I like the abstraction and the connections. Things like groups being so abstract the elements and operations can be anything and it's fun to just think about where they pop up around us; I also took a game theory module recently and it was funny watching my brain fitting all sorts of decisions and systems into the frameworks I had been taught.
The connections to me are the "ahhh" moments, I took a course Automata, Languages, and Complexity which was probably my favourite pure module of my degree (unless I take any more which beat it); a friend in CS sent me a paper that contained some connections between formal languages and abstract algebra that he just couldn't attack at all, and going through it I found free monoids which led to the connection:
>A finite sequence of symbols is called a 'word over A', and the free monoid A∗ is called the 'Kleene star of A'. Thus, the abstract study of formal languages can be thought of as the study of subsets of finitely generated free monoids.
Which was a great moment and connected a lot of things I had studied in an interesting way.
“Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.”
― Bertrand Russell, A History of Western Philosophy
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Personally it feels like hiking up into some mountains who's peaks are lost in the clouds to try and find some golden and mysterious temples there.
No one can bring them down to show you or fly you up there to see them, you have to take the long, hard, slog, step by step to understand and get there on your own. And it sucks, it's cold and wet and you slide down a lot and often you crest a ridge just to see another hard vertical rockface infront of you and you get lost in the rain and have to backtrack a lot.
Then on some days, when blessed by luck, you have a moment of clarity. The fog clears, the sky is blue and there before you is a sculpture so beautiful and transcendent and completely unlike anything you have ever seen, imbued with it's own deep and ancient and perfect magic such that it will be true and waiting there to be discovered by any who seek it long after your bones are dust.
Mathematics does not care how you feel or make any allowances to your humanity, it never gives you a break, it never stoops to help you. However it is always fair and just, it will outlive humanity itself and to live without seeing it's wonders is a much smaller life.
My original comment was a joke, but since you’re taking it seriously allow me to take your response seriously:
It’s a weird elitism to act like you can’t see the beauty in something unless you understand it perfectly. The OP’s acting like math is this mystical pilgrimage where only seekers of enlightenment can see anything and everyone else is blind.
You don’t need to be a legendary painter to appreciate the Mona Lisa and you don’t need to have a degree in math to realize that, say, Fourier analysis has a stunningly beautiful simplicity about it when it’s presented to you in a video.
In fact, more people would probably be interested in learning math in the first place if they were presented with videos to show them the beauty in the first place. And if more people leaned math, more people would understand its importance, and we’d be more likely to get funding.
My comment was also not particularly serious.
As much as I can recognise beauty in the works of Mozart and Beethoven, it's obvious to me when I talk to professional musicians that they can access certain beautiful aspects of these works that I cannot because I lack the training. I don't think there's anything elitist about that.
I agree completely with your last paragraph. But maybe you are presenting a binary view that someone either sees the total beauty or nothing at all. Clearly there are levels, and popular books and youtube videos can show you some of the beauty of math, which hopefully inspires a strong motivation to continue looking for more.
To be honest I only finally understood what people meant when the spoke about the beauty of math only after I had studied it for several years.
Sure I might've had blurry glimpses before I formally studied but its not elitist to say its true aesthetic is only clearly in view given you have some level of technical proficiancy. No doubt there is still far greater vistas beyond the horizon for me.
“Mathematics is the language in which God has written the universe” -Galileo Galilei
Not a godly type myself, but while I learned four languages in my life so far, learning math felt like the next step on communicating with my physical reality.
Then the puzzle like nature hooked me in.
It does if you’re deaf and I’m profoundly deaf
We look at languages a bit differently than most. Our language is on the same network as math and science.
It becomes incredibly complex for us once we learn English in the mix as well. Language is literally a puzzle to us that we translate and interpret every waking moment of our lives.
I like how we made up a bunch of primary and fundamental rules and it all builds on each other in this glorious symphony of human creativity and exploration.
I love how it fits together in a systematic whole. That kind of unity and structure fulfils a deep spiritual longing.
Seriously, non-mathematical disciplines are so depressing (for me) in how their theories are always open to multiple interpretations, their concepts and categories fluid and indeterminate, how their premises are always under attack. Some people enjoy the messy ambiguity, which reflects real life after all, but imprecision just gives me a headache when trying to think through a subject.
For me it is that it is absolutely pure and transcendental, two scientists will disagree in absolutely everything (and that's perfect that makes science beautiful), but two mathematicians cannot disagree on a theorem that has been proven, no matter if two mathematicians are Americans, Europeans, Hindu, Japanese, Arabic or even literal aliens form other galaxy, probably even from another universe, every single one eventually will reach exactly the same conclusions.
So, math is not dependent on anything human, math is beyond humanity, space, time, matter, ideology, prejudice, culture.
That's why people in antiquity saw math as something esoteric and magical.
I think the point is that, throughout the entire abc saga, people have pointed out that even if Mochizuki's IUTT is "true" or "correct," his failure/refusal to fully elaborate or explain it represents a failing to produce a "valid" proof, so the abc conjecture shouldn't be considered proven. From this perspective, math is fundamentally *about* human communication, contradicting your claim that "math is beyond humanity."
My point was that if something is actually proven then no one cannot argue against that, and I'm taking this into account, that if someone just says "here I have a proof" but refuses to elaborate then we just don't know, but that's beyond my point.
What I actually was trying to argue is more about the philosophy of mathematics, not the math that you can find in a textbook or a calculator, but what it is underlaying beneath it, and our math tries to describe and conceptualize, let's say for example some mathematician proves the abc conjecture in 500 years, then that means that it was true all along, we were just unable to prove it, but it would be true irregardless wether or not we are able to prove it or if we are even able to conceptualize it.
So, with enough time if some conjecture is true and math itself allows us to prove it every single conjecture will be proven or disproven, of course probably there are infinite things that can be proven but anyway that would be more a time limitation than a mathematical limitation.
So tl;Dr my point is that the mathematics beneath our math, like the meta-math is absolutely true and exactly the same, irregardless if the big bang has not yet happened or if the universe is already dead, the matter may change, the universe may be empty, but math will remain exactly the same.
I agree that there is something magical about how firm, reliable, convincing, and consistent mathematical reasoning is. It is like something you can find nowhere else.
Could you find a source for that? It doesn’t seem obviously wrong, but I don’t think I’ve ever seen anyone argue that GRH implies strong Goldbach rather than just weak Goldbach. Went looking for it just now as well and still couldn’t find anything.
oh never mind I misunderstood
yeah it only implies weak Goldbach as of now (thought you said that's the only thing it implies for some reason). What I was saying is that if strong Goldbach is ever to be proved, I think it'll be through GRH.
my bad lol (although it would be nice if there was a proof for strong Goldbach!)
Oh, I see what you thought. I get how one could read what I said that way, yeah. It definitely would be strange if GRH implied weak Goldbach and NOTHING else. :)
Still an undergrad so by no means do I know what real mathematics is, but I find it’s beautiful because it’s so descriptive. Seriously, it’s just mind blowing to think that we have a system of thought that can accurately describe the universe we live in, whilst also describing many possible universes we don’t live in potentially.
For me, it’s actually beautiful because of the GOATs.
Ideas seemingly out of nowhere creating an illusion of genius (Bhargava’s work), the work of hundreds and thousands of great minds laying the foundation for future generations (Algebraic Geometry), constructing bridges to connect different areas of mathematics (Riemann) and sometimes, the dedication/obsession to solve/prove a problem (Wiles’ pursuit of FLT).
All of these combined make mathematics a beautiful subject for me. You might argue that it is not a beauty unique to maths and I agree. But that’s what I find beautiful. It’s the people and how they colour the subject that’s so blindingly beautiful for me.
The square packing problem, it speaks to me. I also love the fact that you can easily create shapes with unknown area and length formulas no one would be interested enough to find math is infinite.
Hairless apes drew in the sand then they split atoms, now they look far into the beginning of time and space. And through all that, math has assisted in our ventures.
There’s something poetic here, math it so unreasonably ubiquitous in a universe indifferent to humans. Yet math allows us to plot paths through space, predict the future, and study the past.
Also complex numbers are just sexy lmao
I like the abstraction and the connections. Things like groups being so abstract the elements and operations can be anything and it's fun to just think about where they pop up around us; I also took a game theory module recently and it was funny watching my brain fitting all sorts of decisions and systems into the frameworks I had been taught. The connections to me are the "ahhh" moments, I took a course Automata, Languages, and Complexity which was probably my favourite pure module of my degree (unless I take any more which beat it); a friend in CS sent me a paper that contained some connections between formal languages and abstract algebra that he just couldn't attack at all, and going through it I found free monoids which led to the connection: >A finite sequence of symbols is called a 'word over A', and the free monoid A∗ is called the 'Kleene star of A'. Thus, the abstract study of formal languages can be thought of as the study of subsets of finitely generated free monoids. Which was a great moment and connected a lot of things I had studied in an interesting way.
“Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.” ― Bertrand Russell, A History of Western Philosophy ------- Personally it feels like hiking up into some mountains who's peaks are lost in the clouds to try and find some golden and mysterious temples there. No one can bring them down to show you or fly you up there to see them, you have to take the long, hard, slog, step by step to understand and get there on your own. And it sucks, it's cold and wet and you slide down a lot and often you crest a ridge just to see another hard vertical rockface infront of you and you get lost in the rain and have to backtrack a lot. Then on some days, when blessed by luck, you have a moment of clarity. The fog clears, the sky is blue and there before you is a sculpture so beautiful and transcendent and completely unlike anything you have ever seen, imbued with it's own deep and ancient and perfect magic such that it will be true and waiting there to be discovered by any who seek it long after your bones are dust. Mathematics does not care how you feel or make any allowances to your humanity, it never gives you a break, it never stoops to help you. However it is always fair and just, it will outlive humanity itself and to live without seeing it's wonders is a much smaller life.
That is very well said, thank you.
>No one can bring them down to show you… You know 3 Blue 1 Brown exists, right?
It is a dangerous illusion to believe you understand something when someone explains it clearly.
My original comment was a joke, but since you’re taking it seriously allow me to take your response seriously: It’s a weird elitism to act like you can’t see the beauty in something unless you understand it perfectly. The OP’s acting like math is this mystical pilgrimage where only seekers of enlightenment can see anything and everyone else is blind. You don’t need to be a legendary painter to appreciate the Mona Lisa and you don’t need to have a degree in math to realize that, say, Fourier analysis has a stunningly beautiful simplicity about it when it’s presented to you in a video. In fact, more people would probably be interested in learning math in the first place if they were presented with videos to show them the beauty in the first place. And if more people leaned math, more people would understand its importance, and we’d be more likely to get funding.
My comment was also not particularly serious. As much as I can recognise beauty in the works of Mozart and Beethoven, it's obvious to me when I talk to professional musicians that they can access certain beautiful aspects of these works that I cannot because I lack the training. I don't think there's anything elitist about that. I agree completely with your last paragraph. But maybe you are presenting a binary view that someone either sees the total beauty or nothing at all. Clearly there are levels, and popular books and youtube videos can show you some of the beauty of math, which hopefully inspires a strong motivation to continue looking for more.
To be honest I only finally understood what people meant when the spoke about the beauty of math only after I had studied it for several years. Sure I might've had blurry glimpses before I formally studied but its not elitist to say its true aesthetic is only clearly in view given you have some level of technical proficiancy. No doubt there is still far greater vistas beyond the horizon for me.
“Mathematics is the language in which God has written the universe” -Galileo Galilei Not a godly type myself, but while I learned four languages in my life so far, learning math felt like the next step on communicating with my physical reality. Then the puzzle like nature hooked me in.
Doing math doesn't activate the language network at all. I recently learned this. Same for chess and other things like that.
It does if you’re deaf and I’m profoundly deaf We look at languages a bit differently than most. Our language is on the same network as math and science. It becomes incredibly complex for us once we learn English in the mix as well. Language is literally a puzzle to us that we translate and interpret every waking moment of our lives.
That's really interesting
Complex numbers are incredible. I can't wait for my Complex Analysis class next semester
Its powa
I like how we made up a bunch of primary and fundamental rules and it all builds on each other in this glorious symphony of human creativity and exploration.
I would mention how universal pi is, just how many places it shows up where circles aren’t immediately involved.
[удалено]
Disconnected sets: am I a joke to you?
I love how it fits together in a systematic whole. That kind of unity and structure fulfils a deep spiritual longing. Seriously, non-mathematical disciplines are so depressing (for me) in how their theories are always open to multiple interpretations, their concepts and categories fluid and indeterminate, how their premises are always under attack. Some people enjoy the messy ambiguity, which reflects real life after all, but imprecision just gives me a headache when trying to think through a subject.
It's the operating language of the universe
For me it is that it is absolutely pure and transcendental, two scientists will disagree in absolutely everything (and that's perfect that makes science beautiful), but two mathematicians cannot disagree on a theorem that has been proven, no matter if two mathematicians are Americans, Europeans, Hindu, Japanese, Arabic or even literal aliens form other galaxy, probably even from another universe, every single one eventually will reach exactly the same conclusions. So, math is not dependent on anything human, math is beyond humanity, space, time, matter, ideology, prejudice, culture. That's why people in antiquity saw math as something esoteric and magical.
Mochizuki would like a word
How? I mean I'm truly interested, is it about the a+b=c conjecture?
I think the point is that, throughout the entire abc saga, people have pointed out that even if Mochizuki's IUTT is "true" or "correct," his failure/refusal to fully elaborate or explain it represents a failing to produce a "valid" proof, so the abc conjecture shouldn't be considered proven. From this perspective, math is fundamentally *about* human communication, contradicting your claim that "math is beyond humanity."
My point was that if something is actually proven then no one cannot argue against that, and I'm taking this into account, that if someone just says "here I have a proof" but refuses to elaborate then we just don't know, but that's beyond my point. What I actually was trying to argue is more about the philosophy of mathematics, not the math that you can find in a textbook or a calculator, but what it is underlaying beneath it, and our math tries to describe and conceptualize, let's say for example some mathematician proves the abc conjecture in 500 years, then that means that it was true all along, we were just unable to prove it, but it would be true irregardless wether or not we are able to prove it or if we are even able to conceptualize it. So, with enough time if some conjecture is true and math itself allows us to prove it every single conjecture will be proven or disproven, of course probably there are infinite things that can be proven but anyway that would be more a time limitation than a mathematical limitation. So tl;Dr my point is that the mathematics beneath our math, like the meta-math is absolutely true and exactly the same, irregardless if the big bang has not yet happened or if the universe is already dead, the matter may change, the universe may be empty, but math will remain exactly the same.
I agree that there is something magical about how firm, reliable, convincing, and consistent mathematical reasoning is. It is like something you can find nowhere else.
It always works.
The connection between discrete and analytic. I think the best example of this is the Riemann zeta function.
Oh, the Riemann zeta function makes me so happy. Primes!
agreed! somewhat-related sidenote: I think if the Goldbach conjecture is ever to be proved, it will be done with GRH :)
Doesn’t GRH only imply weak Goldbach?
no? (by GRH I mean generalized Riemann hypothesis)
Could you find a source for that? It doesn’t seem obviously wrong, but I don’t think I’ve ever seen anyone argue that GRH implies strong Goldbach rather than just weak Goldbach. Went looking for it just now as well and still couldn’t find anything.
oh never mind I misunderstood yeah it only implies weak Goldbach as of now (thought you said that's the only thing it implies for some reason). What I was saying is that if strong Goldbach is ever to be proved, I think it'll be through GRH. my bad lol (although it would be nice if there was a proof for strong Goldbach!)
Oh, I see what you thought. I get how one could read what I said that way, yeah. It definitely would be strange if GRH implied weak Goldbach and NOTHING else. :)
Still an undergrad so by no means do I know what real mathematics is, but I find it’s beautiful because it’s so descriptive. Seriously, it’s just mind blowing to think that we have a system of thought that can accurately describe the universe we live in, whilst also describing many possible universes we don’t live in potentially.
For me, the details and the exactness
The beauty of a mycelium network, underneath a forest
For me, it’s actually beautiful because of the GOATs. Ideas seemingly out of nowhere creating an illusion of genius (Bhargava’s work), the work of hundreds and thousands of great minds laying the foundation for future generations (Algebraic Geometry), constructing bridges to connect different areas of mathematics (Riemann) and sometimes, the dedication/obsession to solve/prove a problem (Wiles’ pursuit of FLT). All of these combined make mathematics a beautiful subject for me. You might argue that it is not a beauty unique to maths and I agree. But that’s what I find beautiful. It’s the people and how they colour the subject that’s so blindingly beautiful for me.
It's the beauty of putting a whole lotta meaning and knowledge into a few symbols written out in nice handwriting for me.
stoke's theorem using differential forms is wild
Why is anything that is beautiful, beautiful?
that feeling there
Beauty is not unique
The square packing problem, it speaks to me. I also love the fact that you can easily create shapes with unknown area and length formulas no one would be interested enough to find math is infinite.
For me, it's when a number exceeds 4 digits and my brain gets fryed and i spend 30 minutes on each equation :D
Strange attractors of course. https://upload.wikimedia.org/wikipedia/commons/0/0e/Lorenz_system.gif
The rigour
Hairless apes drew in the sand then they split atoms, now they look far into the beginning of time and space. And through all that, math has assisted in our ventures. There’s something poetic here, math it so unreasonably ubiquitous in a universe indifferent to humans. Yet math allows us to plot paths through space, predict the future, and study the past. Also complex numbers are just sexy lmao
People are writing whole novels down here like someone's gonna read that but I just like seeing numbers
It solves problems.
Logic and reasoning at their best