What's stopping me now? That sweet overpaid SDE salary and the endless obligations that come from being an adult. I suspect I am not alone...
As a field I think math is well past its diminishing returns imho. It's like 'what was the last big philosophical discovery'...yeah...hard to think of one. Maybe the P zombie concept or the simulation hypothesis. But new and important books, fiction and non fiction, are being written all the time.
There are big and hairy problems that are bad investments for a young mathematician. I would steer students clear of the Collatz conjecture. But once you get up to speed in your research area, you usually find interesting problems thick on the ground.
Tenure-track positions are competitive, but I don't think there are a lack of interesting things to work on.
99.99% of everyone won't be remembered for their "contributions" so why not do something you enjoy?
Having said that, if something does come along that interests someone, they should try it. A friend of mine is doing his 2nd PhD 40 years after his first, having found his way into it via a love of jazz music.
It's not easy, but it's certainly very impactful.
Otherwise, I will try to come back to this comment in a few days. I'm a bit busy with an urgent deadline, but I'm glad to expand on this later on!
I wish I just had more time to do math like the OP but the saturation of the field especially with people who can deeply understand the abstraction is very very intimidating.
(apart from saying that the evil demon is a future type computer)
The first incompleteness theorem says that for any consistent formal system T (with a recursively enumerable set of axioms) capable expressing of elementary arithmetic, T can express a statement which it can neither prove nor disprove.
The second incompleteness theorem says that T can't prove the statement "T is consistent". (I've still glossed over a number of technical details here; pick up a book on model theory if you want all the messy internals.)
First order logic is notably not capable of expressing elementary arithmetic. And observers aren't involved in any way.
Sorry if you can’t read deeply into it or something. I’m not posting for grad students. I can sense you just like to correct people. Ahhhh I’m so wrong, you’re right?
It's tempting to try to apply them (or rather the same kind of conclusions) in other (non math) contexts, but it's very not obvious that you'll get something sensible. While you can play with the ideas, invoking Godel's theorem outside of its specific context doesn't make much sense.
I guess you could call a Turing Machine implementing the search algorithm for proofs implied by a logic an "observer", since it produces "reachability observations" i.e. proofs.
The Open University has an AI master that I'm thinking about right now.
It has about 25% of the math that I want to learn, so that would be a good start.
I did some prep work (an official high school math certificate) last few months and I noticed that I need a schedule to keep me going.
One thing that I'm quite certain about, is that doing math is the most important thing. And doing math leads to more doing math.
RE schedule to keep you going: I've had some students use the concept maps as a "world map" (like the stage map in Mario World) and check off boxes as you progress through them (each concept corresponds to roughly one section). See https://minireference.com/static/conceptmaps/math_and_physic... and https://minireference.com/static/conceptmaps/linear_algebra_... I guess you could time-box these and do N of sections each week to make this into a schedule. Make sure you dedicate lots of time for the exercises/problems, because that's when the real learning happens...
I finished pre-Algebra last year and I'm halfway through an Algebra text by Gelfand & Shen now. My friends look at me funny when I tell them I'm re-learning Math from the ground up for fun (esp. with a degree in CS) but it has been so rewarding. I probably won't get to finishing Calculus for another couple years but I'm already having so much fun. Stumbled upon deriving some exponent laws last month by accident and truly understanding the sum and difference of squares has been awesome.
But there is also this vast set of reals that are simply undefineable, non-repeating sequences. These numbers are unmentionable and unknowable. Does it really even make sense to say that this subset of the reals exists in the same way that definable numbers do?
My assertion is that there is something wonky with these u definable numbers and that wonkiness is directly related to how absolutely massive the infinity of reals is.
You’ve only restated the problem without saying _why_ covering the reals is different.
That is, you can cover the rationals with intervals of arbitrarily small total length.
To most people, it just means that when you take the absolute value of a negative number it becomes positive, and the absolute value of a positive number stays positive.
Now there is more to it, but how you might think of absolute value instead is as a distance function, particularly how far away from zero you are on a number line.
This is way over simplified, but an example of how there can be a little more buried beneath the surface.
Actually quite doable as long as one can grit through the Math, some of which do not need a back to back read.
At best school is a syllabus telling you what people think you should know, fairly easy to get a hold of, and maybe a bit of accountability to make sure you actually learn it.
A good starting point could be MIT OCW 18.01, 18.02, and 18.03. Do all their problem sets and get as much understanding you want. It corresponds to a first year university engineering curriculum.
You really can't win :/
I'm still trying to find a part time dev gig so I can just focus on graphs and advanced combinatorics
Clearly you find earning/spending money more appealing than math.
There’s more to mathematics than rigour and proofs - https://news.ycombinator.com/item?id=9517619 - May 2015 (32 comments)
There’s more to mathematics than rigour and proofs - https://news.ycombinator.com/item?id=4769216 - Nov 2012 (36 comments)
Another good starting point (for mathematicians) would be the Lean community group and the Math lib project:
Just to humor you though, I think Deep Operator learning is a vastly exciting new field which combines ideas from functional analysis and deep learning in order to do things like solving PDEs.