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.
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?
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".
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.
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.
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.
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.
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.
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.
I take the ESR view here: math is only unreasonably effective to describe your field because you have chosen to study a field that is effectively described by math.
For example, consistent science-like theories of history are very suspicious to say the least.
By and large, nature doesn't give a damn. E.g. the most natural and mathematically elegant way to describe space is euclidean geometry, which happens to be a "wrong" model.
On some level, model selection is influenced by aesthetic considerations.
With memory, seeing divinity in math probably requires active contemplation, since seeing simple formulas and geometric shapes repeatedly gets old fast
> Many observers report they derive pleasure from discovering simple but novel patterns [...] The observer's learning process causes a reduction of the subjective complexity of the data, yielding a temporarily high derivative of subjective beauty: a temporarily steep learning curve.
> Why are some musical pieces more interesting or aesthetically rewarding than others? Not the one the listener (composer) just heard (played) fifty times in a row without any noticable change. It became too subjectively predictable in the process. Not the weird one with completely unfamiliar rhythm and tonality. It seems too irregular and contain too much arbitrariness and subjective noise. The observer (creator) of the data is interested in melodies that are unfamiliar enough to contain somewhat unexpected harmonies or beats etc., but familiar enough to allow for quickly recognizing the presence of a new learnable regularity or compressibility in the sound stream: a novel pattern! Sure, it will get boring over time, but not yet.
Seeing a Tetrahedron, Euler's identity or a Klein bottle for the first time is much better than the 50th time, because you remember having seen it that many times by then and are not surprised by their meaning and properties anymore. Since you remember, having it remain meaningful requires actively exploring more properties of the object and its relations to other objects
Some simple objects like the yin and yang symbol may remain more easily meaningful because they can have so many interpretations, compare this to say, a simple line, which is so ubiquitous its meanings are "too" unrestricted. Something like a Lie group may be too complex and too hard to understand for most to even consider it divine
That's a pretty weird statement, even if it's true when looking at any math textbook (1% is statements of things, 99% is proving those hold).
Mathematics is a scientific theory built on the most rigorous principles human minds could come up with, while striving for simplicity in the basic assumptions. The entire process of "math building", which permeats all the proofs, is what's useful in plenty other areas of life and science.
I.e. mathematical induction, the core tenet of plenty a proof, is basically function recursion in software development, and underlies all sorts of delegation in any complex undertaking (if I order a part for a machine from someone building those parts, I can trust it's the same as the one I am replacing).
Driving something down to a contradiction is used in many a decision everywhere.
And sure, results of applying these mathematical processes rigorously ensures that we can trust we can apply mathematical results (theorems and such) widely: it's not uncommon to apply processes from the proofs when using a theorem either.
And this is what ensures we can use the same process to add up 3 sheep and 2 sheep or 4 apples and 1 orange to get 5 sheep or 5 fruits. Or to calculate surface area of a circle. Or use integrals to get volumes of complex shapes. Etc.
Mathematics has a language, but it has a bunch of conclusions using said language too.
I don't think you can argue how mathematical theorems are not "scientific theory": the language part comes from all the definitions and axioms, but there is a whole theory built on top of it.
On your other statement, I never said mathematics is the theory of science, but rather, a scientific theory: so of course we had science before we formalized and names mathematics mathematics. It's only loosely based on reality (as a driver for many — but not all — definitions and concepts), just like other scientific theories are based on our perceived reality.
Well yes, doing mathematics consists of trying to find out something new that you didn't know before. You might end up staring at a Klein bottle 50 times or more though, either just for pleasure, or for finding out something new about it.
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!
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.
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.
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.)
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.
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.
"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
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).
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.
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...
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.
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.
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.
The imaginary unit shows up in both quantum mechanics and EE as a way to embed i*i = -1 in the equations. Any differential equation that involves it can be written without it by separating real and imaginary parts into different variables. Furthermore, if you let "one" be represented by the 2x2 identity matrix, the matrix [[0,1],[-1,0]] can act like i in every respect, because that matrix squares to -1 times the identity. In fact, even -i can stand in for i. The "nouns" of math might be real, but whether they are is a metaphysical question; all we observe in science are what you might call the verbs of math.
FWIW, the "crossing out the imaginary part" works like this: if the system of equations is linear and doesn't have complex coefficients, every solution has a counterpart solution that is its complex conjugate, and you can add the conjugate (remember, it's linear) to get another, real-valued, solution. You can do that with time-invariant (definite-energy) solutions to the Schrodinger equation as well, because the energy eigenvalue equation doesn't have complex coefficients.
Descartes called them "imaginary" numbers because he didn't understand them well and didn't like them. He thought they were ugly and therefore false.
If sine, cosine, and tangent had instead been named divine, codivine and tantric we'd be having ridiculous conversations about how magical they are too.
What do you think the universe would look like if things didn't fit together? Sit with that thought for a little while, really let it sink in. Oh, hey, it wouldn't be a universe.
“This is rather as if you imagine a puddle waking up one morning and thinking, 'This is an interesting world I find myself in — an interesting hole I find myself in — fits me rather neatly, doesn't it? In fact it fits me staggeringly well, must have been made to have me in it!' This is such a powerful idea that as the sun rises in the sky and the air heats up and as, gradually, the puddle gets smaller and smaller, frantically hanging on to the notion that everything's going to be alright, because this world was meant to have him in it, was built to have him in it; so the moment he disappears catches him rather by surprise. I think this may be something we need to be on the watch out for.”
-Douglas Adams
Our sense of beauty is formed by the universe. It is not remarkable that the universe is made of systems that seem elegant to the beings that live in it.
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.
> 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.
> but I'm very skeptical of trying to assign metaphysical meanings to it.
Let's leave aside talk of God and such things and think instead of how many of the worlds more prolific mathematicians have talked of their work and their "discoveries".
They feel they explore a landscape outside of that which you can kick and "discover" things that were always true, before they came and after they die.
Many are not approaching the field as toolmakers and engineers seeking a practical means to assist in building want they want but as explorers seeking new ground and hitherto unknown objects and their relationships.
I am not. My comment doesn’t encapsulate the entire idea of mathematics. And, that’s kind of my point.
I agree that math is much more than a tool. Much much more.
Regardless, it is a language. Not just to describe reality, but also abstract ideas. And I believe this encapsulates the entirety of what math is, unless I left out something major.
Category theory has actually had a big recent boom in quantum physics.
Hardy's beautiful number theory math he thought was pure and impractical ended up used heavily in encrypting payment details for online pizza delivery etc.
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.
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?
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.
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.
When you ask “why?” questions you always get religious/philosophical answers. I cannot tell you why stuff amuses me, all I can tell you is that it amuses me.
For all we know the awe and profundity comes from the sense of discovery.
But what if the a sense of discovery and a sense of invention feel exactly the same?
>When you ask “why?” questions you always get religious/philosophical answers
No, that's silly. Most things have a reason. "Why" is a useful word in English outside of religion and philosophy. You only get to the end of the chain when you hit the limits of theoretical physics: https://www.youtube.com/watch?v=36GT2zI8lVA
> I cannot tell you why stuff amuses me, all I can tell you is that it amuses me.
Do not confuse your inability to answer a question with unanswerability.
Nobody is saying it isn’t useful. Sometimes it is. Sometimes it isn’t.
I am saying that in the realm of theoretical science “why?” produces only meta-theoretical (philosophical/religious) answers that may or may not be relevant to the most pertinent issue at hand: “Why are you asking ‘why’?”.
All theories have conceptual dependencies. Necessary mental existents/foundations.
Without dependencies (assumptions) the theory doesn’t work.
Why is there a Big Bang in cosmology?
Why are there elementary particles in QFT?
Why is there spacetime in GR?
The answer to all three is “pragmatic utility.”. If that isn’t the kind of answer you are looking for…
>Do not confuse your inability to answer a question with unanswerability.
Similarly, do not confuse unanswerability with your inability to answer questions.
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.
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.
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.
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).
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.
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 :)
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.
> 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.
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):
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.
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.
Parabolas and ellipses are different conic sections. Objects thrown are actually following a parabolic trajectory and not an elliptic one. It’s not an approximation. They are not in orbit.
Objects are actually in orbit when their trajectory shift from a parabola to an ellipse. If you go even faster, you get another conic section as a solution - an hyperbola - and the object escape.
I don't buy this. A thrown object is in orbit around the earth's center of mass if one were to treat it as a point mass. Since the earth is (roughly) spherical its gravitational pull is equivalent to being a point mass. The thrown object doesn't "know" that it will impact the surface of the earth and thus switch between parabola/ellipse.
Are you contesting the fact that a thrown ball's distance above the ground (y) at a particular time (t) relative to its initial velocity (v) is equal to y = vt + 0.5 * g * t ^ 2, (where g is the direction of acceleration due to gravity) which can very clearly a modeled by a parabola?
I get that the "error" is pretty low, but there are tons of equations in physics used every day that rely on the fact that the difference in some value is negligible (often things like the sine of an angle being approximately equal to the angle itself).
No but I get what they mean. If you view the Earth as a point system, you can use the general expression of gravity between two objects and you will indeed find an extremely elongated ellipse as a solution. Considering a parabola can be viewed as an ellipse with a focal point at infinity, I’m guessing the difference between the two solutions will be so small as to be negligible. They both depend on intentional inexactitudes (g is not a fixed vector, the earth is not a point system) which don’t matter at the scale. I wouldn’t personally view the elliptical solution as truer but I see how you could.
According to their link, the error at a human scale is about 1e-6 units on Earth. You need to be doing the math at velocities such that the object's apogee is over 100k feet / meters from the surface to be "off" by more than 1 unit.
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...
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.
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.
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:
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.
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[ 3.4 ms ] story [ 173 ms ] threadIt'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.
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?
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".
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.
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.
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.
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.
For example, consistent science-like theories of history are very suspicious to say the least.
By and large, nature doesn't give a damn. E.g. the most natural and mathematically elegant way to describe space is euclidean geometry, which happens to be a "wrong" model.
On some level, model selection is influenced by aesthetic considerations.
> Many observers report they derive pleasure from discovering simple but novel patterns [...] The observer's learning process causes a reduction of the subjective complexity of the data, yielding a temporarily high derivative of subjective beauty: a temporarily steep learning curve.
> Why are some musical pieces more interesting or aesthetically rewarding than others? Not the one the listener (composer) just heard (played) fifty times in a row without any noticable change. It became too subjectively predictable in the process. Not the weird one with completely unfamiliar rhythm and tonality. It seems too irregular and contain too much arbitrariness and subjective noise. The observer (creator) of the data is interested in melodies that are unfamiliar enough to contain somewhat unexpected harmonies or beats etc., but familiar enough to allow for quickly recognizing the presence of a new learnable regularity or compressibility in the sound stream: a novel pattern! Sure, it will get boring over time, but not yet.
https://people.idsia.ch/~juergen/creativity.html
Some simple objects like the yin and yang symbol may remain more easily meaningful because they can have so many interpretations, compare this to say, a simple line, which is so ubiquitous its meanings are "too" unrestricted. Something like a Lie group may be too complex and too hard to understand for most to even consider it divine
The rest of the 1% are the incidental useful parts for engineering and science.
I don't know if proofs are divine or not, but they are certainly sublime.
Mathematics is a scientific theory built on the most rigorous principles human minds could come up with, while striving for simplicity in the basic assumptions. The entire process of "math building", which permeats all the proofs, is what's useful in plenty other areas of life and science.
I.e. mathematical induction, the core tenet of plenty a proof, is basically function recursion in software development, and underlies all sorts of delegation in any complex undertaking (if I order a part for a machine from someone building those parts, I can trust it's the same as the one I am replacing).
Driving something down to a contradiction is used in many a decision everywhere.
And sure, results of applying these mathematical processes rigorously ensures that we can trust we can apply mathematical results (theorems and such) widely: it's not uncommon to apply processes from the proofs when using a theorem either.
And this is what ensures we can use the same process to add up 3 sheep and 2 sheep or 4 apples and 1 orange to get 5 sheep or 5 fruits. Or to calculate surface area of a circle. Or use integrals to get volumes of complex shapes. Etc.
You can use mathematics to describe scientific theories, but it is not necessary. (We had been doing science long before we invented math.)
Meanwhile you can use math to describe all sorts of things that aren't and will never be science.
I don't think you can argue how mathematical theorems are not "scientific theory": the language part comes from all the definitions and axioms, but there is a whole theory built on top of it.
On your other statement, I never said mathematics is the theory of science, but rather, a scientific theory: so of course we had science before we formalized and names mathematics mathematics. It's only loosely based on reality (as a driver for many — but not all — definitions and concepts), just like other scientific theories are based on our perceived reality.
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.
The words for numbers were indeed “invented,” but the numbers themselves, like much of math, was discovered.
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.
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.
I have recently been quite interested in the Cantor set, for example - hardly a useful tool...
The theorems follow from the chosen axioms.
Which axioms to choose which are interesting or applicable are discovered.
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.
Bookmarked to use as my just-in-time curriculum for the afternoon coffee break...
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-esse...
FWIW, the "crossing out the imaginary part" works like this: if the system of equations is linear and doesn't have complex coefficients, every solution has a counterpart solution that is its complex conjugate, and you can add the conjugate (remember, it's linear) to get another, real-valued, solution. You can do that with time-invariant (definite-energy) solutions to the Schrodinger equation as well, because the energy eigenvalue equation doesn't have complex coefficients.
There's plenty of things that make me say 'oh, that's neat' but it's all mundane, never sacred.
If sine, cosine, and tangent had instead been named divine, codivine and tantric we'd be having ridiculous conversations about how magical they are too.
What do you think the universe would look like if things didn't fit together? Sit with that thought for a little while, really let it sink in. Oh, hey, it wouldn't be a universe.
“This is rather as if you imagine a puddle waking up one morning and thinking, 'This is an interesting world I find myself in — an interesting hole I find myself in — fits me rather neatly, doesn't it? In fact it fits me staggeringly well, must have been made to have me in it!' This is such a powerful idea that as the sun rises in the sky and the air heats up and as, gradually, the puddle gets smaller and smaller, frantically hanging on to the notion that everything's going to be alright, because this world was meant to have him in it, was built to have him in it; so the moment he disappears catches him rather by surprise. I think this may be something we need to be on the watch out for.”
-Douglas Adams
Our sense of beauty is formed by the universe. It is not remarkable that the universe is made of systems that seem elegant to the beings that live in 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.
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.
You're still thinking of math as a tool, but it is much more than that.
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.
Let's leave aside talk of God and such things and think instead of how many of the worlds more prolific mathematicians have talked of their work and their "discoveries".
They feel they explore a landscape outside of that which you can kick and "discover" things that were always true, before they came and after they die.
Many are not approaching the field as toolmakers and engineers seeking a practical means to assist in building want they want but as explorers seeking new ground and hitherto unknown objects and their relationships.
I agree that math is much more than a tool. Much much more.
Regardless, it is a language. Not just to describe reality, but also abstract ideas. And I believe this encapsulates the entirety of what math is, unless I left out something major.
Hardy's beautiful number theory math he thought was pure and impractical ended up used heavily in encrypting payment details for online pizza delivery etc.
Theorems, constructs and proofs existed in "Math Space" before they were discovered by humans.
Simple rules can lead to complicated and beautiful behavior. Doesn't mean anything beyond that.
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://www.fractint.org/
For all we know the awe and profundity comes from the sense of discovery.
But what if the a sense of discovery and a sense of invention feel exactly the same?
No, that's silly. Most things have a reason. "Why" is a useful word in English outside of religion and philosophy. You only get to the end of the chain when you hit the limits of theoretical physics: https://www.youtube.com/watch?v=36GT2zI8lVA
> I cannot tell you why stuff amuses me, all I can tell you is that it amuses me.
Do not confuse your inability to answer a question with unanswerability.
I am saying that in the realm of theoretical science “why?” produces only meta-theoretical (philosophical/religious) answers that may or may not be relevant to the most pertinent issue at hand: “Why are you asking ‘why’?”.
All theories have conceptual dependencies. Necessary mental existents/foundations.
Without dependencies (assumptions) the theory doesn’t work.
Why is there a Big Bang in cosmology? Why are there elementary particles in QFT? Why is there spacetime in GR?
The answer to all three is “pragmatic utility.”. If that isn’t the kind of answer you are looking for…
>Do not confuse your inability to answer a question with unanswerability.
Similarly, do not confuse unanswerability with your inability to answer questions.
That is a good way to waste a lot of time…
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.
Viz. the whole debate around whether math is "invented" or "discovered" (a debate which, IMO, is just arguing semantics for the same thing).
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 :)
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.
Rule 30, the collatz conjecture and the monster group let me to the conclusion 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.
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
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.
Objects are actually in orbit when their trajectory shift from a parabola to an ellipse. If you go even faster, you get another conic section as a solution - an hyperbola - and the object escape.
You can read some of the answers here for a better idea: https://physics.stackexchange.com/q/373250
I get that the "error" is pretty low, but there are tons of equations in physics used every day that rely on the fact that the difference in some value is negligible (often things like the sine of an angle being approximately equal to the angle itself).
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...
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.
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.
https://vinequai.com/sphericalharmonics
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.