Mathematics is difficult. For any of my peers contemplating a serious study of mathematics, I suggest you contemplate a little more. And if you seek an account of the study of mathematics from a mathematician, it would be best to talk to your mathematics professor, and if you are unwilling, least you can read the following message from Professor Denis Auroux (whose video lectures help my studying calculus III).

Dear Daniel,

Thank you for your kind message - I am glad you enjoyed my video lectures.

> In recent weeks, I've been contemplating about mathematics and possible change of my major to mathematics for few reasons - 1) I hold appreciation for math beyond what I observe from my peers (albeit they're engineers) and 2) I am no longer so attached to the sense of security an engineering degree is said to provide. It is difficult to make certain of my decision, however, because I do reason that mathematics is a trained art. I make analogy to music: Those who have shown aptitude for in such field have been trained since young, and those who are naturally equipped with a talent in the field are visibly better. I have had no such training in mathematics. Nor do I have strong reason to see myself as particularly talented. In sum, I feel ordinary to study math.

Mathematics is not like music: it does require serious work (like many other academic disciplines), and it is not for everyone, but it is NOT something that requires an early childhoold start to get good at it.

Also: there is a big difference in career difficulty and level of commitment between being a math major (which, really, is not harder than being a biological engineering major), and becoming a math PhD with the goal of an academic career in math (which is a pretty challenging career choice and I wouldn't recommend to every math major -- though, for those who make it through the obstacles, it is an extremely rewarding career).

> I constantly think that appreciation of mathematics alone should not be the primary reason for one to pursue math (this thought, I think, was instilled by G.H. Hardy's writing). Also, I am not certain whether what I know as mathematics, which expands up to my understanding of basic linear algebra and differential equations (subjects I am enjoying very much) is what mathematicians would regard as mathematics.

In my opinion, appreciation and interest in math is good enough motivation to give it a try by enrolling in (or, if you're really unsure, auditing) an upper-division math class. I'd recommend a first course either in mathematical analysis or in abstract algebra. (Or perhaps, first, whatever official prerequisites you might be missing). Those are somewhat challenging, but they're the best subjects to find out what it's really like to be a math major. It is only by trying an abstract math class that you will be able to find out whether (1) you do actively enjoy abstract mathematical thinking, or you'd rather stick with the less arcane math you've seen so far, and (2) with a normal amount of effort (to be expected as always when trying to learn something completely new) you end up being competent at it or not.

Obviously you should not take such a course without being willing to make some time commitment to it -- it'll more likely be a more time-consuming endeavor than the math classes you've taken so far. So: don't do this in a semester where you already have a completely full course schedule planned.

I don't know how flexible things are at Purdue, but hopefully there is a way for you to try out an upper-division math class without having to officially switch majors. If you really can't, try sitting in on a few lectures at the beginning of next semester just to get a feel for it? And perhaps take a look at whichever textbooks those classes use, to get a sense of what you'd be getting yourself into.

There is definitely a gap between lower-division and upper-division math classes in terms of the level of abstraction. You may have seen a few fairly simple proofs (or, at least, arguments given to justify theorems) in my video lectures or in other math classes you have taken. If you actively enjoyed those, and wouldn't mind a math course that has many more proofs and fewer concrete examples, and if you find it intellectually challenging in a pleasant way but not overwhelming to follow those kinds of arguments, then math may well be for you. If you enjoy mathematical concepts but would rather not deal with too much abstract proofs, look into whether Purdue has an applied math major that might satisfy your interest in math content without overburdening you with a huge amount of pure abstract math.

And keep in mind that with a math major, you're not making a final commitment to academia (fortunately!) -- as long as you are slightly careful with planning some course work outside of math, there are various careers for math majors in other fields of science, in finance, in consulting, etc. Also: if you decide that it's too scary to make it a full time pursuit but you still enjoy it: see if you can sign up for a math minor?

Finally: in case you do end up being a math major and would like to have a shot at being a mathematician: be aware that it is significantly easier to get a math degree that will lead you to "real-world" jobs than to get into a good math graduate school. So, if you end up liking it so much that you'd like to consider grad school, by the beginning of your junior year you should make sure to talk to your math professors and any mentors / advisors about how to best plan your coursework so that by the time of graduation you'll have taken enough advanced classes, gotten a taste of research, and anything else they think would help further a goal of getting to math grad school. And even then, hard work and perseverance might not quite be enough to get there -- hard work helps a lot but I am not completely sure that it will be enough to get anyone to math grad school.

Fortunately, for now you don't have to worry about such things. My suggestion would be: try taking a math class, see if you like it and whether you're any good at it. If the answer to both seems to be yes, then you should definitely consider switching. If you like it but don't seem to do well, then I'd recommend auditing math classes to satisfy your interests, since it can be fun, but not betting your career on math skills if it seems to not be coming to you naturally.

> Frankly, I think I am afraid of mathematics, for I may not be good at it. I do not want to be a fool and contribute second-rate work for a subject I deem as the most important (I think I can say I don't want to disappoint mathematics, if it has feelings). Do you suppose I am a bit too idealistic and poetic on my conception of mathematics? How do you regard mathematics and what do you suggest for students of mathematics in their view towards mathematics?

My main suggestion would be: if you are curious and would like to give it a try, by all means do so! But keep some backup plan in mind in case it doesn't quite work out (either you end up not liking it as much as you thought you would, or you just find that it's not for you).

Good luck in your studies. Best wishes,

Denis Auroux

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Abstract algebra continues the conceptual developments of linear algebra, on an even grander scale.

Dr. Paul Garrett