Thread by @killahertz_

yo today is depressing af so here's some 🔥 math facts to cheer you up:

the riemann zeta function is universal, which means that somewhere in it (in the complex plane) it contains any other function*

*approximate arbitrary non-vanishing holomorphic functions arbitrarily well.

for any sequence of digits (in any base) you can come up with, they are located somewhere in the expansion of pi (in that base).

for funzies, you can use this to come up with a data compression* algorithm:
github.com/philipl/pifs

*compressed data will be larger than original

noether's theorem means that any symmetry in a lagrangian creates a conservation law. what does that mean?

translational invariance --> conservation of linear momentum
rotational invariance --> conservation of angular momentum
time invariance --> conservation of energy

special relativity can be derived from first principles, assuming a couple simple axioms:

* physics is the same in all (non-accelerating) reference frames
* the speed of light is the same in all reference frames

einstein sat down and derived the rest of SR just from that

any analytic function can be represented by a power series. so by knowing a countably infinite amount of information (all the values of the derivatives of an f at a given point), you know an uncountably infinite amount of info (the value of f at ANY point)

a path integral of a holomorphic function around a closed loop is 0. around any meromorphic function, it is based on the poles inside the loop

that means integrals over entire regions are related to just their values on a boundary

this is a simple version of Stoke's Theorem

there's a natural meromorphic extension of the factorial function to not just R, but almost all of C. its called the gamma function and is really pretty

upload.wikimedia.org/wikipedia/comm…

you can tell the order of the root of a function in the complex plane by how many times the color wheel spirals around it

matematicasvisuales.com/images/complex…

you can analytically continue log(z) to the complex plane, although you need to glue surfaces together. it also looks pretty neato

upload.wikimedia.org/wikipedia/comm…

you can count all the rational numbers. the proof of which uses something called diagonalization, which is really really really neat

homeschoolmath.net/teaching/ratio…

you can use that same technique(diagonalization) to prove that the reals are uncountable

germain.its.maine.edu/~farlow/sec25.…

Cantor literally went insane trying to figure out whether there was a cardinality between aleph-0 (countable infinity, ie the integers) and aleph-1 (uncountable infinity, ie the reals).

Then later some dude (cohen) came along and proved its independent of the axioms of ZFC

i.e. you can add in an axiom making it true or false, theyre both valid, so its literaly impossible to prove one way or another. This proof technique is called "forcing" and is super cool

Speaking of axioms, Hilbert hoped that with ZFC(axiom set) we could sucessfully derive all of modern mathematics from these axioms. The grand text principia mathematica then went on to try, and takes 300 pages to prove 1+1=2 from first principles

Godel then demolished their souls with his incompleteness theorems, which essentially say that any system strong enough to say (in it) that it is consistent (there is no statement s.t. A and not A are true) and complete (all true statements can be proved) isnt. Cant have both

He did this by showing you can encode the statement "this sentence is false" as a statement about the natural numbers, and that statement is true, but unprovable(independent) from the axioms of the system (if it was true, then its false, and if its false, then its true)

This kinda ruined the whole "we can derive all of mathematics from a simple axiom set"

Einstein later took a position at princeton just to work w godel. At godels citizenship hearing he tried to bring up that he found a logical inconsistency in the constitution. AE stopped him

Godel (like cantor, and all good logicians) was batshit crazy, and ended up starving to death after his wife died, bc he refused to eat any food anyone gave him besides his wife, believing it to be poisoned.

Reasoning with set theory breaks your brain. Set theory: not even once

Back to math... Turing used a similar technique as godel to show that the halting problem (tell me if this program will run forever or not) was unsolvable. He did this by showing if a solution existed, you could create a program that halted only if the solution said it didnt.

Turing basically formalized (in that proof) the entire basis of modern computation theory (turing machines). He also devised the system to break the germans encryption system, giving the allies a huge advantage over germany in WW2. He probably saved millions of lives

For his service, the british government chemically castrated him after he was found guilty of being gay, which led to him to killing himself.

Fuck england, and fuck homophobia.

Later on, people would build on his work to define computationa complexity classes, such as P (can be solved by a deterministic turing machine in polynomial time) and NP (can be solved by a nondeterminstic turing machine in polynomial time).

No one knows whether P=NP or not. Roughly, this is asking "is finding a solution to a problem as easy as verifying it". Many good CS theorists have gone mad trying to prove it true or false or undecidable. Many also think if Turing lived longer, he would have figured it out.

One of the crazy things to come out of this "complexity zoo", is the idea of a problem being NP-complete. This means for a problem A that is NPC, any other problem in NP can be "re-encoded" into a version of the A problem. So solving one NP-complete problem in P solves them ALL

This is a method many have tried to prove P=NP. No one has suceeded. Many have gone mad, or at least been very upset.

Chaitins constant is the probability that (for a given definition of a turing machine), a random program on that machine will halt. It is incomputable bc of the halting problem. A program that attempts to calculate it will be more accurate over time, but we cant say how accurate

I once had lunch with Chaitin at a diner. Like all good logicians he was batshit crazy, and kept telling me that the mathematical mafia was out to get him bc they hated/didnt understand his ideas.

Stokes theorem is an extremely general theorem, which has the special cases of the inverse square law (the reason most forces in physics involve r^2), the divergence theorem, greens theorem, and the fundemental theorem of calculus

Even though the reals are uncountable, we can count a lot of them by defining a formal language, then counting reals that can be expressed in this formal language (this gets us pi, e, and most of the other transcendentals). we know these are a but a small slice of the reals tho!

Specifically since all the reals we can define through a formal language are countable (since statements in it are enumerable) they have Lebesgue measure zero of the reals in total. That means that the reals we cant count are "infinitely" bigger than the ones we can

In mathematical terms, thats considered really fucking crazy

In any programming language with "eval", there exists a minimum size program which finds the solution to any problem that halts in N steps if it exists. Thats the program that just tries writing all possible programs and seeing if they solve it in N steps.

Because of this, asking for the smallest program that solves a problem in less than N steps becomes meaningless once the length of the solution is longer than the program that tries to find possible solutions

this means that asking for "the shortest program that solves problem X in <= N steps" becomes meaningless if the "solution" is longer than the program that just tries generating all programs and running them for N steps.

fermat's "last theorem" is that "no three positive integers a, b, and c satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2". he famously wrote that he had a proof of this but could not fit it in the margin of the page he was working on.

Wiles finally solved this problem in '95, using branches of mathematics that were unknown at the time Fermat was alive. so either Fermat's proof was wrong (likely), or there exists some simple proof of this theorem that illudes us to this day.

many have gone mad looking for it

there is a technique called automatic differentiation, which allows any function that can be calculate by a computer program (sorta) to have its derivative calculated perfectly, by abusing the chain rule. ill let WP tell you more en.wikipedia.org/wiki/Automatic…

@newshtwit showed me this

complex numbers were thought to be just the fun abstract nonsense of mathematicians for years. then it turned out they are used heavily by the universe in quantum mechanics and electrodynamics.

these problems can be reformulated w/o complex numbers FWIW, but its often very ugly

likewise, the study of prime numbers was thought to be a purely abstract mathematical exercise for years, until they turned out to form the basis of public key cryptography (RSA).

mathematicians have now retreated to more abstract topics with no military applications (yet)

when i talk about ZFC, i am talking about a set of axioms that are all held to be self-evident, plus one hairy one, the "axiom of choice". people have tried to derive it from the rest of ZF for years, turns out its independent of ZF. we keep it as an axiom bc theorems use it

again, we have Cohen and his method of forcing to thank for showing us that its independent of the rest of ZF (which consists of "self-evident" axioms like if A=B -> B=A)

the axiom of choice: For any set X of nonempty sets, there exists a choice function f defined on X.

going back to zeta functions, bc theyre awesome, the zeta function is used to prove the prime number theorem (a statements about how fast the function that counts the number of primes less than N grows). this is *roughly* bc it encodes the "harmonics" of the prime numbers

a huge open question in mathematics is the Riemann hypothesis: that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. computer cant find a zero that doesnt fit this, but no one can prove it

maybe its independent of ZFC *shrug*

a number of theorems in mathematics and physics assume the Riemann hypothesis is true, as it SEEMS to be, and then you can prove some cool stuff assuming it is.

banach-tarski paradox is that you can decompose a ball (set of all points in R3 with radius=K), and rebuild it into two identical balls, producing "stuff outa thin air". it relies on the axiom of choice.

it is not possible in the real world bc the universe is discrete (c.f. QM)

speaking of QM, a good intro to how wacky it is is the infamous double-slit experiment. basically, a single particle (electron, photon, whatever) can interfere with itself. it doesnt go through one slit or another, it goes through BOTH

en.wikipedia.org/wiki/Double-sl…

in other words, everything is a wave and a particle. its just that (roughly) the bigger something is, the more "definite" its location is, so everything at the macro scale acts like a particle that has a definite position

when we talk about definiteness, we're talking about uncertainty principles, such as the famous one (heisenberg's) that you can't know the exact position and momentum of a quantum object at the same time

the really weird thing is, when you try and figure out things about ambiguous quantum mechanics, it CHANGES the system. this is called the en.wikipedia.org/wiki/Observer_…

it had lead to a lot of people trying to understand QM: en.wikipedia.org/wiki/Measureme… and en.wikipedia.org/wiki/Interpret…

no one really knows WTF wavefunction collapse is. cooper explained it to me once w/ thermodynamics and an argument about entropy, and i swear i understood it for a solid 15-minutes. in the (alleged) words of Bohr: Anyone who is not shocked by quantum theory has not understood it.

einstein famously said "god does not play dice with the universe".

hey, everyone's wrong sometimes.

if you really want to make ur brain hurt, start reading about the en.wikipedia.org/wiki/Quantum_e… and the en.wikipedia.org/wiki/Delayed_c…

and youll realize we're probably in a simulation. i mean we have lazy evaluation, discreteness, and speed caps. seems kinda suspicious doesnt it

okay some last ones. the universe is extremely weird. there are all these constants we dont understand. one of the coolest ones is the fine-structure constant, which is dimensionless en.wikipedia.org/wiki/Fine-stru… and ALMOST BUT NOT QUITE equal to 1/137.

float-precision errors anyone?

for a while, we had these issues with QM where we had calculations that should be infinite. we solved them with "renormalization", which people thought was a mathematical cheat to deal with divergent series (and sometimes uses our fave, the zeta function en.wikipedia.org/wiki/Zeta_func…)

anyway, turns out that renormalization can be formalized well (en.wikipedia.org/wiki/Renormali…). roughly, its the idea that constants can be different at different energy scales. we have no idea what happens when you get to really high energy levels

a lot of physics desires to show that apparently different force are actually the same. you may know that electricity and magnetism are part of the same force. we can actually go further and combine them with the weak (nuclear) force:

en.wikipedia.org/wiki/Electrowe…

people have been trying to figure out how to integrate the strong force into that. integrating gravity into it is even harder. we think that the forces were the same at really high energy levels (right after the big bang)
hyperphysics.phy-astr.gsu.edu/hbase/Astro/un…

the truth is we don't really know what happens at these really high energy levels, or at really small scales. we're pretty sure the electron, quarks, leptons, and gluons have no substructure. here's what we think are the fundemental particles: en.wikipedia.org/wiki/Elementar…

but its really hard to probe at what happens when things get really small, because you need to use REALLY high energies to explore those things.

we only recently proved the higgs boson exists, and we had to built the most powerful particle accelerator ever to do that

we're still not sure how gravity and all these things will fit into our real description of physics. we've been trying to figure out how to integrate SR/GR and QM/QFT (which contradict eachother) for 50+ years now and we're still working on it.

maybe we'll figure it out someday