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Physicist, Startup Founder, Blogger, Dad

Tuesday, January 03, 2012

Mathematical minds

A colleague recommended this beautifully written Quora answer concerning the nature of mathematical thinking. I recommend reading it in its entirety.

The particularly "abstract" or "technical" parts of many other subjects seem quite accessible because they boil down to maths you already know. You generally feel confident about your ability to learn most quantitative ideas and techniques. A theoretical physicist friend likes to say, only partly in jest, that there should be books titled "______ for Mathematicians", where _____ is something generally believed to be difficult (quantum chemistry, general relativity, securities pricing, formal epistemology). Those books would be short and pithy, because many key concepts in those subjects are ones that mathematicians are well equipped to understand. Often, those parts can be explained more briefly and elegantly than they usually are if the explanation can assume a knowledge of maths and a facility with abstraction.

Learning the domain-specific elements of a different field can still be hard -- for instance, physical intuition and economic intuition seem to rely on tricks of the brain that are not learned through mathematical training alone. But the quantitative and logical techniques you sharpen as a mathematician allow you to take many shortcuts that make learning other fields easier, as long as you are willing to be humble and modify those mathematical habits that are not useful in the new field.


You move easily between multiple seemingly very different ways of representing a problem. For example, most problems and concepts have more algebraic representations (closer in spirit to an algorithm) and more geometric ones (closer in spirit to a picture). You go back and forth between them naturally, using whichever one is more helpful at the moment. ...


Spoiled by the power of your best tools, you tend to shy away from messy calculations or long, case-by-case arguments unless they are absolutely unavoidable. Mathematicians develop a powerful attachment to elegance and depth, which are in tension with, if not directly opposed to, mechanical calculation. Mathematicians will often spend days thinking of a clean argument that completely avoids numbers and strings of elementary deductions in favor of seeing why what they want to show follows easily from some very deep and general pattern that is already well-understood. Indeed, you tend to choose problems motivated by how likely it is that there will be some "clean" insight in them, as opposed to a detailed but ultimately unenlightening proof by exhaustively enumerating a bunch of possibilities. In A Mathematician's Apology [http://www.math.ualberta.ca/~mss..., the most poetic book I know on what it is "like" to be a mathematician], G.H. Hardy wrote:

"In both [these example] theorems (and in the theorems, of course, I include the proofs) there is a very high degree of unexpectedness, combined with inevitability and economy. The arguments take so odd and surprising a form; the weapons used seem so childishly simple when compared with the far-reaching results; but there is no escape from the conclusions. There are no complications of detail—one line of attack is enough in each case; and this is true too of the proofs of many much more difficult theorems, the full appreciation of which demands quite a high degree of technical proficiency. We do not want many ‘variations’ in the proof of a mathematical theorem: ‘enumeration of cases’, indeed, is one of the duller forms of mathematical argument. A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way." ...


You are good at generating your own questions and your own clues in thinking about some new kind of abstraction. One of the things I've reliably heard from people who know parts of mathematics well but never went on to be professional mathematicians (i.e., write articles about new mathematics for a living) is that they were good at proving difficult propositions that were stated in a textbook exercise, but would be lost if presented with a mathematical structure and asked to find and prove some "interesting" facts about it. ...

Concretely, this amounts to being good at making definitions and formulating precise conjectures using the newly defined concepts that other mathematicians find interesting. One of the things one learns fairly late in a typical mathematics education (often only at the stage of starting to do research) is how to make good, useful definitions. ...

10 comments:

ben_g said...

A lot of these points can be made to a lesser extent about programming

sineruse said...

There are intriguing stylistic hints as to who the author might be.  

It is a very smart and thoughtful piece of work.   In the comments they mention that Tim Gowers is proposing to publish it in AMS Notices or other higher profile venue.

Robert Sykes said...

On the other hand, mathematicians (and physicists) suffer from their own deformation professionelle. Notice the very large number of mathematicians who are unable to understand the theory of natural selection. (Did that include Bertrand Russell? I forget.) They usually get hung up on the probability end (and make inane estimates of probabilities and rates) and ignore the selection end.

For many years I tried to teach engineering students biology. A significant number, perhaps a majority, could not come to grips with biological concepts. I believe the difficulty was partially due to their mathematically training. Before coming to me, their experience was that a few mathematical concepts and equations were infinitely applicable to all sorts of problems. This is in effect your own claim. But once they encountered a field that was poorly (or even non-) mathematized, they were lost. They could not understand the need to learn large amounts of facts and concepts that could not be expressed in mathematical language.

There was also the lost lambs who had internalized the broader cultural conviction that biology was some how magic. That such irritants and conservation of mass and energy did not apply. But that's a different story.

Edwin said...

Biology is taught by rote memorization,which mathematicians and physicists despise

Dima Klenchin said...

When human skull alone contains 14 different bones and the whole body has ~10X more, there is no way around memorizing a bunch of facts if you want to understand biology.

Robert Sykes said...

While not entirely true (e.g. theory of natural selection, DNA/RNA mechanisms, cell thermodynamics and kinetics), you are largely correct. You cannot be a biologist unless you have memorized large numbers of facts. The same is true of physicians, chemists and geologists. Physics is not normative for other sciences, and it attracts certain kinds of minds and repels others.

Virtually all of the philosophy of science focuses extremely narrowly on physics, and most of that on the quantum mechanics. A broad philosophy of science that considers what the overwhelming (>>99%) of scientists do does not exist.

jooe said...

If all this is true, why do I have such an incredibly difficult time explaining simple things to mathematicians, and have such a hard time getting them to put their answers into someting intelligble ?
in my experience, the more math you know, the less usefull you are to to other people

jooe said...

NO, totally wrong
a) to do high level physics you need to memorize a lot of stuff, it is called math (ok, not precisely memorize, but you have to have at your mental fingertips a lot of math stuff )
b) if you think biology or molecular biology are about memorization, then you know nothing. Thinking about good, clean decisive experiments, paying close (close) attention to the data, keeping your mind flexible.....any of that sound  like rote memorization to you ?

ohwilleke said...

The last point about generating questions and being able to prove interesting facts about new mathematical structures is IMHO not the big divide between the pros and non-pros.  Rather, it is the ability to make bigger leaps of logic, because the intermediary steps between the points are more familiar to the point of being trivial to the future PhD.  Less mathematicans have to take more steps to get from point A to point B, and the more steps you have to take, the more likely it is that you will never get to the end at all.

ohwilleke said...

Memorizing rules and memorizing facts are two quite distinct processes.

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