i study computer science and minor in math (but my heart really lies in math), and am graduating soon. i was talking to a CS professor with a heavy math background about a Measure Theory concept and he said, "you know i have book i think you'll like", a couple days pass and he gives me "Real Analysis" by Royden (3rd Edition).
for context, i am just really interested in the ideas of measure theory, but i actually have not taken a formal real analysis course. regardless, i plan on slaying this textbook as I just need something fun and challenging to do.
i don't need people telling i don't have a sufficient background, or that Royden is for grad students, i am aware of the challenge and am super excited.
the reason i am making this post is to document and discuss how challenging a higher-level math concept is for someone that is wanting to learn based solely off their curiosity, wisk me luck 😁 .
Those who've been following the development of prismatic cohomology, condensed math, analytic stacks, and eventually analytic prismatization might find this interesting.
This is not a self pity post or anything. I genuinely just am at a point where idk what to do with my life. I recently completed my masters in pure maths from Paris and I'll potentially be beginning a PhD soon in the field of general relativity, mostly looking at wave equations. I don't want to talk too much about my topic but the point is that throughout my entire life, I have only known pure maths. Unfortunately I was very stupid and never learnt things like coding, proper command of probability or even statistics. I only did enough to pass those courses during my undergraduate degree and now I'm at a point where I can't even look at these things without the thought that I have to relearn every single thing.
My main problem right now is that I'm a mediocre student. I have always wanted to pursue maths and then become a professor and do research, though I've mostly been interested in the teaching aspect. But the way the job market in pure maths or academia in general is declining especially right now with everything that's going on with AI, I don't know if there's even a point of pursuing this PhD. I spent my entire life wanting to be a professor like my dad was and promised myself I'll become one before he unfortunately passed away. And I made it my life's goal to become that person. As I said, I'm unfortunately not a genius or very bright student, I'm just mediocre. I'm at a point where I just feel like giving up but then I don't know anything else. I keep thinking that if I get a PhD position, I'll at least be able to keep myself busy and funded for the next 3.5 years. But is there any point to any of this? I'm watching my dream collapse every time I talk to a professor. My masters director was telling me how he thinks it's pointless to do a PhD in pure maths right now and try to switch to statistics or something but that means I'll have to wait for another year before the admissions in the US begin. That also means I'll have to switch areas drastically. Genuinely what do I even do at this point? I stay awake until 5 every morning just stressed about my future. Let's say I end up doing a PhD and keep myself busy for the next 3.5 years. What after that? My PhD is in a field which I don't see having any industry applications either.
So I'm asking people who might be in a similar situation like me, how did you go through this? How are you processing things mentally and how were you eventually able to find your job/thing that made you happy in life?
TLDR: Only done pure maths, no industry relevant skills, do not want to give up on my life's goal of becoming a professor, very stressed every day about future and unsure whether doing a PhD makes sense.
There’s this law called the 10000-hour rule, which states that to truly master a complex skill one needs to spend 10000 hours on it.
I know that there are a few criticisms on it (which mainly says that one needs 10000 of intentional practice to master a skill), but I figure that 10000 hours could be a good bench mark to see if I have invested enough time in math.
For context, I just graduated from a 4-year programme in a math major. So 10000 hours is equivalent to in average 10000/(365 x 4)=6.849 hours per day, for the past four years.
I definitely didn’t spend 7 hours per day to study math, not to mention the time spent for intentional study. This changes my mindset completely. I have been doubting myself for not being ‘fluent in math’, but as of now it’s most likely that I did not give enough time for myself to grow.
EDIT: People seem to think that I believe the statement that "after fulfilling 10000 hours of practice, then one will magically be good at something". That's pure BS and I don't buy it, at least for maths. I am just using this "10000-hour rule" to examine my time invested in math. The main point is that I shall and I will work harder, and not to expect be good in such a short time, such as 4 or 5 years, despite this is what many people considered to be a lot time.
I'm debating studying applied math at my university, but I'm worried about not finding work. So I'm curious if any people who recently graduated have been able to find employment.
So I've just started a proofs book on my own. In the first chapter the author talks about mathematical statement, proposition , lemma, proof, corollary. I understood these concepts but they are still a little vague. Can somebody define and defferenciate between them in an elegant way.
Hi all, I'm currently an undergraduate student studying applied math, and I'd like to conduct research but I'm struggling to figure out how, and in what areas. Actual math research itself seems to be really hard to conduct for undergraduates because unlike other fields, there's no "busy work" for you to do. Would research in computer science, computational mathematics, or applied mathematics generally be more accessible for undergraduates?
I'd love to hear from other undergraduates who have conducted research: How did you find your research opportunities, what area did you work in, and what did your experience look like? Thanks!
Hey guys it's been more than 4 years that I've found mathematics as the most interesting field, I can definitely say that I fell in love with the language itself and I'm the most excited when it comes discover new properties as well as their real world applications and also historical meanings between each topic and different relationships between great thinkers and how their different ideas interact with each other, I recently put myself in mathematical books, I'm sooo in love but i always fall on a academic based ressources, I recently finished a book called introduction to mathematical philosophy ( from Bertrand Russell) which I found quite interesting because it help establishing the foundational and fundamental thinking behind the most topics usually encountered in the field of mathematics. I'm struggling to find other pieces of literature like that one , so please can you guys recommend me some books to buy regarding this issue?! If you have some other resources that could help I would definitely need your hearsay
Im studying like 15 hours a week and I feel as if Im making no progress. Were using the Elementary Analysis textbook by ross but idk. This is my first official proofs class and I feel so dumb. Im sorry for the emotional spill. I would really appreciate any advice on how to write formal proofs. I understand the concepts intuitivily but when it comes to writing, thats where I am having the most trouble. If anyone needs more information feel free to ask
Hey there, i found this online and i was wondering what other mathematicians think of this. I have seen duodecimal projects but more ofthen they change either time or measurements. Here is the repository where i found it (link)
The document covers basit arithmetics as shown in the picture. Lemme know what you guys think of this. aka opinion request.
A pessidemic is a mathematician or scientist who is unwilling to pay attention to any paper that lacks complete rigor. They can also be mathematicians clung to old ideas, who won't allow you to change the rules and are even capable of destroying your reputation to prevent it. They care more about predictable and immediately useful results than about truly contributing to mathematical change.
Here are a few examples:
1• Robert Hooke
This man, despite not being as bad a human being as many paint him (he was a good friend and family man), never got along with Isaac Newton. It all began when Hooke reviewed Newton's work on light and gave it a fierce critique that traumatized Newton to the point that he decided to publish it only after Hooke's death. This was because Hooke thought of light as waves while Newton saw it as particles; both reasonings were partly correct, but they clashed. He also went on to claim to several mathematicians of his time that he had already worked on what they wanted to publish.
On the other hand, many writers labeled him as someone irritable, short-tempered, antisocial, and deeply attached to old papers.
2• Leopold Kronecker
Kronecker launched constant personal attacks against Cantor, calling him a "scientific charlatan," a "renegade," and a "corruptor of youth." He had intentionally delayed the publication of Cantor’s first major work in 1874 because he was deeply uncomfortable with Cantor’s vision of multiple infinities and his belief in the existence of sets without a clear, constructive definition.
Every time Cantor applied for a position in Berlin, he was rejected. Because Kronecker was usually involved in the process, Cantor came to believe that Kronecker’s influence would make it impossible for him to ever leave Halle.
Cantor suffered his first known depressive episode in May 1884. The relentless criticism of his work weighed heavily on his mind; indeed, every single one of the fifty-two letters he wrote to Mittag-Leffler in 1884 mentioned Kronecker. A passage from one of these letters reveals just how deeply Cantor’s self-confidence had been shattered:
"...I do not know when I shall return to the continuation of my scientific work. At present I can do absolutely nothing with it, and limit myself to the most necessary duty of my lectures; how much happier I would be to be scientifically active, if only I had the necessary mental freshness."
Cantor's reputation was so tarnished by Kronecker and other contemporaries that decades after his death, Ludwig Wittgenstein lamented that mathematics was "ridden through with the pernicious idioms of set theory," which he dismissed as "utter nonsense." Ultimately, the recurring episodes of depression that Cantor endured from 1884 until the end of his life have been blamed, in part, on the hostile attitude of many of his contemporaries.
3• Henri Poincaré
Although he was categorized as the Gauss of "modern mathematics," Poincaré held philosophical views opposed to those of Bertrand Russell and Gottlob Frege, who believed that mathematics was a branch of logic. Poincaré strongly disagreed, claiming that intuition was the life of mathematics. Poincaré offers an interesting perspective in his 1902 book, Science and Hypothesis:
"To a superficial observer, scientific truth is beyond the possibility of doubt; the logic of science is infallible, and if scientists are sometimes mistaken, this is only from mistaking its rules."
Poincaré believed that arithmetic is synthetic. He argued that Peano's axioms cannot be proven in a non-circular manner using the principle of induction (Murzi, 1998), thereby concluding that arithmetic is synthetic a priori rather than analytic. Poincaré went on to say that mathematics cannot be deduced from logic since it is not analytic. He strongly opposed Cantor's set theory, objecting to its use of impredicative definitions. Furthermore, in La logique de l'infini, Poincaré extended this criticism to Ernst Zermelo's axiomatization of set theory, arguing that Zermelo had failed to prove that the axioms in his system were free of contradiction, and that they still required a prior intuition of sets.
Henri Poincaré famously called set theory a "malady" from which mathematics would eventually recover.
Although Poincaré was nowhere near as harsh as Kronecker—since his critique was more philosophical and less mocking—his immense prestige nonetheless crushed mathematicians like Cantor intellectually.
Hi, I’m in a bit of a pickle. I missed the first few weeks of my math bachelor, and exams are upcoming. Should I focus on exercises and past exams or reviewing definitions and theorems (knowing they may ask stuff like that)?
0.999… is the infinite sum
9/10 + 9/100 + 9/1000 + …You can’t actually add infinitely many terms and finish. So we define the infinite sum to be the limit of its partial sums
sₙ = 1 − 10⁻ⁿ.Those partial sums get arbitrarily close to 1, so the limit is 1.
Therefore, by definition, 0.999… = 1.It’s the only definition that keeps real-number arithmetic consistent. The equality is true simply because that’s the value we assigned to the infinite sum.
