The Exaptation of Ideas
10 June 2010
Science often makes progress when an unphilosophical scientist reads the work of a philosopher, misunderstands it, but nevertheless derives an interesting research program from his misunderstanding of philosophical ideas. In the long run, it is very likely that the scientific misunderstanding of a philosophical idea will have a longer life and prove to be mush closer to the truth than the original philosophical idea, which latter will be immediately disposed by the next generation of philosophers. In the long run, then, it doesn’t really matter where inspiration comes from.
We might call this is exaptation of ideas. Exaptation — the use of anything for a function distinct from the function that defined the initial conditions of the thing so used — has been a theme that I have returned to several times. Today I would like to consider exaptation at its most abstract level.
Scientific method in the broadest sense of the term — patient, careful, and systematic observation of the object under study — has changed our conception of logic. The method of metamathematics has made of logic the object of a science — as it turns out, a science very much like logic. This science has been variously interpreted. We might employ metamathematics to become more acutely aware of the rules by which we reason, so extending the scope and profundity of formalism. But it seems that we are instead following the traces of positivism which are to be found implicit throughout metamathematics, with its scientistic orientation.
Hilbert, the father of metamathematics, was no abstract thinker. His philosophical observations are the occasional remarks of a working mathematician. The formalist program he proposed still inspires philosophers, but this is no surprise as philosophers today are largely inspired by anti-philosophical doctrines. Hilbert’s emphasis upon a misreading of Kantian intuition — a scientific, empirical mistaking of the materiality of intuition for the materiality of the concrete — took on as time passed more and more the character of physicalism, and today we find thinkers who entertain even the physicalization of logic.
The father of metamathematics was generous to his progeny: Hilbert’s praise of his creation is in the same vein as Aristotle’s self-congratulation in his Sophistical Refutations:
“That our programme, then, has been adequately completed is clear …it was not the case that part of the work had been thoroughly done before, while part had not. Nothing existed at all …on the subject of reasoning we had nothing else of an earlier date to speak of at all, but were kept at work for a long time in experimental researches. If, then, it seems to you after inspection that, such being the situation as it existed at the start, our investigation is in a satisfactory condition compared with the other inquiries that have been developed by tradition, there must remain for all of you, or for our students, the task of extending us your pardon for the shortcomings of the inquiry, and for the discoveries thereof your warm thanks.”
Sophistical Refutations, § 34, Works of Aristotle, 183b – 184b
It would seem that Aristotle and logicians since Aristotle have had a fine opinion of themselves, but whether this high estimate is an instance of the sober logical deliberation so carefully cultivated in their discipline, or a failure of the same, is another matter entirely. We can only observe the consistency with which logicians have propounded the finality of their subject, only to have the next generation of dialecticians pronounce the effort corrupt and count their own production the genuine article which at long last delivers on the promise of logic to secure truth and certainty, is perhaps more consistent than the logics themselves.
Wittgenstein’s certainty in the unassailable truth of the doctrines of the Tractatus also comes to mind, as does Russell’s assertion that Wittgenstein, “had the pride of Lucifer.” That so few philosophers have seemed to notice the Hilbertian distortion of Kant suggests that the claim will be controversial, but I find it difficult to imagine anyone who has read Kant in detail believing that what Hilbert calls intuition (“Anschauung”) is what Kant understood by the same term: Hilbert exapted Kant’s conception of intuition. Hilbert’s anachronistic reading of physicalism into Kant darkly heralds further physicalist mischief which was to come. Philosophers today speak of making their theories “physicalistically acceptable,” and in Mechanization of Reasoning in Historical Perspective, Withold Marciezewski offers a “physicalization of logic.” Despite its dubious provenance, physicalism may well be a legitimate theory, though its advocates have yet to indicate that they are willing to deal with the hard questions which it poses.
One might be forgiven for supposing that pride and self-satisfaction are necessary prerequisites for the logicians, marks of character especially suited to systematic and rigorous reasoning. Russell, himself a bona fide member of the Peerage, suggested: “There is a certain lordliness which the logician should preserve: he must not condescend to derive arguments from the things he sees about him.” (Introduction to Mathematical Philosophy, p. 192) The logician is to assume a demeanor of lordly indifference to the surrounding world. This may well be the origin of the role of the arbitrary in rigorous reasoning.
Hilbert’s contribution to the tradition of self-congratulatory exposition is as follows: “I believe that in my proof theory I have fully attained what I desired and promised: the world has been rid, once and for all, of the question of the foundations of mathematics as such. The philosophers will be interested that a science like mathematics exists at all. For us mathematicians, the task is to guard it like a relic, so that one day all human knowledge whatsoever will partake of the same precision and clarity. That this must and will occur is my firm conviction.” (“The Grounding of Elementary Number Theory” in From Brouwer to Hilbert, Paolo Mancosu, New York and Oxford: Oxford University Press, 1998, p. 273) Hilbert’s formalist program did in fact become a relic, but not, one suspects, in the sense to which he aspired. It is now a matter of historical interest, known to specialists, but not a living part of mathematics or an on-going research program. This is not to say that nothing has come of Hilbert’s foundationalist enterprise. The perennial aspects of formalism continue to assert themselves today as they were once asserted in Hilbert’s work. The doctrines unique to Hilbert have enjoyed varying degrees of influence. It is the central theme of Hilbert’s foundationalist program, the inspiration and the motivation, the vision of a finite consistency proof for the whole of mathematics, which was defeated by Gödel. The structure has been demolished, though we may build anew with salvaged bricks.
Hilbert was a visionary in a precisely definable sense of the term: he envisioned something which did not yet exist and sought its realization, i.e., he tried to make the possible actual, only to be shown (by Gödel) that it was in fact impossible. In so far as Hilbert was a visionary, he was a radical, a subversive, a rebel — for a vision of a better world to be realized must be at odds with the imperfect world which is—and stands opposed to classicism. Although Hilbert was in a certain sense the culmination and apotheosis of classical mathematics, he did not put his faith in classical mathematics, but rather in something beyond classical mathematics — in a mathematics yet to be.
Logicism, by contrast, looks frankly reactionary in its elevation of classical mathematics as the end of the successful logicist theory. The only thing that saved logicism from complete hostility to innovation was its willingness to embrace recent tradition, such as Cantor’s set theory and transfinite numbers, as a part of the classicism to be ordained finally with logical certainty. Logicism, despite its tolerance for Cantor’s actual infinite and his non-constructive methods, was inspired in its central program by a constructivist quest for securing mathematics from contradiction piecemeal, one deduction at a time. The constructivism of the intuitionists, by comparison, is always an assumption that their restriction of methods to the apparently safe will simply not issue in inconsistency, though this is by no means guaranteed a priori. Thus constructivism itself is inspired by a non-constructive, top-down conception of how order and consistency are to be imposed upon mathematics by principles determined not in practice, but prior to practices, which are determined, by definition, by the principles adopted to guide them.
Hilbert too had sought a peculiarly logical certainty: consistency. Interestingly, logicism sought the consistency of mathematics through the construction of mathematics slowly and gradually from simple beginnings. Hilbert sought absolute and complete consistency through a consistency proof which would hold good for all that was to follow in the future. This was a top-down effort, in essence non-constructive. Thus the finitist and constructivist strains in Hilbert’s thought conflict with the central vision and inspiration, which was essentially non-constructive: Hilbert, of course, is the one who famously referred to set theory as “Cantor’s paradise.”
Another example of philosophical self-congratulation and smugly self-satisfied logic is to be found in Yehoshua Bar-Hillel’s paper, “A Prerequisite for Rational Philosophical Discussion,” (Logic and Language, Studies Dedicated to Professor Rudolf Carnap, Dordrecht: D. Reidel, 1962, pp. 1-5) in which he sets forth, unblinkingly and without a trace a embarrassment, the reasons he will not even entertain objections to his principles unless they already agree with his principles. This embodies the familiar strategy of logical monism, to argue for monism from the perspective of monism, and employing a monistic logic to prove that logic must be monistic: There is one and only one logic, and (in this context, at least) Yehoshua Bar-Hillel is its prophet.
For Bar-Hillel, there is one and only one way to be rational, and he is completely unwilling to listen to any alternative. He will condescend to discuss his conception of rationality, but only with those who adopt his standards of rationality as the principles by which the discussion is to abide: “. . . I am ready to listen and ague with [a speculative philosopher] only if the meta-language, in which he explains to me his reasons for challenging my standards, itself complies with these standards.” (“A Prerequisite for Rational Philosophical Discussion,” in Logic and Language, Studies Dedicated to Professor Rudolf Carnap, Dordrecht: D. Reidel, 1962, p. 3, italics in original) Clearly, he is not interested in any serious challenge to his views, nor in anything unpredictable and upsetting. Indeed, entering into “dialogue” with Bar-Hillel would be more in the way of reading from a panagyric script in which triumphant reason affirms its own value and veracity. Bar-Hillel is to be congratulated for his honesty, if not for his attitude. Most of convinced of his opinion would not admit as plainly their indifference to any pluralism of reason.
We cannot dictate how others will reason, nor what they will make of our ideas. The exaptation of ideas continues apace. This is an unavoidable aspect of human history. We believe that we are creating a building that will last for the ages, but in fact the materials that we gather and work will be used by others in different constructions. We ought not to fight this. We ought to offer ideas to posterity in the spirit that they will be exapted. Indeed, we ought to consider ourselves fortunate if any of our ideas is exapted, for this is the only way that they will survive. I have argued that the historical viability of an institution only comes with its ability to change intelligently. Ideas are the institutions of the mind, and they too only possess historical viability if they can be adapted and exapted to changing circumstances.
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