And wise men tell us, Callicles, that heaven and earth and gods and men are held together by communion and friendship, by orderliness, temperance, and justice; and that is the reason, my friend, why they call the whole of this world by the name of order (κόσμος), not of disorder (ἀκοσμία) or dissoluteness (ἀκολασία). Now you, as it seems to me, do not give proper attention to this, for all your cleverness, but have failed to observe the great power of geometrical equality amongst both gods and men: you hold that self-advantage is what one ought to practice, because you neglect geometry (γεωμετρίας γὰρ ἀμελεῖς).
— Gorgias, 508a-b translated by W.R.M. Lamb
Thus Michel Serres extends the Socratic vision:
|Et, comme on le voit assez bien dans toute discussion entre un empiriste et un rationaliste — Locke et Leibniz par exemple —, l’empirisme aurait toujours raison si les mathématiques n’existaient pas. L’empirisme est la philosophie vraie des que les mathématiques sont entre parenthèses. Avant que ces dernières ne s’imposent et pour qu’elles le fassent, il faut vouloir ne pas écouter Protagoras et Callicles : et cela, parce qu’ils ont raison. Mais plus ils ont raison, moins on peut les entendre : ils finissent par ne faire que du bruit. L’argument oppose a Locke par Leibniz : Vous ne savez pas de mathématiques, n’est pas un argument ad hominem, c’est la seule défense logique possible.
— La Communication
|And, as is seen clearly enough in any discussion between an empiricist and a rationalist — Locke and Leibniz for example, — empiricism would always be right if mathematics did not exist. Empiricism becomes the true philosophy as soon as mathematics has been bracketed off. Before the latter discipline imposes itself, and for the sake of its doing so, we should want to avoid listening to Protagoras and Callicles: and that, because they are right. But the more they are right, the less we can hear them: they end up only making noise. The argument opposed to Locke by Leibniz: “You do not know mathematics”, is not an argument ad hominem, it is the only possible logical defense. |
— translated by MZ
Empiricists have tried and failed to dispense with the moral and metaphysical baggage of mathematics. Their failure was due to their honesty. Unable to cope with proven inability to delineate meaning and failure to dispense with the dependence of science upon abstract entities, they curtailed the ambitions of their enterprise. Unconstrained by these scruples, sophistry continues to thrive in postmodern endeavors free of scientific scrutiny and uncritical of their content. Thus lack of geometry persists as the defining characteristic of the sophist.
The twentieth proposition in Book I of Euclid’s Elements says that in any triangle two sides taken together in any manner are greater than the remaining one. Proclus Diadochus credits Thales of Miletus with this discovery. He further relates that the Epicureans used to ridicule this theorem as being evident even to an ass and requiring no proof. They claimed that the theorem was “known” (γνώριμον) even to an ass that can invariably be seen choosing to walk towards his fodder in a straight line, in lieu of traversing the two sides of the triangle formed by an outlying point. In his response, Proclus rightly points out that perceiving the truth of the theorem is at a far remove from understanding its proof as a reason why it is true. His focus on proof remained at the foundation of geometrical knowledge, ensuring its dignity with the very means that defined its difficulty.
Within the mediaeval Quadrivium, this difficulty caused general instruction in geometry to stop at the fifth proposition in Book I of Euclid’s Elements, which asserts the equality of angles at the base of an isosceles triangle along with that of the angles formed beyond the base by extending the equal length sides. Pappus (Πάππος) of Alexandria discovered its shortest known proof of the first identity six centuries after Euclid (Εὐκλείδης) codified the mathematics of Plato’s Academy. He observed that that if the triangle is ABC with AB being the same length as AC, then its congruence with its mirror image, the triangle ACB, shows that their two sides and the included angle at A are the same, whereby the fourth proposition implies that the angles at B and C are the same. The mediaeval tutors, frustrated by the resistance of their charges to the idea of a geometrical figure being congruent with its own mirror image, paid an homage to Epicurus by deeming this proposition the pons asinorum, the bridge of asses.
To engage hopefully a mind unfit to follow Pappus across the pons asinorum is a fool’s errand. What is worse, every such engagement is destined to end up in animadversions. For want of reference to the standards of truth and validity, the descendant of Protagoras and Callicles is conduced to a revaluation of values that commences in his repudiation of the standards of truth and validity, and ultimately compels his dependence upon ad hominem appeals.
The sophistical predicament cannot be relieved by following Protagoras in supposing that that man is the measure of all things: of those things that exist as he is; and of those things that do not exist as he is not: πάντων χρημάτων μέτρον ἄνθρωπος, τῶν μὲν ὄντων ής ἔστιν, τῶν δὲ οὐκ ὄντων ής οὐκ ἔστιν. (See Diogenes Laertius, Life of Protagoras, in Lives of the Philosophers 9.) Making man the measure of truth fails to countermand his consideration of truth in that very making. This consideration suggests that all ascriptions if verisimilitude regardless of their qualities or gradations presuppose an underlying postulation of unequivocal truth. Lie us suppose that truth is reducible to verisimilitude, which is graded on a human scale. Then the claim that some proposition has its given human measure of verisimilitude, calls for an interpretation as logically equivalent to claiming as true simpliciter, i.e. true with no constraint by qualification or assessment by gradation, that its human measure is as it has been given. In other words, truth underwrites every claim of verisimilitude.
The sophist does not limit his relativism to a denial of truth. His further denial of its value of truth is refuted by observing that in order to apply any scale of values, one must have already acquiesced in the metric of facts. By parity of the foregoing reasoning, epistemology, deontology, and axiology all depend upon truth. It would be superfluous to belabor the point that ascribing any aspect of knowledge, justice, and beauty is subject to its requirement. The value of truth arises in part from its indispensability for endorsing and applying any scale of values. All value judgments of attraction or repulsion, prudence or recklessness, magnanimity or pettiness, and merit or fault, must rest on the foundation of facts. The persuasive force of sophistical refutations depends on the acquiescence of their audience in dirempting their conclusions from truth. This record of ineluctable prevarication makes sophistry worthless.
There remains the alternative of underwriting values by social custom or individual whimsy. In doing so, the issues of truth might be thought mooted in favor of pragmatic considerations. Unless our moral and aesthetic lucubrations stand on alethic grounds, we must impress our audience by some other means. If truth is not the deciding ground of assertion, there is no argument but ad hominem. Thus the avoidance of mathematics observed by Serres is ineluctably connected in sophistical reasoning to reliance upon special pleading. The rationalist has nothing to gain from engaging his kind in conversation bereft of common ground. Still, men in the frailty of their character conjure their reasons to persevere. The great polemicist of strife, Eric Hoffer, wrote in his notebooks: “Perhaps people throw themselves into heated polemics to give content to their lives, to warm their hearts. What Luther said of hatred is true of all quarreling. There is nothing like a feud to make life seem full and interesting.” (“Sparks: Eric Hoffer and the art of the notebook”, Harper’s Magazine, July 2005, p. 74.)
But there may be a better reason to engage the sophist in a dispassionate discussion uncommitted to a foredoomed effort to change his mind. In Fragment 80, Heraclitus of Ephesus writes: “It is necessary to know that war is common and right is strife and that all things happen by strife and necessity.” (Εἰδέναι χρὴ τὸν πόλεμον ἐόντα ξυνόν, καὶ δίκην ἔριν, καὶ γινόμενα πάντα κατ' ἔριν καὶ χρεών [χρεώμενα]. G.S. Kirk, J.S. Raven and M. Schofield, The Presocratic Philosophers, Second Edition, Cambridge, 1983, pp. 193-194.) And Aristotle quotes Heraclitus in the Nicomachean Ethics 1155b as saying, “Opposition unites” (ἀντίξουν συμφέρον), and “The fairest harmony springs from difference” (τῶν διαφερόντων καλλίστην ἁρμονίαν), and “'Tis strife that makes the world go on” (πάντα κατ' ἔριν γίνεσθαι).
Johannes Moreelse, Heraclitus, c. 1630
Whenever the soul (ψυχή) within it is stronger (κρείσσων) than the body (σῶμα) and is in a very passionate state, it shakes up (διασείω) the whole body from within and fills it with maladies; and whenever the soul ardently (συντόνως) pursues some study (μάθησις) or investigation, it wastes the body (κατατήκω); and again, when the soul engages, in public or in private, in teachings (διδαχή) and battles of words (μάχη ἐν λόγος) carried on with controversy (ἔρις) and contention (φιλονεικία), it makes the body inflamed (διάπυρος) and shakes it to pieces (σαλεύω), and induces catarrhs (ῥεύματα); and thereby it deceives (ἀπατάω) the majority of so-called physicians (ἰατρός) and makes them ascribe (αἰτιάομαι) the malady to the wrong cause (ἀναίτιος).Strife is indeed a disorder in individual bodies, as it is in the body politic. But Heraclitus’ insight cannot be gainsaid. Plato’s Republic is not about to spring forth fully formed from Zeus’ head. There is no order but what comes through disorder. To complain against controversy is not only futile but also harmful, inasmuch as in wasting his effort on inveighing against the inevitable, the complainer debars himself, if not from enjoying its deliverances, then from acknowledging his debt of gratitude therefor. The effort expended on clarifying one’s own thinking is never wasted. The discipline acquired in requiting sophistry with reason and prevarication with patience cannot be derived from conversation with men of honor and integrity. The philosopher must persevere in pummeling his adversary until he has shown himself capable of returning the favor. Outside of the perfect polis, his equality to his fellow man is found only in its proof; and his right to freedom of thought, speech, and action, only in his conquest. This is the disagreeable moral bequeathed to posterity by Charles Baudelaire: « Celui-là seul est l’égal d’un autre, qui le prouve, et celui-là seul est digne de la liberté, qui sait la conquérir. »
— translated by W.R.M. Lamb
Say not the struggle nought availeth,
The labour and the wounds are vain,
The enemy faints not nor faileth,
And as things have been, things remain;
If hopes were dupes, fears may be liars;
It may be, in yon smoke concealed,
Your comrades chase e’en now the fliers —
And, but for you, possess the field.
For while the tired waves vainly breaking
Seem here no painful inch to gain,
Far back, through creeks and inlets making,
Comes silent, flooding in, the main.
And not by eastern windows only,
When daylight comes, comes in the light,
In front the sun climbs slow, how slowly,
But westward, look! the land is bright.
— Arthur Hugh Clough, 1855