October 31, 2004

From politics to mathematics and back

Posted by shonk at 12:54 AM in Geek Talk, Politics | TrackBack

Last night I found myself with an unusually large chunk of time on my hands and, after doing some maintenance work around here that I’d been putting off, decided to catch up on some blog-reading. I read Colby Cosh’s excellent analysis of the ALCS from a week or so ago, enjoyed Billy Beck’s musings on book addiction and rantings on the justice system, caught up on the No Treason/Karen DeCoster/Thomas DiLorenzo shitstorm, uncovered the latest links that appear below in the “External Links” section, and enjoyed Scribbling’s pomegranate pictures. Somewhere along the way, I came across Cosmic Vortex’s “First political diatribe,” which suggests the notion of “political shock levels” as a complement to the future shock levels which extropians go on about. The author lays out a sort of political spectrum, ranging from communist to fascist, and then suggests the following:

Now, its very easy for a socialist and a progressive to discuss issues and come to an understanding, but try to get a socialist and a right wing republican together, and nothing will get accomplished except frustration and anger. Where does this leave us? Not in a good situation really - as theirs no real way to drag anyone more then 1 level away. Even if they did want to try to understand your position, they just couldnt map the concepts over if you jump too far. The cognitive differences would be un-breachable and it would require starting at the begining of their conceptual “tree”, validate every concept along the way, and maybe then something could be worked out.

Interesting idea and stated in a somewhat unique way. What really caught my attention in reading, though, was the sentence I’ve taken the liberty of italicizing. I have to admit, the very first thought that popped into my head upon reading that sentence was: “Sounds like a chain complex!” For those too lazy to click the link, a chain complex is basically a sequence of maps between objects such that moving two steps along always takes you to zero. They arise a lot in topology and homological algebra (for example, I first ran across them while learning about simplicial homology in an algebraic topology class). The connection with shock levels being that if you try to map more than one level down the line you can’t go anywhere but zero, just as the conversation between socialist and republican goes nowhere.

“A nice little analogy,” I though to myself, not quite realizing, for the moment, how loony it would have sounded had I tried to explain it (at this point tenses completely break down, given that I just have tried to explain it). Consider, in addition, how one of my office-mates and I had laughed earlier in the day when she described having just caught herself before asking two of her students (who are identical twins) if they were “isomorphic”. I know I’ve talked about this before (that time when a friend referred to this Strong Bad song as a “canonical techno song”), but I still find the way in which the accumulation of a new vocabulary shapes my outlook either amusing or frightening, depending on the time of day.

Of course, in a sense, the vocabulary is the least important part of what I’ve (hopefully) learned in the last year or so of grad school, but applying the vocabulary outside of its mathematical context is perhaps the most obvious outward sign. Well, one of the most obvious, anyway. Perhaps the other obvious sign of what might be called my increasing mathematical sophistication (or confusion, depending on your perspective) manifests itself in how I answer the questions of my students.

I’m currently teaching four recitation sections of a class innocuously called “Ideas in Mathematics” in the course catalogue, but of which the unofficial course title bestowed by the professor is “Mathematics and Politics”.1 A friend rather uncharitably characterizes it as “math for morons”, in that it’s the only freshman-level non-calculus course that fulfills the college’s math requirement. Anyway, the point is that I spend most of my time answering questions about the homework or the lectures, and, in answering, I often find myself engaging in impromptu monologues about how intuitionists would object to proofs by contradiction or how mathematics only describes the world insofar as it simplifies away the hard bits. And, most importantly, I have a very difficult time answering conceptual questions definitively.

Needless to say, I imagine my students find it frustrating when, for example, they ask “Why is a conditional true when its antecedent is false?” and I have to say something along the lines of the following:

Well, the short answer is because it’s defined that way, and the long answer is still because it’s defined that way, but it’s defined that way because that’s really the most reasonable way to define it. You see, when we’re thinking about whether a logical statement is “true” or “false”, it’s probably best not so much to think actually in terms of “true” and “false”, but rather in terms of compatibility with the world. In other words, can you believe the statement while also believing in reality without contradicting yourself? We only say the statement is “false” if not; otherwise we call it “true” even though it may be counterfactual, absurd, or completely irrelevant to reality.

At this point, I’m usually lucky if the looks I’m getting are merely quizzical. So I try again:

Well, let’s think in terms of an example. Suppose, back in 2002, a friend told you “If the U.N. approves a war in Iraq, France will go to war.” Now, we know that, in reality, the U.N. didn’t approve the war and that France didn’t go to war. So this is a situation where the antecedent and consequent are both false, so, if we’re thinking in terms of logic, we would say the conditional is true. Why? Well, because you can believe what your friend said and also believe in reality. That is to say, you can believe the statement without contradiction. So the assignment of “true” or “false” is more or less like how you treat a friend: because he’s your friend and you trust him not to mislead you, you assume he was telling the truth unless you can definitely prove that he was lying. In this case, the only way you could know he was lying would be if the U.N. had approved the war and France had stayed home (i.e. the antecedent is true and the consequent is false), since that’s not what actually happened, we would say that the statement is “true”.

Having given this explanation more than a half dozen times, it’s been mildly surprising that nobody has actually called me (and, by extension, math) out on it all being a bunch of convoluted bullshit, but I have to imagine some were thinking it. Usually, at this point, seeing the pained expressions on some of the faces staring out at me, I say something along the lines of “Of course, you could just think of this as the definition,” which seems to be a great relief to some. Which is ironic, given that, without the explanation, the notion that things could be this way just because that’s how they’re defined seems entirely unsatisfactory (which, by the way, I completely agree with. Definitions suck without context).

Having spent ten minutes writing about the conditional, I’m not sure it really illustrates the point I’m trying to make. Perhaps more appropriate would be the times that I’ve had to catch myself before I start ranting about epistemology, theories of logic, reductionism and how mathematics education is, essentially, a system of useful lies. Just as a calculus teacher extolls the virtues of the definite integral, talking about how useful it is and how many amazing physical properties it explains without mentioning that, in any actual application integration is not only difficult but usually impossibly difficult, I find myself teaching material which is useful in certain cases but usually too simplistic to be applicable to the real world. I try to point this out as much as possible, but I think it’s still probably misleading.

That having been said, the underlying concepts are, in fact, incredibly deep. It’s difficult, though, to emphasize that what’s important are the concepts, the fundamental ideas which lead us to particular formulas or computations, especially when midterms are looming and homework is due on Friday. I remember one student asking, the night before the midterm, if she ought to memorize a particular counter-example listed on the review sheet. My honest answer was “No, I don’t think you should memorize it; I think you should understand it,” which I don’t think she liked very much.

That question, though, lies at the heart of the topic that I’m apparently (finally) coming around to, which is that there seems to be a fundamental dichotomy in most people’s minds between, say, the humanities and mathematics. I doubt if anybody would ask an English professor, the night before a midterm, if he ought to memorize Joyce’s “The Dead” for the test, but in a math class it seems like a perfectly legitimate question (incidentally, I’m not trying to say that memorizing is completely worthless; in learning a foreign language, for example, unless one is lucky enough to be living in the country where the language being learned is spoken there’s really no way to make progress without memorizing verb conjugations, vocabulary, etc.). The fact that, for whatever reason, mathematics seems to be equated with rote memorization and plugging values into a formula seems to me to be one of the primary reasons that so many people have such a strong aversion to math.

Which is completely understandable, in a way. Memorizing is boring and almost completely lacking in cognitive content, which most people instinctively recognize, and the fact that math is equated with this boring activity is, I think, one of the primary reasons why an aversion to mathematics is considered acceptable even among people who would strongly decry stunted development in other intellectual pursuits. As John Allen Paulos puts it in Innumeracy: “In fact, unlike other failings which are hidden, mathematical illiteracy is often flaunted: ‘I can’t even balance my checkbook.’ ‘I’m a people person, not a numbers person.’ Or ‘I always hated math.’”

As I look back on the above, I hope I’m not giving the wrong impression about my students. They’ve been wonderful, certainly much more perceptive and good-natured than I had any reason to expect, and I hope they’re learning as much as I am. What it comes down to, I think, is that it’s virtually impossible to interact on a daily basis with people whose level of expertise in a given field is significantly less than one’s own without having to think quite a bit both about the nature of that expertise (imperfect though mine still is) and the misconceptions about the field that will inevitably come to light.

Anyway, I’ve now strayed quite far afield of what I originally intended to write, which was a self-deprecatory post about how I’ve become almost stereotypically geeky in grad school, but I guess the above sort of illustrates that point.

1 Actually, a very interesting class. Aside from learning some basic logic and doing some simple proofs, we’ve talked a lot about different voting systems, leading up to the proof of a simplified version of Arrow’s Impossibility Theorem, the full version of which says that there is no voting system (other than a dictatorship, which everybody pretty much agrees isn’t much of a voting system) which satisfies both the Pareto condition (which says that if everybody prefers candidate X to candidate Y, then Y will not win the election) and independence of irrelevant alternatives (i.e. there is no “spoiler effect”). Also, we’ve learned a bit about power indices, namely the Shapley-Shubik and the Banzhaf indices, and are now starting on some basic probability.



It's been a while since I took my last logic course (in a philosophy department, not a math department), but from what I remember, the class dealt with validity and invalidity, not truth and falsehood. An argument was valid if it could not be the case that the premises were true and the conclusion false.

Now, as far as I recall, this was a description of arguments and not individual statements. But it seems to me that arguments and individual statements follow the same rules: just as negating the conclusion of a valid argument negates one or more of the premises, so too negating the consequent of a statement also negates its antecedent. Whereas, negating one or more of the premises of a valid argument tells us nothing about the conclusion, just as negating the antecedent of a statement tells us nothing about its consequent.

So, I guess my question is, why not just use validity to describe both of these relationships, rather than truth? Truth seems confusing in this context, as it implies empirical truth rather than logical possibility.

Posted by: Micha Ghertner at October 31, 2004 07:42 AM

Memorizing is boring and almost completely lacking in cognitive content, which most people instinctively recognize, and the fact that math is equated with this boring activity is, I think, one of the primary reasons why an aversion to mathematics is considered acceptable even among people who would strongly decry stunted development in other intellectual pursuits.

I'd say more central, if maybe more subconscious, is the fact that in same ways there seems to be a fundamental disconnect, or at least a heterogeneity, between the hazy, ambiguous world in which we live and the (at least seeming) precision and clarity of the realm of logic and math, or similarly between them and the ambiguity and imprecision of language. This being the case, logic and math can seem by turns a challenge to our daily reality, pointless escapism from or just different, which given the persistence of nationalism seems to be a sufficient reason for many to shun something.
So, I guess my question is, why not just use validity to describe both of these relationships, rather than truth?

I'll leave this question to my brother to respond since you asked him, but if you were asking me I'd say that when you're addressing the role of logic in the real world, truth and falsehood are much more important than validity and invalidity, which are often irrelevant considerations at best and misleading at worst. Whether logic is in any way prepared to address truth and falsehood is another question, although certainly one of which I'm rather dubious.

Posted by: Curt at October 31, 2004 09:59 AM

So, I guess my question is, why not just use validity to describe both of these relationships, rather than truth? Truth seems confusing in this context, as it implies empirical truth rather than logical possibility.

Well, in a sense, when we say that some sort of statement in logical language is "true", we are really talking about validity. Hence the reason I use the term "compatibility" in the explanation. That is to say, when we're deducing within any particular formal system, we can only talk about our arguments being valid or invalid within that system.

That having been said, I think Curt's right: if we want logic to mean anything, to have any sort of applicability to the world, we need it to speak to truth and falsehood. The ideal, as it were, is for "validity" and "truth" to be one and the same thing. If you make a valid argument based on true premises, you want the conclusion to be true. Otherwise, what's the point? If I make up my own crazy system of logic and, within that system, make valid arguments from true premises and arrive at demonstrably false conclusions, my logical system is going to be worthless or, at any rate, ignored. What we generally recognize today as "logic" is the product of centuries of refinement, the cumulative product of many, many attempts to create a system of rules of inference which, properly applied, will make the conclusions of valid arguments from true premises true.

Now, has that process been successful? That is, if we make a valid argument within the commonly accepted rules of logic from true premises, is the conclusion really true? The answer to that question, I suppose, depends on your theory of epistemology and is, needless to say, a philosophical one. Certainly, as Gödel demonstrated, there are true statements that cannot be arrived at by way of valid arguments from the standard axiom set, but that doesn't necesarily say anything about the converse.

Actually, I tend to side with my brother in that the answer to that question may be irrelevant, anyway, because I think the problem of coming up with non-trivial, true premises is much more difficult than most people like to admit.

I'd say more central, if maybe more subconscious, is the fact that in same ways there seems to be a fundamental disconnect, or at least a heterogeneity, between the hazy, ambiguous world in which we live and the (at least seeming) precision and clarity of the realm of logic and math, or similarly between them and the ambiguity and imprecision of language.

I'd agree with that assessment, although I think one of the problems there is that the "precision and clarity of the realm of logic and math" is itself based on something of a misconception. For example, when we were first introducing the notion of proof in class, many of my students wanted to know "how do I prove something?" Well, there's no good answer to that. There are, of course, certain standard methods (by contradiction, proving the contrapositive, etc.) and certain basic requirements of any proof (if you're proving a conditional, the first thing you need to do is assume the antecedent), but there's no way to say, a priori, how to prove a particular statement. Generally, my answer boiled down to: "Well, first you need to figure out the facts you know and the conditionals (typically premises) you can use. Then, you need to think about how to plug the facts you know into the conditionals you can use to get (i.e. via modens ponens) the conclusion you want."

Another question that came up a lot was in reference to questions of the form: "Evaluate the following statement. If it's true, prove it. If not, find a counter-example." My students wanted to know how to know whether the statement was true or not. Again, there's no good way, in general, to know whether it's true or not. Of course, it often helps to think about particular examples, or to think about why it's probably true or false, but, ultimately, answering the question relies upon intuition, upon "seeing" why it's true or false and then using logic to support that intuition.

What I'm trying to say is that math and logic really aren't algorithmic, that although the statements may be precise, the process of arriving at them is not. I understand why there exists the misconception (primarily because of crappy grade-school and high school teachers who don't understand math very well themselves), but that doesn't mean it isn't frustrating.

Posted by: shonk at October 31, 2004 01:27 PM

I might add that mathematicians and logicians in my estimation have often been their own worst enimies in that regard in terms of setting exaggerated expectations for the precision and certainty of their results (Descartes, for example, essentially blaming the world for failing to live up to the idealized standards of math and logic). In any case, whatever degree of success they've been able to attain within the discipline surely contrasts very greatly with the level of certainty anyone can reasonably expect to attain in defining concepts in the outside world.

Posted by: Curt at October 31, 2004 05:59 PM

Thanks for the mention. I didnt realize anyone actually read my nonsense :)

When I think of political shock levels, it harkens me back to Ayn Rand's constant use of mathematical concepts in relation to language.

See: http://www.objectivistcenter.org/events/advsem04/ThomasARConcepts.pdf

A concept is a mental integration of two or more units possessing the same distinguishing characteristic(s), with their particular measurements omitted.

The units of a concept were differentiated—by means of a distinguishing characteristic(s)—from other existents possessing a commensurable
characteristic, a “Conceptual Common Denominator.” A definition follows the same principle: it specifies the distinguishing characteristic(s) of the units, and indicates the category of existents from which they were differentiated.

Very mathematical perspective indeed. Shes practically discussing math here, even borrowing the terminology. This is why I think A.I.s will be able to make such good headway towards philosophy and "correct thinking". With the ability to syncronize implicit definitions, their discussions will be MUCH more fruitful. Any disagreements could be isolated at the conteptual level where the divergence of viewpoints occurs and a resolution over the issue could be addressed directly at the level needed.

Posted by: CosmicV at November 10, 2004 05:58 PM