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Sorry in advance if this question doesn't belong here. I know it walks the line of what kinds of questions should be asked, but I'm really interested in what the community here has to say on it. Knowing this, I won't be offended if people vote to close it.

This question is meant as a companion to previously asked questions like Proofs every mathematician should know.

More and more I'm beginning to see that there is just too much math to learn. Additionally, there is far more math that I'd like to learn then is covered in my undergrad. So, my question is, what math should every person that titles him/herself as a mathematician, or math-capable know? For example, to say that you play hockey, you should at least be able to skate forwards and backwards, stop, take a pass, shoot, ect. If you say you're a hockey player, but can't stop.. well then..

At the same time, I'd like to have a comprehensive understanding of the math I "know". To give clarification of what I mean by this, I reference the turtles standing on turtles analogy of how any proposition requires justification. In the case of math, I'm hoping that this isn't an example of infinite regress as in the case in science (it's the main reason I study math instead of science). Using the turtle analogy for math, I'd like to know what the stupid turtles are standing on.

Sorry again if this question isn't suitable for this site. I just really respect the amount of knowledge some people in this community have to offer, and would really just like some advice.

Thank you for reading.

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closed as not constructive by Austin Mohr, Grigory M, rschwieb, Michael Greinecker, Eric Naslund Nov 19 '12 at 18:37

As it currently stands, this question is not a good fit for our Q&A format. We expect answers to be supported by facts, references, or expertise, but this question will likely solicit debate, arguments, polling, or extended discussion. If you feel that this question can be improved and possibly reopened, visit the help center for guidance.If this question can be reworded to fit the rules in the help center, please edit the question.

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This is a very broad question. The best answer I can give is "the first two years of graduate level math." That is, the basic grad classes. –  Jesse Madnick Nov 19 '12 at 2:16
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This should probably be made a Community Wiki question. –  Neal Nov 19 '12 at 2:18
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Asking "what are the turtles are standing on" and "what pieces of mathematics should everyone know" are two very different questions. The answer to the first is found primarily in logic and set theory. As others have noted, the answer to the second varies wildly with your particular meaning of "should". –  Austin Mohr Nov 19 '12 at 2:37
    
@AustinMohr I agree on the second part of what you said.. that's why I knew this was a really soft-question. As for the first part, are you saying that there is an answer to the turtle question lying in logic and set-theory? I was under the impression there is some major controversy regarding axiomatic set-theory, for instance the Axiom of Choice. –  user45793 Nov 19 '12 at 2:42
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I vote to close this question as not constructive. "It does not fit well with the Question and Answer format. We expect answers to be supported by facts, references, or specific expertise, but this question will likely solicit debate, arguments, polling, or extended discussion." –  Eric Naslund Nov 19 '12 at 4:53
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8 Answers

up vote 18 down vote accepted

I think that every mathematician should be knowledgeable about the history of mathematics and understand where mathematics comes from and the reasons it is developed.

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The Oxford Advanced Learners Dictionary defines a Mathematician as a person who is an expert in mathematics.

The Cambridge Advanced Learners Dictionary defines a Mathematician as someone who studies, teaches or is an expert in mathematics

I wouldn't take Oxford's definition too seriously but according to Cambridge's

  • If you study Mathematics, you're a Mathematician.
  • If you teach Mathematics, you're a Mathematician.
  • If you're an expert in Mathematics, you're a Mathematician.

Personally, my love for Mathematics came $12$ years ago when I took my first $97\%$ score in Mathematics home to my father and I saw the look in his eyes. Since then, it's been Mathematics everywhere. Mathematics has been my life and even right now, Mathematics is paying my college fees, feeding and clothing me, so I think I can call myself a Mathematician, but

  • Do I deserve to be referred to as one?
  • Do I feel like one?

The first question is something I can't answer because the society has what its own image of a Mathematician. The likes of Euler, Newton, Euclid of Alexandria, Fibonacci etc. that I can't even imagine comparing myself to. So I can't answer that question because the society already has one and it's a NO.

For the second question, the answer is also a NO most probably for the same reason I stated above for the first question. I don't feel like one, I may be in a reasonable percentile for my age but I most definitely do not feel like one because there're Mathematicians I know that I may never measure up to. But does that mean I do not know enough?

Here is another interesting part. What Mathematics should I know so I can feel, or deserve to be referred to as a Mathematician. The answer to this question depends on the question

Why do you love/do Mathematics?

Is it because you want to make a change in the world? Is it because you want to achieve something huge and become famous? Or is it simply because you love it so much, you can't spend a nano-second of your life without it?

Depending on your response to this question you will know whether to try and be the best in one area of Mathematics or learn as much as you can in any area that you can.

In all cases, one cannot always depend on the majority/society to answer our own questions. even though the majority carries the vote, the majority can still be wrong.

In whatever you do, don't forget that

  • To be conscious that you are ignorant is a great step to knowledge.
  • A little learning is a dangerous thing
  • You never know, until you truly know.
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I think if there's one area of mathematics all practicing mathematicians need to know as thoroughly as possible-if you could pick just one-it would have to be abstract algebra. I say algebra because it really is the one language that is used in just about all fields of mathematics, from the fundamental group in topology to the operator rings in functional analysis to the symmetry groups of plane geometry to the tensor products which are the basis for so much of the algebraic coordinate structures in differential geometry to even the Galois groups of differential equations and the spectra of the matricies of graphs. Algebra makes its presence known througout mathematics in so many different ways. So in my opinion, that's the area that's going to give you the maximum effectiveness of insight into the most diverse areas of mathematics.

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I think that some of the wisest people I know recognize and readily admit their utter ignorance...in math as in life itself.

From my experience, the more I learn and the more I know, the more I realize how much I do not know. Like a balloon that's inflating with knowledge and understanding, the surface area at the bounds of an inflating sphere (the frontier of what is yet to be learned) becomes increasingly vast.

Added: I couldn't resist including one of my favorite (of many!) quotes:

Both teaching and rational inquiry, at their creative and inspired best, thus lead us to the very threshold of ultimate mystery and induce in us a sense of profound humility and awe.

~~~ Theodore Meyer Greene

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I can appreciate that –  user45793 Nov 19 '12 at 2:35
    
I agree with you, Carl, that it is better to truly know what you know than it is to chase after every last unturned stone. –  amWhy Nov 19 '12 at 2:36
    
Socrates the philosopher said "i know one thing, that i don't know anything". –  Manos Nov 19 '12 at 2:41
    
@AndréNicolas Oh darn it! You're right! Amended...accordingly, big sigh! –  amWhy Nov 19 '12 at 2:45
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There is none. And the other thread is a lie too. It is better to be best at a small area of math, than to be average at everything.

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I totally disagree with this. It is true jack of all trade is master of none but depending on what you want from Mathematics and "why" one loves Mathematics, neither is better. If I love Mathematics so much that I want to know everything about every possible thing, who are you to tell me to focus (and probably be the best) on just one area of Mathematics. Neither is better, each depends on the individual. –  user31280 Nov 19 '12 at 2:22
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I don't seek to be the best at anything really. I just want to be able to say that I'm capable at math and that I know what I know. But I agree with what you're saying –  user45793 Nov 19 '12 at 2:28
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The kind of math you should know before you can call yourself a mathematician? Well, i don't think there will be jury around judging whether you're elegible for that title. =)

Math is not just about being good at it, right? It's just about doing something you like and enjoy very much! So the answer is: do whatever makes you happy!

If you want to make a profession out of it there's sure basic knowledge of every field that you need to know but since you mentioned you study math i dont need to tell you what that is.

But whoever thinks reading a math book in the sun while sipping cappuchino is a great way to spend a day can call himself a mathematician in my opinion.. If they feel like it.. Or when you're in the library and you can barely stop yourself from screaming in enthousiasm since you just understood why all varieties are birational to a hypersurface... Or .... I could go on. The point is, who cares about titles, just do what interests you...

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I agree totally. But when I meant 'titling', I didn't mean labelling people other people as being 'mathematicians' or not. More in the sense that one can honestly claim to themselves that they know something about math without feeling internally guilty (if you know what I mean). –  user45793 Nov 19 '12 at 2:54
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For example, I'm currently almost done my undergrad in math. But if I scroll through the questions being asked right now on this site...I can't answer the majority of them. And so, at this point in time, I would say that I don't really know anything about math. –  user45793 Nov 19 '12 at 2:58
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Haha that wont change my friend, math is too big.. –  Joachim Nov 19 '12 at 3:00
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The belief I formed on the question: "What mathematics I should know in order to call myself a mathematician" is is that I should have at least a standard graduation curriculum - I'm not trying to say you should have some kind of formal certification given by some university, I'm saying that you should look their mathematics curriculum and study those courses.

For me it works because I'm kinda lost in what I really want to do with mathematics, I had some plans but as I started studying math, new plans were created and some old plans were erased.

Also notice that I'm not saying that universities certifications are not important, they are. But for some of us studying in a university may not be so accessible now, you can obtain some books and study on your own - I'm doing this for some time and I'll enter a math course next month.

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cudos to doing what you enjoy. thanks for the reply –  user45793 Nov 19 '12 at 3:59
    
You can easily find reading lists for finding a way of "entering" mathematics. You could also search some university teacher and ask about what order you should study those courses. One tip: Always remind yourself of asking about agressive/hardcore ways of doing it. ;-) –  Igäria Mnagarka Nov 19 '12 at 4:03
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Listen to the series of Lecture given by Gilbert Strang on Youtube and follow the books of Donald Knuth. Its d universe out there. AGood knowledge of Matrix is very much important ,because exponentiation solves huge problem in matter of logn time Here's a link -Mit Professor Gilbert Strang
Books

  • Donald E. Knuth, The Art of Computer Programming, Volumes 1–4, Addison-Wesley Professional
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It's not really clear what you're saying here, you may want to edit this to make it clear what you mean, and perhaps give some more specific details of which lectures and books you're referring to? –  Tom Oldfield Nov 19 '12 at 16:51
    
have added the link and the name the books compulsory,sorry for my editing being bad –  Queue Tank Nov 20 '12 at 12:57
    
thanks for adding those! –  Tom Oldfield Nov 20 '12 at 15:16
    
There are also free video on Itunes by Gilbert Strang –  Joao Apr 7 at 3:55
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