Abstract
This article canvasses five senses in which one might introduce an historical element into the philosophy of mathematics: 1. The temporal dimension of logic; 2. Explanatory Appeal to Context rather than to General Principles; 3. Heraclitean Flux; 4. All history is the History of Thought; and 5. History is Non-Judgmental. It concludes by adapting Bernard Williams’ distinction between ‘history of philosophy’ and ‘history of ideas’ to argue that the philosophy of mathematics is unavoidably historical, but need not and must not merge with historiography.
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Notes
E.g. Davis and Hersh (1995).
E.g. Butterworth (1999).
For the courtship: Aspray and Kitcher (1988).
Grosholz (2005, p. 337).
Of course, there were historically-minded philosophers of science before Popper, such as Whewell and Duhem. Moreover, French philosophy of science in the inter-war years was heavily historical. For Koyré (1946), the spirit of the time was, “tellement infectée d’historicisme qu’elle ne conçoit pas qu’il puisse y avoir d’elle-même une autre connaissance que la connaissance historique, époque qui n’admet pas qu’elle puisse se comprendre et s’expliquer à elle-même si ce n’est à travers et en fonction de son passé, son histoire.” Quoted in Jorland (1981, p. 72). See also Koyré (1973, p. 17) “le style de notre époque, éperdument théorique, éperdument pratique, mais aussi éperdument historique…”. However, this tradition does not seem to have influenced Popper.
Proofs and Refutations, pp. 152–155.
“One must treat budding programmes leniently: programmes may take decades before they get off the ground and become empirically progressive… there is no refutation without a better theory.” (Lakatos 1978a, p. 6)
That is why Wittgenstein’s suggestion that mathematical statements are expressions of grammar cannot be the whole account. Grammatical expressions (in Wittgenstein’s sense) cannot have counterexamples (that is most of what ‘grammatical’ means here). It is impossible to take seriously the falsehood of a grammatical expression, therefore an attempt to prove one can only be quixotic.
This notion of truth-in-practice is not be confused with the meta-mathematical notion of truth-in- \( {\user1{{\mathcal{L}}}} \) (where \( {\user1{{\mathcal{L}}}} \) is a fully formalised language with an explicit specification of its well-formed formulae).
Reichenbach credits the term rationale Nachkonstruktion to Carnap in Der logische Aufbau der Welt (Berlin and Leipzig 1928).
“Epistemology thus considers a logical substitute rather than real processes. For the logical substitute the term rational reconstruction has been introduced…” Reichenbach (1938, p. 5).
Lakatos (1976, p. 1).
Op. cit. p. 2.
“I maintain that all historians of science who hold that the progress of science is progress in objective knowledge, use, willy-nilly, some rational reconstruction.” (Lakatos 1978a, p. 192; see also op. cit. pp. 120–121).
“My purpose was to distil a methodological message from the history, rather than to write history itself.” (Lakatos 1978b, p. 192).
Some authors have tried to carry the methodology of scientific research programmes (or parts of it, with modifications) from natural science into mathematics (see Hallett 1979, Koetsier 1991, Corfield 2003); for criticism of MSRP see Larvor (1998) esp. chapters four and six; for criticism of Methodologies of Mathematical Research Programmes, see Op. Cit. and Larvor (1997). Kitcher offered another philosophically motivated history of mathematics in his (1983), in which real characters give way to an ideal mathematical agent, just as Reichenbach recommends.
The cows appear in paragraph 16 of the preface to the Phenomenology of Spirit; the owl of Minerva takes flight in the penultimate paragraph of the preface to the Philosophy of Right.
Magnello (2006, p. 220).
Gibbon (1980, pp. 105–106). This remark is often mistaken as an expression of cynicism about mankind, when in fact it makes a point about historiography. “Antoninus diffused order and tranquillity over the greatest part of the earth. His reign is marked by the rare advantage of furnishing very few materials for history, which is, indeed, little more than the register of the crimes, follies and misfortunes of mankind.”
E.g. Laudan, “I believe that the requirement that a methodology or epistemology must exhibit past science as rational is thoroughly wrong-headed.” (1996, p. 195–196).
I lay out this argument in more detail in Larvor (2007).
I am grateful to an anonymous referee for this example and a number of other well-taken points.
Collingwood (1994, p. 115)
Collingwood’s remark needs qualification: a change in the weather can change the outcome of a battle or sink a ship without anyone having to think about it. Nevertheless, his fundamental point survives, because purposes and convictions brought the armies to the field and put the ship to sea, and more thoughts determine the consequences. For example, if the king’s eldest son dies in war or at sea, his death will depend for its significance on beliefs about royal succession.
The universe is expanding, so it did get bigger in the 17th century as it does every century—but not suddenly.
Collingwood (1960, p. 177).
As we do so we should bear in mind that Roman standards were no more clear-cut than ours are today. Whether Julius Caesar was a good emperor is as complex a question as whether Margaret Thatcher was a good prime minister. There is no ideologically neutral standard against which to measure prime ministers or emperors.
See Sorell and Rogers (2005)
In the preface of Williams (1978).
Williams (2006, p. 192). Here and elsewhere, Williams seems to take it that the history of natural science vindicates current scientific practice straightforwardly. The discussion here, and the marital difficulties of history and philosophy of science, suggest otherwise. However, in fairness Williams was principally concerned with ethics, in comparison with which natural science must seem unproblematic.
This is what happened to Thomas Kuhn. See Larvor (2003).
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Acknowledgements
I am grateful to the conveners of this workshop for the opportunity to present these thoughts. Also to Michèle Friend for valuable criticisms and suggestions.
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Larvor, B. What can the Philosophy of Mathematics Learn from the History of Mathematics?. Erkenn 68, 393–407 (2008). https://doi.org/10.1007/s10670-008-9107-0
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DOI: https://doi.org/10.1007/s10670-008-9107-0