What's the best way to measure mathematical ability?

Very soft question I admit, but it's something that's been bothering me for a while.

I've been thinking that being self taught has the problem of accreditation. You can't evaluate a mathematician in a vacuum. You need an accredited mathematician to decide whether or not someone else is also a mathematician worthy of accreditation. Well, who evaluated the other mathematician? Other accredited mathematicians. It's sort of like becoming a member of an exclusive club.

We put the job of accreditation on our universities. But what if some person was discovered, off-the-grid so to speak, who had taught themselves mathematics from library textbooks.

How could such a person evaluate themselves? How do you know if you're making progress when you study?

It's tricky. It's like language learning. Do I speak German more fluently now than I did yesterday? I've no idea. Who can say?

It's like playing with Lego. How do you know if you're getting better with Legos? You build more complicated things. But who's to say one person's Lego helicopter is better than another's Lego Enterprise? What's the goal with Legos? Is there one? Should there be one?

I know already that this question will be deleted almost immediately, but I think these are important questions and many people visiting this site are in fact self-taught and I'm sure these questions show up as massive roadblocks.

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One more or less useful and easy test is: are you enjoying doing mathematics now more than yesterday, no matter what the difficulties found are? If yes then you're very probably making progress. If not you may or may not progressing, but your life is becoming miserable as you're doing a pretty hard thing to do and you are not enjoying it. –  DonAntonio Mar 26 '13 at 1:15
You can usually test yourself with the canonical books. If you're studying, say, Calculus, you can take a shot on Spivak and Apostol's Calc. If you're a little higher, you can look at Rudin, or Apostol's Mathematical Analysis. There are some texts that are usually a very good standard. –  Pedro Tamaroff Mar 26 '13 at 1:20
I think mathematical ability is so impossible to define that there is no good measure for it let alone a best one. –  Alex J Best Mar 26 '13 at 1:24
I'd say the best way to measure mathematical ability is to be able to explain what you learned to others.This not only reinforces the material but it also allows you to learn what areas you need to wor k on. –  user60887 Oct 6 '13 at 0:31
@Adam: It's significantly skewed by overzealous upvoting for joke answers, edits, easy answers to common questions, questions themselves and so on. Popularity, does not, a mathematician, make. Obscure, technically demanding subjects don't get as many votes as, say, homework problems. I could go on. At best, it's a very rough approximation. –  Alice Nov 29 '13 at 14:47

Your question is philosophical. As far as I know, there is no normed linear space of all possible mathematicians on which we have a metric to compare elements in the space!

My personal opinion is that there is an infinite amount of mathematics. If there isn't, then it is extremely large. This means any knowledge you have will be woefully less than what is possible to know. (Put some measure on the infinite set of all possible mathematics?)

Additionally, you will always find a person, either from the past or present, and possibly the future, who knows more mathematics or knows more about some particular branch of mathematics. While it is true that you can make comparisons with other people's mathematical abilities, that is often demoralizing. For most people, anyways, they aren't "at the top".

Of course, there is a threshold I am assuming one is beyond. There is clearly a difference between the majority of calculus students and math PhD students. Here, yes, you should be comparing yourself to see if you can master undergraduate and graduate material to know if you can, say for instance, do research in mathematics.

But all that aside, it is somewhat pointless and self-serving if you do mathematics merely for status. You should enjoy doing it as you should enjoy doing anything difficult. And mathematics is genuinely beautiful. I suggest you are doing well if you continually are surprised at what you learn, if you are able to find satisfaction in meeting non-trivial personal goals, and if you surprise yourself by going beyond your expectations.

In particular, improve your ability to solve problems, improve your ability to propose problems, and improve your ability to explain mathematics.

If you work on a difficult problem and you solve it, then you've surpassed your expectations! Otherwise it would not have been a difficult problem--the doubt of solving it makes it difficult. If you can propose new problems, you can enjoy the excitement of the novelty of new (to you at least) mathematics. If you can visualize the structure of mathematics and express it clearly to others, you can appreciate the beauty of mathematics and in fact share it as well.

In order to do any of this, you do need to know what you don't know. Otherwise you can't envision any improvement. There is some value in making your own personal maps of what the mathematical landscape looks like. This is very much in line with the saying that the wise man knows what he does not know.

That's my 2 cents.

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+1 This is worth $3$ cents. –  Jay Oct 30 '13 at 15:37

"How could such a person evaluate themselves? How do you know if you're making progress when you study?"

I'm reminded of the Dunning-Kruger effect. There's a lot of evidence to suggest that a low level of competence often leads to overestimation in self-appraisal, and vice versa. Succinctly:

"The fool doth think he is wise, but the wise man knows himself to be a fool." Shakespeare.

What does this mean? Well, maybe a rule of thumb for your own mathematical competence in a field of study is the ability to thoroughly recognise your own mathematical shortcomings in that field of study. You'll know just how much you don't know, just how much your proof captures the theorem and so on; but you'd know where to go and what to do to get better. You'll be your biggest critic. This isn't the same as being new to the subject; it's "meta-cognitive ability".

Let me try and use your examples.

Suppose you've been teaching yourself German for a couple of years. [I don't know much about the language so I'm going to use my imagination.] You might begin to notice that you speak with a heavy accent or that you often put umlauts in the wrong place. You might be aware that your repertoire of dialects isn't what it could be or that you get various Latin roots confused, but you know that you do. Then you might be considered competent.

I'm not sure about the Lego. It's very subjective. Maybe that's your point, though, so I don't know . . .

Asking when something is proven seems relevant here too. See the hypothetical dialogue in "The Mathematical Experience," by Philip J. Davies and Reuben Hersh between an "Ideal Mathematician" (IM) and a Student (S) -- quoted in J.Adler and J.Schmid's "Introduction to Mathematical Logic" -- to see why. It starts with the student asking what proof is and continues to:

IM: [. . .] Everybody knows what a proof is. Just read some books, take courses from a competent mathematician, and you'll catch on.

S: Are you sure?

IM: Well, it is possible that you don't, if you don't have any aptitude for it. That can happen, too.

S: Then you decide what a proof is, and if I don't learn to decide in the same way, you decide I don't have any aptitude.

IM: If not me, then who?

This is also relevant: The four stages of competence.

A Much Later Edit: Here's a quote from this wonderful article:

"Only the mediocre are supremely confident of their ability. The better you are, the higher the standards you set yourself - you can see beyond your immediate reach." Sir Michael Atiyah.

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Regarding the Atiyah quote, the inverse is unfortunately false; being humble about one's ability (or having one's work ignored by "the mainstream") does not mean one is spectacularly talented (and/or an unrecognized Galileo, Abel, Tesla, Einstein, ...). –  Andrew D. Hwang Mar 9 '14 at 14:52
Indeed. That's a very important point. Thank you :) –  Alice Mar 9 '14 at 15:12

You can do mathematics in a vacuum, though. It is possible to devise what is necessary without checking whether someone else has done it before. In this way, it's not like learning German or Old English, where one must ultimately access other speakers of it.

Until the advent of the internet, access to material on a subject was quite hard, and often supposes that one is well versed in some leading subjects to understand it. I live in the tropics, where you might be lucky to find a single rack of "pop science" books in the larger bookshops. Getting hold of material was considerably harder then inventing it yourself.

It is interesting now, that while I have discovered many things that have older names, the path to them is different, and the extent of coverage of something is also different.

One discovers from first principles, the nature of non-euclidean geometry, and then proceeds to do perfectly valid dot-products of vectors on an oblique coordinate system. Such is something that comes from the "Coxeter-Dynkin" diagram. Yet in the mathematics one reads on this matter, these diagrams are used of symmetry, and much of Alicia Stott's good work has been for naught.

Mathematics is something that can be discovered, as one might sail ships around the world. Different nations can sail up the coast of australia, and give different things different names. But the landscape does not change, just the names and the mental relations.

I came to something like Mobius geometry through a circuitous path. Because the nature of my geometry is somewhat different, recognising that every circle is a straight line in that geometry is something that I and a famous professor in the USA have chose to disagree on. We are talking of the same thing, but seeing in a different culture.

It is hard then to assess the competence of a mathematician. They may understand the mathematics, but have different words for it. It certainly is true that my endeavours in the geometries of higher dimensions is world-class, and in some parts, well in the lead, but the mathematics is all pre-calculus.

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In principle, in principle. But even if I could I would not choose to go down that path you describe. I think there is great value in the social aspect of mathematics. "Mathematics as a cultural heritage". I think its important to be on the same page. From saying to your friend: remember that old chestnut of xxx? Yeah, thats some problem. What if we do this? Right to research mathematics where a person builds on centuries of research that came before him. –  Adam Nov 28 '13 at 15:58
@Adam While it's all fine to say this, it's a different story if you can't simply find the necessary people or texts. It is easier to get on the same page, if you already know the materials, and just need to learn the names. I know. I have been there. Mathematicians can't tell you how to multiply numbers in base 60 or 120. Their model of infinity leaves a physicist's jaw gaping: it's like saying you don't have to walk to the shop because you're already there if the origion is in Andromeda. I've done useful things with infinity: there are literally thousands of them there. –  wendy.krieger Nov 29 '13 at 8:30

I know something that usually correlates with mathematical skill: vocabulary

The more vocabulary they know and understand usually means that they know and understand mathematics. (It should be noted, knowing vocabulary doesn't mean you smart, it just correlates with it.)

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Take an exam. It is objective and unbiased (usually).

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This might be better as a comment. –  Amzoti Mar 26 '13 at 1:36
Exam technique plays an important role in exams, so it is not an good measure of pure ability. –  L. F. Oct 30 '13 at 14:48
Who makes the exams? People do. Are people biased (whether they admit it or not)? Absolutely. –  bestversionofme Oct 30 '13 at 14:51
This is a good answer. If you can't solve exam-type problems in a subject without looking at the textbook, then you probably don't know the material. (This is a general rule. There are of course exceptions.) I'm not sure why this has so many downvotes. –  Potato Nov 28 '13 at 10:56
Maybe it had downvotes, because its kinda obvious. It doesnt bring anything new to the discussion. –  Adam Nov 28 '13 at 16:20