Three Tiny Essays

I. While the other sciences increasingly have stagnated, or quietly been subsumed into engineering, or crashed and burned spectacularly, or become cynical funding ploys, mathematics has continued to make grand advances even recently, for instance in the work of Tao, Perelman, and Mochizuki. This unfailing progress may be one of the foremost indications that mathematics is not “real” at all.

Unlike science, which depends on empirical verification (however obscure) and is therefore constrained by limitations such as precision, complexity, time and energy, mathematics as a pure construction of the mind is limited only by the bounds of human intelligence, which may still admit an infinite combinatorial expansion much as language allows with words.

To say that “mathematics is still making progress”, then, is like saying “language is still coming up with new sentences” or “musicians are still finding new songs to play”. It is the telltale sign that what we are dealing with in math is not scientific or “in the world”, since the latter things must always bow to materiality and finitude.

Mathematics is instead a kind of pocket-infinity: allowing the experience of infinite extension and possibility, while employing finite resources. One could even say that mathematics, unlike the other pocket-infinities like natural language or music, is distinguished by being purely grammatical, without any other content; it is circumscribed entirely by the working-out of grammatical-type transformations, by the tone of rule-boundedness. (This is like turning generative grammar on its head, in that rule-based systems are seen as instances of grammar instead of vice versa.)

II. The greatest & most valuable effect of philosophy is not that it carries us past illusion to the True Nature of Things by virtue of argumentation and logic–nor that it unmasks this True Nature as itself an illusion caused by a misuse of language and helpfully eliminates it–but rather that it shows these “illusions” are often not illusions at all. For we cannot avoid noticing that said argumentation and elimination fails again and again to disperse them or to rob them of one whit of their power.

We are left to realize that it was the argumentation, now reduced to an absurdity, that was unreal (which is not to say useless) all along. It is as though we shot an arrow which disappeared into smoke when it struck its target, leaving not a trace–and yet we were expected to believe that the arrow had somehow more reality than the target. (Chomsky: “Newton exorcised the machine; he left the ghost intact”.)

The argumentation, and even the elimination, prove in the end to be the only illusions–illusions of the self. This is the full circle through which philosophy rightly brings us round to our real being.

III. What we are seeing with widespread errors and non-reproducibility in science is definitely not of a piece with the normal “two steps back, three steps forward” nature of past scientific progress. Those steps involved the putting forth of paradigms, their replacement by stronger paradigms better matched by observation, and so on.

What we now see is not conceptual improvement, but frank technical error, diminishing returns, and the misunderstanding of even existing concepts–in short, the hallmarks of scientific crisis.

Whether this crisis will eventually break through to a new synthesis that allows more progress, or simply continue to wallow in illusory progress and social privilege, is the single most important question for scientists of our time.

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2 comments

  1. I vehemently agree with the conclusions of tiny essays 2 and 3. I have a comment about #1.

    Of course, the idea that math is a purely grammatical affair is what is usually called nominalism. According to the opposite view, mathematical platonism, mathematical statements are not meaningless strings of symbols arising from a generative grammar, but assertions about mathematical objects – objects that have their own kind of non-physical existence. The nature and properties of these objects are not created, but only discovered, by mathematicians (this is what distinguishes mathematics from the arts). Neither nominalism nor platonism is in current disrepute.

    Your tiny essay amounts to a statement of, not a defense of, the nominalist position (of course no tiny essay could approach the latter). From the platonist standpoint, mathematical progress is not wholly different from scientific progress, but has a subject matter which is more directly accessible to study than is the case for the physical world. My comment is this: direct accessibility explains continued mathematical progress just as well your nominalist account.

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    1. Chris,

      My point isn’t to advocate or restate nominalism (if that’s what it looked like), but to put forward a tentative quasi-empirical hypothesis, which is that limitation or finitude is a necessary criterion of “realness”. All the “physical” things “out there” manifest limits; a certain amount of energy only can be used up so far, a living thing only lives so long, a planet cannot have a limitless surface area, a human being can only read so many words a minute, non-renewable resources don’t get replenished by wishing or digging harder, and so on. These limitations aren’t just incidental, but allow us to lasso the object with definitions and to claim it as physical and “out there”.

      If science is increasingly showing the impact of limits, that is just another sign that however faulty or illusory a “mirror of nature” its theories and terms and results may afford, it is still in-touch with something not wholly dependent on our selves or our viewpoint–something with its own reality, for want of a better term. The scientific encounter with limits, in this sense, is the ultimate empirical finding–the encounter to which all science, if it is science, will tend.

      We could certainly imagine science, as Rescher does, as a kind of limitless silly-putty world with an endless stream of breakthrough new theories, phenomena, and results being discovered, although Rescher at least admits diminishing returns as a purely practical matter. But I propose that our creeping frustration in trying to sustain that limitless-progress view is the very confirmation that what we are doing is science; being in-touch with the “out there”, it needs converge on some something-or-other. In mathematical metaphors, we could say that different scientific theories or paths of progress are like curves in a huge space that don’t overlap, but still converge towards the same asymptote.

      Conversely, the absence of limits or convergence or slowdown, such as we see in mathematics, literature, music, even small talk or pop culture, may be seen as the hallmark of the non-physical, the linguistic, the constructed–what I call the “pocket infinities”. This would be the realm of things that don’t have to be in-touch, don’t have to lead to anything “out there”. They are limited only by imagination, and need not kneel to finitude as “physical” things must. On the contrary, think of Cantor, or calculus: mathematics eats infinities for breakfast.

      I don’t think mathematics’ being “all grammar” makes it merely an empty shuffling of words or symbols. It obviously does not keep it from being useful in many ways, just as spoken language is. With Lakoff, I think math’s grammar is itself highly metaphorical–packed with testimony to our individual-, cultural-, species-, and biosphere-level experience of how this world works, and shaped over billions of years in intimate dialogue with it. That’s pretty impressive to me, and still has a lot to say about the world and ourselves–it’s just not something independently “out there”.

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