Abstract
This paper examines the concepts of infinite multitude and comparability of size in Galileo’s discussion of the paradox of the natural numbers in Two New Sciences, and related points in Leibniz’s philosophy. Galileo’s celebrated denial that ‘greater’, ‘less’, and ‘equal’ apply in the infinite threatens two important mathematical principles: Euclid’s Axiom and the Bijection Principle of Cardinal Equality. I consider two potential strategies open to Galileo for preserving the principles, of which the more promising is due to Leibniz. I find, contrary to the customary view, that Galileo’s denial is limited to comparisons among infinite magnitudes and allows judgments of cardinal equality among infinite multitudes, based on one-one correspondences. The paper’s analysis also reveals how intimately related Leibniz’s definition of ‘infinite’ is to Galileo’s discussion, and illuminates key contrasts between their accounts.
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Notes
- 1.
- 2.
Primary texts are abbreviated as follows. For Galileo: EN = Opere, Edizione Nazionale, ed. Antonio Favaro (Florence 1898). For Leibniz: A = Berlin Academy Edition, Sämtliche Schriften un Briefe. Philosophische Schriften. Series VI. Vols. 1–4. (Berlin: Akademie-Verlag, 1923−99); GP = Gerhardt, Die Philosophischen Schriften, Vols. 1–7. Ed. C.I. Gerhardt (Berlin: Weidmannsche, Buchhandlung 1875–1890); GM = Mathematische Schriften von Gottfried Wilhelm Leibniz, Vols. 1–7. Ed. C.I. Gerhardt (Berlin: A. Asher; Halle: H.W. Schmidt 1849–1863). References to EN, GP and GM are to volume and page numbers; those to A are to series, volume and page. Translations of Galileo generally follow those of Stillman Drake (abbreviated ‘D’), Galileo Galilei: Two New Sciences, Including Centers of Gravity and force of Percussion, 2nd Ed, (Toronto: Wall and Emerson, Inc. 1974), and those of Leibniz generally follow Richard Arthur (abbreviated ‘Ar’), G.W. Leibniz: The Labyrinth of the Continuum: Writings on the Continuum Problem, 1672–1686 (New Haven: Yale University Press 2001). I have sometimes modified translations without comment.
- 3.
A quick note on terminology. In what follows I sometimes use the terms ‘class’, ‘subclass’, etc., for convenience, but without meaning to imply that the many elements of a class thereby form a set or totality or other ‘single object.’ (Mostly here it will cause no harm to read ‘class’ and ‘set’ as equivalent, but sometimes it will lead astray, so beware.) Occurrences of ‘class’, etc., can, with appropriate shifts in syntax, always be replaced by suitable plural expressions—e.g., ‘the natural numbers’ instead of ‘the class of natural numbers’—or by terms such as ‘multitude’ or ‘plurality’ that cancel the implication of one thing formed from many. In contexts in which greater precision is required to convey the intended meaning, and avoid unwanted implications, I shall use unambiguous terms.
- 4.
Cantor writes, “There is no contradiction when, as often happens with infinite aggregates, two aggregates of which one is a part of the other have the same cardinal number” (Cantor 1915, p. 75; noted in Parker (2009)) .
- 5.
Russell (1913, p. 198).
- 6.
For a very illuminating discussion of the history of the idea of measuring the size of the natural number collections, as it has evolved up to Cantor, plus some contemporary alternatives to Cantor, see Mancosu (2009).
- 7.
In the usual formula: S is infinite if and only if there is a one-one map Φ from S into S with some element of S not in the range of Φ. Was sind und was sollen die Zahlen?, Sect. 64; cf. Sect 66.
- 8.
Crew and de Salvio’s 1914 translation renders tutti i numeri as “the totality of all numbers” (cf. p. 31). In the same lines it also inserts ‘number’ in “the number of squares is infinite” and “the number of roots is infinite”, where the corresponding term does not occur in the original. Drake’s translation steers clear of those interpolations.
- 9.
Knobloch (1999), p. 94.
- 10.
My discussion is indebted to Parker (2009), who, defensibly, calls our two principles ‘Euclid’s Principle’ and ‘Hume’s Principle’ . If Galileo had not rejected the one-one maps standard in the infinite case, we should call it ‘Galileo’s Principle’. For reasons to think Archimedes made use of this principle in application to infinite classes, see Netz et al. (2001–2002) .
- 11.
Notably those involved in discussion of a similar principle of equality sometimes called ‘Hume’s Principle’: the number of Fs equals the number of Gs iff there is a one-one correspondence between the Fs and the Gs. The principle is so-called for Frege’s reference, in Sect. 73 of the Foundations of Mathematics, to Hume’s remark, in Treatise I.iii.1, “When two numbers are so combin’d as that one has always an unite answering to every unite of the other, we pronounce them equal.” Yet both of those authors have their sights on slightly more restricted conditions than the ones Galileo considers. Frege takes one-one correspondence between classes to imply the existence of a number that measures them; Hume \t "See Principle" is expressly considering a standard of equality for numbers, where the numbers themselves are conceived as made up of units. It is in this vein that one-one correspondence is sometimes said to be a criterion of ‘equinumerosity’: equality of number.
- 12.
Or he appears to be doing so. Below I shall suggest his account of comparisons of infinite number classes turns out to involve an infinite number after all; that is, if, per impossibile, there were such a comparison, it would have to involve an infinite number.
- 13.
See footnote 3 above.
- 14.
Bolzano, for example, explicitly defended the primacy of Euclid’s Axiom against the Bijection Principle, writing that even two sets that stand in a one-one correspondence “can still stand in a relation of inequality in the sense that the one is found to be a whole, and the other a part of that whole” (Bolzano 1950, p. 98). For discussion of Bolzano, see Parker (2009) and Mancosu (2009). Even Russell acknowledged that “the possibility that the whole and part may have the same number of terms is, it must be confessed, shocking to common sense” (1903, p. 358).
- 15.
On “wielding the big stick”, see Michael Dummett (1994) .
- 16.
It should be noted that Galileo’s analysis is mistaken—at most one of the two circles rolls along the tangent, the other merely revolves continuously along it with the illusion of rolling—but our interest concerns the elements of his analysis, not the quality of his solution. For detailed discussion see Drabkin (1950) , Costabel (1964) , and Knobloch (1999, 2011). See also Mancosu (1996, pp. 121–122).
- 17.
Knobloch (1999, p. 92).
- 18.
This runs parallel to the classical contemporary point-set analysis, which allows unions of infinitely many zero-dimensional points (or singletons) to have any positive measure, though with the proviso, on the contemporary account, that the cardinality of the union be uncountable; countably infinite unions of points would still have measure zero. See Skyrms (1983) .
- 19.
Perhaps an expansion by mere rearrangement of non quanti atoms would seem to violate conservation principles, whereas the interposition of non quanti voids would not, if void is not a conserved quantity, so to speak.
- 20.
For provocative discussion of the bisection principle and its possible denial, see Benardete (1964, pp. 240 ff).
- 21.
- 22.
Another possibility is that Galileo’s denial that ‘greater’, ‘less’ and ‘equal’ apply in the infinite is carefully consistent with his judgments that there as just as many Xs as Ys in some cases of infinite multitudes: perhaps the Xs and the Ys can be just as many without falling under the term ‘equal’. If so, however, it would seem to be only a matter of a word, as no richer notion of cardinal equality seems available for which ‘just as many’ is not sufficient. A more substantial possibility here would be that for Galileo, ‘greater’, ‘less’ and ‘equal’ are essentially metrical notions, and their cardinal counterparts ‘more’, ‘fewer’ and ‘equally many’ cannot be applied on the basis of one-one maps without corresponding geometrical judgments in place as well. I am more sympathetic to this idea but cannot pursue it here; a few related points are discussed below.
- 23.
I am indebted here to Katherine Dunlop.
- 24.
- 25.
See Nicholas ([1440] 1985), De Docta Ignorantia, Book 1, Chaps. 11–23; see especially Chap. 14 for use of the distinction between quanta and non quanta.
- 26.
This so-called Diagonal Paradox was well-known by the seventeenth century; see Leibniz’s use of it at A VI, 3, 199. As noted by Lison (2006/2007, p. 199fn3) , the example goes back at least to Ockham. Note also that similar examples can be constructed even with multitudes containing only quanti parts, provided their total measure remains finitely bounded. Considering a line segment as composed of a sequence of geometrically decreasing non-overlapping subsegments, one can easily construct a one-one correspondence between the subsegments of the side of a square and the subsegments of the diagonal, even though the side can be shown by rotation to be equal to a part of the diagonal.
- 27.
- 28.
Leibniz has a few different lines of argument to offer against infinite number, including, notably for us, a deployment of Galileo’s paradox that expressly says infinite numbers are “impossible” because “it is impossible that this axiom”—Euclid’s Axiom—“fails” (A III, 1, 11). (This passage comes from some remarks by Leibniz on Galileo’s paradox, written in 1673.) Here an infinite number is presumably understood as a whole constituted of infinitely many units or the combination of all natural numbers, themselves taken as wholes composed of units. For some related discussion, see Levey (1998).
- 29.
At Grua 558 Leibniz writes: “God is not the soul of the world can be demonstrated; for the world is either finite or infinite. If the world is finite, certainly God, who is infinite, cannot be said to be the soul of the World. If the world is supposed to be infinite, it is not one Being or one body per se (just as it has elsewhere been demonstrated that infinite in number and in magnitude is neither one nor a whole, but infinite in perfection is one and a whole). Thus no soul of this sort can be understood. An infinite world, of course, is no more one [Being] and a whole than an infinite number, which Galileo has demonstrated to be neither one nor a whole.”(Leibniz 1948, p. 558).
- 30.
For an overview of this dispute in the interpretation of Leibniz, see Levey (2011).
- 31.
See Leibniz (2007), p. 409.
- 32.
Drake suggests that Galileo neither fully accepted nor fully rejected Aristotle’s principle (1974, 42fn26).
- 33.
- 34.
See Ishiguro (1990) , Knobloch (1994, 2002), Bassler (1998), Arthur (2008), (2009) and (2013) and Levey (2008). In the present paper it’s worth noting that the same crossed-out passage that contains the denial of cardinal equality between the evens and odds, Leibniz begins: “There is a syncategorematic infinite or passive power having parts, namely, the possibility of further progress by dividing, multiplying, subtracting, or adding. And there is a hypercategorematic infinite, or potestative infinite, an active power having, as it were, parts eminently but not formally or actually. This infinite is God himself. But there is not a categorematic infinite or one actually having infinite parts formally” (GP II, 314–315). Translated by Look and Rutherford in Leibniz (2007, p. 53).
- 35.
William Heytesbury may have been the first to defend a syncategorematic analysis of ‘infinite’; see sophisma xviii of his Sophismata, in Pironet (1994) .
- 36.
Quoted in Grosholz and Yakira (1998, p. 99 ).
- 37.
See my ‘Leibniz’s Analysis of Galileo’s Paradox,’ ms.
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Acknowledgements
My thanks to Katherine Dunlop , Jeffrey McDonough, O. Bradley Bassler , Richard Arthur , Philip Beeley and Norma Goethe for helpful comments, and to audiences at the University of Oxford and Texas A&M University who heard preliminary versions of this material. My thanks also to Michael Detlefsen for the invitation that eventually led to this work, to Eberhard Knobloch whose skepticism about my claims about Galileo in an earlier paper made me to reconsider the issues more carefully, and especially to Norma Goethe, again, who encouraged me to write the present paper.
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Levey, S. (2015). Comparability of Infinities and Infinite Multitude in Galileo and Leibniz. In: Goethe, N., Beeley, P., Rabouin, D. (eds) G.W. Leibniz, Interrelations between Mathematics and Philosophy. Archimedes, vol 41. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9664-4_8
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