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4 ideas
562 | Axioms are the underlying principles of everything, and who but the philosopher can assess their truth? [Aristotle] |
Full Idea: Axioms are more general, and the principles of all things. If this does not belong to the philosopher, who else will have the job of considering truth and falsity in their case? | |
From: Aristotle (Metaphysics [c.324 BCE], 0997a09) |
573 | The axioms of mathematics are part of philosophy [Aristotle] |
Full Idea: A single science, that of the philosopher, also covers the axioms of mathematics. | |
From: Aristotle (Metaphysics [c.324 BCE], 1005a15) |
9986 | The null set was doubted, because numbering seemed to require 'units' [Tait] |
Full Idea: The conception that what can be numbered is some object (including flocks of sheep) relative to a partition - a choice of unit - survived even in the late nineteenth century in the form of the rejection of the null set (and difficulties with unit sets). | |
From: William W. Tait (Frege versus Cantor and Dedekind [1996], IX) | |
A reaction: This old view can't be entirely wrong! Frege makes the point that if asked to count a pack of cards, you must decide whether to count cards, or suits, or pips. You may not need a 'unit', but you need a concept. 'Units' name concept-extensions nicely! |
9984 | We can have a series with identical members [Tait] |
Full Idea: Why can't we have a series (as opposed to a linearly ordered set) all of whose members are identical, such as (a, a, a...,a)? | |
From: William W. Tait (Frege versus Cantor and Dedekind [1996], VII) | |
A reaction: The question is whether the items order themselves, which presumably the natural numbers are supposed to do, or whether we impose the order (and length) of the series. What decides how many a's there are? Do we order, or does nature? |