Is Math or English Harder to Theory With?

By Graham Peterson

Fabio Rojas and crew got into a discussion on Twitter about whether mathematical theory in social science is more difficult than verbal theory, or as Fabio summed it up:

fixed point theorems

Everyone in the thread agreed that dense verbal theory is much harder to read than mathematical theory.  But I think they’re about the same.  (Andreas Glaeser’s opinion on Foucault is here worth mentioning [insert arms like a symphony]: “you think to yourself, ‘now this is what language can do.’“)

We have a lot of folk assumptions about the difference between “verbal” and “formal” theory in social science, and too much violence between their practitioners, but very little discussion of their actual differences or advantages.  Note quickly: both verbal and mathematical theory are “formal.”  They both aim to generalize formal structures of logic, so I’m ditching the adjective “formal” and will refer to “mathematical” theory henceforth.

Bad verbal theory suffers from the same problems bad mathematical theory does.  If you ever get mad enough at mathematics that you read Why the Professor Can’t Teach, a criticism of mathematical pedagogy and research by Morris Kline, you’ll notice that most of the problems he identifies are exactly analogous for verbal theory.  Kline laments mathematics that generalizes for the sake of generalization, and he laments the presentation of general proofs without intuition and examples.

These are, to my eye, exactly the things that make Foucault et al. extraordinarily difficult to read.  Concepts get generalized for their own sake, until the exercise becomes so meta-theoretic it is only interesting to a handful of specialists, and applicable to nothing.  It might be the case that the material world is merely a realization of the world of ideas, but I really doubt that we’re learning much from “reimagining neoliberal ontologies.”

And where are the examples?  You just know that when you’re reading Bourdieu, there’s some vignette of piano lessons dancing around in his head, while he’s drawing sweeping generalizations about cultural capital.  And he’s probably generalizing from some children’s game where one gains and loses power, while he’s talking about misrecognized exchanges of subconscious power. But without making those examples explicit, the reader cannot extrapolate to generalities in the same way Bourdieu has.

Good theorists present their ideas like recipes or step-by-step instruction manuals, not assertions of propositions and generalities.  That is, good theorists will walk you through exactly those steps they took (usually starting with a rudimentary kernel, case, or example) to arrive at a generality, rather than presenting themselves as if all their brains just trade in dancing abstractions.  We are, though, both mathematical and verbal theorists, tempted to do the opposite.

We induct from one thing or another until we think we’ve found something general.  Then we turn around and assert the generality of that proposition, and try to prove it deductively.  We (sometimes) eventually present the case or example as if it’s just a convenient afterthought or demonstration.  When in fact that kernel drove our logic the entire time!

If we can drop the pomp and pretense, and focus on communicating our thoughts in the way that we actually arrived at them, we will have much clearer and easier to read mathematical and verbal theory.

Also note that good mathematical and verbal theory do pretty much the same things.

Creativity in mathematical and verbal theory is metaphorical and analogical, not deductive.  That is, mathematical creativity comes from (say) writing down a telescoping equivalence into a proof to clean it up, or recognizing a dual from a different subfield.  In verbal theory, analogical creativity comes from (say) writing down an epidemiological metaphor in a new context, like crowd dynamics.

A creative thinker transposes the formal structure of an argument to a domain where she intuits the model will help better comprehend the situation, than whatever story is currently attached.  Full stop.  There is no difference between doing so with a fixed point theorem, entropy function, language game, or model of mutually constitutive social interactions.

Or consider that Bourbaki symbols and the Greek alphabet are not always the most precise and compact language in which to present an idea.  We have intuitions because they are computationally efficient, and it turns out that in groups, intuitive Bayesians make lots of incredibly good predictions.  It is a very strange logic and practice that justifies turning a discussion of expected utility into a derivation of the expectation operator from primitives.

We have and use grammar in natural language that defines hypotheticals and probabilities all the time, “could, should, would, may, might, ought,” and we have and use grammar in natural language that defines quantities and their relations all the time, “most, more, just as much as, lots.”  For many problems, replacing these terms with mathematical symbols would be cumbersome, obfuscatory, and useless.

Both mathematical and verbal theory cannot be reduced to some historical turf war between continental social theory and economics, or some other nonsense about professional identities and territories.  We should rise above these petty disagreements and give young theorists a better guide to which lexicon is useful in which situations, because neither natural language nor mathematics can accomplish all of the goals of theory across all domains.

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One thought on “Is Math or English Harder to Theory With?”

  1. Write a thorough proof on the sum and difference formulas, using epsilons and deltas, then tell me that there is no creative thought. Furthermore, prove that the square root of two is irrational….there is so much creative thought..

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