# What advice would you give a math major that you would like to see when starting math?

Updated on : December 6, 2021 by Alberto Sweeney

## What advice would you give a math major that you would like to see when starting math?

I am not a mathematician so I hesitate to answer but someone asked me to answer and insisted even after I refused.

I like math, I enjoy it in the same way that I enjoy a symphony, not normally as a participant, but as a spectator. I am a huge fan of mathematicians! They constantly and reliably expand the knowledge of humanity, and the usual method they use to do so is by choosing to impose on themselves a prison sentence of 20 years of forced labor in solitary confinement. This is the intellectual equivalent of medieval monks who flogged themselves. I did some flogging in my youth

Keep readingI am not a mathematician so I hesitate to answer but someone asked me to answer and insisted even after I refused.

I like math, I enjoy it in the same way that I enjoy a symphony, not normally as a participant, but as a spectator. I am a huge fan of mathematicians! They constantly and reliably expand the knowledge of humanity, and the usual method they use to do so is by choosing to impose on themselves a prison sentence of 20 years of forced labor in solitary confinement. This is the intellectual equivalent of medieval monks who flogged themselves. I did a flogging in my youth, and I can say that it was rewarding, and it is extremely important because this mathematical thought is the only real thought.

First, the obvious. You should read math! Read great mathematicians, past and present. Read historical work, read current work. Read the original authors, read exhibits if you don't understand. If there is a new idea or method, learn it. Read all the works that interest you, but not the ones that don't interest you. That will be more difficult when you are forced to read things that do not interest you in order to obtain a degree. But always take the time to read the things that interest you. I loved transcendence theory, I loved complicated continuous constructions in analysis, I was bored with group theory, but not so much today.

But I fell in love with math for 10 years. The reason is that I did not understand the foundations well. I was completely shattered as a student by the agony of the foundations, in the third year, I decided that mathematicians were full of rubbish and I stopped studying their work, because I could no longer read it. I did not trust set theory because it kept proving more and more impossibly wrong things, such as the good order theorem (reals do NOT have a good order, this is obvious), the existence of a non-measurable set (there is no set not measurable --- you can pick a random real number between 0 and 1), and the higher-level stuff turned into a swamp of shaky results that it was impossible to keep in order. Was the Radon-Nykodym theorem true? Is it really true? Is it true for some things not for others? Is it usually false? Maybe yes maybe no, there was no way to decide. How about ultrafilters? Do they really exist? Do the networks make sense? Do they really generalize sequences? It is a terrible situation to have to qualify all the theorems in your head in this way, you need to have a solid framework to hang the results.

As an undergraduate student, I actually got to the point where I began to suspect that set theory might be inconsistent with omega (this is not true, but that's why you need to sort out the fundamentals). There are theories that are self-consistent, but that prove lies about the behavior of computer programs, saying that certain programs stop when in fact they do not. Since set theory was proving all these absurdities about the ordering of wells and the continuum that I couldn't understand, I thought I could prove that a program that doesn't stop stops, maybe using some ultrafilter construction, and then the Radon Nykodym's theorem, then a bit of order, and voila, this uninterrupted program has stopped. The worst part is that you would never know Because no matter how long you watch the show, you won't know that you didn't just stop yet. This got me spinning, and I decided I didn't need this kind of heartache, and it's the mathematicians' fault for telling lies, so I don't need to listen to these idiots.

The resolution came a decade later while speaking to a professional mathematician in a coffee shop. He explained his own fundamental struggle to me and learned the axioms of ZF, then Godel's proof of the completeness of logic, and so on. Then he showed me his own work on complex maps, which I really liked, and I got excited about math again, went back and solved the basics. It was actually very fast, it was solved in about a month when I read "Set Theory and the Continuum Hypothesis" by Paul Cohen. The original, no other forcing exposures. What was important were Godel's completeness theorem (an algorithm to make sense of axiomatic systems), the ZFC axioms (the axioms to make sense), Skolem's theorem (that models are really countable), Godel '

The point is that axiom systems describe accounting models, not an abstract universe. Once you understand this, all the uncomfortable results will become obvious. You can immediately interpret any theorem you read in an analysis or topology book as "true on L", this allows you to hang it on your "L" frame. Then you can understand any intuitive probability or measure theory construction as "true in the Solovay universe", and the results that are embedding the measure theory in L, you can hang on the "useless nonsense" shelf.

Then you understand that set theories with powerset are themselves only reflections in the sense of Godel's theorem of set theories without powerset. Set theories without powerset are reflections of arithmetic, and arithmetic is a reflection of its fragments, and this finally hits the groundwork in calculating things with whole numbers. So set theories are NOT inconsistent with omega, they are perfectly fine, they are extensions of Godel's method of previous consistent theories using ordinal strings.

This point of view completely solves the anguish of the foundations, but it rekindles certain politically closed questions. One becomes interested in proving the consistency of set theory by finite means again, using large countable ordinals (these are finitary when they can be represented on a computer). The modern version of Hilbert's program is called "Ordinal Analysis", and it continues in complete isolation within logic, but they demonstrated a lot of things, including the consistency of Kripke-Platek set theory a while ago, and some theories of larger sets most recently. Rathjen has written about this.

The result of this is that all ordinals are countable, the reals are ordinals in any theory just because they are real models, each set of reals is measurable (simultaneously true, but in a different more platonic model of the reals), and the results Mathematics must be classified in the form "L" "not L" to classify the theorems correctly. From this point on, I had no more difficulties with literature, apart from the usual ones of time and difficulty.

Unfortunately, the questions that arise when reading the literature on logic are completely different from mainstream mathematics. But I think this literature is really dealing with all the complexity of mathematics in total generality, whereas a more specific domain, like schema theory, really tries to classify the regularities in more traditional questions about prime numbers, etc. So I like the literature on logic, because it seems more free from human biases about what is important. But the rest of the stuff is also good, not bad, and it seems that is where all the revolutionary things are happening today.

My personal taste in math is testing the obvious things that no one can prove, statistical regularities that are obviously true, but completely unattainable by any known orderly progress method, because the results are statistical, they are not organized. I think the biggest breakthrough here is Appel and Haken's method, their proof of the 4 color theorem, because this seems to be an unexplored and promising path. The surprising thing is that they only needed to use heuristic probabilistic estimates, because then they used a computer program and checked various download algorithms until they found one that worked. Any download algorithm proves a lot of useless stuff about the existence of multiple random subgraphs,

This method looks very promising for tackling superficially insurmountable problems. You can prove many individually useless theorems automatically about subproblems, the theorems only prove the result when the decomposition somehow covers the space of all the examples, and you join these automatically proven subtheorems to prove the result by an automatic search. . All you need are some heuristic estimates about the probability that each subtheorem is automatically provable and covers enough cases to prove the entire theorem. I'd love to try doing some theorems like this when I have some spare time and try to prove something statistically obvious, like the normality of some number. But this is not likely to produce something so widespread,

First and foremost, READ BEFORE THE CLASS. I can't tell you how much clearer it made things to me when I took the time and read the theorems and proofs before the teacher covered them in class. It's okay if there is a step or many steps you don't get, but after viewing the material once before class, you will not only be able to focus on the parts you did not understand, but you will also be able to just sit down. go back and absorb the material. You will not be writing notes desperately because you already know the subject briefly. As a result, you can sit back and take the time to

Keep readingFirst and foremost, READ BEFORE THE CLASS. I can't tell you how much clearer it made things to me when I took the time and read the theorems and proofs before the teacher covered them in class. It's okay if there is a step or many steps you don't get, but after viewing the material once before class, you will not only be able to focus on the parts you did not understand, but you will also be able to just sit down. go back and absorb the material. You will not be writing notes desperately because you already know the subject briefly. As a result, you can sit back and take the time to understand what is happening (since you know what will come next). Things will fit together a lot better.

Second: work collaboratively, even if you think problems are easy. I have often thought that I solved a problem correctly, only to find out by listening to other people solve the problem or help others that A: I did not get it completely right, or B: I missed some subtle element of the test that would have made the solution. was insufficient.

Third: learn to code. Programming opens doors. If you know math and programming, you can get a job in almost any quantitative field, be it finance, technology, or life sciences. Also, if you like math, you will find it absolutely fascinating how programming works.

Fourth: Learn LaTeX! I can't tell you how much time it has saved me writing tests. Unlike my peers, I never have to rewrite my tests in a neat format and spend hours doing that. Plus, once you're super comfortable with LaTeX, you can do all your work in LaTeX. I write my notes from class to class, which makes it so much easier to change my notes, and creates such beautiful documents! Also, I never have to translate written notes in a test because I just started my problem sets in TeX.

Fifth: Be as formal as possible with your tests, when you start writing tests. When you're new to testing, you'll probably make assumptions and logic jumps that aren't necessarily followed. The way around this is to make sure that from the beginning to the end of the test, each sentence is a direct implication of the previous one. This will necessarily result in fairly lengthy tests that are likely to be more detailed than necessary, but it takes time to determine how much is needed to give a convincing proof. The tendency of most students is to give too few details and as a result take a false logical step that makes the test incorrect. When you require of yourself that no statement be made without a direct logical link based on a definition,

This is probably the best place to share my 'wisdom'. Well, when I started math, I thought it would be cool to see numbers every day of my college life. INCORRECT. In fact, it's the exact opposite, most of my time was spent proving theorems and sometimes ... disproving them.

During my math classes, I wish someone had told me: 'PAY ATTENTION AND JUST ASK IF YOU DO NOT UNDERSTAND OR IF IT IS LITTLE' because most of the time, I found that most of the mathematical theorems and proofs are quite trivial, but When I ran into similar problems, I would find them quite difficult to solve.

Keep readingThis is probably the best place to share my 'wisdom'. Well, when I started math, I thought it would be cool to see numbers every day of my college life. INCORRECT. In fact, it's the exact opposite, most of my time was spent proving theorems and sometimes ... disproving them.

During my math classes, I wish someone had told me: 'PAY ATTENTION AND JUST ASK IF YOU DO NOT UNDERSTAND OR IF IT IS LITTLE' because most of the time, I found that most mathematical theorems and proofs are quite trivial, but when I ran into similar problems, I would find them quite difficult to solve. This is because, although the solutions seemed "trivial", some parts were not and required a deep understanding. Lesson: Just because they seem trivial on the surface, make sure you understand them.

I also wish someone had told me that college math is different from high school math. High school math is about beating numbers and finding a final solution. All of that is not important in college, because we have calculators and software to do it for us. But what matters is the understanding of the concepts and the ability to see the problem from different perspectives. The final solution does not matter, it matters HOW TO GET THERE.

Oh, math problems don't take an hour to solve, or two, or three, most of them take half a day or even a day to understand. Do not be discouraged when you get stuck solving something, I advise you to SLEEP AND continue at another time, because you would simply overthink it and start feeling confused, both will not help you.

Don't bother trying to memorize 'solutions' for math tests, because there are an infinite number (roughly speaking) of ways you can be asked, so dig into the concepts first. Also, it's okay to fight, it's okay to feel like nothing makes sense, you're not the only one feeling that way, so I suggest you find a partner or a math study group, because two brains are always better than one. .

Few things on top of my head.

It is worth taking the time to learn some programming.

If you major in math, it probably came easy for you in high school. Unless he is exceptional at some point during his career, it will get tough. Keep an eye on that point, because if you miss it, you will start to fail in classes because you didn't realize you had a job to do, like I did.

Try to find a balance between in-depth courses in the topics that interest you and gaining a broader experience. Don't study too deep and narrow.

But on the other hand, don't take courses that will just bore you.

Mathematics is

Keep readingFew things on top of my head.

It is worth taking the time to learn some programming.

If you major in math, it probably came easy for you in high school. Unless he is exceptional at some point during his career, it will get tough. Keep an eye on that point, because if you miss it, you will start to fail in classes because you didn't realize you had a job to do, like I did.

Try to find a balance between in-depth courses in the topics that interest you and gaining a broader experience. Don't study too deep and narrow.

But on the other hand, don't take courses that will just bore you.

Mathematics is a collaborative subject. The best way to build your own understanding is to work with others and explain your reasoning to them so they can criticize you. But obviously don't work assignments together when that's not allowed.

Read the classic Lakatos Proofs and Rebuttals.

The first time, read only the main text. For the second time, read the text and footnotes. If the last chapter is too difficult or dark, skip it.

Then re-read everything in the middle of your first testing course, when you start to wonder how the hell people on Earth came up with these strange definitions.

Then read it again after you earn your degree, especially if you are graduating in math.

Have fun!

Study some more on the side to get some job skills. Mathematics is great, really, really great, but the job market for pure mathematicians is very bad. Computer science, statistics, economics, and operations research are great options and worth exploring.

Study at least a little (30 minutes minimum) every day. My first Berkeley math teacher advised me. Learning mathematics is like learning a language. Small bites more often instead of larger, less frequent meals. I was surprised that this tip really helped and I spent less total time using this method. The consolidation of mathematics learning occurs in the intermediate periods, such as when daydreaming.

It's a long journey, keep the following in mind:

1. Money-oriented people can never become mathematicians. But have a great lust for knowledge.

2. Read different books for each topic (ask your teacher or the best books on Google).

3. Carry out parallel teaching courses.

4. Tutor children and friends who are weak in math.

5. Share your ideas on math concepts with bright students and teachers.

6. Meditate twice a day (Ramanujan approach)

May God give you the ability to study and understand a lot of mathematics.

Om

Stick to pure math. Maybe do applied mathematical physics or something like that, but keep the stats to a minimum. With pure mathematics you will get a much broader and deeper education. And you can find a branch that you really like.

If you really love math, keep it up! But do an engineering course in addition. If you don't love math, quit as soon as possible and take an engineering course.

Yes, I definitely regret it. He would have been making a lot more money if he had studied economics and made connections through extracurricular activities. All the jobs that I get callbacks for are technical jobs that I'm mostly coding for. It will pay quite well (100 to 200k range) but would have been making the same amount of money if you had been a product manager or done private equity after taking the finance / consulting + MBA route. Both jobs, no offense, don't require as much problem-solving skill as working as a data scientist or software developer. They require social skill

Keep readingYes, I definitely regret it. He would have been making a lot more money if he had studied economics and made connections through extracurricular activities. All the jobs that I get callbacks for are technical jobs that I'm mostly coding for. It will pay quite well (100 to 200k range) but would have been making the same amount of money if you had been a product manager or done private equity after taking the finance / consulting + MBA route. Both jobs, no offense, don't require as much problem-solving skill as working as a data scientist or software developer. They require social skills, which are higher on average among non-mathematicians ...

If you are looking to make money and you are at Harvard or Yale, I would choose CS or economics instead. Math / CS is also an excellent combination. But majoring in math alone is a silly move if you don't enjoy programming because that's probably what you'll do for a job with a math degree. If you are looking to do research in CS / economics, then a math major is helpful but not necessary. Learn additional math if necessary.

From a practical point of view, there are other disciplines that involve a lot of mathematics, such as IEOR, EE, CS, and quantitative economics, that can be traded more immediately ... and even more enjoyable.

A math major is a great foundation, but unless you're determined to teach or research math, you'll need programming skills to make money. Employers will choose a Harvard economics graduate with an internship in mathematics major from MIT for a product manager position with the latter accelerating for software developer / data science jobs. They pay well, but they know that you will work much harder for similar money.

Nobody mentions this because most of the time, math students like to code for fun. But if you take a closer look at who really makes money, you'll see that the venture capitalists and hedge fund managers are the ones who make the bank, not the software engineer making 150k in SF where the cost of living is outrageous.

***

An exception is made if you are good enough to get a job at a place like Citadel or DE Shaw as a quantitative researcher and stay there for a few years. It's not surprising to earn 500k in those places and Chicago and New York are actually a bit cheaper than SF so you would save even more money than if you worked at a tech company in Silicon Valley and made between 150 and 200k ...

Just my observations as someone who worked as a data scientist in technology and had the opportunity to intern at a hedge fund.

It sounded harsh before, so maybe I should clarify that I enjoy studying and learning math. I'm sorry I didn't think more about other topics (mine are economics / finance) that I was interested in to be able to change in that direction.

### Other Guides:

- Which place is good to relocate for an IT job, Bangalore or Pune? They both offered the same package, and I'm from Andhra, please give me details by comparing factors, not just distance.
- How many people have found their dream job?
- I've been looking for a job for two months and I still haven't received any offers. I'm dangerously close to not being able to pay living expenses (more details in the comments). That I have to do?
- What is the best job you can do as a teenager?
- Is it easy to find a job in Australia?
- What should I do after the master's degree in mathematics?
- Are there any genuine websites that offer data entry work from home?
- What if I quit my job after 8 days without notice, also the company has deposited my 10th pass certificate as collateral and now asks me to pay back the 3 months salary or comply with the 3 month notice period?
- What made you leave a high-paying job to take a low-paying job? Do you regret it?
- What is something you find that most people can easily be persuaded to do?
- Is a one-year gap in CAT okay? Should I prepare for the CAT with a job or leave it for a year? Which is the best option?
- I am 15 years old and I really want a job at McDonald's, but I have no experience. What should I do, go in and ask?
- Hydrogen and oxygen combine to form water. Can water be broken down into hydrogen and oxygen? Is this reaction reversible?
- What are the things to say in each interview?
- What are the best courses after mechanical engineer?
- I am 16 years old, what is a good summer job that pays well?
- Should you pursue back-end or front-end development?
- How would a teenager get their first job?
- Does doing an internship put you in a better position to get hired for more internships?
- Will you quit your job for your boyfriend if he asks you to?
- What should I do if I quit but my boss is manipulating me and won't let me quit easily?
- I continually feel bad about myself because my friends are earning so much more than I am. That I have to do?
- Why does my body ache after smoking marijuana?
- I have a LinkedIn profile of stars. But is there REALLY any value in having one for your job search? (my profile is premium, just in case).
- What do the Turks think about the German, Dutch and Austrian ban on campaigning in Europe?
- How long does it take to pay off my 30-year mortgage?