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13190
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I don't admit infinite numbers, and consider infinitesimals to be useful fictions [Leibniz]
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Full Idea:
Notwithstanding my infinitesimal calculus, I do not admit any real infinite numbers, even though I confess that the multitude of things surpasses any finite number, or rather any number. ..I consider infinitesimal quantities to be useful fictions.
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From:
Gottfried Leibniz (Letters to Samuel Masson [1716], 1716)
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A reaction:
With the phrase 'useful fictions' we seem to have jumped straight into Harty Field. I'm with Leibniz on this one. The history of mathematics is a series of ingenious inventions, whenever they seem to make further exciting proofs possible.
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9354
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Why should necessities only be knowable a priori? That Hesperus is Phosporus is known empirically [Devitt]
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Full Idea:
Why should we accept that necessities can only be known a priori? Prima facie, some necessities are known empirically; for example, that water is necessarily H2O, and that Hesperus is necessarily Phosphorus.
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From:
Michael Devitt (There is no a Priori [2005], §2)
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A reaction:
An important question, whatever your view. If the only thing we can know a priori is necessities, it doesn't follow that necessities can only be known a priori. It gets interesting if we say that some necessities can never be known a priori.
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9353
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We explain away a priori knowledge, not as directly empirical, but as indirectly holistically empirical [Devitt]
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Full Idea:
We have no need to turn to an a priori explanation of our knowledge of mathematics and logic. Our intuitions that this knowledge is not justified in some direct empirical way is preserved. It is justified in an indirect holistic way.
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From:
Michael Devitt (There is no a Priori [2005], §2)
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A reaction:
I think this is roughly the right story, but the only way it will work is if we have some sort of theory of abstraction, which gets us up the ladder of generalisations to the ones which, it appears, are necessarily true.
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