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Ideas for 'The Gettier Problem', 'Philebus' and 'Frege's Concept of Numbers as Objects'

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5 ideas

6. Mathematics / A. Nature of Mathematics / 2. Geometry

9867	It is absurd to define a circle, but not be able to recognise a real one [Plato]
	Full Idea: It will be ridiculous if our student knows the definition of the circle and of the divine sphere itself, but cannot recognize the human sphere and these our circles, used in housebuilding.
	From: Plato (Philebus [c.353 BCE], 62a)
	A reaction: This is the equivalent of being able to recite numbers, but not to count objects. It also resembles Molyneux's question (to Locke), of whether recognition by one sense entails recognition by others. Nice (and a bit anti-platonist!).

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers

13861	Number theory aims at the essence of natural numbers, giving their nature, and the epistemology [Wright,C]
	Full Idea: In the Fregean view number theory is a science, aimed at those truths furnished by the essential properties of zero and its successors. The two broad question are then the nature of the objects, and the epistemology of those facts.
	From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], Intro)
	A reaction: [compressed] I pounce on the word 'essence' here (my thing). My first question is about the extent to which the natural numbers all have one generic essence, and the extent to which they are individuals (bless their little cotton socks).

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers

13892	One could grasp numbers, and name sizes with them, without grasping ordering [Wright,C]
	Full Idea: Someone could be clear about number identities, and distinguish numbers from other things, without conceiving them as ordered in a progression at all. The point of them would be to make comparisons between sizes of groups.
	From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 3.xv)
	A reaction: Hm. Could you grasp size if you couldn't grasp which of two groups was the bigger? What's the point of noting that I have ten pounds and you only have five, if you don't realise that I have more than you? You could have called them Caesar and Brutus.

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / d. Counting via concepts

13867	Instances of a non-sortal concept can only be counted relative to a sortal concept [Wright,C]
	Full Idea: The invitation to number the instances of some non-sortal concept is intelligible only if it is relativised to a sortal.
	From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 1.i)
	A reaction: I take this to be an essentially Fregean idea, as when we count the boots when we have decided whether they fall under the concept 'boot' or the concept 'pair'. I also take this to be the traditional question 'what units are you using'?

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic

9865	Daily arithmetic counts unequal things, but pure arithmetic equalises them [Plato]
	Full Idea: The arithmetic of the many computes sums of unequal units, such as two armies, or two herds, ..but philosopher's arithmetic computes when it is guaranteed that none of those infinitely many units differed in the least from any of the others.
	From: Plato (Philebus [c.353 BCE], 56d)
	A reaction: But of course 'the many' are ironing out the differences too, when they say there are 'three armies'. Shocking snob, Plato. Even philosophers are interested in the difference between three armies and three platoons.