### Descartes on Mathematical Truth and Mathematical Existence

This is not so much a post as a note to myself for something I would like to think about in the future.

In the first Meditation, Descartes writes:

"arithmetic, geometry, and other such disciplines, which treat of nothing but the simplest and most general things... areI should look more into this apparent 'truth-independent-of-reference' position, that mathematical truth is independent of the existence of mathematical entities, especially as an alternative to the Quine-Putnam indispensibility argument for the reality of mathematical objects.indifferent as to whether these things do or do not in fact exist, contain something certain and indubitable."

Relevant secondary literature:

- Gregory Brown (in "Vera Entia: The Nature of Mathematical Objects in Descartes"

*Journal of the History of Philosophy*, 1980:23-37) contains a nice discussion of the kind of existence mathematical objects have for Descartes, esp. section III:

"mathematical objects in particular, have a "being" that is independent of their actual existence in (physical) space or time, and that is characterized by what Descartes calls 'possible existence'"(p.36).

- Brown quotes Anthony Kenny ("The Cartesian Circle and Eternal Truths,"

*Journal of Philosophy*, 1970):

"the objects of mathematics are not independent of physical substances; but they do not support the view that the objects of mathematics depend for their essences on physical existents... . Descartes held that a geometrical figure was a mode of physical or corporeal substance; it could not exist, unless there existed a physical substance for it to exist in. But whether it existed or not, it had a kind of being that was sufficient to distinguish it from nothing, and it had its eternal and immutable essence."

## 1 Comments:

I don't like the QP argument. It contains a dangerous epistemic short-circuit. Our best theories rely on the truth of the foundations of mathematics, so any epistemological disconnect in the mathematics implies an ontological disconnect. An example is the idea of limits, introduced to circumvent problems with infinitesimals (e.g. Zeno's paradox, see Huggett in the SEP) for which Hilbert expressed grave concerns because he felt that introducing the epsilon delta definition just moved the problem one step to the left. Present ideas support this caution in that theories of the universe are a deal less problematic if spacetime is discrete. So, where QP say:

(P1) We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.

(P2) Mathematical entities are indispensable to our best scientific theories.

(C) We ought to have ontological commitment to mathematical entities.

I say:

(P1) We ought to have commitment to all and only those entities with a well-founded epistemic basis.

(P2) Neither mathematical entities nor our best scientific theories are currently well-founded.

(C) We ought not commit to mathematical entities or our scientific theories.

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