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Lewis
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Is Mathematics something that humans invent, or is it something that we discover? I like having philosophical conversations with my students here. Questions like these provoke my mind into thinking deeply and usually end up in great conversation. I believe mathematics is the language of the universe and as more constants are discovered, the more the understanding of the universe we have. Einstein actually accidentally discovered a constant for his mathematical equations, and it turned out to be a defining feature of the bending of space-time.
Feb 20, 2018 7:46 PM
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@Lewis It is indeed the same Euler! His work seems to pervade much of modern mathematics.

This is a really fascinating topic to read and think about. You're completely right about our systems of logic never being able to encompass all of mathematics; this  was actually proven by Kurt Gödel in his famous incompleteness theorems. In short, the incompleteness theorems tell us that "no consistent system of axioms whose theorems can be listed by an effective procedure (i.e. an algorithm) is capable of proving all truths about the arithmetic of the natural numbers. For any such formal system, there will always be statements about the natural numbers that are true, but that are unprovable within the system."

(quoted from https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems)

February 20, 2018

As a maths undergraduate I also enjoy thinking about questions like these! I agree for the most part, I think most (if not all) of mathematics is inherent in the universe and is discovered rather than invented, but I think it's less clear in some areas.

With pure/abstract maths (as opposed to maths that can easily be found and applied in the real world) in particular there is room for debate, because there is no physical evidence to verify things, and it's all in our minds. Some branches of maths began with the question of 'what if X was true?' and people just started working under the assumption X and seeing where that would logically take them. There are whole areas of mathematics based on statements that can't be proven to be true or false in our universe, so essentially the entire field is speculation based on an assumption that can't even be confirmed.

My impression is that most people (or most mathematicians at least) think that maths is discovered, but there are a lot of different (and sometimes really bizarre) opinions out there. If anyone wants to read a little about different theories about the nature of mathematics: https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Contemporary_schools_of_thought

Recently a professor of mine used an interesting choice of words during a class; he said something like "...this result was known to Euler in 1765...", and in doing so he avoided entirely the question of whether Euler discovered or 'invented' it (though I suspect he too agrees that it was discovered).

February 20, 2018
Lewis, here, in Russian langauge, we distinguish / separate "humanitarian" sciences, "natural" and "precise" sciences.

It is a bit similar to English 'sciences' ("hard sciences" for what we call "precise") and 'humanities'. Difference is that, for a Russian 'history' is a science too. While some humanities might not be sciences at all. Idea is that... "humanitarian" sciences study things related to humans: language, for exmaple. It is produced by humans. Or history [of humans]. So "lingustics" is 'humanitarian" in our view.Accordingly students and people are divided here to "humanitarians" (a philologist) , "naturalists" (a biologist) "technic-ers" (an engineer). Phisicists and mathematicians stand somewhat above "technic-ers". Mostly division is "humanitarians" and "others". It is supposed that it defines your  mindset. If you graduated from a philological faculty you say about yourself "I'm a humanitarian" and by this you mean: "I'm afraid even to speak about intergrals and, God save, derivatives!"

It is a harmful distinction. I'm a 'multi-class' (in gamers terms) then: you wont' scare me with gauge theory or an ancient manuscript, both are just texts not easy to read which I CAN manage to understand if i work.
It is so much about limitations you put on your brain: not even attempting to think about certain things.
But it is ingrained in Russian students worldview:/

So why I'm telling this: once I asked myself if mathematics is "humanities" or "natural sciences"?
DOes it study the universe - or does it study a human?
February 20, 2018
I absolutely love that last line you wrote there, Tia. This is the same Euler with the Euler-Bernoulli equation, I presume? Thank you for that link as well, I will definitely look into it before I sleep. I also like the word you used in your second paragraph: logically. This, I believe is where our limits as humans are apparent. Is it possible that, since mathematics is a logical way of problem-solving involving known and unknown (sometimes assumed as you mentioned), then our logic is not advanced enough to comprehend some of the most apparent mathematics in the universe?
February 20, 2018
Maths is never invented by any of human beings. It is more like a phenomenon, a natural scene that has been existing for many years long before the earth formed.
February 20, 2018