# Is mathematics for me?

I apologize for the vague title, but I'll attempt to explain better what I mean here.

First of all, my background: I've already completed all of my high school math courses including calculus, and I've been considering to pursue math at university/college starting in September. And yes I know that the way math is done in high school does not reflect what math is really all about in any way; for instance, the focus is on computation and hand-wavy arguments rather than rigorous development of the subject and proofs. However, I really really like rigor and proofs when it comes to math, and I hate using mathematical results which do not come with a proof.

Here are some examples of what I found that I really liked:
1. The proofs of all the arithmetic/algebraic operations we make use of when dealing with integers and rational numbers (from the "field axioms").
2. The proofs of various results from elementary number theory.
3. Euler's proof/solution to the "Seven bridges problem".
4. Proof of irrationality of the square root of 2 (plus, the idea of a proof in general)
5. Proofs of statements from Euclidean geometry.

etc...

Now, here's my problem. Even though I really enjoy the above-mentioned aspects of mathematics, I find it hard (and I don't like) to deal with the mathematical objects which do not really have a tangible or intuitive meaning. Examples of these troublesome objects (to me) include the irrational numbers, complex numbers, infinities, vectors having more than 3 components etc. And, what's even more troublesome is that it seems to me that I'll have to accept more and more of these kinds of objects if I decide to pursue math. (By the way, I should also mention that I read the first few sections of Timothy Gower's "Mathematics - A Brief Introduction". There he says that we shouldn't bother about what these mathematical objects mean, but rather we should be concerned with what they do - that is there properties. And, I found this rather disturbing.)

To wrap it up, I'd say that I really like doing rigorous pure mathematics when dealing with mathematical objects that are meaningful/intuitive to me. But, I hate doing mathematics when dealing with hypothetical/fictional mathematical objects - those objects that I can't imagine or seem completely pointless to me. So, my question is the following: would I enjoy being a pure math major? Or will all those abstraction and fictional aspects of pure mathematics ruin my interest in math?

I'm a little lost, and I don't know what to think about or what to do. I would highly appreciate being advised (especially by those who had once been in the same situation I'm in right now).

Thanks for taking time to read such a long post.

## Answers and Replies

I think you're probably a computer programmer. You get to do things that have to be rigorous in a practical sense -- they must run and produce desired results. You are always working with things that are real -- executing a computer program is a reification of a mathematical abstraction.

If you want to be more abstract than programming, do computer science. If you want to be more concrete, program databases or do computer architecture. There's a wide range, but it acts like math in its rigor and stays in the real world.

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Vahsek, you are not in bad company. Of course, humans were able to count for a long time, but I don't think I'm wrong to say that mathematics existed for roughly 3000 years already. Now, over the course of this 3000 years, the topics you mention (>3 dimensions, infinity, complex numbers, irrational numbers) have been causing great troubles for the mathematicians!

It is only the last 200 years that these concepts have found some reasonable foundation and solution. One cannot deny however, that the resolution and accepting of these concepts have been extremely benefiical on both math and science.

I think it will be very beneficial for you to read about the history of mathematics. Check out the 4 books by Morris Kline called "Mathematical Thought from Ancient to Modern Times". They are a very interested and quick read. This will be good for you because you will see how humans struggled with the concepts you're struggling with now. And you'll see what resolutions they found for things they found weird (often they totally ignored it) and what benefit the acceptance of the objects eventually yielded. I personally don't think you can be a mathematician without knowing the history of mathematics to some extent because a lot of mathematics we are dealing with now eventually comes from historical problems (although this is not always clear from how the courses are taught! sadly enough). But especially for you, reading history would be beneficial.

Second, you might also be interested to some extent in philosophy of mathematics. They also spend time on "with is infinity" or "what is a complex number". So check that out!

Do I think you can be a math major? Sure. But you need to know that the concepts you mentioned pop up everywhere in mathematics and that the professors will use these concepts without even mentioning the possible philosophical and historical issues. I personally think it's a bit of a shame, but it's what it is. If you are prepared to accept these concepts in math class, then you'll be fine. You should definitely read up on history and philosophy though.

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@ IGU : I only find programming so and so, and I do not particularly enjoy it. But, thanks for the advice though.

@ micromass : Reading up on the history and philosophy of mathematics sounds like exactly what I should be doing. Thanks a lot for this precious advice!

PeroK
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To wrap it up, I'd say that I really like doing rigorous pure mathematics when dealing with mathematical objects that are meaningful/intuitive to me. But, I hate doing mathematics when dealing with hypothetical/fictional mathematical objects - those objects that I can't imagine or seem completely pointless to me. So, my question is the following: would I enjoy being a pure math major? Or will all those abstraction and fictional aspects of pure mathematics ruin my interest in math?
In general, I recommend that you keep an open mind. It's easy to go though life saying I like this and I don't like that and close a lot of doors to yourself. Sometimes the things that don't come easily can be the most rewarding.

For example, the set of polynomials of up to degree n is actually an (n +1)-dimensional vector space. And Special Relativity includes the concept of 4-vectors. So, don't think that things with more than 3 dimensions don't exist in some way.

Also, complex numbers are used in many practical branches of mathematics - such as solving differential equations and studying electric circuits - because they allow you to combine cos and sin into the exponential function. And, this leads to a deeper understanding of both the maths and the physics.

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In general, I recommend that you keep an open mind. It's easy to go though life saying I like this and I don't like that and close a lot of doors to yourself. Sometimes the things that don't come easily can be the most rewarding.

For example, the set of polynomials of up to degree n is actually an (n +1)-dimensional vector space. And Special Relativity includes the concept of 4-vectors. So, don't think that things with more than 3 dimensions don't exist in some way.

Also, complex numbers are used in many practical branches of mathematics - such as solving differential equations and studying electric circuits - because they allow you to combine cos and sin into the exponential function. And, this leads to a deeper understanding of both the maths and the physics.
I agree that I should be more patient; maybe it will all make sense once I learn much more math. Thanks PeroK.

I agree that I should be more patient; maybe it will all make sense once I learn much more math. Thanks PeroK.
My favorite mathematician was John Von Neumann. He once said "In mathematics, you don't understand things, you get used to them." It's a bit of a weird statement, but I guess he was refering to your case. You might not exactly understand what irrational or complex numbers are, but you'll get used to them.

Questioning irrational numbers and complex numbers is a good sign and shows you are quite intelligent. But you should certainly keep an open mind, because the use of complex numbers and irrational numbers is undeniable. So yeah, study history and philosophy on the side. I think it will certainly interest you!

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My favorite mathematician was John Von Neumann. He once said "In mathematics, you don't understand things, you get used to them."
Micromass beat me to my favorite quote! I really think this is true though... Take a simple example, sine, cosine and tangent. I still remember initially wondering why we needed these silly functions to represent the ratio of two sides of a right triangle. Later with trigonometry, the answer became obvious.

Other examples of things I was uncomfortable with at first but eventually got used to were Legendre polynomials and Bessel functions. I think this happens with most unfamiliar mathematical objects until you are able to see how they are used in solving a problem for example.

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If you are uncomfortable with it then maybe that is what you should be doing. There is nothing like a challenge. Just a thought. Great admiration often comes from things that seem obscure to begin with.

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Micromass beat me to my favorite quote! I really think this is true though... Take a simple example, sine, cosine and tangent. I still remember initially wondering why we needed these silly functions to represent the ratio of two sides of a right triangle. Later with trigonometry, the answer became obvious.

Other examples of things I was uncomfortable with at first but eventually got used to were Legendre polynomials and Bessel functions. I think this happens with most unfamiliar mathematical objects until you are able to see how they are used in solving a problem for example.
Yes. As somebody teaching math I often feel uncomfortable when students ask me why all of this is necessary or why we define something the way we do. One thing to realize when you're doing math is to have patience. And to trust your teachers, something might feel pointless at first because some concepts just take a while to sink in.

Of course, a lot of this is due to the fact that we teach math in one semester that maybe took hundreds of years to develop. Furthermore, we teach in a style (definition-theorem-proof) that is completely opposite to how math is developed. For example, you can say one of the main applications of calculus was to compute areas. But it takes you a full semester to finally get to that point. They start doing stuff which seem totally unrelated such as limits and continuity. Then derivatives, integrals and only then areas. The way it was developed was always with the application in mind. So limits and continuity actually came last!! I think the way they teach now is very good, but it has as downside that people don't really see the point at times.

This is why I advocate knowing the history of mathematics well. It will help you to see why they do stuff and when it was invented. This way you can always keep the big picture in mind.

Vahsek, you sound a lot like me, especially when I was a bit younger. It sounds like you would like math as an undergrad, but graduate level might be frustrating/disappointing for you. As an undergrad, getting an intuitive understanding AND being rigorous and proving things thoroughly is what it's all about, I think. The lower-level classes will be less rigorous and more computational, of course, but it's good to have some sort of transition.

But at some point, you get to what Terence Tao calls the "post-rigorous" stage, where things actually get more hand-wavy and less rigorous, although you still have a rigorous foundation to buttress it.

In some ways, it can be liberating because you can focus more on ideas than formal stuff. But someone who is really obsessed with rigor can get into trouble because they will be too slow. Or they might be unhappy because they are forced not to be slow. I've known grad students in both of those categories. I think I suffered from that a bit too, but I would say it had more to do with my desire to be thorough, rather than completely rigorous, which is a slightly different thing. I wanted to be able to think about things until they were clear in my mind and there wasn't enough time with such a vast amount of material to learn. One solution could have been not to worry about subjects outside of my research, but you don't want to go too far in that direction and become really narrow. I spent quite a bit of time rethinking subjects that didn't have much bearing on what I was doing in my research, and I think that slowed me down considerably. Some exploration is good, but what I failed to do was make my dissertation the priority (I also had stuff going on in my personal life and became depressed and had too strong of an interest in physics, none of which helped).

Anyway, people like us may suffer a lot when we are bombarded by too much abstract material that we don't have the time to process in a more concrete way. Which is what math grad school seems to be all about these days, pretty much.

Vahsek said:
Examples of these troublesome objects (to me) include the irrational numbers, complex numbers, infinities, vectors having more than 3 components etc. And, what's even more troublesome is that it seems to me that I'll have to accept more and more of these kinds of objects if I decide to pursue math. (By the way, I should also mention that I read the first few sections of Timothy Gower's "Mathematics - A Brief Introduction". There he says that we shouldn't bother about what these mathematical objects mean, but rather we should be concerned with what they do - that is there properties. And, I found this rather disturbing.)
I'm not convinced that irrational numbers are such a big deal. They are just needed to make the real line a continuous one. I'm not sure why the Greeks would have expected any two lines to be commensurable. It's fairly intuitive to me that that wouldn't be the case if you are allowed to have infinitely fine variations in length. As far as complex numbers, if you read Visual Complex Analysis, you'll probably be cured. Once people found a visual representation for them, they started making a lot more sense and a lot of progress was made. Higher dimensional objects come up pretty naturally. For example, if you have a spread-sheet with numbers as entries, you could think of each column as a big vector with many components. It's actually a more intuitive way to think of it than to think of it as just a list of numbers. A related example in physics is a configuration space. If you have one point particle, its configuration space is 3-dimensional space. Just specify its coordinates in some coordinate system. If you have two point particles, you need to give coordinates for each one, so altogether, you get a 6-dimensional space. So, actually, higher dimensional spaces can be very physical.

micromass said:
And to trust your teachers, something might feel pointless at first because some concepts just take a while to sink in.
Some amount of "getting used to things" is part of the process, but often the lack of motivation is entirely unnecessary, and that needs to change. Sometimes, things make sense with experience and practice, but maybe even more often for me, they, instead, make sense after doing a lot of reading and thinking until I finally come to the conclusion that, yes, it makes sense, and why didn't someone just TELL me that from the beginning? Would have saved me quite a bit of work. I think you appreciate that to some degree, given what you've said. You can usually trust the teachers that there's a point, but you shouldn't necessarily trust that it's necessary for them to give poor motivation because a little research usually proves that it's not.

So, yes, read your history of math.

But beyond that, you have to find your list of favorite authors to turn to. For me, that includes Bill Thurston, VI Arnold, John Stillwell, Tristan Needham, Richard Feynman, Morris Klein, David Bressoud, Roger Penrose, John Baez, and Cornelius Lanczos. Once you get to more specialized material, you have to find your own favorite people.

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Whoa that's a lot to digest. But anyway, I'll take it one step at a time and at the right time. For now, I'm going to continue with the reading, and I'll keep on exploring more about math.

I really appreciated all of the pieces of advice that I got in this thread, and I am relieved that my problem is not so bad after all.

Thank you all once more.

Here's another thing I thought of. You should be careful what branch of math you choose to study, if you do. I think maybe some form of combinatorics might be a good choice for you. But the problem is, you are generally required to have a bit broader background, so you have to be sort of okay with all different kinds of math, even though you can choose one branch that fits you the best.

Also, think about what job you can get at the end of it and whether you'd really like teaching. If you wouldn't like teaching, or if there's a chance you wouldn't like it, you might want to come up with a back-up plan.

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Here's another thing I thought of. You should be careful what branch of math you choose to study, if you do. I think maybe some form of combinatorics might be a good choice for you. But the problem is, you are generally required to have a bit broader background, so you have to be sort of okay with all different kinds of math, even though you can choose one branch that fits you the best.

Also, think about what job you can get at the end of it and whether you'd really like teaching. If you wouldn't like teaching, or if there's a chance you wouldn't like it, you might want to come up with a back-up plan.
For now, I find elementary number theory, counting, graph theory (introductory), and euclidean geometry extremely appealing, so yeah I guess I might have an inclination for combinatorics.

In comparison, I am uncomfortable when dealing with any mathematical object that is too abstract; to deal with this kind of math, I always think of it as a game with a set of "rules". But, I'm looking forward to eliminating this weakness.

And yes, I agree with the idea of having a back-up plan for the future; it's generally good to keep an open mind.