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What did Frege take Russell to have proved?

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The last time I saw Frege, as we were waiting at the station for my train, I said to him ‘Don’t you ever find any difficulty in your theory that numbers are objects? He replied, ‘Sometimes I seem to see a difficulty – but then again I don’t see it.’. Wittgenstein, quoted in Geach and Anscombe, Three Philosophers, 1961.

How can philosophical enquiry be conducted without a perpetual petitio principii?

Ramsey, The Foundations of Mathematics, 1931.

Abstract

In 1902 there arrived in Jena a letter from Russell laying out a proof that shattered Frege’s confidence in logicism, which is widely taken to be the doctrine according to which every truth of arithmetic is re-expressible without relevant loss as a provable truth about a purely logical object. Frege was persuaded that Russell had exposed a pathology in logicism, which faced him with the task of examining its symptoms, diagnosing its cause, assessing its seriousness, arriving at a treatment option, and making an estimate of future prospects. The symptom was the contradiction that had crept into naïve set theory in the form of the set that provably is and is not its own member. The diagnosis was that it is caused by Basic Law V of the Grundgesetze (Frege, In: Grundgesetze der Arithmetik: Begrifsschriftlich abgeleitet, Jena: Herman Pohle, 1893/1903. Translated into English as Basic Laws of Arithmetic, also edited by Philip A. Ebert and Marcus Rossberg, with Crispin Wright, Oxford: Oxford University Press, 2013). Triage answers the question, “How bad is it?” The answer was that the contradiction irreparably destroys the logicist project. The treatment option was nil. The disease was untreatable. In due course, the prognosis turned out to be that a scaled-down Fregean logic could have an honourable life as a theory inference for various domains of mathematical discourse, but not for domains containing the logical objects required for logicism. Since there aren’t such objects, there aren’t such domains. On the face of it, Frege’s logicist collapse is astonishing. Why wouldn’t he have repaired the fault in Law V and gone back to the business of bringing logicism to an assured realization? In the course of our reflections, we will have nice occasion to consider the good it might have done Frege to have booked some time with Aristotle had he been able to. By the time we’re finished, we’ll have cause to think that in the end the Russell might well have begged the question against Frege.

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Notes

  1. Gottlob Frege, Die Grundlagen der Arithmetik, Breslau: Koebner, 1884. Republished as Foundations of Arithmetic, J. L. Austin, translator, Oxford: Blackwell, 1950; xiv.

  2. The task is taken up in John Woods and Alirio Rosales, “What logicism could not be”, forthcoming.

  3. A small correction is needed here. For Frege, an axiom is a primitive truth, and a primitive truth is a true proposition that is neither needful nor susceptible of independent proof, whereas an analytic truth is one that lies in the demonstrative closure of first principles. (Grundlagen, p. 4) Accordingly, it is left to axioms to speak for themselves, to disclose their self-evident immunity from overthrow, no matter what.

  4. See here Andrew D. Irvine, “Epistemic logicism and Russell’s regressive method”, Philosophical Studies, 55 (1989), 303–327.

  5. Philip Kitcher, The Nature of Mathematical Knowledge, New York: Oxford University Press, 1984.

  6. James Tappenden, “Geometry and generality in Frege’s philosophy of arithmetic”, Synthese, 102 (1995), 319–361.

  7. See below my remarks on the limitations on application of the proof by contradiction rule.

  8. See again Woods and Rosales, “What logicicism could not be”.

  9. See Frege’s Lectures on Logic: Carnap’s Student Notes, 1910–1914, Reck and Awodey, editors and translators, with an introduction and annotations by Gottfried Gabriel, pp. 17–42, Chicago and La Salle: Open Court, 2004: pp. 29–34.

  10. Richard Kimberly Heck, Frege’s Theorem, New York: Oxford University Press, 2011. Paperback in 2014; pp. 2–3. Originally published under the name of Richard G. Heck, Jr.

  11. Of course, it is doubted by constructivists and intuitionists, but not by Frege or Russell.

  12. Certainly, virtually all post-paradox treatments of sets proceed from a determination to constrain their respective grammars and axioms, lest a well-formed sentence do the damage done to Frege by line (1) and Basic Law V. An exception is Chris Mortensen, Inconsistent Mathematics, Dordrecht: Reidel, 1995.

  13. Bertrand Russell, The Principles of Mathematics, 2nd edition, London: Allen and Unwin, 1937; first published by Cambridge University Press, 1903; xv, xvi, emphasis in the first instance mine, and in the second Russell’s.

  14. My thanks to Bernard Linsky for pointing this out in correspondence. Details can be found in Linksy and Andrew Irvine, “Principia Mathematica”, in the Stanford Encyclopedia of Philosophy, posted on November 3, 2018. Linsky goes on to say that “Frege didn’t say much about PM, except that he couldn’t read it because of the use/mention confusions. That only means that he didn’t think it was logic, not that it didn’t have a proof of Basic Law V in it. How’s them apples?” A referee points out that it could be that “Frege merely says that his English was not up to reading the Introduction to PM.” Perhaps this is so, but for present purposes we needn’t press the matter.

  15. Paul Bernays, “Sur le platonisme dans les mathématiques”, L’Enseignement Mathématique, 34 (1935), 52–69; English translation in Paul Benacerraf and Hilary Putnam, editors, Philosophy of Mathematics: Selected Readings, Cambridge: Cambridge University Press, 1983.

  16. David Hilbert, “Mathematische Probleme”, Vortrag, gehalten auf dem internationalem Mathematiker-Kongress zu Paris 1900”, Archiv der Mathematatik und Physik, 3rd series (1901), pp. 44–63, 213–237.

  17. For further discussion, see Gottlob Frege, Philosophical and Mathematical Correspondence, Gottfried Gabriel, Hans Hermes, Friedrich Kambartel, Christian Thiel, Albert Veraart, Brian McGuiness, and Hans Kaal, editors, Oxford: Blackwell, 1980, and Patricia A. Blanchette, “The Frege-Hilbert controversy”, in Edward N. Zalta, editor, Stanford Encyclopedia of Philosophy, online, Section 4. First posted in 2007; substantially revised on August 9, 2018.

  18. Anthony Kenny, Frege: An Introduction to the Founder of Modern Analytic Philosophy, Oxford: Blackwell, 2000; p. 175.

  19. Posterior Analytics 70b 20–31, 72a 16–18. It is helpful to recall the point that in that work Aristotle thought that the axioms of a discipline implicitly contained its every truth. Under Moore’s influence Russell thought that the axioms of a theory provided the complete conceptual analysis of its subject-matter and that all its provable truths lay in its demonstrative closure. Frege was similarly minded. The basic logical laws are laws that “potentially imply all [the] others.” See Begriffsschrift, eine des arithmetischen nachgebildete Formalsprache des reinen Denkins, Halle: L. Hebert, 1879. Reprinted in Ignacio Angelelli, editor, Darmstadt and Hildesheim: Olms, 1964; p. 25. Also in Jean van Heijenoort, editor, From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, pp. 5–82, Cambridge, MA: Harvard University Press, 1967.

  20. The Stoic term is anapodeiktos, which can be read as “indemonstrable”, but also as “undemonstrated”. One might think this the better translation, since a first principle admits of demonstration but needn’t have one to perform its assigned role. The similarity here to the situation Frege was in is rather striking. I thank Erik Krabbe for instruction on this point.

  21. The idea that Frege had an old-fashioned view of axioms has attracted a good deal of attention of late. Especially important is Patricia A. Blanchette, “Axioms in Frege”, in Marcus Rossberg and Philip Ebert, editors, Essays on Frege’s Basic Laws of Arithmetic, Oxford: Oxford University Press, to appear. The quoted passage is from page 1 of this preprint. See also Michael Hallett, “Frege and Hilbert”, in The Cambridge Companion to Frege, Michael Potter and Tom Ricketts, editors, pp. 413–464, Cambridge: Cambridge University Press, 2010.

  22. It is then puzzling that in his effort to define the truth-values, Frege resorts to stipulation, writing that “it is always possible to stipulate that an arbitrary course of values is to be the True and another the False”, Basic Laws, p. 48 in Furth; see also p. 49 in Ebert and Rossberg. See also Section 64 of Foundations, where Frege shows his readiness to make new concepts by “recarving” concepts presently on hand. Shades of Kant on synthesis and of post-paradox Russell too.

  23. The wording of the claim is Heck’s not Frege’s. See Grundgesetze, last part of ∮ 20, where the claim is made, and ∮ ∮ 29–32 where it is proved. The proof, in any event, is not successful. What concerns us here is that there is documented reason to think that post-paradox Frege believed Basic Law V to be true. See Richard Kimberly Heck, Reading Frege’s Grundgesetze, New York: Oxford University Press, 2012; second printing 2013; p. 34. (Originally published under the name of Richard G. Heck, Jr.) See also chapters 2–4 for further discussion of Frege’s proof.

  24. See again Principia Mathematica at *20.15.

  25. Concerning which, see Simchen: “… the once-true sentence ‘Obama is president’ is carved around ‘Obama’ as an object-expression, where ‘Obama’ refers to the man. But it is about a second-level function from first-level functions to truth-values—a higher-level property true of all and only lower-level properties of Obama—to the extent that the sentence is carved around ‘Obama” as a second-level predicate or quantifier. But nothing in the thought itself betrays how it should be carved.” See Ori Simchen, Necessity and Intentionality: A Study in the Metaphysics of Aboutness, New York: Oxford University Press, 2012; p. xxii. Simchen adds in a footnote “that Frege is using ‘concept’ here as a term for senses in a way that is at odds with the way he normally uses it.”) So, then, what if anything, did “Obama is president” then denote?

  26. Heck to Woods, in correspondence.

  27. Reading, p. 34, note 18. Plumping for the position that Frege had post-paradox doubts about the Law’s truth is, among others, Tyler Burge, “Frege on extensions of concepts from 1884 to 1903”, in Burge, Truth, Thought, Reason: Essays on Frege, pp. 273–298, New York: Oxford University Press, 2005; pp. 290 ff.

  28. David Hume, A Treatise of Human Nature, edited by Ernest C. Mossner, Harmondsworth: Penguin, 1969 p. 119. First published in 1793. Note that here Frege is recurving equivalence relations to yield the concept of identity.

  29. George Boolos, “Saving Frege from contradiction”, in Richard Jeffrey, editor, Logic, Logic and Logic, pp. 171–182, Cambridge, MA: Harvard University Press, 1998. Charles Parsons, “Frege’s theory of number”, in Max Black, editor, Philosophy in America, pp. 180–210, Ithaca: Cornell University Press, 1965. Reprinted with a Postscript in William Demopoulos, editor, Frege’s Philosophy of Mathematics, pp. 182–207, Cambridge, MA: Harvard University Press, 1995. See also Crispin Wright, Frege’s Concept of Numbers as Objects, Aberdeen: Aberdeen University Press, 1983, Richard G. Heck, “The development of Frege’s Grundgesetze der Arithmetik, Journal of Symbolic Logic, 58 (1993), 579–600. Reprinted with a Postscript in Demopoulos 1995, pp. 257–294, and Edward N. Zalta, “Frege’s theorem and foundations of arithmetic”, in Zalta, editor, Stanford Encylopedia of Philosophy, online, Section 6. First posted in 1998 and substantially revised on June 26, 2018.

  30. My thanks to Zalta for correspondence on this point. In response to the question of why FA is consistent if PA is, Erik Krabbe proposes, in correspondence that the text suggests that it is because HP fuels the derivation of PA in FA. Krabbe thinks that what’s needed is the reverse. If FA can be derived in PA, the consistency of PA would show the consistency of FA.

  31. Gottlob Frege, Letter to Russell, in van Heijenoort, pp. 127–128; emphasis added.

  32. Perhaps this is the sort of point that Hilbert was tilting towards in the final passage quoted in section two above.

  33. Gottfried Gabriel, “Frege and the German Background to analytic philosophy” in Michael Beaney, editor, The Oxford Handbook of the History of Analytic Philosophy, pp. 280–297, Oxford: Oxford University Press, 2013; p. 288. I regret that I hadn’t been aware of this paper until Gottfried Gabriel informed me of it; for which many thanks. See also Gabriel and Sven Schlotter, Frege und die kontinentalen Ursprünge der analytischen Philosophie, Münster: mentis, 2017. A considerable value of this work is the clarity it lends to the difference between Frege’s and Russell’s notions of conceptual analysis.

  34. Wilhelm Windelband, “Die Prinzipien der Logik”, in A. Ruge, editor, Encylopädie der philosophischen Wissenschaften, volume 1, pp. 1 to 60, Tübingen: J. C. B. Mohr, 1912; p. 18.

  35. Grundgesteze I, p. 202 in The Frege Reader, Michael Beaney, editor, Oxford: Blackwell, 1997; p. 202.

  36. W. V. Quine, Confessions of a Confirmed Extensionalist, Dagfinn Follesdall and Douglas B. Quine, editors, Cambridge, MA: Harvard University Press, 2008, p. 317. Quine was talking about ontology.

  37. In response to “the thundering heptameter that shattered naïve set theory: the class of all classes not belonging to themselves”, Quine writes that “desperate accommodations were called for.” W. V. Quine, Quiddities, Cambridge, MA: Harvard University Press, 1987; pp. 146 and 148.

  38. Linsky to Woods: “I think I agree with the point that Frege only doubted that Basic Law V is an axiom of logic, although I don’t yet see that it led to a repudiation of courses of values as logical objects.” Indeed; see again the Wittgenstein epigraph to this paper.

  39. See, for example, José Ferreirós, Mathematical Knowledge and the Interplay of Practices, Princeton: Princeton University Press, 2016, especially chapter 7, “Arithmetic Certainty”, and his Labrynth of Thought: A History of Set Theory, 2nd edition, Basel: Birkhäuser, 2007. See also Ivor Grattan-Guiness, editor, From Calculus to Set Theory, 1630–1910, London: Duckworth, 1980, and J. J. Gray, Ideas of Space: Euclidean, Non-Euclidean and Relativistic, Oxford: Clarendon Press, 1989. Mathematicians who are given to philosophical speculation about the work they do might find themselves drawn to anti-realist hesitation. Most working mathematicians aren’t so drawn, but even those who are don’t put philosophical hypotheses to premissory work in their proofs.

  40. As noted earlier, intuitionists among some others, would also quarrel with Excluded Middle, but not Frege. Though not one of the Basic Laws of Arithmetic, Frege thought of Excluded Middle as having the same protection that axioms have. In Principia Mathematica, Excluded Middle appears as Th. 2.08.

  41. As, for example, if it were a truth of logic but not true of a logical object or, as a referee has suggested, wasn’t (or couldn’t be) known to be true.

  42. In Frege: A Philosophical Biography, Cambridge: Cambridge University Press (2019), the late Dale Jacquette argues that the Russell Proof really is invalid. One day at lunch in Istanbul during the 2015 UNILOG conference, Jacquette (a long-time friend) told me of his reservations about the proof, and I took notes:

    1. 1.

      K = the class of those classes that aren’t members of themselves.

    2. 2.

      If K ∈ K, K ∉ K.

    3. 3.

      If K ∉ K, K ∈ K.

    4. 4.

      K ∈ K ∧ K ∉ K.

    Jacquette thought the proof fails at (2). He thought that for (2) to hold, “those” in (1) must mean “all and only the”, but for (3) to hold, it suffices for “those” in (1) to mean “all but not necessarily only the”. I mention this in passing as a matter of interest. Since we are wondering about how Frege himself interpreted Russell’s proof, we can safely say that it wasn’t in the way that Jacquette does in Frege, at pp. 482–490.

  43. Reck and Awodey, “Frege’s lectures”, p. 34. (See two lines below this one.).

  44. Idem., emphasis added.

  45. Gottfried Gabriel, “Fregean connection: Beteutung, value and truth-value”, Philosophical Quarterly, 34 (1984), 372–376.

  46. Russell’s translation of Wahrheitswert as “truth-value” first appeared in Appendix A of Principles (1903).

  47. Gottlob Frege, “Gedankengefüge”, Beiträge zur Philosophie des deutschen Idealismus, 3 (1923), 36–51; p. 51. I have drawn this quotation from Gabriel’s Introduction to Frege’s Lectures on Logic, p. 4.

  48. Translated and edited by Erich H. Reck and Steve Awodey, based on the German text, edited, with introduction and annotations by Gottfried Gabriel, Purdue, IL: Open Court, 2004.

  49. Axiom VI asserts that a = (ιx: x = a). Axiom IV asserts that ¬(P = ¬Q) → P = Q.

  50. Gabriel’s Introduction, pp. 3 and 6.

  51. Gottlob Frege, Posthumous Writings, Hans Hermes, et al., editors, Peter Long and Roger White, translators, Chicago: University of Chicago Press, 1979; p. 176. English translation of Frege, Nachgelassene Schriften, Hans Hermes, et al, editors, Hamburg: Meiner, 1969.

  52. In “Neo-logicism? An ontological reduction of mathematics to metaphysics”, Erkenntnis, 53 (2000), 219–265, Zalta takes what might appear to be the somewhat weaker position that logicism might proceed without “primitive” sets in the manner of ZF, NBG, NF, Kripke-Platek etc., but cannot proceed without logical objects. So seen, primitive sets aren’t logical objects, but sets reconstructed from more basic logical objects in Zalta’s theory of them inherit their logical objecthood. The theory postulates that for any property P there is an abstract logical object encoding all and only properties with the same extension as P. See Edward N. Zalta and James Anderson, “Frege, Boolos, and logical objects”, Journal of Philosophical Logic, 33 (2004), 1–26, where sets are reconstructed as logical objects in this way. So the right thing to say about Zalta is that primitive sets, failing to be logical objects, can’t deliver the goods for logicism, but Zalta sets can.

  53. At Post. An. II, 19 100b, Aristotle speaks of “the noûs of the first principles. This would suggest, contra each of its translations, that noûs is the very grasping of them.

  54. The three inequivalent formulations of LNC can be found at 1005b 19–20, 1005b 20–35, and 1011b 14–15 (Aristotle, 1984). I will come back to 1005b 19–20 ff. below.

  55. Ferreirós, Mathematical Knowledge (2016).

  56. “We learn by epagôgē and demonstration. Demonstration proceeds from from kathekaston (particulars).” (Post. an I 18 81a 40–81b I) At Prior Analytics II, 25 69a 20–36, Aristotle introduces the notion of apaôgē, which Peirce (for a while) would call “abduction”. For Aristotle, an abduction is a syllogism from a major premiss which is certain and a minor premiss which is merely probable to a merely probable conclusion. Some see in this early intimation of reasoning to causes from their effects.

  57. For a further discussion, readers could consult the first edition of my Aristotle’s Earlier Logic, Oxford: Hermes Science, 2001, chapter 8. The book is now in a second and much revised edition in the Studies in Logic series of London’s College Publications, 2014. I regret my decision to omit the original chapter 8, as the first edition is now out of print. Also of interest is the claim that “the action of noûs serves as a bridge or a vehicle to pass from Aristotle’s logic to his cognitive system, from Posterior Analytics to De Anima, since in De Anima we can find a full elaboration of the functions of nous.” See here Christos Pechlivanidis, “Epagôgē, Noûs and Phantasia in Aristotle’s logical system: From Posterior Analytics to De Anima”, 2013 World Congress Proceedings, volume 2, section II: Classical Greek Philosophy, Athens, pp. 251–257; p. 254.

  58. Aristotle, Posterior Analytics, translated by G. R. G. Mure, in Richard McKeon, editor, The Basic Works of Aristotle, New York: Random House, 1941.

  59. For details and discussion, interested readers could consult John Woods and Hans V. Hansen, “The subtleties of Aristotle on non-cause”, Logique et Analyse, 176 (2001), 395–415; pp. 408–412.

  60. At Met. Γ, 1006a 6–8, Aristotle writes, “… for not to know of what things one may demand a demonstration, and of what one may not, argues simply for want of education [= training]. For it is impossible that there should be demonstration of absolutely everything; there would be an infinite regress, so that there would still be no demonstration”.

  61. The other formulations are “No one can believe that the same thing can at the same time be and not be” (1005b 20–25), and “The most certain of all basic principles is that contradictory propositions are not true simultaneously.” (1011b 14–15).

  62. See, for example, Joseph W. Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite, Cambridge, MA: Harvard University Press, 1979.

  63. Alfred Tarski, Introduction to Logic and to the Methodology of Deductive Sciences, New York: Dover, 1995, an unabridged replication of the 9th printing, 1961, of the second edition, revised of the work originally published in New York by Oxford University Press, in 1941; pp. 156–160. Original capital letters suppressed.

  64. See John Corcoran, “Aristotle’s demonstrative logic”, History and Philosophy of Logic, 30 (2009), 1–20; pp. 3–5.

  65. See John Corcoran, “Completeness of an ancient logic”, Journal of Symbolic Logic, 37 (1972), 696–702, and “Aristotle’s natural deduction system”, in Corcoran, editor, Ancient Logic and its Modern Interpretations, pp. 85–131, Dordrecht: Kluwer, 1974. See also George Boger, “Aristotle’s underlying logic”, in Dov M. Gabbay and John Woods, editors, Greek, Indian and Arabic Logic, volume 1 of their Handbook of the History of Logic, pp. 101–246, Amsterdam: North-Holland, 2004.

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Acknowledgements

For stimulating conversation and helpful correspondence, I warmly thank Gottfried Gabriel, Riki Heck, Ed Zalta, Bernie Linsky, Erik Krabbe, Alirio Rosales, Christos Pechlivanidis, Ori Simchen, Carl Hewitt and Harvey Schoolman.

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  1. The Abductive Systems Group, The UBC Honorary Professor of Logic, Department of Philosophy, 1866 Main Mall, University of British Columbia, Vancouver, V6T 1Z1, Canada

    John Woods

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Woods, J. What did Frege take Russell to have proved?. Synthese 198, 3949–3977 (2021). https://doi.org/10.1007/s11229-019-02324-4

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