The World Philosophy Made
Page 40
11. Aquinas thought there are many human souls, which there wouldn’t be if the soul of each human being were the property being human, which is common to all humans. Thus when Copleston says it is matter “which marks off one corporeal thing from other members of the same species” and that “what makes human souls different from one another is their union with different bodies,” he can’t be denying that different human beings differ in both form and matter (Copleston 1955, pp. 95–96). See Book 2, chapter 81 of Aquinas (1975), Summa Contra Gentiles, where he says, “So this soul is commensurate with that body, that soul with that body, and so on with all of them.” For an informative discussion of the problem, see pp. 121–47 of Pegis (1934).
12. Aquinas (1947), Summa Theologica, Ia. 75.IC.
13. Aquinas (1947), Ia. 75.IC, and (1975), Summa Contra Gentiles, II. 65.
14. Aquinas (1947), Ia. 75.2C.
15. Ibid., Ia. 75.2C.
16. The argument for this is discussed on pp. 303–4 of McCabe (1969). This argument is also discussed on pp. 173–74 of Copleston (1955).
17. Kretzmann (1993), p. 132.
18. Ibid., p. 133.
19. Copleston (1962b), pp. 94–98, and Kretzmann (1993), p. 136.
20. Aquinas (1949), Disputed Questions on the Soul IC, my emphasis.
21. Aquinas (1947), Summa Theologica, Ia.3.1, 93.2C, 93.6C.
22. Aquinas (1975), 4.79.
23. See p. 136 of Kretzmann (1993).
24. One of Aquinas’s important early critics was St. Bonaventure, who thought Aquinas had overintellectualized Christianity. More other-worldly and mystical than Aquinas, he once remarked: “[A]mong philosophers the word of wisdom was given to Plato, to Aristotle the word knowledge. The one looked principally to what is higher, the other to what is lower.… But the word both of knowledge and of wisdom was given by the Holy Ghost to Augustine.” (Quoted on p. 15 of D’Arcy 1930).
Apparently seeing no need for a systematic philosophical approach for acquiring worldly knowledge (which would have to be reconciled with Christianity’s inspirational message), Bonaventure offered a kind of Augustinian mystical synthesis, and so saw no need for the Thomistic synthesis. Many others, however, felt the need to engage with Aquinas more directly. See Gilson (1955).
25. Copleston (1962b), p. 151.
26. See ibid., pp. 289–90.
27. See ibid., pp. 56–57.
28. This was Ockham’s general form of argument involving the razor. He says the following (Ockham 1952, p. 37): “Nouns which are derived from verbs and also nouns which derive from adverbs … have been introduced only for the sake of brevity in speaking or as ornaments of speech … and so they do not signify any things in addition to those from which they derive.” The translation is from Copleston (1993), p. 76.
29. Passage from Ockham, Quodlibeta septem, 1, 12, Paris, 1487, translated by Copleston (1993), p. 96.
30. See chapters 1, 9, 10, 18, and section 4 of chapter 24 of Copleston (1993).
CHAPTER 3
1. Although the work, De Revolutionibus Orbium Coelestium Libri IV, in which Copernicus presented his system wasn’t published until 1543, its main ideas were developed around 1512 or shortly thereafter.
2. Preface to Kepler ([1609] 1858–71).
3. Newton’s not dissimilar reaction will be discussed below.
4. Letter to Herwart, Feb. 10, 1605.
5. Galileo was Professor of Mathematics at Pisa from 1589 to 1592 and at Padua from 1592 to 1610, when he was brought by Cosimo II de’ Medici, Grand Duke of Tuscany, to Florence, where he was free to pursue his mathematical, philosophical, and scientific work. It was about that time that he constructed his improved telescope.
6. The impetus that prompted Galileo to conduct his famous experiment at the Tower of Pisa was a paper published by Stevin in 1586 in which he reported the results of his own experiment with lead balls of different weights.
7. Galileo had been an outspoken supporter of the Copernican system—as established truth rather than merely well-supported theory—since at least 1613. This led to decades of tension with papal authorities that culminated in his condemnation by the Inquisition in 1633, resulting in a sentence of what amounted to house arrest—first in the house of his friend the archbishop of Siena and then in his own villa near Florence. During that time he continued to work until going blind in 1637, when he sent a new treatise on physics to Holland that was published in 1638.
8. A strong contemporary case for a physicalist conception of color is given by the articles authored by Alex Byrne and David Hilbert in the volume Byrne and Hilbert (1997).
9. Letter to Mersenne (1:70). See the entry in Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/descartes/.
10. Descartes ([1641] 1991).
11. Descartes ([1644] 1983).
12. The passage is from “The Author’s Letter” (9B:14), of the French translation, in 1647, of Descartes’s ([1644] 1983) (which originally appeared in Latin).
13. The implicit premise here, that what one can coherently conceive must be genuinely possible, is, for good reason, now widely (though still not universally) recognized to be questionable.
14. There are two potential problems with this argument: (i) the idea that which states are possible is independent of which state turns out to be actual, and (ii) the idea that if something is coherently conceivable, then it is genuinely possible. The most promising current explanation of why (ii) is false will be discussed in chapter 9.
15. Newton ([1687] 1934, 1999).
16. Newton’s work is explained in a little more detail in chapter 10, where it is used to introduce the philosophical underpinnings of 20th-century physics.
17. Third letter to Bentley, 1692, in Newton (1959–84).
18. Newton (1999), p. 943, from “General Scholium,” an essay by Newton appended to the second edition of Principia.
19. Newton ([1687] 1934), vol. 1, p. 6.
20. Ibid., p. 11.
21. Could the water’s position relative to some body other than the bucket explain what we observe? This seems unlikely, since we can easily imagine the experiment taking place in an empty universe with the same result. Newton himself invokes this kind of thought experiment not with a bucket of water but with an empty universe consisting of two globes linked by a cord revolving around their common center (Newton [1687] 1934), vol. 1, p. 12).
22. Boyle (1692).
23. Seventh page of “Epistle to the Reader,” in Locke ([1689] 1975).
24. Principle (iii) is sometimes taken to be a consequence of the truth (sometimes called “Leibniz’s Law”): Necessarily, if x = y, then any property of x is a property of y. From this it follows that necessarily, if x has property P, then anything that does not have P isn’t identical with x. However, what is needed to derive (iii) is not that, but rather: If x has property P, then necessarily (i.e., it is a necessary truth that) anything that doesn’t have P isn’t x. This needed principle doesn’t follow without assuming the very thing one is trying to prove. Thus, the case for (iii) fails. See Soames (2014b), pp. 417–19, for an explanation.
25. This point is explicitly endorsed in a letter to Volder, on p. 226 of volume 2 of Leibniz (1875–1890).
26. For Leibniz, these truths are contingent in the sense that we finite beings can’t provide complete analyses that show their subjects to contain the properties predicated of them.
27. See the third, fourth, and fifth letters to S. Clarke in volume 7 of Leibniz (1875–1890), pp. 363, 373, and 400. See also p. 183 of vol. 2. In chapter 10, I return to the dispute between Leibniz and Newton, indicating its relation to developments in modern physics.
28. Hume (1964), vol. 2, page 419.
29. Ibid., pages 428–29.
30. Ibid., page 436.
31. Ibid., page 440.
CHAPTER 4
1. Hobbes ([1651] 1994), p. 113.
2. Second Treatise, Chapter 2, A
rticle 6, in Locke (1987).
3. Ibid., 8, 95.
4. Ibid., 9, 131.
5. Ibid., 11, 138.
6. Copleston (1964), p. 151.
7. Hume’s A Treatise of Human Nature, book III, part I, section 2 (Hume 1964, vol. 2, p. 246).
8. Hume’s Treatise, book III, part II, section 2 (ibid., pp. 262–63).
9. Ibid., p. 263.
10. Hume’s Treatise, book III, part II, sections 3, 4, and 5 (Hume 1964, vol. 2).
11. Hume’s Treatise, book III, part II, section 1 (ibid., p. 258).
12. Hume’s Treatise, book III, part II, section 2 (ibid., pp. 269–71).
13. Hume’s Treatise, book III, part II, section 2 (ibid., p. 271).
14. See Adair (1957).
15. Herman (2001), p. 173.
16. Smith ([1776] 1997), book IV, chapter 7, part 3. A thorough account of Adam Smith’s far-reaching influence on American founders, including Madison and Jefferson, is presented in Fleischacker (2002).
17. A nod should also be given to two other philosophers—to Jeremy Bentham (1748–1832), who evaluated the rightness or wrongness of individual actions on the basis of the total value of their consequences (for human happiness), and to John Stuart Mill (1806–1873), some of whose writings can be read as suggesting evaluating the rightness or wrongness of actions in terms of the consequences (for human happiness) not of individual actions themselves but of the universal adoption of rules mandating or prohibiting such actions.
CHAPTER 5
1. Also prominent among these philosophers was Charles Sanders Peirce, an American contemporary of Frege who independently developed a system of logic comparable to Frege’s, which, though not without influence, did not have the historical impact that Frege’s did.
2. Frege ([1879] 1967).
3. The definitions of model, of truth in a model, and of logical consequence in terms of truth in a model were not made explicit until they arose from the work of Alfred Tarski in the 1930s. Nevertheless, for the most part, Frege’s logical and semantic principles implicitly tracked them, in part, perhaps, because his logical and semantic ideas gave birth to the practices from which the later definitions were abstracted. This point is taken up in more detail in chapter 6.
4. Frege ([1884] 1950).
5. Frege ([1893, 1903] 1964).
6. Although for Frege concepts are associated with predicates, they are neither ideas in the mind, nor anything psychological. Rather, they are functions that map objects of which the predicates are true onto truth, and those of which the predicates are not true onto falsity. In this way they are more like sets, of which there are infinitely many, than they are like the finitely many ideas present in any human mind.
7. The system, popularized in Giuseppe Peano (1889), is standardly attributed to Richard Dedekind, who is said to have come up with it in 1888, though Frege also seems to have come up with it independently.
8. The discovery of the contradiction, its application to the systems of Frege and Russell, and the lessons to be drawn from the contradiction are discussed in Soames (2014b), pp. 120–29.
9. Bertrand Russell and Alfred North Whitehead (1910, 1912, 1913), Principia Mathematica. Simplified explanations are given in Soames (2014b), pp. 474–88, 500–11.
10. See Soames (2014b), pp. 488–91, 494.
11. There are two versions of Russell’s theory of types—the original ramified theory used in Principia Mathematica and the simple theory in Ramsey (1925). The axiom of reducibility is required by the former but not the latter. As Kamareddine, Laan, and Nederpelt (2002) report, that axiom was “questioned from the moment it was introduced.” (See also Gödel 1944). Although the replacement of the ramified theory by the simple theory, and the resulting elimination of reducibility, was a mathematical step forward, it highlighted a philosophical worry that Russell had managed to blur. Using a theory of logical types to block his 1903 contradiction requires a questionable theory of intelligibility. Just as it is impossible for a set to be a member of itself, so Russell maintained, a proper logical analysis would show it to be meaningless, not just false, to say that a set is a member of itself. But that’s not obvious. To say that no set is identical with any of its members seems true; if so, its negation should be false, not meaningless. It is also virtually impossible to explain simple type theory involving real sets without saying things which, once one has the theory, are declared to be meaningless. Surely there is something wrong with any theory the strictures of which pronounce the best descriptions of the theory to be incoherent. Although this doesn’t undercut the claims the theory makes about sets, and the identification of the natural numbers with some of them, it does undercut the idea that the type theory is nothing more than a system encoding logical principles needed to reason about any subject.
Landini (1998) and Klement (2004) plausibly argue that in adopting his type theory Russell had come to think there is no more reason to suppose that one who says “Some sets are so and so, while others are such and such” is committed to the real existence of sets than there is to think that one who says “The average man has 1.7 children” is committed to there being some really existing person who is the average man. The issue involves what is now called the substitutional analysis of the quantification into predicate position employed in Principia Mathematica. On this analysis, Russell’s type-theoretic constraints on intelligibility are, arguably, justified, but—as shown in Hodes (2015), Soames (2014b), pp. 511–31, and Soames (2015a)—the reduction of mathematics to logic, along with other aspects of Russell’s philosophical logic, are threatened. The reduction of mathematics to set theory, which posits the real existence of sets, isn’t threatened, but set theory isn’t logic.
12. Zermelo (1904, 1908a, 1908b), Fraenkel ([1922] 1967).
13. These essentially are the arguments given in two classic articles in recent philosophy of mathematics, Benacerraf (1965, 1973).
14. This way of thinking of natural numbers grows out of two pathbreaking articles—one, Boolos (1984), by my former teacher, George Boolos, and the other, “Arabic Numerals and the Problem of Mathematical Sophistication,” forthcoming by my former Ph.D. student Mario Gomez-Torrente.
15. Page 2, section 1 of Wittgenstein (1953).
16. Ibid., pp. 2–3.
17. Thanks to Jing He for helpful discussion of this point.
18. See pp. 435–436 of Boolos (1984) for a fuller discussion.
19. For more detail, see Gomez-Torrente (forthcoming).
CHAPTER 6
1. See chapter 1 of Soames (2014b).
2. Tarski ([1935] 1983) and ([1936] 1983). Although the contemporary notion of truth in a model is based on these papers, it is an abstraction from the concepts explicitly defined there.
3. Gödel had already proven results relating provable sentence (of a given logical system) to logical truth and logical consequence, using informal understandings of those concepts. Even Frege had an implicit grasp of them. According to him, the truths of logic were entirely general, and did not depend on any special subject matter. This suggests that any sentence that counts as a logical truth should remain such no matter how its names, function signs and predicates are interpreted. Since a model is a formalization of the idea of such an interpretation, it shouldn’t be seen as entirely foreign to Frege.
4. Gödel (1930).
5. Section 4 of chapter 1 of Soames (2014b) explains the difference between first and higher-order quantification in Frege.
6. Hahn, Carnap, and Neurath (1929).
7. The theorem is formally proved in Tarski ([1935] 1983). It is often called the Gödel-Tarski theorem because it is an obvious corollary of Gödel’s first incompleteness theorem, presented in Gödel (1931).
8. A free occurrence of a variable in a formula is one that is not bound (i.e., governed) by a quantifier containing that variable. For exampl
e, the three occurrences of ‘x’ within the parentheses in the formula every x (x = x & x ≠ y) are bound by ‘every x’, while the occurrence of ‘y’ is free.
9. A more detailed explanation is given in section 3.3 of chapter 8 of Soames (2018).
10. Here, ‘P’ is a predicate variable, occupying predicate positions in sentences, that ranges over of sets of objects.