This morning a story on slashdot linked to A Mathematician’s Lament by Paul Lockhart as well as a blog post discussing the issue. This is the first time I’ve read Lockhart’s rant but little of what he said was new to anyone who has listened to any of the mathematicians vocally crusading for better mathematics education. As usual most of his piece was the same unrealistic claptrap about how if we could only show the children the joy and beauty of math all would be well. It’s a pleasant fiction to believe, and it’s seductive to think that with just one little nudge all children could discover the pleasure we take in mathematics, but it’s about as reasonable as believing everyone would love to garden, read Shakespeare or anything else if they were only exposed to it.

To be fair I was quite impressed with Lockhart’s brutally frank analysis of what currently passes for mathematical edification in K-12. Apart from teaching kids to sit still and signaling social status¹, for 95% of the class everything after multiplication (and a good bit before) is totally pointless. Go talk to some doctors (GPs), lawyers, managers, etc.. to solve a simple algebra problem (say two linear equations in two unknowns) and see just how few have the slightest clue. No one’s benefiting by making them rotely memorize some rules they promptly forget. However, the notion that if we only taught children real math they would gobble it up is simply absurd.

I remember taking a differential geometry class in college that was taught in what was (at least for me) an abominable way. No rigor, just vague comments about pictures and twisting so unsurprisingly I kept putting that homework off and gradually falling further behind. At that point understanding became nigh impossible. Just doing the problem sets made me feel frustrated, angry and perhaps a bit inferior and I’m unreasonably over confident about my mathematical abilities. Naturally one then puts them off and when you force yourself to work you just grind through the problems without any curiosity or hope of understanding the bigger picture. Psychologically you just can’t force someone to be curious and deeply thoughtful about a subject that makes them feel bad and that’s what understanding math requires. So certainly a sufficently bad teacher (who won’t follow a book) can discourage real learning but could better teaching really significantly encourage real mathematical understanding in K-12?

Sure if you somehow eliminate the social significance of mathematical ability and turned math class into a non-threatening fun activity like most HS art classes you might make some progress. That, however, is simply impossible. Nothing the teacher says can erase the knowledge that actually showing interest and talent in mathematics opens up many lucrative doors and signals intelligence. So long as the mathematically gifted are financially rewarded students (and their parents) will care about how they perform in the subject. Unlike art or literature math also has right and wrong answers and can often leave one feeling lost and frustrated so unavoidably half your student body will resent math for making them feel stupid and inferior even if they would never admit it. No matter how excellent the teacher they can do no more than try to distract the under performing students from inevitable comparisons with those who are doing better. Worse, any attempt to discourage people who dislike the subject from taking the courses will simply increase the incentive for them to camouflage themselves as someone who does like math to future schools and bosses. The problem would be a lot easier if it was just that some people weren’t smart enough.

I don’t really know what we should do about this situation. However, I suspect one reason people are so reluctant to face this possibility is that it would require us to explicitly consider how we want to trade off the benefit to the small fraction students who could benefit immensely from non-rote proof based mathematics and in turn contribute disproportionately to our economic growth against the interests of the larger number of students who are too intimidated by the subject to do anything but rote work. I think we ought to consider using programming, with it’s more video game/slot machine pace of rewards, as the means to teach logic and quantitative thought but that still doesn’t answer the math question.

So browsing the web this morning I came across this amazing blog largely focused on the author’s (apparently a philosophy grad student somewhere) continentalist approach to Godel’s incompleteness theorem. Rather than describe the content I’ll just include his last post.

Perhaps this will be my last post here? A simple reiteration of negative Platonism, situating its significance in the context of awakening from the wrong expectations performed so thoroughly and unconsciously in the second Critique.

To put it once again with maximal simplicity: The diagonal is what relates, without religious/imaginary synthesis, our mathematical/cognitive and ethical/existential lives.

We already live in both places: in consistency through calculation and consciousness, in completeness through care and the unconscious. What we suffer from, as both theoretical inadequacy and ethical alienation, is an inability to relate these in a way that makes sense and is good.

Thus it has suddenly become possible, after long stagnation, to say something rigorous and suggestive, something that opens logoi both mathematically lucid and existentially thick (again without synthesis: it’s a matter of bridges and transitions, not of sovereign unities or systems) about the fundamental Socratic question: which knowledge, which part, of knowledge, would do us any good?

At stake here is exactly what gets talked about, prephilosophically, as “the meaning of life”. It is good philosophical practice to avoid this question until one has something real to say about it, and instead, to work the problem from either side. But it is not good practice, once the relation has become clear, to remain squeamish about naming it: Idea of the Good, Diagonalization.

Note, if you read the rest of the blog it’s totally clear that he really means diagnolization in the sense of the mathematical technique employed by Godel. Moreover, he seems to genuienly understand the mathematics (Godel’s theorem is a result in a meta-system describing provability in some formalized system) so what’s going on here is surely not mathematical confusion. It’s the philosophy that’s totally fucked (I’m pretty confident now that it’s not a hoax).

However, to be fair to this blogger, he isn’t some isolated crank, but rather a participant in a ‘respectable’ philosophical tradition. Indeed, one of the famous philosophers he references, Alain Badiou is even more incoherent. While he would almost certainly quibble with the description given on wikipedia if the following is even remotely accurate he might as well be spouting gibberish.

Badiou’s use of set theory in this manner is not just illustrative or heuristic. Badiou uses the axioms of Zermelo–Fraenkel set theory to identify the relationship of being to history, Nature, the State, and God. Most significantly this use means that (as with set theory) there is a strict prohibition on self-belonging; a set cannot contain or belong to itself. Russell’s paradox famously ruled that possibility out of formal logic. (This paradox can be thought through in terms of a ‘list of lists that do not contain themselves’: if such a list does not write itself on the list the property is incomplete, as there will be one missing; if it does, it is no longer a list that does not contain itself.) So too does the axiom of foundation — or to give an alternative name the axiom of regularity — enact such a prohibition (cf. p. 190 in Being and Event). (This axiom states that all sets contain an element for which only the void [empty] set names what is common to both the set and its element.) Badiou’s philosophy draws two major implications from this prohibition. Firstly, it secures the inexistence of the ‘one’: there cannot be a grand overarching set, and thus it is fallacious to conceive of a grand cosmos, a whole Nature, or a Being of God. Badiou is therefore — against Cantor, from whom he draws heavily — staunchly atheist. However, secondly, this prohibition prompts him to introduce the event. Because, according to Badiou, the axiom of foundation ‘founds’ all sets in the void, it ties all being to the historico-social situation of the multiplicities of de-centred sets — thereby effacing the positivity of subjective action, or an entirely ‘new’ occurrence. And whilst this is acceptable ontologically, it is unacceptable, Badiou holds, philosophically. Set theory mathematics has consequently ‘pragmatically abandoned’ an area which philosophy cannot. And so, Badiou argues, there is therefore only one possibility remaining: that ontology can say nothing about the event.

For any readers familiar with set theory the part about drawing ethical maxim’s from Cohen’s method of forcing might be even more amusing. Sure, he is hardly the first continental philosopher I’ve read who should be properly regarded as a crackpot but when it’s about my subject (mathematical logic) it just makes the point all the more clearly.

Now reading this sort of BS is kinda amusing but I do have a broader point. Despite being essentially indistingushable from the sort of crank theories that pop up from physics crackpots all the time the people publishing this stuff are still seen as respectable, even acclaimed, philosophers. If philosophy wants to be a serious intellectual discipline it needs to take the same hard line that they physicists do about crackpots, even if it means tossing out entire university departments.

The physicists wouldn’t simply sit quietly and say nothing about a crank being allowed to teach physics courses, nor attend conferences or journals that treated them as respectable researchers. Moreover, were they to do so the progress of the discipline, and certainly the public understanding of physics, would be greatly harmed. My point is ultimately that it’s not enough for analytic philosophers (particularly tenured ones) to sit back and privately dismiss all this crap as rubbish. They have a positive duty to denounce these people as cranks and eliminate them from the field. Failing that they have a duty, even if it imperils funding, to demand departments be split and otherwise clearly distingush what they do from what the continental crankpots do.

To be clear not everyone one might classify as a ‘continental philosopher’ should be deemed a crank. Despite being notoriously confusing Kant surely is not. Mere error or poor writing is not enough to be a crank. However, neither the blurriness of the line or our inclinations to charity are an excuse for tolerating obviously incoherent gibberish as valid philosophy. Since it’s notoriously difficult to conclusively establish that some convoluted continental style ‘argument’ lacks any reasonable interpretation the burden should be on the person presenting the apparent gibberish to convince others they are merely really poor writers with a meaningful point.