Is math independent of our sensory experience?

Question

I've been asking myself this and other questions in the field of philosophy of mathematics. Could we, if we were isolated from any kind of sensory experience, be able to learn mathematics?

Also, what does it take to learn math?, is there a 'module' or a 'structure' (like the one theorized by Noam Chomsky in his studies on linguistics) in our brain that lets us apprehend mathematics?, do we need a language (be it natural, be it symbolic) to learn mathematics?

PD: I'm a beginner to SE, if this question if off topic please do tell and do recommend me where should I ask it.

Answer 1

I think this is a very interesting question, which is hard to formulate precisely. It's also controversial. You might be interested in Misha Gromov's theory on the "Ergobrain." Here is an attempt at a summery of this idea.

Gromov partly attempts to first pose and then answer questions similar to yours. Roughly speaking, Gromov thinks that much of mathematics that we are capable of is highly dependent on (limited by) the structure of how our minds think. He defines the "ergobrain" as a system that takes raw information and tries to form structures from it. I am not an expert in category theory but to me his idea is to try to categorize the way systems can build mathematical-looking structures. The ultimate conclusion is that, at least when it comes to human minds, we think we know what mathematics is but in reality, we don't. Putting it more mildly, it takes a lot of effort to transcribe our thoughts into mathematics language. This is not saying that mathematics lacks rigor. The issue is that we cannot perfectly dissociate ourselves from our sensory experience which corrupts the fact that mathematics is fundamentally rooted in axioms. We think we know that $2+2=4$ based on our daily experience but in reality, proving such a statement from fundamental axioms is highly nontrivial but doable.

In other words, our physical confirmation that $2+2=4$ falls woefully short of rigorously defining what it means mathematically. Here's a typical example of this, where one wants to prove that $\sqrt{5}$ exists (there is some real number equal to $\sqrt{5}$), a seemingly ridiculous question for our sensory experience (it's somewhere between 2 and 3!). Yet, you do not need to know Peano arithmetic to teach a baby how to add numbers. This is precisely the distinction between the ergobrain and mathematics. In fact, it's not unreasonable to guess that the baby, after enough exposure to the world around her, will develop her own sense of $2+2=4$. It already takes a leap of faith to assume that two apples plus two apples equals four apples carries over to oranges, clouds and sand. Now you're unconciously developing the concept of equivalences, isomorphisms etc. So, that must be something innate to your mind, the capacity to declare equal and not equal. Unless you axiomatize it though, it's still just sensory, not mathematics.

Other examples of this include Boltzmann's original definitions of what entropy is, based on somewhat loose physical ideas and principles (here are his thoughts on this). A rigorous mathematical definition is considerably more difficult to formulate and took another half century until Kolmogorov and others provided rigorous foundations for this field. I hope all this doesn't sound disparaging. Afterall, great ideas are not born overnight, especially in mathematics.

Answer 2

I think that for math to be a possibility for me, I need the ability to conceive of an object. Symbols take the form of prototypical objects for most of math, in the sense that we can differentiate between two instances of a symbol and judge two instances of a symbol to be the same. Our use of symbols takes advantage of our latent ability to differentiate our sensory experiences into 'objects', but builds upon this ability with norms for judging symbols to be the same or different. Now, if we were to isolate a child from birth from as many senses as we could, it is not clear to me that it could develop a concept of object, since it would not have undergone the kind of interactive sensory training that we all did as children to deal with the world as though it consisted of objects. So I do think that sensory experience is a prerequisite for mathematics, but on the other hand, I think that sensory experience is a prerequisite for organized thought of any sort. Now, if we were to deprive someone already trained by experience to conceive of objects, then this complaint doesn't apply. It depends when we are depriving the person of senses, and what it means to have a thinking and experiencing being that does not sense.

On the topic of language: mathematics is a semantic activity, both formally and informally, in that there are syntactic regularities in the symbols through which we can predict the 'effect' of a symbol within the syntax, and by which we can make analogy to natural language, which has a much messier, richer syntax and looser bounds on successful prediction. So if you define language loosely as a semantic activity, then it seems that mathematics would constitute a language, and so doing math would be using a language. But my definition of 'semantic activity' was so loose that it could basically be applied to anything, which was no accident. We can ask "in what sense does math occur 'in the wild', beyond human experience?" After all, is it not addition when one rock rolls off a hill into another? If this seems pedantic to you, I think it is because our usual way of talking about math presumes the language systems in which we do math and talk about math. To ask about 'math itself' is strange to me, because I am not clear what 'math itself' could even be. If the universe turns out to be formal, as the scientific realists believe it is, then is the universe itself just math? And if so, it would seem that language is unnecessary would be unnecessary for math, since certainly a universe could exist without language. But if the universe were formal, would that mean instead that everything is just language, and so we haven't escaped the necessity of language?

I think most of these issues arise when we ask of the 'true' nature of the 'referents' of our terms. The question of what math really is, and what are the 'necessary' (by what logical calculus?) conditions for it, presume, in my understanding, a very strong analogy between the semantics of our natural language and the 'objective' state of the world. This is the 'picture theory of language': each word picks out a thing, and the organization of words in language mimics the organization of things in reality. Assuming that language works in this way, it would make perfect sense to ask what math really is; it is what 'math' refers to, and how it is placed in the world. But this way of viewing meaning has many drawbacks: it really only handles propositional statements, it privileges particular meanings of words as 'real' beyond the extent to which such realness is suggested empirically, and it has trouble dealing with our easy ability to talk fiction, among other problems. Most poignantly, it generates questions that have no good solutions, since it relies on an assumed but unspecified correspondence between language (understood syntactically, and idealized without vagueness) and reality (understood as consisting of objects that behave, in some sense, like the symbols they correspond to).

So to conclude, I think that the question of the necessary prerequisites for mathematics is the same question as the necessary prerequisites of objects in general, with the caveat that mathematics distinguishes itself by the formality of its objects. I hope this has been some food for thought. I am being kicked out of the library, but I suggest late Wittgenstein and Judith Butler for thoughts about the nature of objects and meaning.

Answer 3

The short answer is a strong NO. Mathematics is a product of sensory experience. Mathematical and philosophical learning has happened over a long period of human evolution that some of the sensory experience has been taken for granted. Hence, on the face of it, it might appear that mathematics is independent of sensory experience. If just the mind existed without other senses or input would it create mathematics? Obviously it won't. The mind learns from different sensory inputs over a long period of time and eventually abstracts the pattern. Though abstraction is very powerful, few mathematicians for some reason (including ego, inferiority complex, etc.), like to believe abstraction is the most important thing in mathematics and disregard input and other "lowly" experimental/computational/applied mathematics. A true mathematician will appreciate that mathematical reality is the union of sensory input, experimentation, computation, abstraction, rigor, intuition etc.