- For the ‘indivisible’, see Ann. Sci. 1972 28 261 261 245, 272 (footnote), and 275. T. L. Heath (Euclid's Elements: Dover, vol. i, p. 156) misunderstood the Aristotelian ‘indivisible’. Hence, he failed to grasp the significance of the ‘point’ as a division, and so the grounds for Aristotle's rejection of Plato's ‘point’ (as the extremity of a line) as unscientific.
- All references to Aristotle will be to the Oxford Translation, and will be abbreviated as in this instance. The Thirteen Books of Euclid's Elements 1926 by T.L. Heath in the Cambridge Edition and in the Dover Edition of 1956, will appear as Elements (Cambridge), and Elements (Dover). Galileo's Two New Sciences translated by Henry Crew and Alfonso de Salvio, in the Dover Edition of 1914, will be referred to as T.N.S. The Philosophical Works of Descartes by E. S. Haldane and G. R. T. Ross in the Cambridge Reprint of 1968, and the Geometry by D. E. Smith and M. L. Latham in the Dover Edition of 1954 will appear as Descartes, Works (Cambridge), and Descartes, Geometry (Dover). The History of the Calculus and its Conceptual Development by Carl B. Boyer, Dover Edition, 1949, will appear as Boyer: Calculus (Dover).
- In one way, the ‘limit’ can be a third thing. This depends on our thinking. We can think of the ‘limit’ (i) of the water, or (ii) of the container, or, of neither the one nor the other, but simply (iii) of where the two meet. If he had understood Aristotle on ‘place’ here, Descartes could not have confused ‘body’, ‘extension’, ‘internal place’, and ‘space’ in the way he did Descartes Works Cambridge i 259 259 But, neither did he understand ‘combination’ and the Aristotelian ‘indivisible’ which, like everyone else, he took to be what (because of its smallness) was beyond the power of any creature to divide further (ibid., Principle XX, p. 264). Hence, he resorts to Pope Urban VIII's irrational method of silencing Galileo, and finds some explanation of the paradox of division ad infinitum in God's omnipotency (ibid.). See Galileo's Two Chief World Systems by Stillman Drake, 1962, p. 464 and note, p. 500.
- There is a unification here similar to that in compounds and combinations found throughout Nature, though it has escaped the notice of geometers. As I have said Ann. Sci. 1972 28 280 280 we can ignore the line, qua line, and divide the line, qua matter; but, even here, division must come to a halt at the constituent ‘parts’ of the compound. ‘Parts’ in this sense find no place whatever in Euclid's Elements. This indivisibility of a straight line segment makes Dedekind's so-called Postulate groundless and indefensible, and its application (by Killing) to elementary geometry erroneous, though Killing's position throughout is quite untenable on its own account. See: Heath, Elements (Dover), vol. i, pp. 234–240, and Boyer, Calculus (Dover), 291. This will receive attention in a subsequent paper. The discussion in Heath (ibid.), on the Principle of Continuity, with special reference to Dedekind's Postulate, makes the continuous purely geometrical, and composed of discretes. No account is taken of the continuous which has its origin in Nature, in the movement at work in organic and inorganic combination—the doctrine on which is central not only to Aristotle's definitive criticism of the Atomism of Democritus and Leukippus, but also to an understanding of his Physics, and so to an understanding of his philosophy as a whole (Ann. Sci., 1972, 28, 247 ff).
- There is a real problem here, but it will have to wait until I come to deal with the Unit and the Theory of Number. We must have a Unit of measure, and this unit must be ‘homogeneous with the thing measured’ (Ar. Metaphysics: 1053a 25). This choice is ours, but, having chosen it, we cannot divide it up, as we mistakenly divide our ‘foot’ length up into 12 inches, or the inch into tenths, eighths, etc. That is impossible (See Ann. Sci. 1972 28 273 273 and 276). However, we can use our Unit to measure aggregates, and our measure will be accurate because, as I have already said, the limits of things in contact are coincident. But, any aggregate of such units will be contiguous. The unit, being indivisible, measures similar indivisibles, and these as ‘many’ cannot as ‘parts’ combine to form a ‘one’, i.e. a new unit—a fact recognised by Democritus as being equally true of his indivisible magnitudes (Ar. Metaphysics: 1039a 10). Lines as units cannot be continuous or form the perimeter we need to make a ‘figure’. The truth is that we make a ‘line’ continuous simply by drawing it, and it is this that creates the problem of division, and that arising from the junction of geometrical ‘sides’.
- The Development of Mathematics second edition Bell E.T. McGraw-Hill 1945 To be referred to as: Bell, Mathematics (McGraw-Hill).
- 1938 . “ The Unity of Philosophical Experience ” . In Etienne Gilson: Sheed & Ward 138 – 138 .
- In his philosophy, Descartes betrays no knowledge at all of the Aristotelian indivisible, or of the continuous, or of the infinite, or of the importance of the doctrine of organic and inorganic combination to an understanding of the Physics. But, there is an error of more fundamental importance in all Descartes' thinking, as a mathematician. In falling into this error, he is in distinguished company for, like all mathematical philosophers since, and like all philosophers in the western philosophic tradition, Descartes failed to make the radical distinction between the ‘one’ the unit of measure, and the ‘one’ that is the starting-point of all number. Now, this was precisely the mistake made by the Neo-Platonist Porphyry, and prompted him to ask the fatal question which started Boethius, and then the medievals on the fruitless and misguided search for the meaning of the ‘universal’. (See Etienne Gilson, opus cit. Ch. 1, and Selections from Medieval Philosophers McKeon Richard New York i 67 67 and 91 (Boethius).) Aristotle's solution of this problem will be dealt with in a subsequent paper on the Unit and the Theory of Number.
- Cp. Descartes Geometry Dover 10 10 footnote where in a letter to the Princess Elizabeth of Palatine, Descartes says that in the solution of a geometrical problem, he reduced the question to such terms as made it depend on these two theorems.
- Other instances like this of Galileo's mental confusion have already been referred to See Ann. Sci. 1968 24 322 322 and 328 ff; 1970, 26, 140 ff; and 1972, 28, 257 ff.
- See Bell E.T. Mathematics 57 57 83, 154, 272–281. Galileo's misreasoning on the infinite has been acclaimed like a message from a new John the Baptist heralding the dawn of a new branch of mathematics. For the refutation of Galileo's position here, see Ann. Sci., 1970, 26, 143 ff.
- See Heath T.L. Elements Dover ii 39 42 for the 16th century controversies on the angles between circumferences of circles touching one another, internally or externally, and the angles made by ‘the contact of a straight line with a circle’. Since Galileo held the same opinion as Vieta here, it is surprising that the point in question is passed over in silence.
Annals of Science
Volume 31, 1974 - Issue 3
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