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.