Other Should I Become a Mathematician?

[Deveno]

one can look at mathematics as a type of game people play. as with any game, there are differing strategies:

a) one can play by "gut feeling"
b) one can make an exhaustive analysis of the rules
c) one can devise a "toolkit" which covers most common situations
d) one can adapt strategies from some other game, and hope they work
e) something else entirely

there are certain attractive, and unattractive features of every strategy, including: how much information has to be internalized, how efficient the application of the strategy is, how successful it is.

the "theoretical" approach aligns most closely with (b). this requires a long "learning curve" and a good deal of retained information. it is highly successful and efficient in application. most people in point of practice go with (c), which represents a compromise between (b) and (d). it should be noted that people who stick with (d) usually resort to (a) if their approach doesn't work. (d) doesn't require a great deal of retained information, because adaptation is certainly easier than assimilation.

(a) is arguably the worst explicit strategy listed, because it relies extensively on internal inductive reasoning (unconscious pattern recognition). some people use it reasonably well, arguably because they are better at recognizing relevant information without first "translating" it into some other area.

in any case, there's an inherent tension between abstract/instance. how deeply does one examine the particular example of an interesting case with "nice" properties (like, for example, the real numbers, instead of an arbitrary field), versus examining the shared characteristics of a wide variety of disparate examples (like abstract linear operators in a hilbert space, rather than complex matrices)?

in one sense, topics like category theory, and differential equations lie at "opposite ends" of this spectrum. one studies structures so general they seem removed from anything remotely "real" at all, while the other studies things so rooted in reality, and particular in nature, that the methods are tailored to the distinct case in hand (this type of function, with these types of numbers, subjected to this constraint, under these sets of assumptions). what it means to get a "result" and what is meaningful, is very different for these two areas.

i argue that a well-rounded individual needs both: an orderly set of cupboards to organize the ideas (abstract), and plenty of food in them (particular and interesting examples). for example: the abstract properties of a determinant aren't needed if you never actually calculate any determinants, and for a particular determinant calculation, knowing the abstract properties can make the calculation easier (short-cuts). the abstract gives mathematics shape, and the concrete gives mathematics flavor and texture.

 

[homeomorphic]

You guys should never look at Bourbaki's or Hoffman/Kunze's definition of the determinant so

Not sure what you're getting at, but if it's the exterior-algebra approach, I don't see it as being fundamentally different from the visual definition as the signed volume of the parallelepiped spanned by the column vectors. It's basically the same thing, except it's more general. But, then, I don't know their definition. I don't think there is a definition of determinant that I can't interpret geometrically or at least intuitively in one way or another.

 

[homeomorphic]

i argue that a well-rounded individual needs both: an orderly set of cupboards to organize the ideas (abstract), and plenty of food in them (particular and interesting examples).

I agree, but your cupboard analogy makes it seem like the cupboards should come before the food, which is backwards. Design the cupboards with the food in mind. The cupboards are not an end in themselves.

It's not so much abstraction that I object to. It's unmotivated definitions (i.e. not laying the groundwork for the abstractions), and gruesome calculations that obscure the concepts. I'm not against all gruesome calculations, but I see it as extremely unfortunate if they should impinge on the theory. Ideally, they ought to be more like the end result of a theory, not part of the theory itself. For example, you can understand curvature very well theoretically, and then compute some curvature in some example and it will be horrific, and you won't understand it conceptually. But you just want the answer, so it's fine. Also, maybe you can compute the curvature of a sphere because it's good practice, even though it can be understood conceptually. So, as practice or as a way to get answers, I have no problem with calculation. But it irks me when it REPLACES conceptual understanding. That is the real shame.

 

[sponsoredwalk]

Not sure what you're getting at, but if it's the exterior-algebra approach, I don't see it as being fundamentally different from the visual definition as the signed volume of the parallelepiped spanned by the column vectors. It's basically the same thing, except it's more general. But, then, I don't know their definition. I don't think there is a definition of determinant that I can't interpret geometrically or at least intuitively in one way or another.

All I meant by that comment was that these books give the most "rigid, formal" definitions
of the determinant that I've seen (well, except for one in a book called Linear Algebra &
Group Representations) with such a lack of motivation that it caused me to stop reading
both despite the fact I was pulling my hair out trying to understand the concept at that
abstract a level

 

[mathwonk]

here are free class notes on linear algebra (intended as a second course), with determinants discussed starting on page 62.

https://www.math.uga.edu/sites/default/files/inline-files/4050sum08.pdf

 

[mathwonk]

happy thanksgiving! (sorry, off topic)

 

[kramer733]

What kind of textbook do you guys recommend with a course description of the following
:

http://www4.carleton.ca/calendars//ugrad/current/courses/MATH/2000.html

Calculus and Introductory Analysis II (Honours)
Higher dimensional calculus, chain rule, gradient, line and multiple integrals with applications. Use of implicit and inverse function theorems. Real number axioms, limits, continuous functions, differentiability, infinite series, uniform convergence, the Riemann integral.

Would i still use spivak calculus?

I'm using spivak now for the following course:

http://www4.carleton.ca/calendars//ugrad/current/courses/MATH/1002.html

Elementary functions. Limits. Continuity. Differentiation. L'Hôpital's rules. Indefinite and definite integrals. Improper integrals. Sequences and series, Taylor's formulae. Introduction to differential equations.

I'm supposed to use stewart's calculus book but I've heard bad things about it and instead bought spivak for $50.00. Compared to what they were selling stewart's book for, spivak was 3-4x cheaper.

 

[deluks917]

Spivak "Calculus" doesn't have any multi-variable stuff in it. Spivaks calculus on Manifolds is quite good and covers Higher Dim Stuff. If you haven't already taken it I really suggest you learn some linear algebra before taking that multi-variable class.

 

[bpatrick]

Granted, it has been a while, but I used the 5th edition of Stewart's single variable and multivariable calculus books when I learned the majority of my calculus, and for what it's worth, I thought they were wonderful books ... not sure why they'd have a reputation of being bad.

 

[Nano-Passion]

Granted, it has been a while, but I used the 5th edition of Stewart's single variable and multivariable calculus books when I learned the majority of my calculus, and for what it's worth, I thought they were wonderful books ... not sure why they'd have a reputation of being bad.

Who said they were bad? I saw them recommended a few times. I use stewart's 4th edition as a supplement with Larson's Calculus.

 

[homeomorphic]

Granted, it has been a while, but I used the 5th edition of Stewart's single variable and multivariable calculus books when I learned the majority of my calculus, and for what it's worth, I thought they were wonderful books ... not sure why they'd have a reputation of being bad.

They are okay, but they could be better. I can't recall specific examples, but there are many cases in which there are better explanations of things out there. They aren't terrible. You can learn calculus from it and you'll get the main concepts, but you won't have the deepest possible understanding of everything. Which, maybe you can't expect the first time you learn it, anyway, but it could be improved upon.

The last chapter about Stokes theorem and that stuff has some particularly bad sections in it.

 

[Sina]

What kind of textbook do you guys recommend with a course description of the following
:

http://www4.carleton.ca/calendars//ugrad/current/courses/MATH/2000.html

Calculus and Introductory Analysis II (Honours)
Higher dimensional calculus, chain rule, gradient, line and multiple integrals with applications. Use of implicit and inverse function theorems. Real number axioms, limits, continuous functions, differentiability, infinite series, uniform convergence, the Riemann integral.

Would i still use spivak calculus?

I'm using spivak now for the following course:

http://www4.carleton.ca/calendars//ugrad/current/courses/MATH/1002.html

Elementary functions. Limits. Continuity. Differentiation. L'Hôpital's rules. Indefinite and definite integrals. Improper integrals. Sequences and series, Taylor's formulae. Introduction to differential equations.

I'm supposed to use stewart's calculus book but I've heard bad things about it and instead bought spivak for $50.00. Compared to what they were selling stewart's book for, spivak was 3-4x cheaper.

Marsden, vector analysis. I think it has Newton on its cover.

 

[kramer733]

Marsden, vector analysis. I think it has Newton on its cover.

Is the rigor on par with spivak's calculus book?

 

[nucl34rgg]

Granted, it has been a while, but I used the 5th edition of Stewart's single variable and multivariable calculus books when I learned the majority of my calculus, and for what it's worth, I thought they were wonderful books ... not sure why they'd have a reputation of being bad.

Anyone who tells you "Stewart is bad" because it is handwavy is being a math snob. Yes, it lacks rigor. That doesn't matter. The point of the first course in calculus is to wrap your head around the fundamentals and teach you to calculate things. If you want rigor, there are better books that firmly set you up for more courses in analysis, such as Fitzpatrick, Rudin, or Wade. Honestly, I can't imagine skipping Stewart and simply starting Wade. I'm sure I'd be able to explain the theorems nicely, but I sure as hell wouldn't be able to solve any problems! :P I'm almost positive that everyone goes through something like Stewart first to become familiar with the computations of calculus.

 

[lurflurf]

^That is not true. Stewart is like many popular calculus books in that instead of actually teaching calculus it teaches one to solve trivial problems without any understanding like a defective calculus robot. It is true that many people enjoy and benefit from a first calculus book that is theoretical. Theorems are problems, so it is nonsense that there is some risk of understanding so much that one cannot solve trivial problems. Stewart is not a particularly good book of its type. There is not much to recommend it, even if cookbook calculus is desired.

 

[Dougggggg]

I used Stewart for Calculus, then Lay for Real Analysis. I liked Stewart for like nucl34rgg said, showed me what something was and how it worked in a way very pleasing to my intuition. Now that I am in Real Analysis, I'm appreciated that slight bit of Calculus intuition to help give me an idea of where to start trying to understand the actual foundations of it all. However, with the exception of Calc III, I had teachers that went through at least a watered down version of different theorems and their proofs. I still do use the Stewart text for reference whenever I am doing something else and I'm not 100% sure on something.

 

[Nano-Passion]

^That is not true. Stewart is like many popular calculus books in that instead of actually teaching calculus it teaches one to solve trivial problems without any understanding like a defective calculus robot. It is true that many people enjoy and benefit from a first calculus book that is theoretical. Theorems are problems, so it is nonsense that there is some risk of understanding so much that one cannot solve trivial problems. Stewart is not a particularly good book of its type. There is not much to recommend it, even if cookbook calculus is desired.

Stewart is a good introductory book. I think it has some pretty challenging problems [non plug and chug] in the book, you just have to look for them. I would personally recommend many to uses Stewart to grasp the basic concepts and problem solving and hit a more rigorous book such as Spivak, that seems like the best thing to do.

What I do is that as soon as I feel like that the plug-and-chug problems are too easy I will stop. Its practice to me so that I know that I won't mess up on the test, which is then followed by some harder problems that require thinking.

 

[Sina]

Is the rigor on par with spivak's calculus book?

I have not seen spivak's book but I wouldn't think that the level fo rigour on Marsdens book to be inadequate. But he also puts much importance to intuitevely and geometrically grasping the topics. That might be disturbing if you happen to develop good feelings toward Bourbaki

ps. just because it has Newton in its cover doesn't mean it is non-rigorous

 

[Sina]

Okay I will ask a question my self.

Is there any good books on functional analysis that goes parallel with application to quantum mechanics?

Remark:
1- I am already reading von neumann's book but ofcourse its scope is limited
2- I actually like von neumanns approach where he builds resolution of identity as a measure
so that approach would be a bouns
3- I know reed and simons book but I think it is mathematics first applications later right? I like it better when ideas are immedieatly applied to some physical problems.

Best wishes

 

[homeomorphic]

Okay I will ask a question my self.

Is there any good books on functional analysis that goes parallel with application to quantum mechanics?

Remark:
1- I am already reading von neumann's book but ofcourse its scope is limited
2- I actually like von neumanns approach where he builds resolution of identity as a measure
so that approach would be a bouns
3- I know reed and simons book but I think it is mathematics first applications later right? I like it better when ideas are immedieatly applied to some physical problems.

Best wishes

I'm not sure if this is what you want, but since no one else answered, I would recommend the last chapters of Robert Geroch's Mathematical Physics. Best intro to functional analysis I have ever seen by far. Despite the book's title, there isn't that much physics, though.

 

[Mathguy15]

Hey guys! I'm new here, and I am strongly considering becoming a mathematician, but, like most people, I don't know if I am smart enough. I'm fifteen in the tenth grade, and I have taken tests like the AMC10 and 12, but I haven't done very well. I have dabbled some in abstract algebra and I have done some linear algebra(at least more than abstract algebra) and some number theory, but I'm not sure if I am smart enough. Are earlier posts in this thread going to be helpful?

 

[mathwonk]

Let me remind you that the voting is open in the PF lounge for awards. There is a math award, and I urge you to consider the very valuable work of those people who answer actual math questions day in and day out in the math forum for your vote.

 

[mathwonk]

To go out on a limb here, I want to suggest that for most of us there is such a thing as too much math. I.e. think about whether solving a famous problem is worth so much to you that you are happy to live like a hermit the rest of your life and only come out every few years for air, or whether you would rather be (if you are a guy) sort of a cross between a top math genius, brad pitt or jet li, segovia, umberto eco, david beckham, picasso, and the world's strongest man.

At some point in this journey you are at least going to want to know something about art, music, literature, politics, sports and psychology, even if only to get a date with someone other than "Watson". So take some courses in college that are not all math and science. I.e. there are skills courses and enrichment courses. Enrich your life a little, so you don't come across as a total nerd, like me. Note I have virtually never mentioned anything in this gargantuan thread except geeky stuff.

(Nonetheless, at least until recently, I could play pool fairly well, sing falsetto in the car, ride a bicycle, converse about wine, deal from the bottom of a deck, make an almost unguardable hookshot layup, a swan dive from a height slightly above my head, and the occasional three pointer. These accomplishments took years of dedicated practice mostly outside the library.)

Since a mathematician is also a person, and a happily adjusted person can actually do more math, becoming a mathematician includes these extra curricular topics too. Try not to become too narrow to relate to the rest of the society entirely. Just a suggestion.

 

[Sina]

no only take mathematics and physics courses while you still can you can read philosophy etc later

 

[Sina]

The small fonts were meant to signify that it was a joking statement :p

 

[mathwonk]

i logged back into delete my latest, and saw this. another thing about nerds, we never get the joke.

 

[Nano-Passion]

To go out on a limb here, I want to suggest that for most of us there is such a thing as too much math. I.e. think about whether solving a famous problem is worth so much to you that you are happy to live like a hermit the rest of your life and only come out every few years for air, or whether you would rather be (if you are a guy) sort of a cross between a top math genius, brad pitt or jet li, segovia, umberto eco, david beckham, picasso, and the world's strongest man.

At some point in this journey you are at least going to want to know something about art, music, literature, politics, sports and psychology, even if only to get a date with someone other than "Watson". So take some courses in college that are not all math and science. I.e. there are skills courses and enrichment courses. Enrich your life a little, so you don't come across as a total nerd, like me. Note I have virtually never mentioned anything in this gargantuan thread except geeky stuff.

(Nonetheless, at least until recently, I could play pool fairly well, sing falsetto in the car, ride a bicycle, converse about wine, deal from the bottom of a deck, make an almost unguardable hookshot layup, a swan dive from a height slightly above my head, and the occasional three pointer. These accomplishments took years of dedicated practice mostly outside the library.)

Since a mathematician is also a person, and a happily adjusted person can actually do more math, becoming a mathematician includes these extra curricular topics too. Try not to become too narrow to relate to the rest of the society entirely. Just a suggestion.

Top notch advice every time. You see the bigger picture of things mathwonk.

 

[lisab]

Top notch advice every time. You see the bigger picture of things mathwonk.

I agree, that was excellent advice.

I'm sometimes torn when I encounter a bright, ambitious young person seeking advice here. They're often so brilliant and willing to sacrifice to achieve their goals. Of course, I want to help them on their path - that's what PF is for.

But I also want to tell them, go hike in the woods! Learn to ski! Fall in love! There is more to life than academic achievement, and life is so short!

 

[eliya]

mathwonk is correct, of course, but it's not just that. I sometimes feel like people think that mathematicians sit in a study and work on mind boggling problems to a candle light. People have this very romantic (not in the sense of falling in love) view of mathematicians, or scientists in general. I think it's quite different than that. A lot of the time people are stuck on a problem and pull hairs out trying to figure it out. You need to vent and occupy your mind with other things every once in a while. You also want to be able to hang out with people, because human beings, whether you like it or not, are social animals. Living alone in an attic and doing math for fifty years will probably drive you insane. Do math, lots of it, but also know how to be a human being, because you are one.

 

[Nano-Passion]

I agree, that was excellent advice.

I'm sometimes torn when I encounter a bright, ambitious young person seeking advice here. They're often so brilliant and willing to sacrifice to achieve their goals. Of course, I want to help them on their path - that's what PF is for.

But I also want to tell them, go hike in the woods! Learn to ski! Fall in love! There is more to life than academic achievement, and life is so short!

This really influenced me actually.

Edit: Especially because I'm kind of a 'romantic' and love all the things you listed above.