Rightly termed, witcraft

Eric Schliesser’s note on translation put me in mind of one of my favorite books. Early in my graduate career I discovered in the Stanford stacks a series of reprints, mostly in English, from the sixteenth and seventeenth centuries, published by Scolar Press, now sadly defunct. Among the many wonderful facsimile volumes in the series was Ralph Lever’s The arte of reason, rightly termed, witcraft teaching a perfect way to argue and dispute (1573). This is said to be, after Thomas Wilson’s Rule of reason (1552), the second logic book written in the vernacular. And boy is it…

To give you a taste, here is an excerpt from the chapter which in our English would be entitled “Of the subjects and predicates that are in a simple proposition”. The “storehouses” are the categories of Aristotle; a “naysay” is a negation.

Of the foresets and back sets that are
in a simple shewesaye. Chap. 3.

1 The foreset is a nowne placed afore the verbe, and the backset after, as, man is juste: man is the foreset, and just, is the backeset.
2 Sometymes a whole sentence or a clause of a sentence is a backset, or a foreset: as to rise earely is a holesome thing: in this shewsay, to rise earely, is the foreset, and a holesome thing is the backsette, they both supplying the roome and office of an nowne.

To what use foresettes and backesettes serve.

3 The storehouses serve to shew the nature of wordes as they are taken and considered by themselves alone.
4 The foreset & backset of a shewsay declare the respecte that wordes have one to [71|72] an other, as they are coupled and linked together in a perfect saying.

To knowe what respecte the backset hath to the foreset
in every simple shewsaye of the seconde order.

5 If the backset is deuided and parted a sunder from the foreset by a naysay, then doth it but eyther differ from it, or els it is a gainset to it.
6 What differing words and gaynsets are, we have shewed afore in the. 12. Chapter of the first booke.
7 If it be affirmed & coupled to the foreset by a yeasay: then muste the foreset and backset be such as either may be saide of other turne for turne, or not saide.
8 If either may be said of other turne for turne, then is the one of them the kindred, and the other his saywhat: or els one the kindred and the other his propertie.
9 For onelye the saywhat and the propertie compared to the kindred, maye bee said of it, and it of them, turne for turne: but the saywhat expresseth what the kindred [72|73] is, and the propretie doth not.

Lever was very deliberate in his coining of terms; the new terms are intended to replace the Latin-derived terms hitherto in use, to the advantage of those “english men” not versed in the Classical languages. There is no question but that English is up to the task of teaching logic; the only problem will be that some new words will have to be used.

The truth about infinitesimals

In what follows I am not pushing any conclusions; I’m offering an extended example with which to think about questions of truth and historical interpretation.

Russell knew how things stood with infinitesimals. Weierstrass and Cantor have solved the problem of placing the calculus on a rigorous basis. The infinitesimals of an earlier age are banished; the contradictions entailed by their use can now safely be ignored because it has been shown we can do without them. (NB: My guide in this discussion is John Bell’s The continuous and the infinitesimal, esp. ch. 4.)

Leibniz in particular is censured for having given a “wrong direction to speculation as to the Calculus”.

His belief in the actual infinitesimal hindered him from discovering that the calculus rests on the doctrine of limits, and made him regard his dx and dy as neither zero, nor finite, nor mathematical fictions, but as really representing the units to which, in his philosophy, infinite division was supposed to lead (Russell, The Principles of mathematics 325).

Leibniz’s—and his successors’—attempts to save infinitesimals were bound to come to naught, because “infinitesimals as explaining continuity must be regarded as unnecessary, erroneous, and self-contradictory” (Principles 345: his verdict on Hermann Cohen’s Neo-Kantian interpretation of the calculus).

Celebrating the accomplishments of Weierstrass, Russell makes his work a fulfillment of Zeno’s:

After two thousand years of continual refutation, these sophisms [Zeno’s paradoxes] were reinstated, and made the foundation of a mathematical renaissance, by a German professor, who probably never dreams of any connection between himself and Zeno. Weierstrass, by strictly banishing all infinitesimals, has at least shown that we live in an unchanging world, and that the arrow, at every moment of its flight, is truly at rest (347).

What Russell knows is that the key notion of the calculus—that of limit—has been defined by nineteenth-century mathematicians in terms that make no reference to other than “normal” finite quantities. In the classical ε, δ definition of convergence, those variables range over ordinary numbers like .01 or 1/π. The dx and dy of Leibniz’s calculus are thus eliminated; every proposition containing such symbols can be replaced, salve veritate and also salve deductive relations, by a proposition not containing them. Elimination by definition became a familiar strategy in analytic philosophy, carried forward in extremis by “nominalists” whose aim was to eliminate by similar means all abstracta, including the sets or classes to which, in the late nineteenth century, all other mathematical entities then conceived had been reduced. The model and origin of all such reductions was, directly or indirectly, that of Weierstrass.

The importance of that model would be difficult to exaggerate. It was seen to be the successful solution, ending all dispute, to a longstanding philosophical and mathematical problem, the problem of infinitesimals. It was Progress with a capital P.

Translate: carry across

The metaphor implicit in translation is that of something being carried over from one “place” to another. The most fortunate, the most deserving of saints might be translated into heaven; their earthly remains were often translated from one church or monastery to another, along with the prestige of possessing them. Translate in this older sense bears a clear relation to transfer, of which it is merely the irregular past participle.

In those cases a thing was carried over or across: a person, a relic. In the now most common case, that of translation between languages, what’s carried across is not at all obvious. Decades ago, when theories of meaning—or rather speculation about theories of meaning—were all the rage, one would have said that translation consists in the construction and use of a systematic mapping of the sentences of one language to those of another, a mapping that preserves meaning or truth. The first attempts at computer translation worked from a similar definition.

Source: John Macnamara, Journal of Social Issues; 23.2 (1967) 59.

Translation so conceived “carries across” only abstracta: the meaning or truth-value of the source. For anyone who has taken seriously the task of translation, that is a sort of caricature. The truth in it—what makes it caricature and not outright falsehood—is that something can be captured by the algorithms employed by Google and other automatic translation services. Call it the “gist”.