A Preliminary Field Guide to Linguists, Part Two--Athanasious Schadenpoodle SpecGram Vol CL, No 2 Contents Linguistic Topology--I. Juana Pelota-Grande

Point: Why Linguistics is Not a Science

[Editor’s Note: This opinion piece is the first of a contrasting pair discussing the relationship between Linguistics and Science. The opposing piece will run in the next issue of SpecGram.]

While many have claimed, and probably rightly so, that Linguistics suffers from a bad case of Physics Envy, it is Mathematics, the Queen of the Sciences, which is best suited to provide a role model for bringing some sorely needed rigor to the field.

Any practicing mathematician will speak of the crucial role intuition plays in the formulation of ideas--the mysterious spark, the gut feeling, the leap of faith that points the way to a difficult but elegant theorem cannot be replaced by any mechanical means. However, the intuitive leap is inspiration, and has no place in the final proof of a claim. For decades every mathematician has known, instinctively, that both Fermat’s Last “Theorem” and the Four Color Map Hypothesis must be true. While there was little doubt among practitioners, all agreed that neither was true until rigorously proven.

Linguistics as a whole lacks such a tradition of rigor, and linguists sometimes show their incomplete appreciation of formalism in interesting and even disturbing ways. Two anecdotes from my own experience provide a sense of how things can go bizarrely wrong.

Formalism for the Sake of Formalism

First, I recall a paper that I read long ago, in which the author introduced considerable and complex machinery, including advanced topics in group theory and discrete mathematics, to say precisely what is meant by the notion that “lines in diagrams should not cross.” The end result was not appreciably distinguishable from the simple statement that lines shouldn’t cross--perhaps with the added caveat of “no cheating with weird lines going around underneath or over the top of the diagram.”

The mathematics in this case was in fact correct, but was also off-putting and out of place. The entire line-crossing-prevention construct was in no way related to the main theory being presented, the notions of line and crossing used were no different from the well known entities used by tree diagram drawers from decades gone by, and the entire enterprise was utterly abandoned once fully explicated.

The only value provided by all of this mathematics was several additional pages in length, and a misleading air of formalism and competence. Tellingly, while I recall the showy but misguided mathematics, I have no recollection of the supposed main content of the paper.

Trusting the Theory

I also recall another incident in an introductory linguistics course concerning formal systems. The instructor discussed a gavagai-brained interpretation of imaginary numbers I had never encountered before, namely that i (the square root of -1) did not have a fixed value, but rather oscillated between +1 and -1.

On the surface this can seem quite clever. i 2 = i · i = (-1) · (+1) = (+1) · (-1). The class was made up mostly of humanities majors, and they seemed quite impressed with the intuitive sense that this system makes. However, there was also the the usual smattering of engineering types--mostly programmers with the prototypical hackerish proclivity for linguistics--and they objected strenuously.

They raised objections that ranged from the pointed but fairly trivial to the directly devastating, such as:

  • Why not oscillate between 1/2 and -2? No one ever asked that before.
  • What is the speed--or even the medium--of the oscillation? I don’t know.
  • How do you know if you are on the upswing (-1 to +1) or the downswing (+1 to -1)? There have to actually be two distinct values of i: i1 (+1/-1) and i2 (-1/+1). So i1 · i2 = (-1), but i1 · i1 is always 1. So i12 is just 1 again, which is clearly incorrect, unless 1 = -1. It just works out if you choose the details correctly.

These are questions in the Mathematical tradition--the devil is in the details, so let’s beat them to death once and for all and be done with it. The answers are very much in the Linguistic tradition--squishy answers with good, intuitive interpretations: symmetrical seems better, so i obviously oscillates between -1 and +1, not 1/2 and -2; the speed is irrelevant, since the numbers just sit there on the page and there is no medium of oscillation; i · i has to be aligned right to work so it aligns right and works. The reader can be trusted to do the right thing.

Continuing in the Mathematical tradition, I refused to let the matter go. i 3 is, at best, going to oscillate between (-1) · (+1) · (-1) and (+1) · (-1) · (+1), or, between -1 and +1, which is the original description of i. But in the classical theory of complex numbers, i 3 = -i. Now i = -i, implying i = 0 !

Still this was not enough to convince the professor. Even the computer science students, undergrads all, were losing their nerve to fight with a distinguished professor. I dropped the matter in class, but wrote up my final reductio ad absurdum (as if the previous two weren’t enough) in our next written assignment.

It is possible, you see, to do dark, foul things with complex numbers. If you are mathematically masochistic enough--or if you are an electrical engineer (which implies the previous condition, now that I think of it). Horrid, evil things, like taking the complex cube roots of regular numbers, such as 8. It turns out that the complex cube roots of 8 are:

-1 + i · √3      and      -1 - i · √3

Oscillating values of i mean that each root actually oscillated with the other--an almost elegant result on the surface. Multiplying through with alternating values, though, we get the result that vibrating at an unknown speed in an unknown medium between the values of -1.464101... and 5.464101... is the same thing as being 8--a clear absurdity.

That short paper was the only one in my grad school career to get a completely unacceptable grade. I knew better than to try to pursue the physical interpretations of complex numbers that arise in electrical engineering!

Perhaps this all seems irrelevant to the day-to-day working of linguistics, but I see a frightening analog between this incident and another relayed to me by a colleague. At a conference, a linguist presenting a theory made some startling but potentially powerful theoretical claims about a particular language. Another linguist, incidentally a native speaker of the language in question, stood up and contradicted some of the obvious conclusions of the theory with simple statements in his mother tongue. The presenter acknowledged that the data did seem contradictory, but concluded that, “at some point, you just have to put your trust in the theory.”

There is no coherent reply to such an incoherent assertion.

Rolling Eyes

At this point, I’m sure there has been a lot of eye rolling. Most mathematically inclined readers, I expect, will be shaking their heads at the lack of rigor and the lack of understanding of rigor demonstrated above. Prototypical linguists, I fear, will be shaking their heads at the excruciatingly boring mathematical detail, which they’ve skimmed without really internalizing. That’s the problem!

This is the same problem that hampers much work in linguistics. Intuition without rigor is used to fill in the gaps in a theory. In mathematics, this can lead one terribly astray when one’s intuitions aren’t up to the challenge. In linguistics, the situation is actually worse: what is essentially a theory of linguistic intuition has gaps, which are silently filled in by the reader, using their intuitions. I’d say that the recursion is frightful, but no one other than old, grey LISP programmers has really good intuitions about recursion.

Optimism for the Future

Despite the pessimistic tone so far, there is a cause for cautious optimism. The best hope for remedying this situation is some sort of redemption through the growing composite field of computational linguistics. You have to remove all traces of intuition when instructing a computer in how to perform any task, linguistic or otherwise--or you must at least explicitly formalize those intuitions! There can be no magical jumping of gaps with transparent, unconscious ‘just knowing’, at just the right time--not when your mind is crisp, cold silicon.

Alpberta Cedium-Ndelemeiyerov Siberian Holographic Institute of Technology

A Preliminary Field Guide to Linguists, Part Two--Athanasious Schadenpoodle
Linguistic Topology--I. Juana Pelota-Grande
SpecGram Vol CL, No 2 Contents