/mg/ - Mathematics General

[AIpiqiA]

A thread for stuff related to Mathematics and Mathematicians. To start off the thread, here are some questions.

>What do you like about Mathematics?
>Who are some of your favourite Mathematicians?
>What are some of your favourite pieces of Mathematics?

[AIpiqiA]

To answer my own prompts,

>What do you like about Mathematics?

1. I'm attracted to it's utter straightforwardness and the axiomatic truth systems. What I'm describing here applies to most disciplines that uses formal systems, but I find the applications of it in Mathematics to be some of the most interesting and alluring.

>Who are some of your favourite Mathematicians?

2. There's a lot of cool Mathematicians out there, but one whose lifestyles I find really interesting is Erdős Pál.

>What are some of your favourite pieces of Mathematics?

3. I'm really a newbie so I only know elementary Mathematics, but I really like the very simple geometric proof of Thales' Circle Theorem and De Moivre's Formula/Euler's Formula connecting the Complex Numbers and Trigonometry. The algebraic/geometric proofs relating to Trigonometry in general are some of the most accessible pieces of Mathematics, and thus enjoyable can be enjoyed by most people who have atleast some understanding of Highschool Mathematics (or earlier levels depending on your education system).

[Alrune]

Thanks for starting a math thread
I was just about to do the same after seeing how lively the discussion was in the Introduction Thread but you beat me to it.

>What do you like about Mathematics?

What initially attracted me to mathematics was the underlying unity of the subject. It just scratched an itch in my brain to feel the a satisfying click when a concept slid into place and suddenly exposed bridges to other domains. Lately I have been been more interested in category theory and topos theory for this reason.

I am also very interested in the ontological status of mathematics and its role in describing our world, especially as it relates to our perception (since mainstream cognitive science and neuroscience primarily operate within a computational framework); Bergson and Deleuze has some interesting critiques of this, but it would derail the thread

>Who are some of your favourite Mathematicians?

I find Alexandre Grothendieck's abilities of abstraction aspirational; He also lead a very interesting, if tragic life.

>What are some of your favourite pieces of Mathematics?

In the interest of time I will start with just one:
One theorem I love is the Eckart–Young-Mirsky theorem from linear algebra. It states that if you compute the singular value decomposition of a matrix, then the best* rank-k approximation to that matrix can be obtained simply by truncating the decomposition to its first k singular values and singular vectors. In a sense, the matrix already “contains” the information about its own optimal approximation, which I’ve always found remarkably elegant.

Geometrically the SVD describes how a matrix transforms spheres into ellipsoids, with the singular values giving the lengths of the ellipsoid’s principal axes. I also like that it resembles ideas such as Fourier series or Taylor expansions, where complicated objects are decomposed into simpler pieces. But unlike Fourier or Taylor expansions, whose basis elements are fixed in advance, the SVD adapts its basis directions to the matrix itself.

* “Best” here means optimal in either the Frobenius norm (least-squares sense) or the spectral/operator norm.

2. There's a lot of cool Mathematicians out there, but one whose lifestyles I find really interesting is Erdős Pál.

Did you watch the documentary "N is a number: A Portrait of Paul Erdős"? It's nothing groundbreaking as a documentary, but it is quite comfy

[netdoll]

I find Alexandre Grothendieck's abilities of abstraction aspirational; He also lead a very interesting, if tragic life.

In context it would be remiss not to mention his 1972 CERN speech where he went on at length about why he left the mathematical establishment and retreated from the affairs of the world more generally. https://github.com/Lapin0t/grothendieck-cern

[AIpiqiA]

Did you watch the documentary "N is a number: A Portrait of Paul Erdős"? It's nothing groundbreaking as a documentary, but it is quite comfy

Yes I believe so, but it had been quite a while since I had last seen it so maybe I'll give it a rewatch some time.

I find Alexandre Grothendieck's abilities of abstraction aspirational; He also lead a very interesting, if tragic life.

Grothendieck certainly seems to be a very interesting figure I wish to learn more about outside of just the wikipedia pages.
One of the things I'm interested in getting some level of understanding in the future as I do my studies is Algebraic Geometry, since it comes up or gets mentioned so often from the maths spaces I've lurked on online.

[drummyfish]

>What do you like about Mathematics?

Mathematics to me is the ultimate intellectual endeavor, like the mount Everest of thinking where we climb the mountain of abstraction to heights that disconnect us from the real world and we reach something like the "outer space" of thinking, an alien world that can be as mysterious as planets in distant galaxies. I do also find comfort in the clarity and no space for opinions to fight over, something either holds or it doesn't -- it's like a world with a language that cannot be misunderstood, unlike our natural human language that's extremely fuzzy and constantly changing. Mainly I love the freedom and "minimalism" however, the fact that it's something that cannot be taken away from anyone, that all this beauty and exploration of the vast mathematical landscape can only be done with pure thought with no equipment needed. It is one of very few certainties in this world I can hold on to. To put it in very lame terms: it is like an infinite number of video games for me to play anywhere that cannot be owned or fucked up by any corporation

Also there is the aesthetic beauty that's very apparent when you start programming, and doubly so if you're doing something like computer graphics, shaders, fractals, procedural textures, bytebeat and all this kind of stuff. I totally understand how people hate math when all they remember is the struggle to memorize the formula for solving a quadratic equation without knowing what the fuck it's even useful for, but once you see the utility (which programming is excellent for), you just cannot but fall in love with it.

>Who are some of your favourite Mathematicians?

I never really thought about this much, but maybe I'd highlight Perelman. Of course there have been quite a few great minds in history, but among them it's still rare to find some who also so strongly held on to their principles and moral beliefs with their actions.

>What are some of your favourite pieces of Mathematics?

I think stuff about computability. I remember a small brain implosion moment when my uni prof told us that a complement of a formal language can be computationally simpler than the original, i.e. that a computer may be literally theoretically unable to recognize exactly which words belong to a certain language (i.e. follow a certain pattern), BUT it may very well be able to tell exactly which ones are NOT in that language. It is as if you were being shown random animals and you weren't smart enough to reliably tell "this is a bird", but you'd ALWAYS be able to reliably tell "this is NOT a bird".

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That said I was surprised about how many people here, being mostly into philosophy, are actually into math too. I wonder why this is and would love to hear your opinions. My initial guess is that all these intellectual disciplines actually meet when you follow the road long enough, as if "all roads lead to Rome", with Rome here being the edge of knowability. We know for certain that some theorems in math are unprovable, some function uncomputable and a lot is simply beyond our comprehension, and philosophy (epistemology?) knows this too, so these disciplines meet at this point coming from different directions. I think it's no coincidence that physicists are now all discussing "interpretations" of quantum theories and basically get into metaphysics, I think they arrived at the same spot. I would like to ask the philosophers here: is this how you arrived to mathematics? Like I said I'm a noob at philosophy, as a programmer I came from the math side but think I am now seeing I might naturally continue down the philosophy road as well.

[AIpiqiA]

That said I was surprised about how many people here, being mostly into philosophy, are actually into math too. I wonder why this is and would love to hear your opinions. My initial guess is that all these intellectual disciplines actually meet when you follow the road long enough, as if "all roads lead to Rome", with Rome here being the edge of knowability. We know for certain that some theorems in math are unprovable, some function uncomputable and a lot is simply beyond our comprehension, and philosophy (epistemology?) knows this too, so these disciplines meet at this point coming from different directions. I think it's no coincidence that physicists are now all discussing "interpretations" of quantum theories and basically get into metaphysics, I think they arrived at the same spot. I would like to ask the philosophers here: is this how you arrived to mathematics? Like I said I'm a noob at philosophy, as a programmer I came from the math side but think I am now seeing I might naturally continue down the philosophy road as well.

Mathematics is like an art made out of Logic, and Logic has always been a part of and even central to Philosophy (well, atleast in the way Modern Academic Philosophy saw it, at least for the English speaking ones).

Maths and Logic are more connected to Philosophy in some traditions more than others, such as the Analytic Philosophy tradition which good ol Wittgenstein is a big figure in. Since many folks here probably have seen Amlux's Tractatus Logico-Philosophicus lecture, it's probably no surprise atleast some of them would be really into Mathematics.

Philosophy studies quite a big range of subjects, especially in regards to what tools to use to answer questions and why to use them for subjects where its not clear what the way to go is (once we have an established methodology, it typically moves on to science or other disciplines), and Logic is some of the strongest tools we have, and also some of the most convincing seeing how predominant Science and Rationality is. So we'll probably keep having Philosophers interested in Mathematics as long as this is the case.

[Amlux]

When I was a kid I hated mathematics. And it's interesting in that I hated it for precisely the reason that many seem to love it. There's no room for ambiguity or vagueness. That made it unappealing to me. I felt like the pure abstraction of it made it irrelevant for me, as anything beautiful to me required a margin of indefiniteness and uncertainty.

But learning about the application of mathematics to philosophy, as Aipiqia said, made it much more appealing eventually. Certainly it would be impossible to understand Wittgenstein without a decent background in logic. Reading drummy's comments are interesting and have made me more understanding of a different form of appeal in mathematics: the pursuit of extreme levels of abstraction and disconnection from the material world. I suppose it's for this reason that Pythagoras and many of his followers thought that mathematics was divine. Of course, coming to know more about computers and programming has given me more and more respect for it as well.

At the end of the day though, I do think that we should ultimately desire there to be indefiniteness and uncertainty in our understanding and experience of the world. This is a point that always made me resonate with Heidegger and which seems to not be reflected in a lot of the modern western philosophical tradition. A world with no mystery would be a very joyless one indeed. So while I originally thought that mathematics was an enemy to this pursuit, I've increasingly come to see how mathematics is helpful in this regard. The whole moral of the Tractatus in my reading is that logic, when rigidly applied to all the "problems" of philosophy, doesn't do nearly as much as we think it might and that there is always something shown which can never be said. It's all the more remarkable that this work predated Gödel's incompleteness theorems and Tarski's undefinability theorem. I'm not enough of a mathematician to understand these theorems in their details, but my impression is that they come close to formalizing insights that are remarkably close to what Wittgenstein said.

A fun aside: My mom was strict with video games as a kid and I didn't play a lot of vidya at all until I was in university. I've also always been far poorer at mathematics than my friends. But since I've been playing vidya, math has become a lot easier for me. Maybe video games do really help with your math skills. Take that, mom!

[Aris]

The aesthetics of mathematics are really incredible to me. They're most obvious in geometry and set theory, but even outside those areas I think beauty is a great way of getting into the subject. I remember taking a class sophomore year on ancient Greek mathematics which was really transformative for me; we read Euclid, Apollonius of Perga and Archimedes the most. They represent aesthetically perhaps the most extreme version of the "mathematics as closed axiomatic truth" model, but you can also see in their writings attempts to grapple with the ambiguous and strange. One of my favourite theorems of Euclid's is the "horned angle theorem," in which he shows that the "angle" made by a circle and its tangent (if you can call it an angle . . .) is smaller than any angle possible from two straight lines, while the angle of the circle and its diameter is larger than any acute angle. The proof is in showing that it's impossible to make a straight line which fits between a tangent and a circle, which means the hypothetical smaller or larger acute angles are impossible to construct. That you can't fit a line between a tangent and a circle is now a commonplace observation, but the framing of that observation through a proof for something being the greatest acute angle is just strange enough that it made me remember how difficult of a truth it really is to imagine, physically or maybe metaphysically, and it made me remember that someone had to come up with the proof, see it as something that could be proven. And I find that very beautiful.

[drummyfish]

So I've got a small discovery I just made, wondering if we can figure out what's going on. I've taken the Ulam Spiral and scaled it down WITH point sampling (i.e. without antialiasing) and now I'm seeing a checkerboard:



What's this about? It looks like a pattern in the distribution of prime numbers, definitely too simple not to be known by now, but I don't know this. Does anyone know this phenomenon? I have also noticed the same checkerboard when resizing primes that I placed on Hilbert Curve. It looks like it could be used as a quick primality test? If the number lies on the "dark square", it's not a prime, however if it's on the "light square", it may or may not be. The size of the checkerboard also changes when you fiddle with the scaling, so there could be more.

You can try this out yourself in GIMP or any similar editor, just remember to turn of antialiasing (use nearest neighbor interpolation).

EDIT:

Spoiler

I mean thinking about it the checkerboard makes sense if you imagine a one dimensional case, it's really just divisibility, but there seem to be more cool patterns if you play around:



Could be worth exploring other types of transforms without antialiasing, combined with other ways of placing primes in 2D, like some space filling curves and so on. Might result in nice pictures. Share if you get any :-)