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How Mathematics Changed Me

130 points by benbreen 4y ago

qsort 4y ago
I hear this a lot and I'm happy for the author, but the "divinity" of math is not something that has ever resonated with me, not even when I was younger. It's a tool. It's by far the most powerful tool man has ever made, because it's a tool for your mind as opposed to a tool for your hands, but it still is a tool.
- bambataa 4y ago
  I think people use different terms imbued by their own value system to describe the same thing. Sometimes people talk about code being “beautiful”. I’ve never felt beauty from code, but I’ve felt pleasant satisfaction with well-designed code. Maybe just the same thing?
  - c0mptonFP 4y ago
    That's because these things elicit an emotional response, and people all react differently. Also, describing emotions is difficult. When I see an instance of good problem solving, I feel a rush of happiness. It sometimes makes me want to physically get up from my desk and cheer as if I was watching a sports event. All I can think of is "this code/solution is fucking awesome".
  - vsareto 4y ago
    You're going to struggle to find beauty in some areas (like business software and a lot of other tooling). The subject matters as much as its expression. The same is true with art - you're not likely to find paintings of burning tire mountains in the Louvre. But software has another kicker like math in that you'd likely need to be familiar with the subject to know the hard problems, present or previous, and why a solution is elegant or beautiful.
- auggierose 4y ago
  It's both a toy and a tool. A tool for seeing the world as it is, beyond your immediate mental and physical capabilities. If you believe in god or something similar, it makes sense to see something divine in math.
  - PartiallyTyped 4y ago
    What people find as "divine" is the "unreasonable" and "unexpected" effectiveness of mathematics in describing complexity and complex phenomena abstractly. Part of me thinks that we may be very close to reaching what the brain of the average mathematician can comprehend and grasp. After all, not all of us are Gausses, Newtons and Galois.
    - auggierose 4y ago
      Math is not limited to what your brain can come up with on its own. The most important part of math, in my opinion, is that other people (or, other entities, if we include AI), can come up with insights (because of their superior brain power and/or expertise) you would never be capable of conceiving yourself, and package them up in a nice theory for you.
      - PartiallyTyped 4y ago
        While I do agree that mathematics are not limited by what either of us can conceive, there is a fundamental barrier in understanding that one may not be able to process due to sheer volume. A trivial example here, though a bit absurd, is 15000 page long proofs, or machine generated proofs that span hundreds of pages. It is one thing to conceive of something new, and another to fully grasp all that we have built as a species. Certain pieces of work required hundreds of mathematicians to verify, which to me suggests that we have hit the limit of what our brains can do.
        auggierose 4y ago
        The proof of the Kepler conjecture has been formalised, because the original proof was just too large and contained computer calculations that mathematicians could obviously not verify just with their brains in any amount of time. So what? The high-level structure of the proof is quite understandable, and the statement of the Kepler conjecture is very easy to understand as well. What I am saying is that understanding something does not necessarily mean you understand it in all of its details down to the lowest level. And machines can help us understand the things we need to understand. But even with machines we will never understand everything in all of its detail down to the lowest level. We won't even know all the theorems of something as simple as Peano arithmetic. What you can do is not limited to what you can do without machine help, and just with your brain. I would say what we can do now will pale to what we will be able to do in the future, but of course this is speculation.
- linspace 4y ago
  It's a tool for you. Being also a tool is a happy accident.
  - yamtaddle 4y ago
    Accident? We originally constructed math specifically so it would relate to reality and be a useful tool. Even "pure math" topics have trouble shedding that legacy & foundation completely, which is why I think it's weird that people are surprised when, sometimes, those fields still end up relating to reality in ways we hadn't expected. It'd be surprising if they didn't!
- aaaaaaaaaaab 4y ago
  Take the natural numbers. They were invented for counting things, right? One sheep, two sheep, three sheep, simple as. Yet there’s an immense complexity and structure in natural numbers, starting with primes, and ending in algebraic- or analytic number theory! Why is it that this thing we use for counting objects has such rich structure? Where does it come from? Why does it have to be this way? It certainly feels something divine to me. It certainly feels as something that humans didn’t invent, rather was always “out there” to be discovered. Physical laws (which are mostly just OK approximations) feel much more arbitrary and down-to-Earth compared to mathematics.
  - Koshkin 4y ago
    > invented The words for numbers were indeed “invented,” but the numbers themselves, like much of math, was discovered.
    - ivanbakel 4y ago
      That's a philosophical view, and I don't think there's any basis for stating it as fact. One of the great debates of mathematical ontology is whether mathematical concepts are discovered or invented.
      - Kinrany 4y ago
        Like most great philosophical debates, it probably comes from humans being bad at understanding their own language.
        Koshkin 4y ago
        Indeed, the word ‘invented’ is often abused by being used in place of ‘discovered’ or ‘created.’ (No, Faraday did not “invent” the laws of electromagnetism; nor did Linus Torvalds “invent” Linux.)
- Koshkin 4y ago
  > It's a tool. It’s also an art, and a science, and as such it is often seen as beautiful. Beauty, of course, is in the eyes of the beholder, but people tend to ascribe it the objective status, sometimes that of a divine nature.
  - gausswho 4y ago
    Your response made me think of a fundamental bugbear I have with English, and maybe other languages: the word 'is' has a tendency to imply exclusivity unless coddled by further explanation. If someone says something is a tool, that doesn't technically imply there is nothing else it 'is'. But in colloquial use, that's sometimes the intent of the author and sometimes not.
- lkozma 4y ago
  "Seeing an astronomer using a telescope to observe a galaxy, no-one will confuse the telescope with the galaxy. Mathematics differs from science in that there is no clear distinction between the tools and the objects of study." -- D.Aldous
  - anon_123g987 4y ago
    I don't think there's a difference here between Science and Mathematics. Scientists also study their tools (e.g. telescopes are studied by Optics), and use the objects of their studies as tools (e.g. galaxies can be used as telescopes, see: https://en.wikipedia.org/wiki/Gravitational_lens https://en.wikipedia.org/wiki/Gravitational_lens). Edited to add: The author of the quote tried to find an example that evokes vivid imagery, and also very extreme: it's obviously absurd to confuse a telescope with a galaxy, thinks the layman. While this turns out to be literally false, the point of the quote may still stand. The discipline that studies the real tools of Science is not Science itself, but the Philosophy of Science. This shouldn't be like this, and it wasn't until lately. Philosophy and Science used to be one and the same, but around a hundred years ago they separated, and this hurt both tremendously, in my opinion.
- leto_ii 4y ago
  I get that, but I have to say I also see merit in the position that maths is discovered, not invented. I'm currently doing some self-study into more rigorous mathematics and I do often get the feeling that some of the structures are just there, not human inventions made for human purposes. I have recently been quite interested in the Cantor set, for example - hardly a useful tool...
  - LanceH 4y ago
    The axioms are chosen. The theorems follow from the chosen axioms. Which axioms to choose which are interesting or applicable are discovered.
- lizknope 4y ago
  How much math have you learned? When I first saw Euler's Identity I kind of laughed and said WTF? We "discovered" Euler's number while calculating compound interest continously. We knew pi from grade school. Learned about square root of -1 around 9th grade. But when I learned Euler's Identity I didn't even understand what it was trying to say. Then someone took an hour to derive it through a Taylor series and my mind was blown. I still find it amazing. Then in quantum physics class we learned about the wave function and Schrödinger equation. Those "imaginary" numbers influence the real world. I'm not religious but when someone sees some kind of divine beauty in math I kind of feel it too.
  - yw3410 4y ago
    If you haven't done it, then the statistical proofs for the gas equations and the proofs for Maxwell's equations also rank up there for me as well.
    - ivansavz 4y ago
      Thanks for the recommendations, here are the three first links that come up for the "statistical proof" of the gas equations: https://www.one-tab.com/page/UhuMmBwCTma12IwYdMb54Q https://www.one-tab.com/page/UhuMmBwCTma12IwYdMb54Q Bookmarked to use as my just-in-time curriculum for the afternoon coffee break...
  - ianai 4y ago
    Partially, those imaginary numbers offer ways to embed more geometric/topological reasoning into the number system than being imaginary. Like how adding the point at infinity makes the 2d plane more like a sphere.
    - lizknope 4y ago
      I'm an electrical engineer so we did a lot of math in circuits class where we used Euler's identity. We did stuff like e^(i * theta) = cos(theta) + i * sin(theta) Then our teacher said "that part is imaginary" and just crossed out the isin(theta) part so e^(itheta) = cos(theta) and use that to solve circuit problems. I kind of laughed but some other people have told me that it is more of a math shortcut rather than "i" having a real world impact. As I said I have taken a quantum physics class but I'm not a physicist. But it seems like "i" really is necessary for quantum mechanics and manifests itself in the real world. https://www.quantamagazine.org/imaginary-numbers-may-be-essential-for-describing-reality-20210303/ https://www.quantamagazine.org/imaginary-numbers-may-be-esse...
- koprulusector 4y ago
  Math isn’t a tool, and man didn’t make it. Math is the secret language of the universe that describes existence. Man is a toddler who’s beginning to pick up and use the language.
  - Kinrany 4y ago
    A language is a tool for communication. Who is using math to communicate, except for people?
  - ksd482 4y ago
    I disagree. Math is a language completely invented by humans to describe reality as they perceive it. It's an extremely sophisticated language, beautiful even but a language nonetheless. For e.g., an apple existed before the word or even language was invented. Similarly, the physical reality around us existed without the language to describe existed.
    - mr_mitm 4y ago
      There are entire fields of math that have nothing to do with describing reality. As an example, category theory is completely detached from reality. You're still thinking of math as a tool, but it is much more than that.
      - qsort 4y ago
        > category theory is completely detached from reality One could very easily object to that, computable structure theory sounds very concrete indeed. But even taking that statement at face value, you are not really disproving that math is a tool. It is a tool for describing formal thinking. For example, a forgetful functor from Grp to Set is the formalization of the idea that you may take the underlying set from a group "forgetting" about the algebraic structure. There is value in having language and tools to describe that kind of thinking, even if it is very abstract. The choice of which mathematical objects to study is basically arbitrary. This is not to say that math is useless, quite the opposite. It's probably the most consequential human invention after writing, but I'm very skeptical of trying to assign metaphysical meanings to it.
- the_only_law 4y ago
  I’ve managed to catch glimpses of that sort of thing, but way too late.
- nyc111 4y ago
  I don't think mathematics is a tool invented by humans. We see that who ever designed the shell of the snail knew about math and used it. The more interesting question for me is, did the designer of the snail shell use the mathematics like humans are using mathematics, or if that designer also invented the rules of mathematics.
- dTal 4y ago
  When Benoit Mandelbrot wrote a simple computer program to map z -> z^2 +c and to his amazement discovered something intricate and beautiful that nobody knew was there, where was the "tool"? Why should such an abstract structure, with no apparent practical application, speak so profoundly to the cortex of a primate evolved for hunting animals and gathering fruit?
  - injidup 4y ago
    Because "shiny"
    - dTal 4y ago
      You have moved the question into the word "shiny". What is "shiny"?
  - ahelwer 4y ago
    I like to think of it this way. Compare mathematical axioms and rules of logic to the starting configuration and rules of a board game, let's say chess. All moves in chess follow from the very simple rules of chess. Every possible board position, no matter how complicated, can be reduced to iterated applications of the simple chess rules. It can result in beautiful play that exhibits extraordinary human ingenuity reaching (arguably) the same heights as any art form! Yet, we don't think the rules of chess are a product of the divine simply because we can see something transcendent from their application. Simple rules can lead to complicated and beautiful behavior. Doesn't mean anything beyond that.
  - lizknope 4y ago
    In high school we used Newton's method to solve polynomials. Then we got an equation and just a slight change in the starting point gave a different solution. We started plotting everything on graph paper. Then we wrote a program in MathCAD to calculate the numbers for us and plot it. Then we wrote a program to plot the solutions. Then we added color to the program. It was then that our teacher told us that we had created a fractal. We then played around with fractint, learned about some real world cases of fractals, a little about chaos theory and weather prediction. I loved it all. The best part was rather than our teacher just saying "here is a fractal" we "discovered" it on our own each day over the course of a week. https://en.wikipedia.org/wiki/Newton_fractal https://en.wikipedia.org/wiki/Newton_fractal https://www.fractint.org/ https://www.fractint.org/
- WastingMyTime89 4y ago
  I don’t think describing mathematics as a tool do them any justice. Mathematics are an endeavour in and of themselves. It’s just an interesting subject. I think the surprising part of mathematics, where people are tempted to see the divine at least since the Pythagorean, is in how much unexpected order emerge from so few rules. Small axiomatic systems give rise to very advanced theorems. Abstractions which looked very different join in unexpected and perplexing way. I think that where the beauty of mathematics lie: they reveal connections.
- vouaobrasil 4y ago
  PhD in math here...I disagree that it is just a tool. I don't really give two cents about the usefulness of math. I only studied it as an art because I appreciate its artistic beauty. For me, math is only interesting as an art, and practical problems aren't any motivation for me in that regard.
  - ukj 4y ago
    Sounds like you’ve found (psychological?) utility in aesthetics and beauty…
- hackerlight 4y ago
  Philosophy of mathematics is a thing. You either have very strong views on that field of study, or you don't know about it.
  - qsort 4y ago
    If you are so inclined, you may take this as a rejection of platonism on my part. Hardly an unusual position, especially among people with a CS background.
- lake_vincent 4y ago
  It's a tool and a language, just like a programming language. While I don't subscribe to any religious interpretation, I sympathize with the feeling of divinity in math because it is, by design, the language of reality. It is humanity's best attempt to define, abstract and encode what we perceive to be objective universal truths. Viz. the whole debate around whether math is "invented" or "discovered" (a debate which, IMO, is just arguing semantics for the same thing).
- vinceguidry 4y ago
  Feels more like a statement about divinity than it is about math, particularly given the scare quotes.
jp0d 4y ago
Unfortunately for me, I realised the beauty of it after graduating from university. Mathematics can be really beautiful and painful at the same time.
- zaik 4y ago
  It's painful while you're doing it.
  - Cthulhu_ 4y ago
    I can visualize a meme, "thinking about math" and "actually doing math"
    - jp0d 4y ago
      hahaha.. Put it on one of the latest AI algorithms and generate a meme! :D
- criddell 4y ago
  Same here. The way it was taught did not resonate at all with me at the time. I could memorize enough to pass the exam and then it would float away because it wasn't anchored to anything. I need examples and applications and concreteness and that way of looking at seemed to be discouraged. The professor was far more interested in teaching proofs.
necovek 4y ago
Mathematics, in all its "purity" and "abstractness", is, to me, always a reminder of the limit of the human brain. How we've got to have a starting point of assumptions and concepts to be able to construct any theory, or really, to think about anything: we can never really start from nothing and fully understand things. In that sense, instead of divine, it's painfully humane to me! And that's why I respect and love it even more: it's the nicest thing our constrained brains could come up with :)
- mr_mitm 4y ago
  Funny, I see math as a tool to overcome our shortcomings. Most fundamental concepts like spacetime or quantum mechanics are absolutely mind bending and our limited brains are hopeless at understanding them. Math enables us to break down and translate hugely complex phenomena into things we can understand.
- ivansavz 4y ago
  > math is the nicest thing our constrained brains have come up with WOW, What a nice idea and well worded! I've already paraphrased you statement to make it more slogany, but I think it can be improved further if we s/thing/structure/ or some other more precise word like that. Or to reuse your description from earlier: > math is the nicest structure our constrained brains have been able to construct This is a nice one-phrase explainer of what math is. So many people I know confuse the idea of math (abstract thinking and modelling) with arithmetic, so I'm always on a lookout for ways we could re-associate the pointer "math" to it's correct meaning.
Gehinnn 4y ago
> [...] mathematics can lead one to the conclusion that behind the veil of life there is a structure and an order. Rule 30, the collatz conjecture and the monster group let me to the conclusion that even math is chaotic.
- leto_ii 4y ago
  > that even math is chaotic. Math chaos is actually a kind of order, just not the kind that is intuitively easy to grasp by humans.
  - NegativeK 4y ago
    That's a different definition of chaos than mathematical chaos.
guerby 4y ago
A striking fact for me is that we found historical evidences of the mathematical concept parabola 3d or 4th century BC: https://en.wikipedia.org/wiki/Parabola#History https://en.wikipedia.org/wiki/Parabola#History But people started noticing that throwing a ball made a parabola if you measure the trajectory more than two thousand years later, in 17th century (1): https://en.wikipedia.org/wiki/Parabola#In_the_physical_world https://en.wikipedia.org/wiki/Parabola#In_the_physical_world Discovering the power of measuring what happens in the physical world and going back to mathematics! (1) May be some discovered that earlier but we have no traces yet.
- ahelwer 4y ago
  Only tangentially related to your comment but it was recently pointed out to me that the trajectory of thrown objects is really an ellipse, not a parabola, since they're in a (very short!) orbit while airborne. Of course parabolas are a very, very good approximation over short distances and much easier to deal with mathematically than ellipses. Still, wish they'd mentioned this as an aside in physics classes.
  - injidup 4y ago
    A parabolic trajectory would occur if the earth was flat.
    - guerby 4y ago
      Context is 17th century so ball is not going very far and parabola will be perfect with the measuring technology of the time I think :)
    - chongli 4y ago
      Yes and since the earth is large enough to be locally flat a parabola is a great approximation of what happens when we throw a ball.
    - ahelwer 4y ago
      I think it would also require that gravity doesn't obey the inverse-square law, so stays constant in magnitude throughout the entire vertical component of the trajectory. An infinitely-flat earth would also mean the force vector points straight down, instead of shifting slightly to point to the earth's center of mass across the horizontal component of the trajectory.
ThinkBeat 4y ago
For me math represents a peek into the Platonic plane.
sylware 4y ago
I was able to code real-life assembly in my teens. But I felt something was missing, that's why I went the engineering path: maths, physics and chemistry. I still believe it was the right choice. This is were the real stuff happens. 99% of programming out there is high-school level, the remaining 1%, you need at least a college degree, because it is real maths (not high-school calculus). That said, based on my experience, be very careful when using the word "mathematics", because in the mind of tons of ppl, "mathematics" = calculus...
- gausswho 4y ago
  Thanks for sharing this. I went through a similar arc, substituting Assembly with late 90's html/js/css. I initially enjoyed it but ultimately found it all a pretty shallow pool. Lacking the sublime. So in college I cold-shouldered the CS department in favor of the Math. I ended up deeply enjoying abstract algebra and cryptology. I then left college with a larger student debt payment than rent. I didn't see a financially stable path with the tools I'd honed. In desperation I took a basic web shop job. A software engineering career transpired. A couple decades later, on strange days, I sometimes wonder how I ended up stuck in the shallow pool, or if that's even such a bad thing.
  - sylware 4y ago
    if that path taught me something: when dealing with the web, resist the Rube Goldberg Machine temptation: noscript/basic (x)html only!
Decabytes 4y ago
It seems to me you fall into two camps about "Mathematics". 1. You consider Mathematics to be the symbols, words, and equations I.E the language which we used to describe the fundamentals of reality. Therefore it's human made and Mathematics are a tool. 2. Mathematics are the fundamental concepts of our reality which we discover and then record using a notation that makes it easy to express those concepts. Therefore humans don't "make" Mathematics but discover it. I fall into the second camp.
kkwteh 4y ago
A concrete example of how math unveils a hidden structure and order is how any rotated spherical harmonic can be expressed as a linear combination of other (unrotated) spherical harmonics within the same family. It is one of many examples of an amazing fact so wild, that no one could have dreamed of it before it was discovered. I chose this as an example, because it can be expressed purely in terms of geometry, is straightforward to understand, and yet stunningly beautiful (at least to me). Here's a link to some visualisations: https://vinequai.com/sphericalharmonics https://vinequai.com/sphericalharmonics
NegativeK 4y ago
Hopefully the author is never told of how there are "more" incomputable than computable reals, that our system of mathematics can never be both slightly useful and complete, and the whole host of other ignorances that are spread out in the field of mathematics that humanity can never even pretend to overcome. For me, those gaps make math even more beautiful; for others, I suppose they could destroy the order they find or present a gap in which you could place god.

130 points by benbreen 4y ago

I hear this a lot and I'm happy for the author, but the "divinity" of math is not something that has ever resonated with me, not even when I was younger. It's a tool. It's by far the most powerful tool man has ever made, because it's a tool for your mind as opposed to a tool for your hands, but it still is a tool.

bambataa 4y ago
I think people use different terms imbued by their own value system to describe the same thing. Sometimes people talk about code being “beautiful”. I’ve never felt beauty from code, but I’ve felt pleasant satisfaction with well-designed code. Maybe just the same thing?
- c0mptonFP 4y ago
  That's because these things elicit an emotional response, and people all react differently. Also, describing emotions is difficult. When I see an instance of good problem solving, I feel a rush of happiness. It sometimes makes me want to physically get up from my desk and cheer as if I was watching a sports event. All I can think of is "this code/solution is fucking awesome".
- vsareto 4y ago
  You're going to struggle to find beauty in some areas (like business software and a lot of other tooling). The subject matters as much as its expression. The same is true with art - you're not likely to find paintings of burning tire mountains in the Louvre. But software has another kicker like math in that you'd likely need to be familiar with the subject to know the hard problems, present or previous, and why a solution is elegant or beautiful.
auggierose 4y ago
It's both a toy and a tool. A tool for seeing the world as it is, beyond your immediate mental and physical capabilities. If you believe in god or something similar, it makes sense to see something divine in math.
- PartiallyTyped 4y ago
  What people find as "divine" is the "unreasonable" and "unexpected" effectiveness of mathematics in describing complexity and complex phenomena abstractly. Part of me thinks that we may be very close to reaching what the brain of the average mathematician can comprehend and grasp. After all, not all of us are Gausses, Newtons and Galois.
  - auggierose 4y ago
    Math is not limited to what your brain can come up with on its own. The most important part of math, in my opinion, is that other people (or, other entities, if we include AI), can come up with insights (because of their superior brain power and/or expertise) you would never be capable of conceiving yourself, and package them up in a nice theory for you.
    - PartiallyTyped 4y ago
      While I do agree that mathematics are not limited by what either of us can conceive, there is a fundamental barrier in understanding that one may not be able to process due to sheer volume. A trivial example here, though a bit absurd, is 15000 page long proofs, or machine generated proofs that span hundreds of pages. It is one thing to conceive of something new, and another to fully grasp all that we have built as a species. Certain pieces of work required hundreds of mathematicians to verify, which to me suggests that we have hit the limit of what our brains can do.
      - auggierose 4y ago
        The proof of the Kepler conjecture has been formalised, because the original proof was just too large and contained computer calculations that mathematicians could obviously not verify just with their brains in any amount of time. So what? The high-level structure of the proof is quite understandable, and the statement of the Kepler conjecture is very easy to understand as well. What I am saying is that understanding something does not necessarily mean you understand it in all of its details down to the lowest level. And machines can help us understand the things we need to understand. But even with machines we will never understand everything in all of its detail down to the lowest level. We won't even know all the theorems of something as simple as Peano arithmetic. What you can do is not limited to what you can do without machine help, and just with your brain. I would say what we can do now will pale to what we will be able to do in the future, but of course this is speculation.
linspace 4y ago
It's a tool for you. Being also a tool is a happy accident.
- yamtaddle 4y ago
  Accident? We originally constructed math specifically so it would relate to reality and be a useful tool. Even "pure math" topics have trouble shedding that legacy & foundation completely, which is why I think it's weird that people are surprised when, sometimes, those fields still end up relating to reality in ways we hadn't expected. It'd be surprising if they didn't!
aaaaaaaaaaab 4y ago
Take the natural numbers. They were invented for counting things, right? One sheep, two sheep, three sheep, simple as. Yet there’s an immense complexity and structure in natural numbers, starting with primes, and ending in algebraic- or analytic number theory! Why is it that this thing we use for counting objects has such rich structure? Where does it come from? Why does it have to be this way? It certainly feels something divine to me. It certainly feels as something that humans didn’t invent, rather was always “out there” to be discovered. Physical laws (which are mostly just OK approximations) feel much more arbitrary and down-to-Earth compared to mathematics.
- Koshkin 4y ago
  > invented The words for numbers were indeed “invented,” but the numbers themselves, like much of math, was discovered.
  - ivanbakel 4y ago
    That's a philosophical view, and I don't think there's any basis for stating it as fact. One of the great debates of mathematical ontology is whether mathematical concepts are discovered or invented.
    - Kinrany 4y ago
      Like most great philosophical debates, it probably comes from humans being bad at understanding their own language.
      - Koshkin 4y ago
        Indeed, the word ‘invented’ is often abused by being used in place of ‘discovered’ or ‘created.’ (No, Faraday did not “invent” the laws of electromagnetism; nor did Linus Torvalds “invent” Linux.)
Koshkin 4y ago
> It's a tool. It’s also an art, and a science, and as such it is often seen as beautiful. Beauty, of course, is in the eyes of the beholder, but people tend to ascribe it the objective status, sometimes that of a divine nature.
- gausswho 4y ago
  Your response made me think of a fundamental bugbear I have with English, and maybe other languages: the word 'is' has a tendency to imply exclusivity unless coddled by further explanation. If someone says something is a tool, that doesn't technically imply there is nothing else it 'is'. But in colloquial use, that's sometimes the intent of the author and sometimes not.
lkozma 4y ago
"Seeing an astronomer using a telescope to observe a galaxy, no-one will confuse the telescope with the galaxy. Mathematics differs from science in that there is no clear distinction between the tools and the objects of study." -- D.Aldous
- anon_123g987 4y ago
  I don't think there's a difference here between Science and Mathematics. Scientists also study their tools (e.g. telescopes are studied by Optics), and use the objects of their studies as tools (e.g. galaxies can be used as telescopes, see: https://en.wikipedia.org/wiki/Gravitational_lens https://en.wikipedia.org/wiki/Gravitational_lens). Edited to add: The author of the quote tried to find an example that evokes vivid imagery, and also very extreme: it's obviously absurd to confuse a telescope with a galaxy, thinks the layman. While this turns out to be literally false, the point of the quote may still stand. The discipline that studies the real tools of Science is not Science itself, but the Philosophy of Science. This shouldn't be like this, and it wasn't until lately. Philosophy and Science used to be one and the same, but around a hundred years ago they separated, and this hurt both tremendously, in my opinion.
leto_ii 4y ago
I get that, but I have to say I also see merit in the position that maths is discovered, not invented. I'm currently doing some self-study into more rigorous mathematics and I do often get the feeling that some of the structures are just there, not human inventions made for human purposes. I have recently been quite interested in the Cantor set, for example - hardly a useful tool...
- LanceH 4y ago
  The axioms are chosen. The theorems follow from the chosen axioms. Which axioms to choose which are interesting or applicable are discovered.
lizknope 4y ago
How much math have you learned? When I first saw Euler's Identity I kind of laughed and said WTF? We "discovered" Euler's number while calculating compound interest continously. We knew pi from grade school. Learned about square root of -1 around 9th grade. But when I learned Euler's Identity I didn't even understand what it was trying to say. Then someone took an hour to derive it through a Taylor series and my mind was blown. I still find it amazing. Then in quantum physics class we learned about the wave function and Schrödinger equation. Those "imaginary" numbers influence the real world. I'm not religious but when someone sees some kind of divine beauty in math I kind of feel it too.
- yw3410 4y ago
  If you haven't done it, then the statistical proofs for the gas equations and the proofs for Maxwell's equations also rank up there for me as well.
  - ivansavz 4y ago
    Thanks for the recommendations, here are the three first links that come up for the "statistical proof" of the gas equations: https://www.one-tab.com/page/UhuMmBwCTma12IwYdMb54Q https://www.one-tab.com/page/UhuMmBwCTma12IwYdMb54Q Bookmarked to use as my just-in-time curriculum for the afternoon coffee break...
- ianai 4y ago
  Partially, those imaginary numbers offer ways to embed more geometric/topological reasoning into the number system than being imaginary. Like how adding the point at infinity makes the 2d plane more like a sphere.
  - lizknope 4y ago
    I'm an electrical engineer so we did a lot of math in circuits class where we used Euler's identity. We did stuff like e^(i * theta) = cos(theta) + i * sin(theta) Then our teacher said "that part is imaginary" and just crossed out the isin(theta) part so e^(itheta) = cos(theta) and use that to solve circuit problems. I kind of laughed but some other people have told me that it is more of a math shortcut rather than "i" having a real world impact. As I said I have taken a quantum physics class but I'm not a physicist. But it seems like "i" really is necessary for quantum mechanics and manifests itself in the real world. https://www.quantamagazine.org/imaginary-numbers-may-be-essential-for-describing-reality-20210303/ https://www.quantamagazine.org/imaginary-numbers-may-be-esse...
koprulusector 4y ago
Math isn’t a tool, and man didn’t make it. Math is the secret language of the universe that describes existence. Man is a toddler who’s beginning to pick up and use the language.
- Kinrany 4y ago
  A language is a tool for communication. Who is using math to communicate, except for people?
- ksd482 4y ago
  I disagree. Math is a language completely invented by humans to describe reality as they perceive it. It's an extremely sophisticated language, beautiful even but a language nonetheless. For e.g., an apple existed before the word or even language was invented. Similarly, the physical reality around us existed without the language to describe existed.
  - mr_mitm 4y ago
    There are entire fields of math that have nothing to do with describing reality. As an example, category theory is completely detached from reality. You're still thinking of math as a tool, but it is much more than that.
    - qsort 4y ago
      > category theory is completely detached from reality One could very easily object to that, computable structure theory sounds very concrete indeed. But even taking that statement at face value, you are not really disproving that math is a tool. It is a tool for describing formal thinking. For example, a forgetful functor from Grp to Set is the formalization of the idea that you may take the underlying set from a group "forgetting" about the algebraic structure. There is value in having language and tools to describe that kind of thinking, even if it is very abstract. The choice of which mathematical objects to study is basically arbitrary. This is not to say that math is useless, quite the opposite. It's probably the most consequential human invention after writing, but I'm very skeptical of trying to assign metaphysical meanings to it.
the_only_law 4y ago
I’ve managed to catch glimpses of that sort of thing, but way too late.
nyc111 4y ago
I don't think mathematics is a tool invented by humans. We see that who ever designed the shell of the snail knew about math and used it. The more interesting question for me is, did the designer of the snail shell use the mathematics like humans are using mathematics, or if that designer also invented the rules of mathematics.
dTal 4y ago
When Benoit Mandelbrot wrote a simple computer program to map z -> z^2 +c and to his amazement discovered something intricate and beautiful that nobody knew was there, where was the "tool"? Why should such an abstract structure, with no apparent practical application, speak so profoundly to the cortex of a primate evolved for hunting animals and gathering fruit?
- injidup 4y ago
  Because "shiny"
  - dTal 4y ago
    You have moved the question into the word "shiny". What is "shiny"?
- ahelwer 4y ago
  I like to think of it this way. Compare mathematical axioms and rules of logic to the starting configuration and rules of a board game, let's say chess. All moves in chess follow from the very simple rules of chess. Every possible board position, no matter how complicated, can be reduced to iterated applications of the simple chess rules. It can result in beautiful play that exhibits extraordinary human ingenuity reaching (arguably) the same heights as any art form! Yet, we don't think the rules of chess are a product of the divine simply because we can see something transcendent from their application. Simple rules can lead to complicated and beautiful behavior. Doesn't mean anything beyond that.
- lizknope 4y ago
  In high school we used Newton's method to solve polynomials. Then we got an equation and just a slight change in the starting point gave a different solution. We started plotting everything on graph paper. Then we wrote a program in MathCAD to calculate the numbers for us and plot it. Then we wrote a program to plot the solutions. Then we added color to the program. It was then that our teacher told us that we had created a fractal. We then played around with fractint, learned about some real world cases of fractals, a little about chaos theory and weather prediction. I loved it all. The best part was rather than our teacher just saying "here is a fractal" we "discovered" it on our own each day over the course of a week. https://en.wikipedia.org/wiki/Newton_fractal https://en.wikipedia.org/wiki/Newton_fractal https://www.fractint.org/ https://www.fractint.org/
WastingMyTime89 4y ago
I don’t think describing mathematics as a tool do them any justice. Mathematics are an endeavour in and of themselves. It’s just an interesting subject. I think the surprising part of mathematics, where people are tempted to see the divine at least since the Pythagorean, is in how much unexpected order emerge from so few rules. Small axiomatic systems give rise to very advanced theorems. Abstractions which looked very different join in unexpected and perplexing way. I think that where the beauty of mathematics lie: they reveal connections.
vouaobrasil 4y ago
PhD in math here...I disagree that it is just a tool. I don't really give two cents about the usefulness of math. I only studied it as an art because I appreciate its artistic beauty. For me, math is only interesting as an art, and practical problems aren't any motivation for me in that regard.
- ukj 4y ago
  Sounds like you’ve found (psychological?) utility in aesthetics and beauty…
hackerlight 4y ago
Philosophy of mathematics is a thing. You either have very strong views on that field of study, or you don't know about it.
- qsort 4y ago
  If you are so inclined, you may take this as a rejection of platonism on my part. Hardly an unusual position, especially among people with a CS background.
lake_vincent 4y ago
It's a tool and a language, just like a programming language. While I don't subscribe to any religious interpretation, I sympathize with the feeling of divinity in math because it is, by design, the language of reality. It is humanity's best attempt to define, abstract and encode what we perceive to be objective universal truths. Viz. the whole debate around whether math is "invented" or "discovered" (a debate which, IMO, is just arguing semantics for the same thing).
vinceguidry 4y ago
Feels more like a statement about divinity than it is about math, particularly given the scare quotes.

jp0d 4y ago

Unfortunately for me, I realised the beauty of it after graduating from university. Mathematics can be really beautiful and painful at the same time.

zaik 4y ago
It's painful while you're doing it.
- Cthulhu_ 4y ago
  I can visualize a meme, "thinking about math" and "actually doing math"
  - jp0d 4y ago
    hahaha.. Put it on one of the latest AI algorithms and generate a meme! :D
criddell 4y ago
Same here. The way it was taught did not resonate at all with me at the time. I could memorize enough to pass the exam and then it would float away because it wasn't anchored to anything. I need examples and applications and concreteness and that way of looking at seemed to be discouraged. The professor was far more interested in teaching proofs.

necovek 4y ago

Mathematics, in all its "purity" and "abstractness", is, to me, always a reminder of the limit of the human brain. How we've got to have a starting point of assumptions and concepts to be able to construct any theory, or really, to think about anything: we can never really start from nothing and fully understand things. In that sense, instead of divine, it's painfully humane to me! And that's why I respect and love it even more: it's the nicest thing our constrained brains could come up with :)

mr_mitm 4y ago
Funny, I see math as a tool to overcome our shortcomings. Most fundamental concepts like spacetime or quantum mechanics are absolutely mind bending and our limited brains are hopeless at understanding them. Math enables us to break down and translate hugely complex phenomena into things we can understand.
ivansavz 4y ago
> math is the nicest thing our constrained brains have come up with WOW, What a nice idea and well worded! I've already paraphrased you statement to make it more slogany, but I think it can be improved further if we s/thing/structure/ or some other more precise word like that. Or to reuse your description from earlier: > math is the nicest structure our constrained brains have been able to construct This is a nice one-phrase explainer of what math is. So many people I know confuse the idea of math (abstract thinking and modelling) with arithmetic, so I'm always on a lookout for ways we could re-associate the pointer "math" to it's correct meaning.

Gehinnn 4y ago

> [...] mathematics can lead one to the conclusion that behind the veil of life there is a structure and an order. Rule 30, the collatz conjecture and the monster group let me to the conclusion that even math is chaotic.

leto_ii 4y ago
> that even math is chaotic. Math chaos is actually a kind of order, just not the kind that is intuitively easy to grasp by humans.
- NegativeK 4y ago
  That's a different definition of chaos than mathematical chaos.

guerby 4y ago

A striking fact for me is that we found historical evidences of the mathematical concept parabola 3d or 4th century BC: https://en.wikipedia.org/wiki/Parabola#History https://en.wikipedia.org/wiki/Parabola#History But people started noticing that throwing a ball made a parabola if you measure the trajectory more than two thousand years later, in 17th century (1): https://en.wikipedia.org/wiki/Parabola#In_the_physical_world https://en.wikipedia.org/wiki/Parabola#In_the_physical_world Discovering the power of measuring what happens in the physical world and going back to mathematics! (1) May be some discovered that earlier but we have no traces yet.

ahelwer 4y ago
Only tangentially related to your comment but it was recently pointed out to me that the trajectory of thrown objects is really an ellipse, not a parabola, since they're in a (very short!) orbit while airborne. Of course parabolas are a very, very good approximation over short distances and much easier to deal with mathematically than ellipses. Still, wish they'd mentioned this as an aside in physics classes.
- injidup 4y ago
  A parabolic trajectory would occur if the earth was flat.
  - guerby 4y ago
    Context is 17th century so ball is not going very far and parabola will be perfect with the measuring technology of the time I think :)
  - chongli 4y ago
    Yes and since the earth is large enough to be locally flat a parabola is a great approximation of what happens when we throw a ball.
  - ahelwer 4y ago
    I think it would also require that gravity doesn't obey the inverse-square law, so stays constant in magnitude throughout the entire vertical component of the trajectory. An infinitely-flat earth would also mean the force vector points straight down, instead of shifting slightly to point to the earth's center of mass across the horizontal component of the trajectory.

ThinkBeat 4y ago

For me math represents a peek into the Platonic plane.

sylware 4y ago

I was able to code real-life assembly in my teens. But I felt something was missing, that's why I went the engineering path: maths, physics and chemistry. I still believe it was the right choice. This is were the real stuff happens. 99% of programming out there is high-school level, the remaining 1%, you need at least a college degree, because it is real maths (not high-school calculus). That said, based on my experience, be very careful when using the word "mathematics", because in the mind of tons of ppl, "mathematics" = calculus...

gausswho 4y ago
Thanks for sharing this. I went through a similar arc, substituting Assembly with late 90's html/js/css. I initially enjoyed it but ultimately found it all a pretty shallow pool. Lacking the sublime. So in college I cold-shouldered the CS department in favor of the Math. I ended up deeply enjoying abstract algebra and cryptology. I then left college with a larger student debt payment than rent. I didn't see a financially stable path with the tools I'd honed. In desperation I took a basic web shop job. A software engineering career transpired. A couple decades later, on strange days, I sometimes wonder how I ended up stuck in the shallow pool, or if that's even such a bad thing.
- sylware 4y ago
  if that path taught me something: when dealing with the web, resist the Rube Goldberg Machine temptation: noscript/basic (x)html only!

Decabytes 4y ago

It seems to me you fall into two camps about "Mathematics". 1. You consider Mathematics to be the symbols, words, and equations I.E the language which we used to describe the fundamentals of reality. Therefore it's human made and Mathematics are a tool. 2. Mathematics are the fundamental concepts of our reality which we discover and then record using a notation that makes it easy to express those concepts. Therefore humans don't "make" Mathematics but discover it. I fall into the second camp.

kkwteh 4y ago

A concrete example of how math unveils a hidden structure and order is how any rotated spherical harmonic can be expressed as a linear combination of other (unrotated) spherical harmonics within the same family. It is one of many examples of an amazing fact so wild, that no one could have dreamed of it before it was discovered. I chose this as an example, because it can be expressed purely in terms of geometry, is straightforward to understand, and yet stunningly beautiful (at least to me). Here's a link to some visualisations: https://vinequai.com/sphericalharmonics https://vinequai.com/sphericalharmonics

NegativeK 4y ago

Hopefully the author is never told of how there are "more" incomputable than computable reals, that our system of mathematics can never be both slightly useful and complete, and the whole host of other ignorances that are spread out in the field of mathematics that humanity can never even pretend to overcome. For me, those gaps make math even more beautiful; for others, I suppose they could destroy the order they find or present a gap in which you could place god.