Examples in Mathematics

Another example is that Saharon Shelah, perhaps the most accomplished living set theorist/logician, is known to disdain examples. Here's one expression of his view:

I have always felt that examples usually just confuse you (though not always), having always specific properties that are traps as they do not hold in general.

My opinion is that Grothendieck, Kontsevich and Shelah are not fooling themselves, but their advice is wrong for most people who are not them. Personally, I'm annoyed at myself at not remembering more often to approach unfamiliar claims through small and simple examples. Whenever I do, I end up understanding the general claim more clearly, nonwithstanding the warning about the specific properties of examples. When I do not, I often end up feeling as if I'm in a fog, grasping perhaps the literal meaning of the claim but not being able to see its significance or what it implies.

Perhaps those exceptional people who dislike examples (and I don't think that this view is typical among even the most accomplished mathematicians) get that clarity of understanding from the claim itself, and don't need to unfog their brain through looking at examples. I could believe that in the case of Shelah, anyhow. I took his advanced course in set theory once, a long time ago. It was the closest I've ever come in my life to feeling that I've encountered not just someone much smarter than me, but a truly superior intellect from a whole different level. His atomic inferential step was unbelievably wide - that is, he (genuinely and humbly, without any attempt at showing-off) saw as an immediate consequence something it took a hard effort of several minutes for others to work through, again and again. It was incredible to watch, and I've never seen anything like that with any other mathematician.

I feel reminded of the discussion whether you can actually see something you imagine. Turns out, some people can see a vivid image in front of their eyes and some are simply incapable of this. Maybe this is a similar case: Some people work well with examples, some are better with general abstract concepts.

P.S.: I feel this is a good time to say this: In my limited experience teaching and learning I found that people are massively inhomogenous with regard to how they grasp concepts and there is no telling which way is obviously better. I assume that proper didactics would address different ways of learning and working.

I suspect Groethendieck was an alien.

For myself, I agree with Scott, I run into trouble without a ton of examples. You eventually get a feel for when the list of examples is "good enough" for the general case.

I don't think it is strictly true that Grothendieck didn't rely on examples. Here is a quote from a letter from Luc Illusie, who was a grad student under Grothendieck. Quote:

“In his filing cabinets, located behind his desk, Grothendieck kept many handwritten notes, where he had studied specific examples: he sometimes told me that he was weak on surfaces, but as everybody knows, he was not so weak in local algebra, and he knew enough of curves, abelian varieties and algebraic groups to be able to test his ideas. Also, his familiarity (and constant interest) in analysis and topology was a strong asset. All these examples appeared when you discussed with him."

Slightly related ... there is a recent philosophy of mathematics book (albeit a work of continental philosophy, but it is still interesting). I say slightly related, because firstly it has an entire chapter on Grothendieck and creativity, and secondly the entire book is about the interplay between examples and theory. It goes into the way mathematics moves up and down a ladder of abstraction (from particulars to universals, local to global, specialization to generalization). It focuses on the last 100 years of mathematics, concentrating on algebraic geometry, category theory, model theory, and others. The author splits up the abstractions into:

• Eidal mathematics, or "transfusions of form." This is mathematics that moves up the ladder from the particular to the universal. In this part of the book it focuses on the work of Serres, Langlands, Lawvere, Shelah.
• Quiddital mathematics, or "transfusions of reality." This is abstractions made concrete moving down the ladder. Here the book focuses on Atiyah, Lax, Connes, Kontsevich.
• Archeal mathematics, or "decantations of the universal." This is mathematics that studies the invariants while one moves up and down the ladder. Here it focuses on Freyd, Simpson, Zilber, Gromov.

Now you could say this philosopher is just dressing up stuff already known to mathematicians (Both Kevin Houston's How to think like a mathematician, and Mason/Burton's Thinking Mathematically talk about generalization and specialization). But it is still interesting, as I think it may be the only philosophy of mathematics book out there that does an indepth treatment of 20th and 21st Century mathematics that goes beyond Russell and Frege.

Of my two friends who I talk the most math with, I explain to one of them with examples and the other with general statements. I don't know why it helps, I just know from experience that I have to give examples to one and general statements to the other.

It reminds me of the witch powers from Alicorn's Twilight fanfic. Some witches have power over the same thing, experienced through different senses.

Such witches were especially powerful working in pairs. I wonder if there's anything special about collaborations between example thinkers and general thinkers? I'm an example thinker, and with my general thinker friend, a common pattern is that he'll conjecture, and I'll search for counterexamples.

Messing about with actual matrices never gave me the slightest grasp of linear algebra, and the fourier series formulae seemed completly pulled out of thin air, but as soon as I saw the expression of those concepts using abstract linear operators on general vector spaces, all the results and methods seemed obvious. I still feel really pleased when something that's true in my geometrical picture actually works when you stick numbers into matlab.

On the other hand, I first ran into group theory abstractly presented, and it meant nothing to me. I needed to play with lots of examples before I even cared about it, and before I came upon the cycle representation it was all just completely opaque.

They two seemed to be similar in content, introductory first-year maths, similarly presented, and both lecturers were clear and gave beautiful notes, and yet they spoke to me in very different ways. I'm still very happy with linear algebra and rather mystified by groups.

I think in my case the difference is that linear algebra is intrinsically geometrical, and I'm much better at visualizing pictures than at manipulating symbols, but given that one use of groups is to talk about physical symmetry, whereas linear algebra is all about vast tables of numbers, maybe that should be the other way round.

Anyone get the reverse feeling?

In both of quotes the non-example users are presented as unusual, right? So that doesn't really seem to contradict Aaronson's advice.

Anecdote: I'm not especially talented at math compared to the Lesswrong average, I've done math up to diffEq and linear algebra, but it's not my main interest.

I realize that the level of abstraction involved in the sort of math I am familiar with isn't very high in the first place, and perhaps if the level of abstraction were higher I'd change my tune. But I've found that the best way for me to learn math is to continually stare at the definitions and the most general theorems until they make sense, and then work through the examples. The examples are a useful check to make sure I understand, but they are never the mechanism by which I understand - unless I'm dealing with something completely unfamiliar, in which case examples do help.

I've had a few professors who lead with a few examples, and then lay down the basic definitions and general theorems. This style of teaching causes me to become completely and utterly lost...my mind keeps sounding alarm bells every time something that doesn't make complete sense appears, and this prevents me from just going ahead and processing it anyway. Or, I'll just instinctively ignore the example itself entirely, and instead allocate my attention to trying to glean an underlying principle within the example.

is there something to be learned?

I think yes, from both sides.

From the "example" learning style, I need to learn the value of being able to work through a problem procedurally, even if you don't completely understand absolutely everything that is necessary to derive what you are doing...you can always figure out the more abstract stuff later. I've performed poorly in some of the more applied math classes, as well as organic chemistry, as a result of this mental block in the past - O-chem, especially as traditionally taught, seems like a subject where you need to master the example-based learning style, and I wasted an embarrassing amount of time trying to learn the theory hoping that it would eventually allow me to derive things without ever working through examples.

I think the value of the "abstract" learning style is the more intense focus on first principles, and the ability to differentiate whether one really understands something, or is just going through the motions which get the right answer - perhaps related to a refusal to "guess the teacher's password" even when doing so would actually lead to a correct and useful answer.

I don't think these two styles are necessarily dichotomous - some people seem to be able to do both.

This passage by Grothendieck (source) seems potentially relevant:

What my experience of mathematical work has taught me again and again, is that the proof always springs from the insight, and not the other way round – and that the insight itself has its source, first and fore- most, in a delicate and obstinate feeling of the relevant entities and concepts and their mutual relations. The guiding thread is the inner coherence of the image which gradually emerges from the mist, as well as its consonance with what is known or foreshadowed from other sources – and it guides all the more surely as the “exigence” of coherence is stronger and more delicate.

To me, examples are to mathematics as experiments are to physics.

Although every example is particular in a way that a "general theory" is not, it is usually possible to "twiddle the experimental knobs" of the example such that you a feel for the more "general theory". Related to this is solving a problem by considering a simpler sub-problem which is a particular case (i.e. a special example) of the general problem you want to solve.

For example: if you have trouble solving a geometry problem in 3D, look for similar problems in 2D and 1D. Are they easier to solve? How does the solution to the 1D and 2D problem shed light on the possible 3D solution?

But this probably also depends on which field of math you study.

Feynman's greatest strength as a problem solver was bringing a different set of mathematical tools to a problem. Even if bringing along a big fat stack of examples is the best way for you to solve a problem, I fully expect some prominent mathematicians to solve problems from building a theoretical framework. If for no other reason, they'll be successful at the problems that framework-building is the best strategy at and earn respect for solving those problems, even while the majority of problems get done easier with examples.

In other words, I expect framework building to be good at solving at least some problems that example-pulling are bad at. That's enough for a mathematician to earn success by having a mind that's particularly well suited for the former method. This success and kind of mind are completely independent of your best research strategy.

Grothendieck's mind was indeed extremely strange. The levels of abstraction upon abstraction he achieved in algebraic geometry boggles the mind.

But I don't think you can really make meaningful comparisons between thought processes based on self-reporting. One complication is that different fields of mathematics work differently in this regard. In things like statistics, analysis, and geometry, you rely heavily on examples. In things like algebra, examples can indeed be cumbersome and hindering, because the point of algebra is to simplify things to symbol manipulation. Of course, it might also be the case that people with more abstract-type thinking are naturally drawn to algebra.

It would be useful to look at the 'information content' of storing examples vs. storing symbolic representations, and see how that compares across different mathematical subjects.

If we assume that humans use different brain structures to learn/reason symbolically and sub-symbollically, then I think it is natural to assume the following:

• Examples are better suited to train a brain part that deals with sub-symbol representations of the world.
• Existing symbols are better suited to operate on by a brain part that deals with symbols.

And it seems plausible that there are persons with brains better suited the the former and other to the latter. So there is no conflict but a 'it depends'.

I couldn't quickly find a good reference for symbol vs. sub-symbol learning in humans. You have to take this from AI: http://en.wikipedia.org/wiki/Artificial_intelligence#Sub-symbolic

A long time ago I wrote something about this distinction which explains why and when examples help here: http://c2.com/cgi/wiki?FuzzyAndSymbolicLearning

I suspect part of the problem might be that the sort of minds that do well in theoretical math tend to be the sort of people who are pedantic, and pedants are the sort of folks who are more likely to get caught up on specific details instead of using a few examples and some squinting to get a sense of the general picture. I've done that many times myself

I don't know about Grothendieck. But, Kontsevich's statement is telling:

For myself sometimes I work on one or two examples

Halmos (quoting Hilbert) captures this very well:

What mathematics is really all about is solving concrete problems. Hilbert once said (but I can't remember where) that the best way to understand a theory is to find, and then to study, a prototypal concrete example of that theory, a root example that illustrates everything that can happen. The biggest fault of many students, even good ones, is that although they might be able to spout correct statements of theorems, and remember correct proofs, they cannot give examples, construct counterexamples, and solve special problems.

I shall discuss many concepts, later in the book, of a similar nature to these. They are puzzling if you try to understand them concretely, but they lose their mystery when you relax, stop worrying about what they are, and use the abstract method.

Timothy Gowers in Mathematics: A Very Short Introduction, p. 34