There have been a number of Scientific discoveries that seemed to be purely scientific curiosities that later turned out to be incredibly useful. Hertz famously commented about the discovery of radio waves: “I do not think that the wireless waves I have discovered will have any practical application.”
Are there examples like this in math as well? What is the most interesting “pure math” discovery that proved to be useful in solving a real-world problem?
Strangest? Functional analysis, maybe. I understand it’s used pretty extensively in quantum field theory, although I don’t actually know firsthand.
That’s a body of mathematics about infinite-dimensional spaces and the operations on them. Even more abstract ways of defining those operations exist and have come up as well, like in Tseirlson’s problem, which recently-ish had a shock negative resolution stemming from quantum information theory.
There’s constructions I find weirder yet, but I don’t think p-adic numbers, for example, have any direct application at this point.
As far as I know, matrices were a “pure math” thing when they were first discovered and seemed pretty useless. Then physicists discovered them and used them for all sorts of shit and now they’re one of the most important tools in in science, engineering and programming.
Huge in 3d graphics and AI.
Integration.
Imaginary numbers probably, they’re useful for a lot of stuff in math and even physics (I’ve heard turbulent flow calculations can use them?) but they seem useless at first
The invention of the number 0, the discovery of irrational numbers, or l the realization that base 60 math makes sense for anything round, including timekeeping.
60 was chosen by the Ancient Sumerians specifically because of its divisibility by 2, 3, 4, and 5. Today, 60 is considered a superior highly composite number but that bit of theory wouldn’t have been as important to the Sumerians and Babylonians as the simple ability to divide 60 by many commonly used factors (2, 3, 4, 5, 6, 10, 12, 15) without any remainders or fractions to worry about.
12 is the most based number in that respect IMO.
But then…hey, we use that for hours!
and in parts of the world for inches to a foot. pretty useful for carpentry for example
Having watched all the veritasium math videos I feel like all the major breakthroughs in math were due to mathemicians playing around with numbers or brain teasers out of curiosity without a concrete use case in mind.
It’s crazy how engaging and well done Veritasium videos are and they’re just free to watch on YouTube.
And on spotify nowadays
The math fun fact I remember best from college is that Charles Boole invented Boolean algebra for his doctoral thesis and his goal was to create a branch of mathematics that was useless. For those not familiar with boolean algebra it works by using logic gates with 1s and 0s to determine a final 1 or 0 state and is subsequently the basis for all modern digital computing
Shoutout to Satyendra Nath Bose who helped pioneer relativity as a theoretical physicist because he didn’t want to study something useful that would benefit the British.
Same thing with early studies on prime numbers
George Boole introduced Boolean algebra, not Charles. Charles, according to this site on the Boole family, he had a career in management of a mining company.
Was he trying to dunk on his professors?
Yes and no
So no.
But also yes
A brain teaser about visiting all islands connected by bridges without crossing the same bridge twice is now the basis of all internet routing. (Graph theory)
freaking freaky little Russian outpost that one is. Bridges galore
If I recall correctly, one mathematician in the 1800s solved a very difficult line integral, and the first application of it was in early computer speech synthesis.
the man you’re thinking of is, I believe, George Boole, the inventor of Boolean algebra.
IIRC quaternions were considered pretty useless until we started doing 3D stuff on computers and now they’re used everywhere
This talk by Freya Holmer on Quarternions is awesome and worth anybody’s time that like computer graphics, computer science, or just math.
That was a cool watch. Thanks.
I wonder if complex numbers predate the discovery of electromagnetism
Yes, mathematicians first encountered equations which could only be solved with complex numbers in the 16th century.
Complex numbers. Also known as imaginary numbers. The imaginary number
iis the solution to√-1. And it is really used in quantum mechanics and I think general relativity as well.A complex number is just two real numbers stitched together. It’s used in many areas, such as the Fourier transform which is common in computer science is often represented with complex numbers because it deals with waves and waves are two-dimensional, and so rather than needing two different equations you can represent it with a single equation where the two-dimensional behavior occurs on the complex-plane.
In principle you can always just split a complex number into two real numbers and carry on the calculation that way. In fact, if we couldn’t, then no one would use complex numbers, because computers can’t process imaginary numbers directly. Every computer program that deals with complex numbers, behind the scenes, is decomposing it into two real-valued floating point numbers.
I don’t think this is really an accurate way of thinking about them. Yes, they can be mapped to a 2d plane, so you can represent them with their two real-numbered coordinates along the real and imaginary axes, but certain operations with them (eg. multiplication) can be done easily with complex numbers but are not obvious how to carry out with just grid points. (3,4) * (5,6) isn’t well-defined, but (3+4i) * (5+6i) is.
That’s not quite accurate because the two numbers have a relationship with each other. i^2 = - 1, so any time you square a complex number or multiply two complex numbers, some of the value jumps from one dimension to the other.
It’s like a vector, where sure, certain operations can be treated as if the dimensions of the vector are distinct, like a translation or scale. But other operations can have one dimension affecting the other, like rotation.
It’s used extensively in electronic circuit design (where it’s called “j”, as "i’ already meant electronic current).
Also signal processing has i or j all over it.
I’m the akshually guy here, but complex numbers are the combination of a real number and an imaginary number. Agree with you, just being pedantic.
Sure, but 1 is a real number. 😜
Yes, and 1 is also a complex number.
Of course, but 1 is the loneliest number.
2 is as bad as 1: it’s the loneliest number since the number 1.
Electromagnetics as well.
Not math but the discovery of Thermus aquaticus was seemingly useless but later had profound applications in medicine. There’s a good Veritasium video on it
I’ve read that all modern cryptography is based on an area (number theory?) that was once only considered “useful” for party tricks.
prime number factorization is the basis of assymetric cryptography. basically, if I start with two large prime numbers (DES was 56bit prime numbers iirc), and multiply them, then the only known solution to find the original prime numbers is guess-and-check. modern keys use 4096-bit keys, and there are more prime numbers in that space than there are particles in the universe. using known computation methods, there is no way to find these keys before the heat death of the universe.
DES is symmetric key cryptography. It doesn’t rely on the difficulty of factorizing large semi-primes. It did use a 56-bit key, though.
Public key cryptography (DSA, RSA, Elliptic Curve) does rely on these things and yes it’s a 4096-bit key these days (up from 1024 in the older days).
RSA mostly uses 4096 bit keys nowadays. DSA is no longer used (or shouldn’t be lol). Ed25519 uses 256 bit keys.
thank you
How do you define “pure math discovery”?
A math discovery unmotivated by research in other fields; just discovering math to see if it works out
With that as the definition I would say Boolean Algebra.
Non-Euclidean geometry was developed by pure mathematicians who were trying to prove the parallel line postulate as a theorem. They realized that all of the classic geometry theorems are all different if you start changing that postulate.
This led to Riemannian geometry in 1854, which back then was a pure math exercise.
Some 60 years later, in 1915, Albert Einstein published the theory of general relativity, of which the core mathematics is all Riemannian geometry.
This won’t make any sense to any of you right now, but: E = md3
Oh god, the cringe.
That’s a perfect example of a typical interaction between a Technology Management Consultant and somebody from a STEM area.
Techies with an Engineering background who are in Tech and Tech-adjacent companies are often in the receiving end of similar techno-bollocks which makes no sense from such “Technology” Management Consultants, but it’s seldom quite as public as this one.
