Wednesday, September 17, 2008

Approximation in mathematics

The New York Times has an interesting article arguing that imprecise approximational intuitions--the ability to quickly and roughly reckon things--are crucial to success at abstract mathematics.

To someone whose mathematical work was in real-number based areas--analysis and probability theory--this is very plausible on introspective grounds. But I wonder how true it is in more algebraic fields. I've never been very good at higher algebra--things like the Sylow theorems were very difficult for me (I still passed the algebra comp, but it came noticeably less naturally to me than the analysis comp), perhaps in part because my approximational intuitions were close to useless.

Anecdotal data suggests to me that there are two distinct kinds of mathematical skills. There are the skills involved in analysis, skills tied to problems that are real-number based (complex numbers are real-number based, of course, since C is just the cross product of R with itself), often visualizable, and where approximation and limiting procedures may be relevant. And then there are the skills involved in more algebraic fields, where (as far as I can) approximation gets you nowhere, and while visualization is helpful, the visualization is much more symbolic (visualizing a path of a brownian particle is pretty straightforward, one visualizes quotient groups either explicitly in symbols like "A/H" or perhaps in some strange and highly abstract diagrams). I don't know where to put the combinatorial--it may somewhat straddle the divide (a lot of visualization is involved), but I think is very algebraic in nature.

It is quite possible for a person to be really good at one of these, without being very good at the other. There are fields of mathematics that call upon both sets of skills. And there is an asymmetry: I think the analysis-type skills may be of very little use to mathematicians working in very algebraic areas, but just about every mathematician working in an analysis-type area needs to be able to do algebraic manipulation (though I have a strong preference for proofs in analysis-type fields where the algebraic manipulation is just a way of making precise what is intuitively obvious).

16 comments:

mrcontrarian said...

I myself have always, until recently, had a very weird problem with maths. When I was a kid I was quite mathematically precocious (if I say so myself): I could do calculus and complex numbers with relative ease when I was 10. But for some reason, I was never particularly good at arithmetic. I knew people who could add up long lists of numbers quicker than I could, but those people weren't as good as me at any other field of maths. I always found that quite a perplexing fact.

Alexander R Pruss said...

My memory for numbers (other than easy numbers like -1, 0, 1, 2^n, sqrt(2), e, pi and i) is pretty poor.

My arithmetical skills aren't that good. Certainly, they aren't as fast as those of other people, because I rely as little as possible on memory. Thus, while some people remember what 8+7 is, I visualize two sticking out of the seven, being added to the eight to give ten, and then the remaining five yielding 15. This is pretty quick, but probably a fraction of a second slower than for someone who had it memorized.

I once failed a question on a group theory exam (I think it was a 3rd year undergrad class) because I factored 96 incorrectly. The professor had forbidden calculators on the exam on the ground that we wouldn't need them. Hah!

My base 12, 24 and 60 arithmetic, which alas one needs for working with times (what is three hours after 10 am? when does a 1.5 hour class starting 2:45 end? when will the J-B Weld that I applied at 19:45 be cured, given that it takes 15 hours?) is embarrassingly poor.

mrcontrarian said...

And I thought I was the only one!

mjl said...

I find this fascinating. I've long since gone over to the dark side of software engineering, but as an undergraduate math major in the late 60s, my interest grew exponentially when introduced to abstract algebra. I also loved linear algebra (as long as we weren't working with anything so mundane as numbers). While I did reasonably well in my analysis courses, I found them boring compared to algebra.

Perhaps that's what attracted me to computing in the first place. Developing software, even numeric software, seems to call on the abstraction mechanisms of algebra much more than analysis.

Alexander R Pruss said...

There may be a similar distinction in programming. I prefer lower levels of abstraction. I've not managed to learn any OO language, for instance. Instead, I do serious programming in C (and in the past did some in F77), and quick and dirty stuff in perl (but not using any OO features, except canned ones). One of the programming projects I am happiest with was writing much of a low level library for the Sharp 7xx electronic organizer series (with associated reverse engineering), a lot of it in Z80 assembler. Another thing I was really pleased with was a DOS utility that converted .com executable files into text files that were still executable (an example is here). This required writing decoding code in x86 assembly under the constraint that all of the opcodes and numeric constants had to be in the decimal 32-126 range. That was fun.

I can appreciate the beauty of highly abstracted methods. But I think I am most at home producing optimized code at a low level of abstraction. Likewise, I prefer analysis, and in lower dimensions.

Clark Goble said...

It's an interesting question. One could argue that even classic domains like geometry come out of approximations. Yet often being able to prove something even when you know it is true is quite different. And the methods of proof needn't always be tied to approximations.

Indeed I think this is why physicists taking math classes often struggle in certain places since the skills don't translate.

Enigman said...

My initial suspicion is that there is a sociological explanation, e.g. that those who had good innate maths skills came to like maths in their early years, and so were more likely to be attracted to abstract maths (and would have a few less problems there too); and (more strongly) that those who had weaker innate skills were more likely to dismiss higher maths out of hand (and so forth). (I presume that the researchers compensated for such facts as those from good families being relatively good at both on average.) Still, the brain is very complex, and there may well be apposite connections between the two indicated areas...

Nichoteh said...

Hi Alex, I agree that the ability to approximate is crucial to analysis whereas it rarely helps in algebra. (But then again, there is functional analysis, which is much "softer" and requires more algebraic/topological methods.) I would note, however, that analogous skills are necessary in algebra -- one needs to develop an intuition for how "good" or "tame" algebraic objects behave, so one can handle more complicated cases. (Think of the methods of homological algebra, spectral sequences, etc for instance.) That's a *kind* of approximation...

Alexander R Pruss said...

I think you're right about the algebraic kind of approximation. But I wonder if there may not be a difference in the intellectual skills involved in approximation in things that involve real numbers (that may involve spatial reasoning, for instance) and kinds of approximation where real numbers are not involved. For instance, when I did algebra, I had very little in the way of approximative intuition.

(I think what I say about real numbers also works for non-standard reals.)

Heath White said...

Really interesting. Anecdotally, I have noticed that people who are good at geometry are also good at physics, and vice versa. I think it's because they both use a form of spatial reasoning. Calculus is all about space: areas under curves, slopes of lines, rotating curves around axes, etc. And limits are just approximations taken to the nth degree. I am fairly good at all of that.

OTOH the skills involved in formal logic seem quite different to me, spatial reasoning is not much use, and the skill-set comes less naturally to me. Sometimes I can visualize diagrams of possible worlds and that helps to an extent.

But now I'm curious, because when I was a software engineer, the higher the level of abstraction, the better I liked it. I absolutely hated writing assembly code. And I think it was because you have to obsess over nitpicky details at that level; you can't "approximate" to anything.

Alexander R Pruss said...

I really liked doing Z80 assembly. I've done years of low-level C programming for Palms, too, and that's been fun.

I've only very recently learned object oriented programming--it was just becoming prominent when I was finishing up as an undergraduate and only now did I get into it. There is something really lovely about the level of abstraction, but it's also annoying. I feel I don't have as good a handle on the details of program flow.

I find abstraction beautiful, but hard.

Alexander R Pruss said...

Heath:

Your comment suggests a triple distinction:
1. Higher-order abstraction (the sort that get used in comp sci or algebraic abstraction)
2. Detail-based low-level abstraction (like the low-level algebra one does in high school, or the non-geometric parts of calculus, or assembly programming)
3. Spatial reasoning.

Here would be an interesting experimental philosophy question. See what correlation there is between abilities in 1-3 and philosophical views.

One might conjecture, for instance, that philosophers good at 1 are going to be attracted to functionalist or normativist accounts of various things. Philosophers good at 2 might have a pull towards reductionist and/or dualist programs. (I think there is a connection between these two tendencies.) A natural conjecture is that philosophers good at 3 would like mereological ontologies.

My own abilities rank probably as follows from best to worst: 1.5,3,2,1, where 1.5 is something in between 1 and 2 in level of abstraction. I have reductionist and dualist leanings. I recognize the power of functionalist and normativist accounts, but generally think they're not going to be right. But I strong disagree with mereological ontolgoies.

Heath White said...

Fascinating.

FWIW, my abilities would be roughly 1 > 3 > 2. I am quite attracted to functionalist (often normativist, but that is just functions of norms) accounts of many things. I generally think dualisms of all kinds need to be overcome. Reduction is okay, when it works, because it’s a form of functionalism, but it generally doesn’t work. Mostly I despise eliminativism and fictionalism.

I have no settled views in ontology except I have an animus against views which give me weird answers, which seems to be all the views—so I am if anything a kind of pragmatist or ordinary language philosopher about ontology. Again, you could think of this as a kind of functionalism about ontology (what there is, is a function of how we talk). Probably I could be talked out of that—it doesn’t sound promising when I put it in pixels—but it is my natural instinct.

Alexander R Pruss said...

So here's the question with bite. How many of our philosophical views V are driven simply by the fact that we are better at the sort of thinking that would be called for if V were true than at the sort of thinking that would be called for if some relevant alternative were true?

I think this is the sort of thing we can correct for to some degree by two means. First, sometimes we can withhold assent to a view V, but think: "I am exploring V-type views, because V-type views are what I am best at exploring. I'll let those who are better at exploring V*-type views explore those." Second, and perhaps better, we can compensate by admiring and respecting philosophers who have abilities different from our own, and increasing the weight that their opinion plays in our deliberation. But it's hard to get this right.

We don't do much of this in intellectual endeavors we're not actually engaged in. I'm no good at the sort of mathematics that string theory involves. But I don't feel much temptation to think that a theory in physics that uses that sort of mathematics is less likely to be true than a theory that uses mathematics that is easier for me. However, if I were a physicist, I might feel such a temptation.

Heath White said...

Yes, that question does bite. Essentially, it is a suggestion that we suffer from a heretofore unnoticed form of wishful thinking. I can imagine a quite interesting program of experimental philosophy excavating all this.

I wonder if other fields would be equally susceptible.

Thomas Larsen said...

Another C programmer - yay! :-)