Abstract
Do mathematical expressions have intensions, or merely extensions? If we accept the standard account of intensions based on possible worlds, it would seem that the latter is the case – there is no room for nontrivial intensions in the case of non-empirical expressions. However, some vexing mathematical problems, notably Gödel’s Second Incompleteness Theorem, seem to presuppose an intensional construal of some mathematical expressions. Hence, can we make room for intensions in mathematics? In this paper we argue that this is possible, provided we give up the standard approach to intensionality based on possible worlds.
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Notes
- 1.
Well, perhaps it says, over and above this, that there is one and only one tallest bald man; however, this again does not seem to be too nontrivial.
- 2.
- 3.
This is quite obvious if, instead of PrfR, we use still another predicate, namely PrfC(x, y) ≡Def. Prf(x, y) ∧ ¬Prf(x, ⌈0=1⌉). Given Prf(x, ⌈0=1⌉) for no x, PrfC(x, y) holds for the same x’s and y’s as Prf (x, y); while ¬∃xPrfC(x, ⌈0=1⌉) is ¬∃x(Prf(x, ⌈0=1⌉) ∧ ¬Prf(x, ⌈0=1⌉)), which is obviously provable. In fact, if we consider a further modified version of PrfC(x, y), Prf\(_{ {\mathrm{C'}}}\)(x, y) ≡Def. Prf(x, y) ∧ ∀x¬Prf(x, ⌈0=1⌉), we may return to our example (1*) vs. (2*), for we may think of (1*) as roughly capturing the sense of the standard consistency sentence based on the predicate Prf, and of (2*) as capturing that of its variant based on Prf\(_{ {\mathrm{C'}}}\) – for ∀x¬Prf(x, ⌈0=1⌉) clearly amounts to the consistency of PA. (Cf. Auerbach 1992).
- 4.
We may think about a “non-intensional” content of the Second Incompleteness Theorem: for example we may take it as saying that we cannot prove Pr(⌈0 = 1⌉) for any predicate Pr fulfilling Löb’s derivability conditions (Löb 1955; cf. also the thorough discussion of Boolos 1995). But then the question is why this result should be very interesting; unless we show that there is a reason to think that fulfilling Löb’s conditions amounts to being the provability predicate, it is a far cry from what is usually taken as the Second Incompleteness Theorem (see Detlefsen 1986).
- 5.
Put forward by Montague (1974) and others.
- 6.
This is an oversimplification; technically, the systems of intensional logic tend to be somewhat more complicated. See Peregrin (2006).
- 7.
See Peregrin (2014, Chapter 7).
- 8.
I discussed this in great detail in Peregrin (1995).
- 9.
To be sure, there may be expressions, especially definite descriptions, that have an intension but do not have an extension – in the case of mathematical expressions an example might be the greatest prime. But this is because they are composed out of meaningful subexpressions in such a way that they are themselves meaningful, though they do not pick up any extension.
- 10.
See Peregrin (2014).
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Work on this paper was supported by Grant No. 17-15645S of the Czech Science Foundation.
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Peregrin, J. (2018). Intensionality in Mathematics. In: Piazza, M., Pulcini, G. (eds) Truth, Existence and Explanation. Boston Studies in the Philosophy and History of Science, vol 334. Springer, Cham. https://doi.org/10.1007/978-3-319-93342-9_4
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