Examples in Mathematics
After reading Luke’s interview with Scott Aaronson, I’ve decided to come back to an issue that’s been bugging me.
Specifically, in the answer to Luke’s question about object-level tactics, Scott says (under 3):
Sometimes, when you set out to prove some mathematical conjecture, your first instinct is just to throw an arsenal of theory at it. (..) Rather than looking for “general frameworks,” I look for easy special cases and simple sanity checks, for stuff I can try out using high-school algebra or maybe a five-line computer program, just to get a feel for the problem.
In a similar vein, there’s the Halmos quote which has been heavily upvoted in the November Rationality Quotes:
A good stack of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one.
Every time I see an opinion expressing a similar sentiment, I can’t help but contrast it with the opinions and practices of two wildly successful (very) theoretical mathematicians:
One striking characteristic of Grothendieck’s mode of thinking is that it seemed to rely so little on examples. This can be seen in the legend of the so-called “Grothendieck prime”. In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. “You mean an actual number?” Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, “All right, take 57.” But Grothendieck must have known that 57 is not prime, right? Absolutely not, said David Mumford of Brown University. “He doesn’t think concretely.” Consider by contrast the Indian mathematician Ramanujan, who was intimately familiar with properties of many numbers, some of them huge. That way of thinking represents a world antipodal to that of Grothendieck. “He never really worked on examples,” Mumford observed. “I only understand things through examples and then gradually make them more abstract. I don’t think it helped Grothendieck in the least to look at an example. He really got control of the situation by thinking of it in absolutely the most abstract way possible. It’s just very strange. That’s the way his mind worked.”
(from Allyn Jackson’s account of Grothendieck’s life).
Saito: This is one typical point of your work. But I find that in much of your work, by hearing one symptom you capture the central point of the problem and then give some general big framework. That’s my general impression of what you are doing.
Kontsevich: Yeah, I really don’t work on examples at such a level.
Saito: How can you work in that way?
Kontsevich: For myself sometimes I work on one or two examples, but...
Saito: You already keep some examples in mind, but still you construct theory.
Kontsevich: Yes. And generally I find examples sometimes to be misleading. [Laughter]. Because often the properties of examples are too special, you cannot see general properties if you constantly work too much on concrete examples.
(from the IPMU interview).
Are they fooling themselves, or is there something to be learned? Perhaps it’s possible to mention Gowers’ Two Cultures in the answer.
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:
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.
Be careful about using these wide inferential steps as an example. It is much easier to see certain consequences from a model than it is to prove consequences generated by a different model. It is a much better idea to practice deriving a (possibly different) set of consequences from a result you are comfortable with. This will give you a better idea of his intellect. Leading mathematicians often seem so much farther ahead than others because they are less constrained by the paths of other people.
I can see where Shelah is coming from, but if you’re quarter-way decent at compartmentalization and maybe work from two radically different examples (prime and composite, say, or if functions, odd and even functions, etc), it shouldn’t be a serious problem.
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 don’t see it in front of my eyes. I see it very clearly in an entirely different space. I kind of get where the whole ‘third eye’ idea comes from.
However, when I imagine a sound, I sometimes confuse it for the real thing.
Whereas I can (somewhat) make sense of thinking with examples, it seems hard to describe just what exactly does it mean to think with general abstract concepts.
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.
Was? He’s (probably) alive, if hard to find.
We gave him honorary Terran status in 1957.
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:
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.
Thanks for the piece of counter-data!
I might look into the book, but the naming convention is a big turnoff.
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.
Speaking as a abstract thinker, examples itch. I can’t work with someone throwing out examples on an idea I’m not fully clear about. The examples are too irritating for me to maintain my attention on the problem and I get stuck shooting down the parts of the examples that are too specific. I’ve learned to tolerate examples as a check, but I am not be able to work too deeply with example oriented thinkers.
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.
No, of course not, but it still might make sense to wonder why it’s so.
Yeah, fair point.
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.
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:
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.
I’m skeptical that the relevance of the two modes of thinking in question has much to do with the mathematical field in which they are being applied. Some of grothendiek’s most formative years were spent reconstructing parts of measure theory, specifically he wanted a rigorous definition of the concept of volume and ended up reinventing the Lebesgue measure, if memory serves, in other words, he was doing analysis and, less directly, probability theory...
I do think it’s plausible that more abstract thinkers tend towards things like algebra, but in my limited mathematical education, I was much more comfortable with geometry, and I avoid examples like the plague...
Maybe the two approaches are not all that different. When you zoom out on a growing body of concrete examples you may see something similar to the “image emerging from the mist”, that grothendiek describes.
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:
Halmos (quoting Hilbert) captures this very well:
I already mentioned what Halmos’ stance was. What I’m more interested in is how is it possible to work without examples.
The point I was trying to make is that it may not be necessary to have “a large stack of examples”. It might instead be much more useful to have a couple of “protoypal concrete examples...a root example”. Kontsevich seems to have similar thought patterns.
Timothy Gowers in Mathematics: A Very Short Introduction, p. 34
It is important to keep in mind that this was written for laypeople, not for working mathematicians. What is “concrete” for a working mathematician can be very abstract for an average reader. For example, thinking of a 5-dimensional space is very concrete for a mathematician but very abstract to other people.
That seems somewhat surprising coming from Gowers.