Back to the Night of 1610
Chapter Fifteen's close made a promise: the epilogue would return to the night of 1610. Now we go back.
The night of January 7, in Padua. Galileo pointed a wooden tube with a lens mounted in each end toward Jupiter. The introduction told the physical facts of this night: beside Jupiter stood three small stars arranged in a straight line, recorded in no star catalog ever made; on the nights that followed they moved sometimes eastward, sometimes westward, appearing in due course as four, and they circled Jupiter — not the Earth. The introduction also told the intellectual situation of this night: for nearly two thousand years the heavens had been "eternal" in the consensus of the West — the fifth element, the perfect circular orbits, the two worlds; and this consensus was not maintained by laziness: behind it stood a complete physics, cosmology, and theology, and every night of naked-eye observation supported it — the naked eye could see no satellites about Jupiter.
Now the reader comes back carrying a whole book's ledger, and sees several layers more than the introduction could give.
The wooden tube was the extension of observation — the instrument, as an extension of observation, drew a resolving power the human eye did not possess into one and the same process of convergence. The three small stars were an observational divergence — the consensus of the naked eye said there was nothing there, and the lenses said there was; the divergence did not remain a quarrel between two modes of observation, for the tracking of the nights that followed compelled any re-verifier to book it. And the loosening of the old consensus was a re-convergence — not the heavens collapsing in a night, but local surgery: Jupiter had moons, the path "all heavenly bodies circle the Earth" was reopened, while the other hundreds of millions of deposits in the sky's account stood untouched. Chapter Fifteen, four hundred years later, said of this dialogue-machine with the world that it has been built ever finer — the telescope pushed observation farther, the inclined plane made it one measure cleaner, the photographic plate handed it to the whole world for re-verification, the probability amplitude wrote its incompleteness into grammar. And on that January night four hundred years ago, the machine had just had its first lens mounted.
The introduction set that night and Wigner's perplexity of 1959 inside one and the same set of problems: a tool ground from glass, a tool written in symbols; one pushed observation farther, the other folds observation into structure — why do the tools with which human beings come to know the world work at all? Reason is a miracle, and a miracle demands explanation. The account can now be rendered.
Three Accounts
The introduction laid down three puzzles; the book answered them in five parts. Let them be settled one by one.
First account, Wigner's question: why mathematics is effective for the world. Part Three gave the answer in four entries. A common history of formation — mathematics differentiated out of such observational activities as counting, surveying, and calendrical reckoning; its basic structures carry the shape of the environment of their formation, and the ground of the fit is the ground of common origin; to say that the world "obeys" mathematics inverts the causality. The grammatical freedom of expansion — mathematics' expansion proceeds along the laws of combination and transformation of structures already locked, and the small handful that hit physics could hit precisely because they remain within this generative grammar: Riemann's geometry is no language fallen from the sky but a direct descendant of the Euclidean matrix. The survivors' ledger — the overwhelming majority of pure mathematics never meets any application at all, and this face proves that the fit is not a universal miracle, while the selection effect magnifies it. The suturing of re-convergence — between mathematics and physics there is no once-for-all contract, only one explicit re-joining after another. The four entries together: the effectiveness of mathematics = the long history of dialogue between convergence paths and the world. It owes no debt to miracle — it owes the debt of a common history of formation and the wages of one re-convergence after another. Wigner's spreading of hands was performed over a bill on which only the column of expenditure had been printed.
Second account, Hume's question: why repetition breeds the illusion of necessity. Chapter Two's answer is a genesis: from event to "always so" passes through four stages — observational consensus, causal laws, the rule framework, and the reverse presentation as a priori; and it is maintained by two mechanisms — the continuous feeding of dissipation and the closed loop of consensus. Hume saw the generation and stopped at the door, because he found only two roads: either experience delivers no necessity, and one stops in skepticism; or, like Kant, one freezes the contribution of the subject's pole into innate forms. RC gives the third: the feel of necessity has a history of formation — it was saved up by the consensus reinforcement of Axiom A6, trodden solid by hundreds of millions of mutually verifying walkings; Hume could not find the "necessary connection" because it has never been inside any single observation — it is a standing accumulated in the mutual reinforcement of observation upon observation. The projection of habit cannot be justified, but the stability with a mechanism can be explained: the bridges built upon it truly do not fall because they stand on real deposits in a real account. The a priori feel is not an illusion but the self-report of stability; and the word that reports its rank wrongly — "always so" should be read as "has never yet failed" — is precisely the thing this book has rewritten chapter by chapter.
Third account, Gödel's question: why a system cannot prove itself. Chapter Seven's answer turned this shadow entirely over: Gödel is not the failure of reason but the strictest proof, inside the world of symbols, of the thesis that determinacy equals what is locked. The territory of truth is strictly greater than the territory of proof, because "being proved" means being taken in by a chosen procedure of locking, and whatever aperture is chosen, there are things it cannot take in — locked determinacy necessarily carries margin, Axiom A7. The undecidable propositions and the undefinable truth predicates are not defects of the system but the mode of existence of the act of locking; a system that truly locked away every margin would be unable to lock out even its own freedom from contradiction; it could not so much as exist as a consistent system. The proof of the boundary is precisely RC's internal evidence: this one, the most fearsome-looking of the three puzzles, turns out to be not a knife pointed at reason but the medical report reason hands over concerning itself — what has no boundary is not omniscience but nonexistence.
The three accounts together redeem, at the close, the programmatic sentence of the introduction: the three "apriorities" are appearances cast by sufficiently stable convergence paths. The absoluteness of logic (to question it one must use it), the necessity of mathematics (self-consistency inside the snapshot), the universality of science (the account saved up in unbroken dialogue with the world) are three kinds of stability run out by one and the same mechanism of convergence in three directions of dialogue. Once this is understood, reason for the first time becomes something that can be explained, maintained, and kept evolving, rather than a gift of the gods with no ascertainable origin.
Modesty Is Strength
One last question remains, the one Chapter Sixteen's close left hanging: since all three forms are limited, fallible, and dependent on cross-subject re-verification, by what title does reason yet deserve to be trusted?
Chapter Seventeen gave the measure: the health of reason is measured by the capacity to keep converging, not by the stock of true theorems held. Chapter Sixteen gave the position: all three forms are cognitive relay stations, whose value lies not in being forever true but in converging vast numbers of observations into workable determinacy at the least cost, and in not burning the instruments when the transfer is made. Chapter Eighteen gave the outlook: the future of reason depends on keeping margin for what is not yet locked.
Lay the three answers one upon another, and there is the final reading of this book's title. Convergent rationality — in these words, "reason" is no longer a judge looking down on the world but a party to the dialogue, four hundred years old (far more than four hundred, in truth; only the earlier ages left no ledger); and "convergent" is no longer a shortcoming but reason's very mode of being. Galileo's lenses, Frege's symbols, Euclid's boundaries, Eddington's plates, Born's probability amplitude — not one of these was an a priori gift; every one is an achievement of convergence; and they still work, are still reliable, still deserve to have the next bridge, the next theorem, the next star catalog entrusted to them — not because they are eternal, but because the accounts behind them are real, and the channel is open.
The introduction said: take hardness for the nature of water at every temperature, and one will be seized with panic on the day of the thaw. What the epilogue wants to say is the other half: it is the one who knows that ice has a history of temperature who dares truly to build the bridge upon the ice — for he knows at which temperature ice bears its load, at which temperature it cracks, and, before the cracking, which accounts ought to be moved away. So it is with reason. To know that determinacy is locked is to know where it may be relied upon, where it needs re-locking, and where the interface should be kept. This is not a weakening of reason — it is the most important lesson reason has learned in four hundred years, and the only guarantee that it can go on learning. The modesty of reason is the strength of reason: modesty is not a posture; it is the interface kept ready for the next convergence.
On the night of January 7, 1610, a mathematician of Padua pointed a wooden tube at Jupiter and saw stars that had no right to exist. What he saw was not a miracle; what he saw was that observation could be extended one step farther. In the four hundred years since, humanity walked that one step into the telescope, the interferometer, the formal system, the axiomatic edifice, and the probability amplitude — into the long story this book has told. The story is not finished: the question Chapter Eighteen delivered to the door, the words "to be continued" on the last page of RC's General Outline, and the margin in the Ground of Possibility that is never locked through — all of these are its continuation. But one sentence can be set down now, with the pen at rest: reason is not a set of perfect equipment that fell ready-made from the sky, but the skeleton left behind after observation, through history, extended itself and converged, again and again — it is still extending, and the one who knows how it extended knows how much farther it can go.
The telescope did not abolish the naked eye, just as formalization has not abolished intuition, nor the model the experiment. The night that wooden tube pointed at Jupiter was not the beginning of this history, nor will it be its end — only, in the ledger, the first page to have been read through clearly.