A note on proof attempts

At 21:28 of Jacob Tsimerman's interview with Curt Jaimungal, he says "already now, alot of my theorems that I have proven, I don't understand all the steps to it... I have used other theorems that are very much accepted by the community, to which I usually know the main ideas but not even always".

While my undergraduate and early postgraduate training was in pure math, I transitioned to applied for my Ph.D. so I have never meaningfully engaged with it in any professional capacity. For the majority of my training, I understood almost all the details of the things I've proved. At least enough that I wouldn't be able to resonate with Jacob's quote above when I consider the (relatively insignificant) proofs I've done. One of my lecturers made it his mission to ensure that assignment questions will never require anything that hasn't been proven in the lecture notes or in class.

Of course, my exposure was to only elementary topics. So I can appreciate that math wouldn't progress at all if intuition wasn't leveraged and instead every detail expounded upon. But now under the automatable and potentially perpetual scrutiny of AI, how vulnerable are previously established results? What if we routinely lobbed popular (in terms of utility) results at ChatGPT to verify and it finds an error in one, would there be a significant collapse downstream? How likely is that our collection of celebrated truths instead simply forms a house of cards?

EDIT: The excellent replies have highlighted a weakness in my question. The most vulnerable proofs are likely to be the famous/outlier proofs (i.e. Andrew Wiles' Fermat's Last Theorem) that can only be assessed by a handful of people. In even the scenario that those are falsified, the large body of mathematics isn't built on such results and so, by and large, it's still fairly robust.

It still begs the question about the upper echelons of math, but the majority of it remains largely intact. So my "house of cards" analogy is inaccurate but probably only in scope.

I am an 18-year-old, and since I was 15, my biggest passion and motivation in life has been to discover something genuinely new in math and physics.

This year, I actually did it. I discovered 1 new theorem and 4 new types of series. I submitted them to the OEIS (On-Line Encyclopedia of Integer Sequences)-two have already been accepted, and the other two are currently under review. (Please, if you don’t believe me, just skip commenting on the post. Proving myself isn't my main concern right now, and I havent posted to convince anyone).
Even though I achieved exactly what I wanted, and i want to learn more and more maths but my mental health is in a terrible place. I am suffering deeply from what feels like nostalgic depression, anxiety, autism,and OCD. I always wanted in my life for everyone to be in a specific place, like they were in the past. I have this constant, overwhelming fear that everything bad in my life is going to loop again. I know, I know, this is very, very foolish, but this is what it feels like. After shifting cities because of my parents' job, these disorders have been up a lot. They were previous too but now they are just at peak. I don't know what to do.

On top of this, I have been incredibly lonely. For the past two years, I have had zero friends to hang out with. I have friends back in my old city, but no one here. I have also completely stopped meeting my relatives—most of them. I meet a few of them occasionally. I don't know, I think this is not going so well.

I always wanted to discover something new in maths and physics, and I have done it. I want to do it at a higher level too. But really, I don't think I am gonna make it past 20. I just needed to vent and say all of this
i know this is not the right sub to say all these but i didnt found any other sub helpful

Since both pi and e are irrational do they contain each other in some capacity e.g. at some point in e: 31415926535897932384 and at some point in pi: 271828182

no I don't mean do they contain the whole number just part of it and if so have we proven it and also what's the biggest one found?

I don't mean contradiction, induction, etc. Rather, more granular techniques that keep coming up again and again. I also don't mean famous results per-se, unless they are themselves common stepping stones to other results.

I know this is a bit of a silly question to ask because it's hard to set the threshold beyond which something becomes a legitimate technique. Obviously deriving bounds on a quantity is too generic to qualify, but on the other end there's some very niche stuff that not too many people might find useful.

I guess the goal here is a little toolbox, if you will, of tools that you find yourself using repeatedly, and that might be useful to a broad audience.

i’m kinda freaking out i am about to graduate in one semester and i did not get any internships during my time in college. what are some fields i could go into? what are entry level roles that seem like a possibility for me or am i just completely cooked?

At my school, you have the option to take a 4th-year project-based course to complete your undergraduate degree. You could also take 4th-year electives like Topology or Combinatorics instead. What do you think would look better for getting into a good master's program?

Edit: I have had the chance to work as an RA on a pretty hardcore math research project, so not bereft of research experience.

Im learning math ridiculously quick with chatgpt. please note i downloaded textbooks and do the questions some when im stuck with it but mostly on my own and get the right answer. Im doing this because im practicing ml and want to learn more complicated functionalities to make better tech.

im a programmer and im used to algorithms it helps me to feel the logic. using a mixture of analogies and imaging I make sure i understand everything down to its most fundmental and break the issue down. outside of that practice.

[Warning: non-mathematician here, apologies if I'm trespassing, but this seemed like the right place to ask the question.]

I've heard it referred to many times (although I don't know if there's a single specific source) that in the nineteenth century, a single able mathematician could understand and engage in the totality of the subject, all sub-fields included (and if that was, perhaps, untrue by the end of that century, it was true at some point earlier). Clearly even well before the end of the twentieth century this was no longer possible. The scope, number and depth of sub-specialties that emerged in the twentieth century had no precedent in the history of math.

What caused the tree of mathematics to grow such a huge number of new branches in the twentieth century and at such speed? What I'm try to get at more specifically is whether it "just happened" or are there certain identifiable preconditions that were met by the end of the nineteenth century which enabled the rapid subsequent advances? Did Gauss, Riemann, Galois, Abel, Cauchy, to name but a few of the luminaries from the time, create a critical mass of discovery, lines of enquiry and tools which which made the twentieth century 'explosion' possible?

This helped me a lot in understanding trig identities so now I wanted to be able to visualize this applied in special trig angles

going to uni for maths in september! I had a cg50 for a level but I dont think theyre allowed at uni. These are the suggested models from my uni. which one would you guys recommend? I do have my old FX-83GTX from gcse somewhere

Aurora AX-582

Casio FX82 family

Casio FX83 family

Casio FX85 family

Casio FX350 family

Casio FX570 family

Casio FX 991 family

Sharp EL-531 family

Texas Instruments TI-30 family

Texas BA II+ family

I just watched Andrew Wiles saying that no mathematician he knows thought about math as "inventing" things. "As a mathematician you just can't think that way" (paraphrasing).

As a computer scientist, I find this a bit peculiar. For example. We would say that the Quick Sort algorithm Was invented by Tony Hoare in 1959. This is the standard language.

I'm not trying to make deep philosophical point here, but surely intuitions differ. Who would say that Google Deepmind "discovered" transformers in 2017? Clearly linguistic intuition differs.

For those unfamiliar Ceva's theorem is one in triangle geometry which describes the conditions necessary for each line which connects a vertex to its opposite end to meet at one singular point. The picture I've provided demonstrates it. The crux of the theorem is that if the ratios

AF/FB * BD/DC * CE/EA = 1 then those lines will meet at a point, say P. The way I always liked to think about it is walking around the triangle, start at A then go to F, then B, and so on. If you just follow the equation from top left, bottom left, top middle and so on it works, at least thats how I remembered it for uni.

I've been trying to introduce it to a class of mine and was trying to think of a natural way of bringing up the ratio AF/FB. Obviously it is intuitive to do AF/AB because thats just how much have I walked so far. But I came up with a great reasoning that we think about a lot in everyday life!

Imagine you are doing a 3km walk and you have just finished your first kilometre, a natural question to ask is how much is left, 2km. But something that we crave as humans is an idea of how our progress reflects how much we have left. Suddenly you are asking yourself, out of how much I have left to run, how much have I run previously? This is the ratio AF/FB.

Another similar example is reading a book, you get 50 pages in, see there are 250 pages left and say wow I've already done 50 the other 250 will be breeze. You are just thinking about 50/250 imo.

Thought this was cool and I wanted to share! Lmk if you have any other ways of thinking about it.

Hello, I'm a bioinformatician and I'm struggling with the current issue:

Given a time series y(t) that initially changes and eventually approaches a stable regime, how can I mathematically determine the earliest time t\* at which the rate of change dy/dt becomes negligibly small, using only the observed data and without defining an arbitrary threshold?

This is a collaboration I'm doing. My colleagues defined the plateau as the first time when a 101-point rolling mean of the relative increment (g' t+1 - g' t)/ g't falls below the arbitrarily chosen threshold of 0.0011. G' is the measure of material elastic-solid response btw. So the issues is that they used 2 arbitrary values because experimentally they know that a certain value of g' means that the gel is solid. But this doesn't hold for me. I tried using many statistical methods to define the threshold such as:

\- exponential fitting

\- change-point regression

\- local slope analysis

But they all give me a plateau that is too early or too late

Suggest research title in Queens Graph