Abstract
I sketch an application of a semantically anti-realist understanding of the classical sequent calculus to the topic of mathematics. The result is a semantically anti-realist defence of a kind of mathematical realism. In the paper, I begin the development of the view and compare it to orthodox positions in the philosophy of mathematics.
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Notes
- 1.
For an enlightening historical account of the development of the mathematical sciences and logic, along with with the rise of the ‘Semantic Tradition,’ see Coffa’s The Semantic Tradition from Kant to Carnap [5].
- 2.
Let us not beg the question in favour of multiple conclusion arguments at this point.
- 3.
Consider the changes in the meaning of the terms force and mass in physics [11].
- 4.
- 5.
In what follows, we will use the following shorthand notation. For a formula B with some occurrences of a term t marked out, we will write ‘Bt’, and we will write ‘Bs’ for the result of replacing the selected instances of t in Bt by s.
- 6.
Or, as Schütte calls them, free object variables [25].
- 7.
We require the restriction to arbitrary names a, since proving that 0 is even (on the basis of no assumptions) should not be enough to prove that every number is even.
- 8.
However, consider \(\frac{1}{0}\) or \(\lim_{x\to 0}\frac{1}{x}\). If these are not eliminated from the vocabulary, the something like Beeson and Feferman’s logic of ‘definedness’ seems appropriate [7].
- 9.
In sequent form, using X as an arbitrary predicate, the rules are \(\frac{\varGamma{,}Xs\Rightarrow Xt{,}\varDelta} {\varGamma\Rightarrow s=t{,}\varDelta}\) and \(\frac{\varGamma\Rightarrow Bs{,}\varDelta}{\varGamma{,}s=t\Rightarrow Bt{,}\varDelta}\).
- 10.
For example, consider an approach that uncovers the ‘meaning’ of a mathematical statement in terms of a conditional (if there is an ω sequence, then …). You must show, for example, that the statement (if there is an ω sequence then ¬ A) should either be equivalent to the negation ¬(if there is an ω sequence then A) or there should be an explanation of the divergence, for the mathematical statement ¬ A appears to be the negation of A, but on the conditional analysis of mathematical statements, appearances are deceiving.
- 11.
Think about it: the formal language itself provides us with an omega sequence of formulas. We already have enough ontology when we have a language to speak. Numbers add no more.
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Acknowledgements
This paper has a webpage: http://consequently.org/writing/antirealist. Check there to post comments and to read comments left by others. Thanks to Dan Isaacson for hospitality during my visit to Wolfson College and the Philosophy Faculty at Oxford University, and to Allen Hazen, Dan Isaacson, Øystein Linnebo and Alexander Paseau, the audience at the (Anti)Realism conference in Nancy, and two anonymous referees, for helpful comments and discussion. This research is supported by the Australian Research Council, through grant DP0343388.
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Restall, G. (2012). Anti-realist Classical Logic and Realist Mathematics. In: Rahman, S., Primiero, G., Marion, M. (eds) The Realism-Antirealism Debate in the Age of Alternative Logics. Logic, Epistemology, and the Unity of Science, vol 23. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1923-1_14
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