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A Note on the Relation Between Formal and Informal Proof

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Abstract

Using Carnap’s concept explication, we propose a theory of concept formation in mathematics. This theory is then applied to the problem of how to understand the relation between the concepts formal proof (deduction) and informal, mathematical proof.

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Notes

  1. See Bassler (2006), and Tymoczko (1979).

  2. See e.g. the example of Lakatos in section three below.

  3. See Tymoczko (1979), reprinted with some comments in Tymoczko (1998), for an argument of the empirical character of the proof of the Four-Color Theorem. In Indispensability, the Testing of Mathematical Theories, and provisional Realism (submitted paper) I argue against Tymoczko on this issue. As examples of ‘experimental’ mathematics one could mention simulations in non-linear dynamics, and numerical support for e.g. Riemann’s hypothesis. See e.g. Horgan (1993) and the introduction of Chaitin (1974). There are also some comments on numerical experiments in mathematics in my paper mentioned above. See also Baker (2008) for an overview and a characterization of experimental mathematics.

  4. See e.g. Avigad (2006), Bassler (2006), Black (2000), Ernest (1998), Folina (1998), Hersh (1997b), Mendelson (1990), Rav (1999), Shapiro (1993), and Thurston (1994).

  5. Hersh (1997b); see also Hersh (1997a) pp. 208–216 on the relation between formal and informal proof.

  6. Compare with the relation between Turing computable functions and functions, where the set of Turing computable functions is a proper subset of the set of functions, and the Turing machines are syntactical objects.

  7. See Carnap (1950), chapter one. See also Beaney (2004) on the development of Carnap’s ideas on rational reconstruction and explication, and also for a comparison with Frege’s, Kant’s, and Husserl’s notions of explication.

  8. Carnap (1950), p. 3.

  9. Carnap (1950), p. 3.

  10. Carnap (1950), p. 7.

  11. One common misunderstanding of Carnap’s conception is that the explicatum must be formulated so precisely that it is expressible in e.g. FOL. See Boniolo (2003), Strawson (1963), and Carnap’s reply to Strawson in Carnap (1963). See also Maher (2007) for a defense of Carnap against his critics.

  12. This process of explicating is, however, not uncontroversial. Giovanni Boniolo compares Kant’s and Carnap’s version of explications and comes out in favor of Kant’s; Carnap refers explicitly to Kant in his discussions (Boniolo 2003).

  13. This position of Carnap has been questioned by Georg Kreisel in Kreisel (1967a) and Kreisel (1967b), and indirectly by Joseph Shoenfield in Shoenfield (1993), p. 26. We will return to this problem below.

  14. Quine (1960), §53.

  15. This distinction comes close to what Joseph F. Hanna calls explication 2, and explication 1 in Hanna (1968). Hanna, however, seems to mean that with explications of type 2 one clarifies what an object really is, and we think this is a mistake.

  16. A proof of the thesis appears in Dershowitz and Gurevich (2008). It is necessary for a proof that the explicandum is exact enough, and we regard the paper of Dershowitz and Gurevich as a clarification of the explicandum. The process of explicating a concept consists of two steps, where the first is to clarify the explicandum. This is what Carnap does, separating two concepts of probability, in Carnap (1945); the first paper in which he uses the term ‘explication’. See also Carnap’s reply to Strawson in the Schilpp volume on the philosophy of Rudolf Carnap (1963). The second step is the formulation of the explicatum. With a precise enough explicandum, a proof of the thesis may be possible, and this is what Dershowitz and Gurevich accomplish using some extra postulates on algorithmic computation, following Shoenfield’s idea in Shoenfield (1993).

  17. Euler, Introductio in Analysin Infinitorum, 1748. The quotation is from Kleiner (1989).

  18. See Pedersen (1974).

  19. Euler, like other 18th-century mathematicians, did not distinguish between the concepts function, and continuous function (Kleiner 1989).

  20. See Carnap (1950), pp. 5f., and Hanna (1968), who is not inclined to call this kind of introduction of terms or concepts into a scientific language explication at all. In chapter 1, §4 Carnap (1950), Carnap discusses explications of classificatory, comparative and quantitative concepts, which he all reckons as legitimate.

  21. See Toulmin and Goodfield (1965), especially chapter VIII. Not even Darwin formulated any definition of species in On the Origin of Species (Darwin (1859), especially chapter two). In modern textbooks species is defined as

    a group of actively or potentially interbreeding populations that is isolated reproductively from other such groups.(Ricklefs and Miller (2000), p. 384. This definition originates from Ernst W. Mayr in 1942.)

    A species, in the modern view, is a genetically distinctive population, a group of natural demes that share a common gene pool and are reproductively isolated from all other such groups. A species is the largest unit of population within which effective gene flow occurs or can occur.(Gould and Keeton (1996), p. 488.)

  22. This issue is further developed in Mathematical Concepts as Unique Explications, a paper jointly written with Christian Bennet (submitted paper).

  23. See Edelstein-Keshet (1988), pp. 218f.

  24. These ideas of concept formation and explication are more fully discussed in Sjögren (2006, 2008), and in the two submitted papers mentioned in footnotes above.

  25. See e.g. Mendelson (1997), Troelstra and Schwichtenberg (2000), and Prawitz (1965).

  26. Obviously, explicatum is exact, and it is also simple enough for the intended purpose as is obvious from results in proof theory.

  27. The concepts need not have the same meaning, i.e. coincide intensionally; the only demand is that explicatum is similar enough to replace explicandum in relevant contexts, and this is surely the case.

  28. Lakatos (1978), chapter 4, What does a Mathematical Proof Prove? Euler’s theorem can be proved with another, straightforward induction argument over the number of edges in a connected simple planar graph. See also Alama (2008), where Alama proves the theorem with computer-assisted methods.

  29. See Bridges (2009) and Richman (1990).

  30. See e.g. Shapiro (1991), Boolos (1998), Väänänen (2001), and the survey of structuralism in Reck and Price (2000).

  31. See e.g. Folina (1998), and Black (2000). Black names the thesis that every proof in mathematics can be formalized in FOL “Hilbert’s thesis”.

  32. See e.g. Bridges (2009).

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Acknowledgements

I want to thank Christian Bennet and Dag Westerståhl for valuable comments on earlier versions of this paper, and Christian Bennet for fruitful discussions of the issues at stake. Finally, I want to thank an anonymous referee for suggestions that have clarified and improved the text.

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  1. School of Life Sciences/Mathematics, University of Skövde, Box 408, 541 28, Skövde, Sweden

    Jörgen Sjögren

  2. Dept. of Philosophy, Linguistics, and Theory of Science, University of Göteborg, Göteborg, Sweden

    Jörgen Sjögren

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Sjögren, J. A Note on the Relation Between Formal and Informal Proof. Acta Anal 25, 447–458 (2010). https://doi.org/10.1007/s12136-009-0084-y

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