How can a mathematician interested in the foundation of mathematics be satisfied by an always partial knowledge of the mathematics?

Question

I don't study Mathematics at university, and I probably will never understand of what mathematics consists of in all its aspects. But I love to find structures and links betwen ideas, and to ask myself philosophical questions about foundations of mathematics. If I have understood well is what is happening with the "categorical" point of view, there is some kind of great unification of the mathematics from a structural point of view.

Obviously I understand that for humans it is impossible to reach a total understanding of something, and that what moves me is a mirage. I must give up until I'm sane and study modestly mathematics, in a relaxed way: playing with easy problems, reading books, and learning from the basics all I can, without excessive demands on myself.

More I understand and more new interesting mathematical topics I discover... but the facts that overwhelm me are the acceleration of this process and the fact that there are a lot of new theories; and that this implies that this growth will be even faster.

Probably I, as an amateur mathematician, don't have a chance to come at new ideas in the foundational research (or even understand the actual point of view since it is changing so fast)... but

Q₁: From your experience how can mathematicians deal (or how they usually chose to behave) with this huge mathematical universe and be satisfied from a philosophical point of view?

Q₂: How can a mathematician interested in the foundation of mathematics be satisfied by an always partial knowledge of the mathematics?

Is maybe possible that in 100 years tha amount of mathematical knowledge will be so huge that even Mathematicians will be totally overwhelmed from it? In other words, is possible that will be impossible for anyone to research new things because the knowledge required will take like 80 or 90 years of hard study? I say this because I think that humans has a limit to the speed of learning.

If this is possible, should mathematicians abandon the hope of understanding mathematics?

I remember a quote of John von Neumann:

"Young man, in mathematics you don't understand things. You just get used to them."

I have read many stories about his math skills, and how he was a genius... anyways from my point of view... I feel very sad when I read this quote.

Is really this the ultimate nature of mathematics?

Notes

I'm searching for human experience of real mathematicians, that is very important for me in this moment, I asked this question on SEMath but it was closed in 40 seconds...is a soft question, as many others questions, but it has philosophical contenents too and I think it deserves at least a chance, thanks in advance.

(Link to a closed crosspost on SE Mathematics)

Answer 1

Consider that quotation by John von Neumann.

"Young man, in mathematics you don't understand things. You just get used to them."

What does it mean to "understand" something? Consider a simple example, of understanding what someone has said, or written: you have taken some sensory data, and incorporated it into your body of experiences — you develop your idea of who the person is, or the subject that they describe; and you connect it to other things that you know, securely enough that you will remember. Understanding consists of having those links in place.

When you learn something entirely new, there is little that you can link it to in your mind, precisely because it is entirely new. There is nothing to do but to reinforce by repetition, until it becomes something which remains by sheer force of weight — or possibly because eventually you manage to retain it long enough that you can start to connect it to other things that you have been learning in the meantime.

It's a question of quantity over quality. If your primary aim is to understand a piece of mathematics, you may have to spend a lot of time on the mathematics which motivated it, to build the links. It is not an efficient way to learn as much mathematics as possible; it is at best a route to understand all of the mathematics as well as possible, as one is learning it. The alternative is to learn things by rote — which we often do; the names of the numbers 1, 2, 3, our addition and multiplication tables, how to multiply matrices and vectors, and so forth. We may come to understand them later, but the understanding only secures the knowledge, while the memorization allowed us to use the knowledge before we had a secure foundation to understand why it could be useful. (Of course, knowing why it might be useful is not necessarily important, so long as one knows that it is useful.)

This is true of any subject. What does it mean to understand that tomatoes start green, and then turn red? It's just a simple fact; one learns it and moves on. Perhaps if one studies horticulture or biochemistry, one can actually come to understand why tomatoes turn red; but there will be other simple facts of the world which are left unexplained. Can a gardener be interested in biology, and still be content to plan and plant their garden without understanding every cellular process in each plant that they cultivate? Yes. However, understanding the plants better may give them more insight as to better approaches to gardening. Because they can never obtain complete knowledge, they will always have some things about which they are curious — something more to work on to understand.

The world is big with many things in it, and mostly must be taken as it is; perhaps with some investigation if one is curious about details, if one has the time. This is true of mathematics, music, history, poetry — and when for some reason one needs to know something quickly and use it without necessarily understanding it (which is often the case), getting used to it is the best one can hope for, until it has embedded itself into your brain so that you forget what it was like not to know it, or until you can forge connections with other things that you know.

Answer 2

The planet Earth is too large for one man to explore fully in one lifetime. Yet it's relatively straight forward to get some idea of what it looks like from an atlas; and similarly its geology; its weather; and its flora and fauna.

Similarly although the mathematical universe is too large for one mathematician to explore fully its possible to get a feel for the shape of it, what are the currently important research programmes and movements and thus to find an important question which can advance knowledge and secure his reputation.

For example, an important research 'idea' is the Langlands programme - it occupies the energies of a large number of mathematicians. What is it? Class Field Theory in Number Theory which is about describing Abelian Galois extensions is now seen as beginning of a ladder of such theories. The interest is describing this ladder more properly. This is the Langlands programme.

A similar strategy is seen in higher-dimensional category theory where ordinary category theory is seen as the simplest case. Again the interest is in describing this ladder of theories.

Without understanding any of the technicalities there, one should be able to see that there is a common pattern here. An important theory is described as the simplest case, and one climbs and describes a ladder of such theories.

The simplest place where this idea occurs is of course the integer: 1,2,3... What I've described is simply doing this with more sophisticated objects. It's already been done for vector spaces and manifolds - the higher dimensional objects reveal new phenomena unavailable in lower dimensions.

One could say mathematicians are always learning to count!

Answer 3

Q1: From your experience how can mathematicians deal (or how they usually chose to behave) with this huge mathematical universe and be satisfied from a philosophical point of view?

Generally, the same way most people deal with a huge universe; they don't think about the philosophical underpinnings. Almost everyone handles reality perfectly comfortably without confronting questions of whether possible worlds, universals, temporal parts and so on exist, or whether their conception of the world corresponds in a meaningful way with reality. Mathematicians have the structures they work with, and on the whole don't necessarily interrogate their assumptions about in what sense those structures exist and so on.

Q2: How can a mathematician interested in the foundation of mathematics be satisfied by an always partial knowledge of the mathematic?

I think someone interested in the foundation of mathematics actually has a good reason not to be troubled by their incomplete knowledge of mathematics: they are working towards a theory of the foundations, not what is built above them. Just in the same way a particle physicist concerns themselves with the fundamental constituents of matter and doesn't worry about their inability to predict the flow of chaotic systems, a foundational mathematician has no reason specifically to worry about their inability to solve some complex problem elsewhere in mathematics.

Answer 4

Re Q1: I don't think this is any more of a problem for mathematics than for any other modern discipline.

Information overload is the same everywhere. People focus narrowly and try to guess what will and what will not be important to others. There is a basic problem with the acceleration of the accumulation of ideas and the elaboration of nuances, but it is everywhere, not especially here.

Re Q2: I really don't think they can, and I really don't think they try.

It is said "Logicians are Platonists on Weekdays and Formalists on Sunday." The two positions do not undermine one another. I think what is really going on in mathematics is a kind of poly-alethian compromise, where we have strikingly different standards for truth, and we admit much more is true in one of them than we allow to be true in the other.

These are two different 'games' in the sense of Wittgenstein. We know by intuition that we are far enough from the boundary between productive "Platonic" mathematics and the deep "Formalist" concerns, that we can always tell which matters more. As long as that continues to be the case, the games need not converge into a single game. And when they do, I am betting that instead of finding a common basis, we will evolve a third game to decide when one is playing each of them.