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Introduction

I don't study Mathematics at university, and probably I don't have any chances to have a little understanding of what mathematics 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 moves me is a mirage. I must give up until I'm sane and study modestly mathematics, in a relaxed way: playing with my easy problems, reading books, and learning from the basics all I can, without eccesive expectations maybe.

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 young 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)...


Questions

$\mathcal Q_1$ From your($*^1$) experience how can mathematicians deal (or how they usually chose to behave) with this huge mathematical universe and be satisfied from a philosophical($*^2$) point of view?

$\mathcal Q_2$ How can a mathematician interested in the foundations of mathematics be satisfied by an always partial knowledge of the mathematics?


Personal observation and conclusion

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, is in the destiny of mathematicians to abandon the hope of understanding of 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.


Notes

($*^1$) I ask this question HERE on SEMath because I'm looking for human experiences of real mathematicians, that is very important for me in this moment, and I do not think is fair that this question 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.

I must quote Andres Caicedo that was able to express way better than me, maybe, the meaning of my question:

"I think that there may be insights that only working mathematicians could provide (as opposed to philosophers), and even if there are wildly differing points of view, seeing them described may be useful. [...] Of course, it may be that answering the question in detail, considering as many of its subtleties as possible would just be too long and unfeasible. That's fine; even providing a few references and ideas that can potentially be fleshed out would be more useful that simply dismissing it."

Improved version 05/14/'13:

($*^2$) Following the advice of Gustavo Bandeira I explain better the meaning of "satisfied from a philosophical point of view" in my fisrt question; What I mean is only linked to the mathematicians interested in the foundations, in other words who is interested in that process of generalization of all the mathematics to easier concepts ( like "$\in$" for set theorists or the structuralists for example)

Here the same question on Phylosphy.SE

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

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This doesn't look to me like a "practical, answerable question" and I'm not sure if even an entire book could answer this question... –  A.P. May 8 '13 at 8:34
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Anyway, as far as $Q_2$ goes, you don't need full knowledge of something for it to be useful. Think of physics, for example (which IMHO has worse chances to reach "full knowledge", whatever that could be). –  A.P. May 8 '13 at 8:34
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How can [insert any academician] be satisfied by an always partial knowledge of [insert any academic area]? –  Arthur Fischer May 8 '13 at 8:36
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$\large 1.$ I don't believe there is an answer to "what makes you satsified?" - We somehow look at that and we think that is beautiful, but there are plenty of answers to such questions based on free association.| $\large 2.$ What do you mean with philosophical point of view?| $\large 3.$ Mathematics is already huge, I posted a quote from Sawyer in the chat some months ago.| $\large 4.$ In the future, we will work as we always did: Get the big stuff, divide the work for thousands of people. –  Vladimir Putin May 13 '13 at 9:58

2 Answers 2

up vote 4 down vote accepted

@MphLee - May I add some new comments ?

You speak of math as human experience : this is a good point to start with. You quote also von Neumann statement's: it looks a little bit "wittgensteinian".

Try to link them; math is a game where you play with simbols, prove theorems, explore new concepts... If you live inside the math community (if you play the math game) you know well how it works.

What is the "sense" of this human activity or experience ?

You can find the answer in the field of philosophy, where you can find reflections on other human experiences : art, religion, ... Are we satisfied with available answers to this kind of reflections and questions ? I don't think so.

But math is also part of a bigger human activity: science. With "science game" we were able to build rockets and travel to the moon, to bulid atomic bomb (alas !), to calculate the earth's circumference a lot of time before Columbus (ignoring ancient Greek's calculation and using a wrong estimate) discovered the Americas.

Again, we can ask ourself what is the "sense" of science as human experience, but we may not ignore the fact that science (with the central role played in it by mathematics) is a "big game".

Both "games" grows with an impressive rate and you are right : how can I understand a field of knowledge like mathematics (or physics) if I'm not able to "manage" the total amount of mathematical knowledge ?

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Certo che puoi! Thanks for the comment. I need time to think about your answer. –  MphLee Dec 12 '13 at 17:52

I'm a student of mathematics at the third year undergraduate level.

Lets begin with what's unsatisfying. Example are when:

  • you perceive that a branch of math "isn't set up right", or
  • a theorem is clumsily worded or clumsily proved, possibly because the writer hasn't had enough exposure to adjacent fields, or
  • an author's definition is suboptimal for ease of expression, or
  • terminology/notation in the literature is inconsistent and there seems to no way to fix the problem - its like you can't win,

etc.

On the other hand, the existence of a huge mathematical universe is not unsatisfying. It's wondrous. I love the fact that there's always more to explore, always harder problems to face... in fact, at any level of difficulty, there's always progress to be made - its like a video game set in an enormously massive world that is just full of secrets, and it just stretches forever.

It can become pretty addictive. I'll leave you with the following quote.

Yeah, I used to think it was just recreational... then I started doin' it during the week... you know, simple stuff: differentiation, kinematics. Then I got into integration by parts... I started doin' it every night: path integrals, holomorphic functions. Now I'm on diophantine equations and sinking deeper into transfinite analysis. Don't let them tell you it's just recreational.

Fortunately, I can quit any time I want.

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Thanks for the answer. +1 –  MphLee May 14 '13 at 16:39

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