# The Art of Mathematics



## Gruvian (Feb 6, 2014)

I wanted to ask the NTs, have you ever found yourself thinking ''this equation is beautiful''? Or generally speaking, have you ever found mathematics to be beautiful?

I always thought there was some kind of beauty in it. Some equations are so beautifully, carefully crafted that look so simple yet hold the entire world in them. A great example of that is Euler's Identity, or Pythagoras. I could go on and on, to me, they look like the finest bits of art, they're as beautiful as van Gogh's paints in my eyes. 

Do any of you NTs relate to this?


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## lemony snicket (May 21, 2014)

Yes I relate to this! I especially like working out the proof behind a theorem or writing out my process as I solve a difficult problem, I like looking back at all the logical steps building up to this perfect concise conclusion, there's definitely a lot of beauty in that


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## Tzara (Dec 21, 2013)

Yes, but I wouldnt go too far and call math an art. Much like how I wouldnt call chess a sport.


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## HAL (May 10, 2014)

I think it's beautiful. Maths is one of the few things that indisputably _makes sense_.


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## nuut (Jan 13, 2014)

Yes, mathematics is beautiful. The language of gods.


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## an absurd man (Jul 22, 2012)

Sometimes, but not really. I'm more apt to admire it's power; the fact that we can develop a language to model and manipulate phenomena is profound.


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## Protagoras (Sep 12, 2010)

For those who are interested: there have been studies on this. I believe there have also been phenomenological studies on the subject, but this book adopts a naturalistic account of mathematical beauty.


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## HypoTempes (Nov 25, 2013)

Yes, there's an inherent beauty to it. 

Especially Complex system analysis for me personally.

On a side note, do things that don't sit right "annoy/irritate" you? Even if you haven't found "it" yet?

Still not sure if it should be considered a form of art though, seeing how: 

"Art is a representation of reality according to the person who made it" (picasso if I'm not mistaken please correct me if I'm wrong" 

Then again, Maths seem to be a way for humans to understand and define/represent the rules of the universe itself so if that's not a form 
of Art....then I don't know what is.


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## knife (Jul 10, 2013)

I adopt a Kantian position and consider math in the realm of the sublime. The feeling, the whoosh I get when I understand the genius behind a proof is much more comparable to a thundercloud scudding across the plains than a piano piece or van Gogh painting.


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## Mr. Meepers (May 31, 2012)

Well, I am not an NT. I am an INTP, but actually earlier today someone had asked me in a PM why I liked math (I think I was slightly tired when I answered it), and I am feeling lazy right now, so, I'm just gonna copy and paste my response:


Well, I can't say I truly appreciated math for its own sake until I was a sophomore in college. Before that I only appreciated math because of its relation to physics and how I could view physics geometrically. I still like it for that as well, but when I was a sophomore, I got to see how beautiful it was without the lens of physics. I had seen the elegance of how the sqrt of 2 is irrational, how unrelated definitions (such as i,e, pi, 1, and 0) could relate to each other in such a simple way (e^(i*pi) + 1 = 0), and I saw that the "size" of infinities are not all the same (for instance, the set (0,1) has "more" elements in it than the set of all natural numbers ... the natural numbers and the rational numbers have the same "number" of elements). I found that mathematics could completely shatter my intuition with curious oddities, such as a function that is continuous at one and only one point, and create new intuition by creating a world that is more and more abstract to create a new, and bigger playground for my imagination. I loved seeing why everything was true (and not just taking it for granted all the time) and I loved seeing the consequences of how we define our words lead to interesting, and unintended, ideas. Mathematics also had helped me shape my own notion of what is means to be beautiful and made me see how interconnected language is with abstract thoughts.


Edit: Here are some links that are interesting.
Mathematical beauty activates same brain region as great art or music
Mathematical beauty
Mathematics: Why the brain sees maths as beauty


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## Protagoras (Sep 12, 2010)

hammersklavier said:


> I adopt a Kantian position and consider math in the realm of the sublime. The feeling, the whoosh I get when I understand the genius behind a proof is much more comparable to a thundercloud scudding across the plains than a piano piece or van Gogh painting.


Then don't you mean the Kantian realm of the beautiful? If I remember correctly, Kant's sublime is characterized by being incomprehensible, overpowering, grander than you; beyond you in a terrifying way. But if it is comprehension, harmony, etc.; if you mean a feeling of things 'coming together' and just making sense (understanding/comprehension), and so on, then it seems to me that you are firmly in the realm of what Kant called the beautiful, which is _not_ the same as the sublime.


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## knife (Jul 10, 2013)

Protagoras said:


> Then don't you mean the Kantian realm of the beautiful? If I remember correctly, Kant's sublime is characterized by being incomprehensible, overpowering, grander than you; beyond you in a terrifying way. But if it is comprehension, harmony, etc.; if you mean a feeling of things 'coming together' and just making sense (understanding/comprehension), and so on, then it seems to me that you are firmly in the realm of what Kant called the beautiful, which is _not_ the same as the sublime.


It is a subtle distinction, yes. Part of the point of the imagery I used is that -- at least in the way I was taught the _Critique of Judgment_, so many years ago -- a good way of separating the two is the kind of imagery it evokes. The distinction between the two is that things of beauty are entirely within your control, while those of the sublime are not. To me, when I really _get_ a concept in math, it's more like I've tapped something that's more like a force of nature, beyond my control, and in fact slightly terrifying. "Math" is, after all, merely a notation we use to describe relations we find _in_ nature.


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## Protagoras (Sep 12, 2010)

hammersklavier said:


> "Math" is, after all, merely a notation we use to describe relations we find _in_ nature.


I agree that math can be used like that, but I am not sure if that is what math is. Math is often a description of forms, bodies, lines, relations, etc. that are nowhere to be found in the natural world. So, unless you think Platonic Ideas of forms, bodies, lines, relations, etc. are a part of nature, or something like that, I think that is not quite right. However, I think I understand what you mean when you think of this as slightly terrifying.


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## knife (Jul 10, 2013)

@Protagoras I disagree slightly. I think that the (usually geometrical) simplifications you're thinking of are simply a useful abstraction. So Platonic Forms arose in large part in a sort of confusion, that mathematical abstractions somehow represented something "more" real.

We focus on the figure today, but that wasn't the crux of Greek mathematics. No, what the system _really_ focused on was one of finding a relationship. Indeed, think of the Pythagoreans' insistence that all numbers were ratios: what they were really saying was that everything in the universe was related to everything else. (Of course, the demonstration that root-2 isn't a ratio really mucked that up).

So in math we look to find the relationships, because the constants are the things left over. And we still don't really know why.


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## Protagoras (Sep 12, 2010)

@hammersklavier:

My point was that we describe abstract entities (which can be relations, numbers, classes, etc.) in mathematics that do not exist in the natural world. So, you would either have to do away with abstract entities entirely or say that they _do_ exist in the natural world in order to say that mathematics is "merely a notation we use to describe relations we find in nature." 

However, I have never seen a number in the natural world, I have only seen representations of numbers in the natural world. I have also never _seen_ a mathematical relation in the natural world, this is itself an abstraction. Simply put: 3:1 does not exist 'out there' in nature, there are merely 3 sheep and 1 farmer and their relation is an abstraction. So, the relation does not exist in nature, but it does exist. Hence, we also cannot do away with abstract entities altogether.

So, basically, my point was that saying that mathematics is a description of relations we find in nature is problematic, because it raises the question of the ontology of these relations. Do they really exist in nature or are they abstractions that exist in another way than natural entities exist, and are they therefore something like Platonic Ideas? I am saying that I think the latter option seems more plausible to me at this moment. I think it is a mistake to treat abstractions or abstract entities as though they are natural entities, if you understand my point.


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## Tezcatlipoca (Jun 6, 2014)

they found that the same part of the brain activates when looking at a beautiful equation as when one enjoys a beautiful sunset. also there are mathematical rules that undergird our perception of beauty, though I think at higher stages you appreciate flaws because they are indications of entropy patterns of beauty


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## knife (Jul 10, 2013)

Protagoras said:


> My point was that we describe abstract entities (which can be relations, numbers, classes, etc.) in mathematics that do not exist in the natural world. So, you would either have to do away with abstract entities entirely or say that they _do_ exist in the natural world in order to say that mathematics is "merely a notation we use to describe relations we find in nature."


Ah. I see I was mistaken. I originally interpreted your argument as a claim _to_ formalism.

To me, numbers are merely instantiations of values. Yes, they are abstract, but the heart of mathematics is more a way of saying so-and-so behaves--indeed, has to behave--in such-and-such a manner for this thing to make sense.

Consider, for example, the problem of doubling a square--finding, from one square, another with double the area. The relationship we're looking for here is quite explicit. We want to go from _x_ to 2_x_ and do so while maintaining the essential properties of a square. 2_x_ here is merely a shorthand: _x_ + _x_ = 2_x_.

We find that by dividing the square in half diagonally, we get two triangles. The triangle that was half of the original square is a quarter of the doubled square -- _again_, this is a pure statement of relationship -- which means that the line diagonally halving the square is the side of the square with double the area.

You can see how numerical values are unimportant here. However, if you apply the Pythagorean Theorem to get the length of the new square's sides (_c_) from the original one's (_a_), you get:

_a^2 + a^2 = c^2
2a^2 = c^2
__√(2a^2) = c
a√2 = c_

You can see how a _constant_ (in this case, the Pythagorean constant, or _√2_) drops out. Numbers are merely input values. _Constants_ are arbitrary _outputs_, values that turn out to be _required_ for the thing to work. Quite a lot of mathematics works in this manner -- taking an (admittedly abstracted) problem, then thinking it through and applying related problems, prodding it along until things like constants that show that something arbitrary has crept in somewhere.


> However, I have never seen a number in the natural world, I have only seen representations of numbers in the natural world. I have also never _seen_ a mathematical relation in the natural world, this is itself an abstraction. Simply put: 3:1 does not exist 'out there' in nature, there are merely 3 sheep and 1 farmer and their relation is an abstraction. So, the relation does not exist in nature, but it does exist. Hence, we also cannot do away with abstract entities altogether.


I agree with you. We can see instantiations of numbers; we can also see instantiations of mathematical relations in the real world. For example, the Fibonacci sequence is quite subtly instantiated all throughout our biota.

3:1 is a Leibniz relation, and so postdates the kind of relations the Greek were fond of. The Greeks would have stated the relation as 1/3 (or "there is one farmer for three sheep"). In fact, fractions and the fractional representation of division are Greek ratios; the _mathematical definition_ of rational numbers (i.e. "a number that can be expressed as a ratio") is { _R_ | _R = P/Q_ } -- i.e. a rational number _must_ be a quotient (it does not matter, at least not today, whether that number is an integer).

Continuing from our example, the Pythagoreans, once they found the rather odd constant _√2_, wondered whether or not it was rational. (They hypothesized it must be.) The proof that it is not is quite ingenious, and of course cracked open a whole new world of ideas to explore.

You are starting to see where I come from, I trust, when I say the heart of mathematics is the statement of relationships -- though they may not be obvious. These relationships keep compounding as mathematicians blunder down the next rabbit hole, and before you know it, you've got things like hypercubes and Galois number theory and Boolean algebra and all these interesting things that numbers can do that -- incredibly -- _still_ have practical application (i.e. are instantiatable in meaningful ways).


> So, basically, my point was that saying that mathematics is a description of relations we find in nature is problematic, because it raises the question of the ontology of these relations. Do they really exist in nature or are they abstractions that exist in another way than natural entities exist, and are they therefore something like Platonic Ideas? I am saying that I think the latter option seems more plausible to me at this moment. I think it is a mistake to treat abstractions or abstract entities as though they are natural entities, if you understand my point.


Ah. Maybe I was not so mistaken after all. (Indeed, many mathematicians are Platonists, for the very good reason that they interpret math as form.) The metaphysical problem for me is, I find formalism untenable in many other respects, and so I don't find the impulse to it persuasive, but I don't really have a very good countertheory, either.


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