Honestly I don't really care whether statement is true or not, what I want to know is the reasoning, the inspiration, the trials and errors. I think Lean is cool but if the futur of math is unintelligible code written by AI, what's the point ?
(Sorry for my english)
I’ve been thinking about the epistemological boundary of what it actually means to know or understand a specific domain of mathematics. Excluding elementary arithmetic (which relies heavily on procedural computation), higher branches like abstract algebra, differential geometry, or algebraic topology demand a completely different level of mental modeling. Even just calculus, when can we truly claim to understand a subject?
At what point do you consider yourself to have crossed into genuine "comprehension" of a subject?
Is it when you:
For those specializing in pure or applied math whether it's topology, analysis, abstract algebra, or PDE what is your personal threshold for claiming true fluency in a subfield? Where does mechanical competency end and genuine mathematical understanding begin for you?
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?
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 Tsimerman'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.
EDIT 2: Another interesting point brought to me by the comments is the idea of repairability. A commenter mentioned that most of the errors encountered are easily fixed. At a high-level, this suggests that the direction offered by intuition is powerful enough to render errors insignificant. Maybe instead of AI destroying math from the foundations, it instead works to validate the strength of intuition by perpetually exposing errors and instantly fixing them. Wouldn't it be wonderful if AI shows that the fix-rate of errors was near 100%?
To begin with, I apologize if this sounds stupid. I already feel like an idiot writing this.
I'm studying at a university in Germany, bachelors, and I used to think I disliked mathematics. But I've realized that I actually enjoy it once I sit down and start trying to solve problems.
My main problem is that after a few months, I tend to forget what I've learned. I know repetition is important, but I feel like the bigger issue is that I often learn how to use an equation without really understanding what's behind it. That is the problem.
I come from Central Asia, where I feel like I've always been surrounded by people who are extremely good at mathematics. Because of that, I've often felt like I'm simply not naturally good at it.
Recently, though, I've realized that I actually enjoy the process of solving mathematical problems and puzzles. I want to develop that way of thinking rather than just memorize formulas and procedures.
I've found these books:
Would these be a good approach for someone in my situation? Are there other books or resources you'd recommend for developing mathematical intuition, reasoning, and problem-solving ability?
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
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?
Okay so I have a relatively “simple” math problem in relation to no longer being able to afford something (Being vague on purpose, the item is personal and I don’t want help with the item itself just the math. Let’s call it bread just to call it something) I had 8 slices of bread and I cut them all in half to try to stretch the bread until next Thursday, the 27th, so now I have 16 halves. I want to have the maximum amount of bread each day, and they cannot be cut anymore, so I have 16/9 basically. (I’m very bad at math so I am not even sure where the help I need starts). So if you break that down it’s like 1.77777, it means that I can have 1.5 slices of bread each day? But when I go back and try to confirm my math, I got a different answer all together. When I did 1.5 times 9, I only got 13.5 not the 16 I was expecting it to be. I am genuinely so stumped that I couldn’t even say for sure what calculator stage I’m making a mistake at. Thank you in advance for help with figuring this out!
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.
[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?
For those curious: The equation is at the top of the first picture.
So the equation is:
B=(|x|/2)*V(+-)n*V(of(n))/(a(of(steps))*(j *((x/(a))^2)+(x(of(a))+(x(of(a-1))^2)^0,5
B: NewPoint
|x|: Distance
V: 0.9170411968
n: counts for every V on x = 0.9170411968 (n element Z[whole numbers])
a: divides distance in equal parts [a is never 0]
x: x(ofn)=Functionvalue ; eg. n=2 f(x)=25x , so x=25-|(25|/2)
j: Variable [stays Variable] to draw a new line that bends
B element M ; f(x) element a,x,j element R[Real numbers]
M:= ( f(x) ; x ; n ; a ; j )
Those who are really curious, the equations are from me, not AI, i just used Gemini to draw alot.
It helped to refine the equation with pictures/graphs only.
And i won't just post how i got the number correct V: 0.9170411968 ; it was the most difficult part.
Everything took me 2 days of constantly working actually, to get to this point, because the number wouldn't mean anything without achieving any geometrical shapes.
I spoil it only a little, the number has to do with 3D-Objects. And the equation to calculate them/the number, is a little big.
Please tell me, where the equation doesn't work and i go back to number work.
Let's call the number: "The number"
Have fun, let's hope the equation works for everything and bends everything to a Circle/Sphere.
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.
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
This helped me a lot in understanding trig identities so now I wanted to be able to visualize this applied in special trig angles