorem's underlying axioms. In other words, our mathematics and their mathematics would not just be dressed up differently, but they would entail different sets of theorems.

You might argue that these aliens are unrealistic, to which I agree. The true aliens will likely be even stranger and less comprehensible to our patterns of thought. Probably, their mathematics would not hit quite so many human-comprehensible milestones. The point is that we have no reason to believe that their mathematics would look any more similar to our mathematics than what I have described above, and in fact it would likely be substantially more different.

The moral of this story? Mathematics is not and has never been an absolute fixed point in "thought-space." This does not refute Platonism, of course, because we might suppose that the aliens have their own Platonic forms for their own mathematics, but it certainly puts a dent in the "alien mathematicians" argument for Platonism.

Answer 2

Why would someone bring in an altogether different notion of their abstract reality, when the argument about numbers being a projection of reasoning (mental faculty/intelligence) is the most logical explanation?

There is no such a thing as "the most logical explanation". An explanation is logical or it is not (and, most often, it is not). So, I will guess that what you mean is that the explanation you propose is the most obvious one; or perhaps obviously the only one possibly correct; or the one more likely to be the correct one; of the one which is supported by science, etc.

Let's suppose that God created the world such that numbers are the fundamental reality: Whatever we believe exists outside numbers is in fact epiphenomenonal, or perhaps even illusory. As broadly outlined, this is a logical explanation, certainly just as logical as the idea that numbers only exist as a concept in the human mind. Perhaps the God-did-it explanation is less likely, less scientific, less intuitive, less obviously correct than your explanation, but it is, or could be made, formally valid.

Logic has nothing to do with preferring one logical explanation over another logical explanation. Thus, one possible solution would be to assume that the problem is an empirical one: Which of all logical explanations is empirically correct? We don't seem to be able to prove God exists, but we probably can't prove He doesn't either. So, we can't decide empirically that the God explanation is not correct.

So, perhaps we could decide on the basis of which logical explanation is the more reasonable explanation? But then, we would need to define "reasonable", and more to the point, many other people won't stop submitting explanations just because we think they are not reasonable people.

All we can do maybe is to agree among self-selected reasonable people which logical explanation is the more reasonable.

However, you can in fact replace God, in the God-did-it explanation, by reality itself, i.e. maybe it is in the nature of reality that numbers are at the foundation of all existent beings, in which case your opinion that numbers "don't influence the world" falls by the wayside. Not only numbers exist by themselves, but they decide on the existence of everything else. Whether reasonable people will see this explanation as reasonable, I don't know, but it can be argued.

Answer 3

It seems to me that the way you explain the formulation of numbers is off-target. The concept of 'countability' rests on two basic cognitive orientations:

1. That the universe (or perceptual world) is primarily composed of discrete, independent objects.
2. That these discrete, independent objects can be arranged into 'kinds' (or 'categories', or 'classes') according to in-common properties.

The first is necessary because discrete objects are what we actual count; the second is necessary because we only count within like kinds. Any conscious entity that structures the world perceptually in this way will develop at least the basic elements of counting. Human infants and many lower species all show an innate understanding of countability, in the sense that they show surprise when the numbers of things change mysteriously (the classic and often repeated experiment in which some object is added to or removed from a collection behind a screen...).

Platonism takes the two orientations listed above as features of the natural world, not as constructs imposed on the natural world, and thus 'countability' is itself a feature of the natural world. It must be an abstract feature of the natural world (we don't find specific instances of 'countability' lying around in fields) and it seems to be a one-off ('countability' is not itself countable), and that leads Plato to the thought that it is an ideal form.

The real question here is whether or not the world is naturally construed in terms of discrete objects that group into kinds. That concept has intuitive power: neither you nor I are physically connected to anything, and we share many similarities of form and function, and the same can be said of many of the things we experience in the world. But that is shaky ground. Intuitive power is not the same as established fact, but seeing past that intuitive formulation takes a great deal of philosophical insight. We cannot dismiss the Platonic intuition out of hand, even if we find it displeasing.

Answer 4

It is WE who began the process of counting : A process which generates numbers. (We defined 1 as: single instance of counting, 2 as: an instance of counting and again an instance of counting, and then formalized them). If these abstract numbers cannot exert influence on world on their own (which I am sure they don't), does it even makes sense to argue of their independent existence?

You're begging the question against the Platonist here by already defining 1 in terms of the outcome of a naming process.

Let's draw a parallel with physical science. What does it mean for us to have identified a particular subatomic particle? Well, we can point to the sequence of measurements we took that led us to draw a model of how the current state of affairs is at the subatomic level.

It is one thing to say that our process of measuring tells us that there is a subatomic particle there, and another to say that our process of measuring is compositionally involved in that particle's existence. In the latter case, we would say that the particle exists abstractly - in the former, it exists concretely.

It seems reasonable for the Platonist to say "your process is a measurement, not a definition". Mathematically, in fact, this would be right - there is nothing in the definition of the Dedekind-Peano Arithmetic system that demands that its objects are "generated" except in as much as we apply a loaded interpretation of the Successor operator.

(I'm a formalist - I don't actually think the Platonist is right to say that PA determines a domain of objects in their own right - but I would agree with them on this point)

This has always been Platonism's key selling point - by allowing the axioms of mathematics to determine its objects as independent ontologies, Platonism turns statements of standard mathematical practice into Realist claims about the world. We don't just say that "there is an object that satisfies abstract qualified conditions that allow it to serve the zero role in an overall framework realizing the system of Peano Arithmetic and a complicated algebraic relation of succession tying it to other objects that usefully capture our intuitions about counting", but rather we talk about how "zero and its successors exist" and can stop trying to paraphrase what we mean when it is in fact perfectly obvious what we're talking about when we're doing maths.