Proof vs. Essay: Impossible Convergence?
John Arley Writes in Facebook:
We learned how to do proofs in geometry class, then pretty much that all got ignored for ten years and we focused only on calculations. Why?
My Response (too long to fit in Facebook):
Well, actually, this kind of step-by-step rigor is also supposed to be taught in your English, History, and Economics classes in the form of argumentative essay-writing (ie -- argue a thesis based on a magical synthesis of facts that leads to a larger epiphany not contained within the set of individual facts; which is arguably the concept of a mathematical proof as well). However, the English language, while being a vastly richer and more subtle language than symbolic arithmetic (and thereby capable of communicating an equivalent, if not superior, amount of rigor and complexity) often gets abused by lazy students and teachers who Just Don't Care™ and who instead turn that expressive richness upon itself to create a fertile loam of BS.
There are places where the rigorous essay survives, but very few. It's particularly telling that most complex theoretical mathematics proofs are actually written out longhand, essay-style, and not in the two-column approach taught in Baby Geometry. And in Economics right now, we're having to write essays stepping logically through a sequence of graphical descriptions (basically supply & demand curves) of the interactions of macroeconomic forces to arrive at broader truths about how a stimulus package or spending reduction affects unemployment rates, GDP, price levels, wages, etc. But our professor is pissing everyone off by giving out B-‘s to sloppily argued analyses. “It’s just opinion and BS!,” cries the indignant thought slacker. I wonder how much longer before our professor just switches over to multiple choice questions and gives up on essays altogether.
The real problem is that most everyone seems to have given up on the potential of longhand, non-symbolic language (well, I guess all languages are technically symbolic, but you know what I mean) to construct rigorous arguments and discover higher truths, which is arguably the root cause of the demise of the liberal arts education. These days, a well-written essay is almost universally seen as "opinion" and not "fact," even if it begins with rock-solid assumptions and contains an airtight chain of logical arguments leading to a singular conclusion (or a constrained distribution of possible outcomes). Whereas, an inferior argument, coded in a bare-bones symbolic language, underlain by a basic vagueness of assumptions and bogus constraints, can, as long as it is free of obvious symbolic manipulation errors, oftentimes pass as truth for decades before the underpinning flaws are revealed. And even then, an overturned equation is usually seen as an honest mistake rather than a “gotcha” detection of BS.
So proofs are potentially all around us in high school and as undergraduates (even in -- gasp -- MBA School!), just not necessarily in the same form that we're taught to associate with "proofs." But as our usage of spoken and written language continues to deteriorate, and our confidence in it as a medium of argument declines, we find ourselves only trusting the most barren and simple symbolic languages to derive greater truths, identifying equation derivations as “proofs” in stark relief against the BS of the Essay. The two approaches should be convergent methods of truth discovery but aren’t treated as such and therefore don’t function as such in real-world practice. I’m not surprised, then, that you don’t see proofs everywhere in school. It’s not just for lack of looking, or looking in the wrong places; but that the English-language version of your prey has given up and taken up a postmodern camouflage of vagueness, masquerading as opinion instead of fact, BS instead of logical proof.
The real question is: how did education become all about getting assignments in on time and checking off task boxes instead of critically arguing towards greater understanding? If we could understand that, we might be able to understand how it was that students and teachers stopped taking essay-format proofs seriously. As it is, we only seem to have faith in arguments expressed in the absurdly simplistic language of symbolic logic. Which means that we really only have faith in that which is computationally calculable. A depressing thought indeed for the human species: that our own thought is constrained by the discrete workings of machines. That our larger-than-life thoughts are made in the image and likeness of neural firings collected within synaptic groups instead of being weak representations of infinite truths.
Or is this all just more BS, conjured up within the vagaries of the English language, cheap rhetoric, and sneaky sophisms? If only I could make my argument using a two-column proof, you just might buy into it.
Maybe next time.
John Arley Burns
Perhaps some of this is brought about the by legal profession: teaching students to argue any point with equal rigour regardless of its underlying veracity.
Perhaps some is the adoption wholesale of the Prussian education model, complete with bells, rigidly structured class periods, rows of desks, training the bureaucratic soldiers of tomorrow.
Perhaps some is the triumphal march of a globalized Hollywood across the dead corpses of the Literati, where fame and fortune is awarded to those most able to debase themselves before a jeering crowd.
Perhaps some is the Rise of the Computer, which cares not for our well-worded responses, which understands not a word of our speech not specifically altered for its limited understanding, which crashes for the slightest of misplaced characters.
But among this I see a ray of hope, a banner of progress, a march to a different drum. The Continent, "Old Europe", especially France, as imbued as it is with McDonalds and Celebrity-worship and Monster Truck Rallies, has an intelligentsia that seems less interested in the complete Anglo-Saxon subjugation of all effort and intellect to the service of profit, and more interested in Art (with a capital A) and its effects. Only when you have enough vacation time and cultural acceptance of living in sub-million-dollar apartments is there possibly enough time to think about things. It always saddens me that NYC, our supposed intellectual heartland, America's answer to our hypothesized lack of culture, is the city more than any other given over to the complete worship of material success, with a price level to match. You can be an artist, but no starving artists here: bring a reference letter from your trust fund manager when applying for the apartment lease.
You see a little kernel, maybe growing smaller each year with the Californication of Culture, in Austin, where you can Think Different and you don't need to do the New New Thing and work for an Investment Bank to be worth listening too. Where it's okay to sit back and read some obscure work by a Basque philosopher Just Because You Want To, and then shoot the breeze about it until 2am over some joe.
But every time I go back I see fewer and fewer of the Free Thinkers, and more and more of the Douchebags with BMWs.
We could sum this up with: if you ever need anything more in life than a bookstore, a cafe, and an apartment, you've got your priorities wrong.
Anyhow, enjoy MBA school. Maybe you'll come out unscathed and work for a non-profit doing micro credit. The soul is a terrible thing to waste. :-)
posted 1:18 PM