Combining Philosophers

Ideas for Hermarchus, Ren Descartes and Noam Chomsky

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5 ideas

6. Mathematics / A. Nature of Mathematics / 1. Mathematics
2252 Surely maths is true even if I am dreaming? [Descartes]
     Full Idea: Surely whether I am asleep or awake, two plus three makes five, and a square does not have more than four sides.
     From: René Descartes (Meditations [1641], §1.20)
2430 I can learn the concepts of duration and number just from observing my own thoughts [Descartes]
     Full Idea: When I think that I exist now, and recollect that I existed in the past, and when I conceive various thoughts, the number of which I know, then I acquire the ideas of duration and number which I can thereafter transfer to all the other objects I wish.
     From: René Descartes (Meditations [1641], §3.44)
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
13445 Descartes showed a one-one order-preserving match between points on a line and the real numbers [Descartes, by Hart,WD]
     Full Idea: Descartes founded analytic geometry on the assumption that there is a one-one order-preserving correspondence between the points on a line and the real numbers.
     From: report of René Descartes (works [1643]) by William D. Hart - The Evolution of Logic 1
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units
24035 Unity is something shared by many things, so in that respect they are equals [Descartes]
     Full Idea: Unity is that common nature in which all things that are compared with each other must participate equally.
     From: René Descartes (Rules for the Direction of the Mind [1628], 14)
     A reaction: A lovely explanation of the concept of 'units' for counting. Fregeans hate units, but we Grecian thinkers love them.
24036 I can only see the proportion of two to three if there is a common measure - their unity [Descartes]
     Full Idea: I do not recognise what the proportion of magnitude is between two and three, unless I consider a third term, namely unity, which is the common measure of the one and the other.
     From: René Descartes (Rules for the Direction of the Mind [1628], 14)
     A reaction: A striking defence of the concept of the need for the unit in arithmetic. To say 'three is half as big again', you must be discussing the same size of 'half' in each instance.