How Mathematics Changed Me
newyorker.com
newyorker.com
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.
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.
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.
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.
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.
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.
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.
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.
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…
Simple rules can lead to complicated and beautiful behavior. Doesn't mean anything beyond that.
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.
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 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.
Theorems, constructs and proofs existed in "Math Space" before they were discovered by humans.
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.
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.
> 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.
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.
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.
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.
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.
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.
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.
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).
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.
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.
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.
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.