Combining Texts

All the ideas for 'fragments/reports', 'Sources of Knowledge of Mathematics' and 'Modes of Extension: comment on Fine'

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4 ideas

6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique

9545	Late in life Frege abandoned logicism, and saw the source of arithmetic as geometrical [Frege, by Chihara]
	Full Idea: Near the end of his life, Frege completely abandoned his logicism, and came to the conclusion that the source of our arithmetical knowledge is what he called 'the Geometrical Source of Knowledge'.
	From: report of Gottlob Frege (Sources of Knowledge of Mathematics [1922]) by Charles Chihara - A Structural Account of Mathematics Intro n3
	A reaction: We have, rather crucially, lost touch with the geometrical origins of arithmetic (such as 'square' numbers), which is good news for the practice of mathematics, but probably a disaster for the philosophy of the subject.

9. Objects / C. Structure of Objects / 6. Constitution of an Object

6019	If someone squashed a horse to make a dog, something new would now exist [Mnesarchus]
	Full Idea: If, for the sake of argument, someone were to mould a horse, squash it, then make a dog, it would be reasonable for us on seeing this to say that this previously did not exist but now does exist.
	From: Mnesarchus (fragments/reports [c.120 BCE]), quoted by John Stobaeus - Anthology 179.11
	A reaction: Locke would say it is new, because the substance is the same, but a new life now exists. A sword could cease to exist and become a new ploughshare, I would think. Apply this to the Ship of Theseus. Is form more important than substance?

18. Thought / E. Abstraction / 7. Abstracta by Equivalence

18883	Any equivalence relation among similar things allows the creation of an abstractum [Simons]
	Full Idea: Whenever we have an equivalence relation among things - such as similarity in a certain respect - we can abstract under the equivalence and consider the abstractum.
	From: Peter Simons (Modes of Extension: comment on Fine [2008], p.19)
	A reaction: This strikes me as dressing up old-fashioned psychological abstractionism in the respectable clothing of Fregean equivalences (such as 'directions'). We can actually do what Simons wants without the precision of partitioned equivalence classes.

18884	Abstraction is usually seen as producing universals and numbers, but it can do more [Simons]
	Full Idea: Abstraction as a cognitive tool has been associated predominantly with the metaphysics of universals and of mathematical objects such as numbers. But it is more widely applicable beyond this standard range. I commend its judicious use.
	From: Peter Simons (Modes of Extension: comment on Fine [2008], p.21)
	A reaction: Personally I think our view of the world is founded on three psychological principles: abstraction, idealisation and generalisation. You can try to give them rigour, as 'equivalence classes', or 'universal quantifications', if it makes you feel better.