Are there any mathematicians on the forum?
What’s going on…? Any cool developments in your field of focus, or the larger umbrella of mathematics? Any drama?
I always have a fascination with math, but I’m not very good at it… I feel like there’s an intuition that lets me half-understand stuff sometimes. The whole field seems kinda interesting, though. It doesn’t get the same amount/degree of press as, say, physics or computer science.
Well there’s @mathbrush and @Spike; I suspect there are others too (Graham Nelson I suppose, though he’s not much of chit-chatter on the forum!)
for obvious reasons, I study many branches of mathematics & physics (mainly ballistics, of course…) up to having coded appropriate programs in the appropriate language (Fortran…)
Best regards from Italy,
dott. Piergiorgio.
I am not in any way a mathematician, but some of those near me have been enjoying the ‘hat’ and ‘spectre’ aperiodic monotiles.
(A friend’s take on it, and their playable puzzle.)
In my school days I had specialization in math and bio. But I’m not as good in math now as I was back then.
Recently I tried to understand the proof of Euler’s theorem. [Edit: It was Fermat’s theorem.] But it’s too sophisticated/advanced for me.
A little bit more light/easy is a book with collected math riddles and wonders by Martin Gardner.
I got a “B” in maths at ‘O’ Level. 
I don’t have any history of being great at math, but I had a ton of fun with the Project Euler website, where you are given problems that you must write a computer program to solve because they are way too computation-heavy to be done by pencil. Believe it or not I used TADS for this, as it was the only language I knew at the time… did learn some Python as the problems got harder though, because TADS’ speed really started showing up.
Of the top of my head, the big advances in math in the last like 5-8 years that I can remember are:
Drama? Right now on math Wikipedia someone’s making an article about AI and math but for examples they’re just listing some basic algorithms which have nothing to do with AI, so there’s some back and forth.
Edit: one really big source of drama is respectable mathematicians thinking they solved the Riemann hypothesis and writing long papers with lots of thanks and congratulations to themselves in it and dedicating each chapters to a different loved one, then having the whole paper be wrong. It’s really embarrassing. One of my professors did it once. 120 years ago they made a lis of the hardest unsolved math problems and it was #1. 20 years ago they made another list and it was still #1.
Not strictly maths, but at least maths-adjacent: the recent Conway’s Game of Life result that anything buildable can be built with 15 gliders was quite good.
I googled it unsuccesfully. Do you have a link?
Like many math dilletants (I am one) I’m interested in prime numbers (and other number theory).
Wiles made a mistake, too, in his first publification on the proof of Fermat’s theorem.
The guy is Yitang Zhang and his paper is on “bounded gaps between primes”, and it’s related to the Twin Prime conjecture.
That’s true about Wiles! His theorem was another one that took forever to solve…they even had Star Trek predict it wouldn’t be solved in the future!
I’m very much not a mathematician myself, but my sister is, and to the best of my understanding (which is not very far) she works with combinatorics and Schubert varieties. She tried to explain pipe dreams to me recently and the diagrams are cool but I still don’t really get what they mean. (But I can relay questions to her if this is meaningful to others.)
My own mathematical journey was mostly on the CS side of things, with proving the correctness and efficiency of algorithms and all that. But there were two moments I remember that felt like glimpses into the field as a whole. First was reading Hofstadter’s Gödel, Escher, Bach, which introduced me to the idea of Peano arithmetic and axiomatizing everything. We don’t actually need to take it on faith that the natural numbers work the way we expect them to. We can actually build them from the ground up! We can build an entire system of number theory this way! We can show from first principles that 1+1=2 (or rather S0+S0=SS0)!
The other was when I took a class on computer graphics taught by a famously eccentric math professor, whose goal for the class was for all of us to find something mathematical that interested us and then somehow make graphics about it. I ended up looking into geometric algebra and then somewhat falling down the rabbit hole.
If you’ve done any physics or engineering, you’re probably used to the right-hand rule for rotations. Apply a force F at a distance r and the resulting torque is the cross product r × F, a vector pointing along the axis of rotation.
But this always felt a little weird. What if you’re in two dimensions? Obviously rotations can happen in two dimensions. But there’s no longer an axis of rotation, there’s a point of rotation. And the torque is a scalar, not a vector. What about in four dimensions? How do you even do that?
It turns out there’s a beautifully simple and elegant solution. Instead of treating rotations as vectors, you can treat them as bivectors. If a vector is a directed length, a bivector is a directed area. And the dimensionality of the space of bivectors, in n spatial dimensions, is (n choose 2).
It’s pure coincidence that in three dimensions, (3 choose 2) is also three, and we can create a mapping between bivectors and standard vectors. But this mapping requires imposing a handedness on the universe (the right hand rule instead of the left hand rule) that can lead to wrong intuitions in physics. If you instead consider rotations to be bivectors then you don’t need a handedness and your intuitions will work in any number of spatial dimensions!
And then you can start working with multivectors, which are a linear combination of scalars and vectors and bivectors and so on. And once you do that (through a proof that’s absolutely beautiful and too long for me to derail this thread with) it turns out that both the complex numbers and the quaternions are special cases of this general system, in 2 and 3 dimensions respectively. And there’s a clear and intuitive reason why there’s not an equivalent of the quaternions in four dimensions (which would have seven elements in its basis). It’s really cool.
…so yeah, that’s why I like mathematics, though it’s not the path I ended up taking in life!
Okay, in hindsight I should have assumed that @mathbrush did not put “MATH” in his name for the sake of being whimsical. That’s entirely on me.
Ohh!! This could cause improvements for 3D graphics! I think? I’m pretty sure camera transformations are done with 3x3 matrices.
(adds to lookup list)
Yikes.
HA!
This is actually something that has made it’s why to Numberphile, I think, so I remember hearing that this happens. That’s really funny that they do the whole “Directed by Johnny Blaze, Lead Actor: Johnny Blaze, Produced by Johnny Blaze, Written by Johnny Blaze, Assistant Grip: Johnny Blaze” across the paper!
I’m sorry to hear about the loss of your professor to the grueling Riemann hypothesis. /lh /j
This is another one of the few moments where I’m like “Aha!! I actually remember this one!!” It’s absolutely fascinating to see how basic assumptions are structured at lower levels.
Gosh, I got a few flickering moments of clarity in that. Should probably add this to my lookup list as well.
I probably could have presented animation. I very rarely use keyframes when animating stuff; I usually use time-based functions instead.
I am always excited for Conway’s Game of Life!
(also adds to lookup list)
I think I remember hearing about this. I should look into this more, too!
If you’re ever hanging out on ifMUD, Olly has a mathematics degree and is usually pretty keen to chat about various math topics. When he does, some of the more STEM peeps (usually more physics or CS leaning) will often chime in too.
Olly (and most of the ifMUD regulars for that matter) is a pretty friendly and interesting fellow in his own right, so you can’t go too wrong striking up a conversation in general.
I don’t like this. Can’t get more axiomatic than 1+1=2 IMO. (Maybe 0+1=1.) ONE and TWO are semantic primes after all.
It’s like how lots of research maths is all about set theory. With no numbers anywhere. Ugh. Just because you can generalise beyond numbers doesn’t mean that you’ve actually found the foundation for numbers.
Maybe we need to more legitimise calling this type of maths a creative activity? Then anyone can create their own foundations of maths, but without pretending that what they do is somehow more axiomatic than putting 1 and 1 together.
Numberphile is what has been teaching me most of the new maths I know since I finished uni. They have two videos on the new aperiodic monotile which I haven’t watched yet. Aperiodic tessellations are fascinating!
You should look at Goedels Incompleteness Theorem! It was written when people were really trying hard to axiomatize stuff. It basically says that any attempt to axiomatize math perfectly will fail (specifically any kind of formal proof system that includes natural numbers will have a true statement about the system that it can’t prove)
Isn’t that what Peano arithmetic is? Start with a number (usually 1) and an increment axiom (1+1=2), show that those rules give you all the properties of natural numbers? And it’s not like they’re new, Peano was late 1800s IIRC…
Edit: though yeah, I see your larger point
See, before learning about Peano arithmetic, I didn’t know about any axiomatization of number theory. I’d learned about Euclid’s postulates at one point, but everything I learned about number theory started with “so we have the natural numbers and the operators that you learned back in elementary school”. Whether 1+1=2 is an axiom or a provable theorem, either of those was a big step beyond “it works that way because everyone learned that as a kid”!
Whether it uses set theory or Peano arithmetic or lambda calculus or whatever, the idea that you could actually axiomatize numbers instead of relying on intuition was mindblowing for me. That you could actually explicitly define what a natural number was rather than taking it on faith.