Abstract
It will be argued that the overcoming and avoiding of dogmatism is a decisive feature of the change in Wittgenstein’s thinking that takes place in the beginning of the 1930s when he starts to emphasise the autonomy of the grammar of language and to talk about grammatical pictures and language games as objects of comparison. By examining certain crucial features in this change in Wittgenstein’s thinking, it will be shown that he received decisive impulses and ideas from new developments in mathematics and natural science in the early twentieth century. These ideas include the axiomatisation of geometry and, in general, Hilbert’s axiomatic method, but also relativity theory and the so-called method of ideal elements. By drawing upon certain analogies rather than theory-constructions, these ideas affected not only his thinking about mathematics, but also his thinking about language and the nature of philosophy in general.
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Notes
- 1.
The importance of the ‘Middle Period’ for understanding Wittgenstein’s philosophy of mathematics was emphasized by Gerrard (1991).
- 2.
In PR, p. 186 we find the following sharpened statement of this feature of mathematical symbolism: “Let’s remember that in mathematics, the signs themselves do mathematics, they don’t describe it. […] You can’t write mathematics [in the sense in which you write history], you can only do it.” (References to printed editions of Wittgenstein’s works are given using the abbreviations indicated in the bibliography. Such an abbreviation followed by a decimal number refers to the remark or section with that number. An abbreviation followed by a roman numeral indicates a part of the abbreviated work. References by MS and TS number are from the Bergen electronic edition of Wittgenstein’s Nachlass, Oxford University Press, Oxford, 2000).
- 3.
LFM, p. 150.
- 4.
The nature of this realm is the subject matter of the ‘ontology of mathematics’, in the philosophical discourse.
- 5.
By ‘modern mathematics’ will be meant, in this paper, the new trends in mathematics that emerge in the late nineteenth and the beginning of the twentieth century when the old conception of mathematics as the science of quantity is experienced as outdistanced and inadequate in regard to the internal change and development of the discipline of mathematics, in particular in regard to the new emphasis of formal rigor, and of pure mathematics as an independent and autonomous discipline. See Mehrtens (1990), Epple (2003), and Gray (2006). Mehrtens focuses in particular on the foundational crisis as a symptom of this change and of the need to revise and redefine the identity of mathematics as a discipline. He presents Hilbert as the main advocate of the modern mathematics, and as the main champion for the autonomy of mathematics and mathematical discourse. In the beginning of the 1930s Wittgenstein (unlike many mathematicians and logicians) was obviously sensitive to this cultural aspect of mathematics and aware of this scientific change as being to some extent part of a broader cultural change.
- 6.
Thomae writes: “The formal conception of numbers sets itself more modest limitations than does the logical conception. It does not ask, what are and what shall the numbers be, but it asks, what does one need about numbers in arithmetic. For the formal conception, arithmetic is a game with signs which one may call empty; by this one wants to say that (in the game of calculation) they have no other content than that which has been attributed to them concerning their behaviour with respect to certain rules of combination (rules of the game). Similarly, a chess player uses his pieces, he attributes to them certain properties which condition their behaviour in the game, and the pieces themselves are only external signs for this behaviour. To be sure, there is an important difference between the game of chess and arithmetic. The rules of chess are arbitrary, the system of rules for arithmetic is such that by means of simple axioms the numbers can be related to intuitive manifolds, so that they are of essential service in the knowledge of nature. – The formal standpoint relieves us of all metaphysical difficulties, this is the benefit it offers us.” Quoted from Epple (2003), p. 301.
- 7.
PR, p. 51.
- 8.
DL, p. 8.
- 9.
DL, p. 119.
- 10.
PR, p. 186.
- 11.
DL, p. 8.
- 12.
PG, p. 319.
- 13.
Wittgenstein knew of course that the picture and the talk of geometry as being about ‘ideal objects’ is very common in mathematical work, where it may be harmless and even a stimulus for mathematical inventions. But he is concerned with the way in which this pictorial talk tends to deceives us in the philosophy of mathematics. This kind of deception often takes place when philosophers are inclined to talk about “possibilities in principle”.
- 14.
Hilbert (1902).
- 15.
PG, pp. 311–312. The question of finding the six- or seven-place relation is connected with the problem stated in TLP 5.554–5.5542.
- 16.
PG, pp. 312–313.
- 17.
See Mülhölzer (2005, 2008), for other interesting similarities (and differences) between Hilbert’s and Wittgenstein’s thinking about issues in the foundations of mathematics. On the whole one might say that Wittgenstein was much more stimulated by Hilbert’s ideas than what has been generally recognized among Wittgenstein scholars. This is not to say that Wittgenstein endorsed Hilbert’s foundational program but I think one could say that he found it interesting and useful, not least for the philosophical mistakes it contained.
- 18.
AL, p. 4.
- 19.
PG, p. 52.
- 20.
Hilbert (1902, pp. 5–6).
- 21.
In The Blue Book Wittgenstein writes: “We shall also try to construct new notations, in order to break the spell of those which we are accustomed to.” (BB, p. 23).
- 22.
- 23.
- 24.
- 25.
LFM, p. 150.
- 26.
Frege (1980, p. 40).
- 27.
About this abstract algebraic trend, inspired by Hilbert’s axiomatics, Corry (2000, p. 52), writes: “[..] in the years immediately following the publication of the Grundlagen, several mathematicians, especially in the USA, undertook an analysis of the systems of abstract postulates for algebraic concepts such as groups, fields, Boolean algebras, etc., based on the application of techniques and conceptions similar to those developed by Hilbert in his study of the foundations of geometry. There is no evidence that Hilbert showed any interest in this kind of work, and in fact there are reasons to believe that they implied a direction of research that Hilbert did not contemplate when putting forward his axiomatic program. It seems safe to assert that Hilbert even thought of this direction of research as mathematically ill-conceived.” I am inclined to agree with Corry here, and it seems to me that the view of Hilbert as a formalist, has been based very much on the view that this abstract algebraic trend was in harmony with aims of Hilbert’s axiomatic program. This trend did not have the epistemological and clarificatory aims of Hilbert’s axiomatics. Alfred Tarski’s conception of metamathematics as ‘methodology of the deductive sciences’ is also an example of a work in the spirit of this abstract algebraic trend, but which is incompatible with the epistemological aims of Hilbert’s program. Van der Waerden’s book Moderne Algebra is another work in the abstract algebraic trend. It appears to have been an important source of inspiration for the Bourbaki programme.
- 28.
If Hilbert was a formalist, I think that the most appropriate sense of the word ‘formalism’ is the one explained by Detlefsen (1993a, b).
- 29.
WVC, p. 162.
- 30.
Einstein (1973, p. 233).
- 31.
Einstein (1973, p. 235).
- 32.
In his talk “Geometry and Experience” given at the Berlin Academy of Sciences in 1920. The ‘certainty’ that Einstein speaks of here is presumably the certainty that comes from the necessity of a proved mathematical proposition.
- 33.
Einstein (1973, p. 273).
- 34.
See Penco (2010).
- 35.
PG, p. 459.
- 36.
MS 108, p. 271. Quoted from Hilmy (1987, p. 146). The sentence on relativity theory is repeated in The Big Typescript (TS 213, pp.355–356).
- 37.
MS 109, p. 199.
- 38.
RFM VI, 28.
- 39.
TLP 5.634.
- 40.
But Hilbert did not reject Kant’s apriorism altogether. I think he was very serious when he continued the same sentence by saying: “afterwards only the apriori attitude is left over which also underlies pure mathematical knowledge: essentially that is the finite attitude which I have characterized in several works (Hilbert (1930), the quotation is from Ewald 1996, p. 1163). Commentators of Hilbert’s foundational program tend to ignore this remark and similar ones. I don’t think that he insisted on this point so often only in order to “pay a due tribute to the towering figure of his fellow Königberger”, Corry (2006a, p. 159).
- 41.
BB, pp. 57–66. See also PR 58.
- 42.
Hilbert (1926, p. 195).
- 43.
Jahnke (1993, p. 279) reports that even the romantic poet Novalis alludes to the method of ideal elements in an attempt to characterise the mathematical genius. Another somewhat surprising allusion to modern mathematics is the following, written by Wittgenstein in his diary in 1947: “Weierstrass introduces a string of new concepts to bring about order in the thinking about the differential calculus. And in that way on the whole, it seems to me, I must bring about order in psychological thinking through new concepts. (That it concerns a calculus in the first case, but not in the second, is not important.) (MS 135, p. 115–116, my translation).
- 44.
As the preface to the Philosophical Remarks indicates, Wittgenstein did not share the strong faith in modern science that Hilbert often expressed.
- 45.
Corry (2006b).
- 46.
Baker (1988). See also Barker (1980), Wilson (1989), Visser (1999) and Kjaergaard (2002).
In November 1946 Wittgenstein spoke to the Cambridge Moral Science Club. His talk had the title Philosophy, and was occasioned by a talk three weeks earlier by Karl Popper in which Popper criticized Wittgenstein as the leading figure of the “Cambridge School of linguistic philosophy”. In a letter to G.E. Moore Wittgenstein described the content of his talk as follows: “I’m giving a talk, roughly, on what I believe philosophy is, or what the method of philosophy is” (PPO, p. 338). According to the Minutes of the meeting, Wittgenstein was not happy about the description of his philosophy as ‘linguistic philosophy’. He quoted Hertz with great approval, and presented Hertz as his main source of inspiration for his view of philosophical problems and his method for dealing with them. He also mentioned Mach. But no other philosopher, not even Frege, appears to have been mentioned in his talk. (PPO, p. 342)
- 47.
Hertz (1956, pp. 7–8).
- 48.
Barker (1988, p. 245).
- 49.
Hertz (1956, p. 8).
- 50.
Hertz (1956, p. 9).
- 51.
MS 213, p. 421.
- 52.
Hertz (1956, p. 4), my emphasis.
- 53.
I agree with Marie McGinn when she writes: “One of the central themes of the Investigations is to try to show that, insofar as a sentence is used to express a grammatical proposition, it does not represent (cannot be compared with reality for truth or falsity): there is nothing that grounds or justifies grammar. […] the concern to reveal the autonomy of grammar echoes Wittgenstein’s earlier concern to show that logic does not represent – that ‘logic takes care of itself’ – and it represents a deep continuity of philosophical purpose”. McGinn (2006, p. 295).
- 54.
See Barker (1980), for an elaboration of this similarity between the Tractatus and Hertz’Principes.
- 55.
Hertz (1956, p. 7).
- 56.
PI 132.
- 57.
Hertz (1956, pp. 6–7).
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Acknowledgements
This article is a revised and abridged version of my original article Le “Wittgenstein-intermédiaire” et les mathématiques modernes to be published in the Canadian journal Philosophiques. The article appears here by permission of the editors of Philosophiques. I am indebted to Kim-Erik Berts, Juliet Floyd and Kim Solin for helpful comments on an earlier version of this article.
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Stenlund, S. (2012). The “Middle Wittgenstein” and Modern Mathematics. In: Dybjer, P., Lindström, S., Palmgren, E., Sundholm, G. (eds) Epistemology versus Ontology. Logic, Epistemology, and the Unity of Science, vol 27. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4435-6_7
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