8 ideas
17611 | We want the essence of continuity, by showing its origin in arithmetic [Dedekind] |
Full Idea: It then only remained to discover its true origin in the elements of arithmetic and thus at the same time to secure a real definition of the essence of continuity. | |
From: Richard Dedekind (Continuity and Irrational Numbers [1872], Intro) | |
A reaction: [He seeks the origin of the theorem that differential calculus deals with continuous magnitude, and he wants an arithmetical rather than geometrical demonstration; the result is his famous 'cut']. |
10572 | A cut between rational numbers creates and defines an irrational number [Dedekind] |
Full Idea: Whenever we have to do a cut produced by no rational number, we create a new, an irrational number, which we regard as completely defined by this cut. | |
From: Richard Dedekind (Continuity and Irrational Numbers [1872], §4) | |
A reaction: Fine quotes this to show that the Dedekind Cut creates the irrational numbers, rather than hitting them. A consequence is that the irrational numbers depend on the rational numbers, and so can never be identical with any of them. See Idea 10573. |
17612 | Arithmetic is just the consequence of counting, which is the successor operation [Dedekind] |
Full Idea: I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetic act, that of counting, and counting itself is nothing else than the successive creation of the infinite series of positive integers. | |
From: Richard Dedekind (Continuity and Irrational Numbers [1872], §1) | |
A reaction: Thus counting roots arithmetic in the world, the successor operation is the essence of counting, and the Dedekind-Peano axioms are built around successors, and give the essence of arithmetic. Unfashionable now, but I love it. Intransitive counting? |
18087 | If x changes by less and less, it must approach a limit [Dedekind] |
Full Idea: If in the variation of a magnitude x we can for every positive magnitude δ assign a corresponding position from and after which x changes by less than δ then x approaches a limiting value. | |
From: Richard Dedekind (Continuity and Irrational Numbers [1872], p.27), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.7 | |
A reaction: [Kitcher says he 'showed' this, rather than just stating it] |
16616 | Substances 'substand' (beneath accidents), or 'subsist' (independently) [Eustachius] |
Full Idea: It is proper to substance both to stretch out or exist beneath accidents, which is to substand, and to exist per se and not in another, which is to subsist. | |
From: Eustachius a Sancto Paulo (Summa [1609], I.1.3b.1.2), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 06.2 | |
A reaction: This reflects Aristotle wavering between 'ousia' being the whole of a thing, or the substrate of a thing. In current discussion, 'substance' still wavers between a thing which 'is' a substance, and substance being the essence. |
16585 | Prime matter is free of all forms, but has the potential for all forms [Eustachius] |
Full Idea: Everyone says that prime matter, considered in itself, is free of all forms and at the same time is open to all forms - or, that matter is in potentiality to all forms. | |
From: Eustachius a Sancto Paulo (Summa [1609], III.1.1.2.3), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 03.1 | |
A reaction: This is the notorious doctrine developed to support the hylomorphic picture derived from Aristotle. No one could quite figure out what prime matter was, so it faded away. |
13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
Full Idea: In a single day there lies open to men of learning more than there ever does to the unenlightened in the longest of lifetimes. | |
From: Posidonius (fragments/reports [c.95 BCE]), quoted by Seneca the Younger - Letters from a Stoic 078 | |
A reaction: These remarks endorsing the infinite superiority of the educated to the uneducated seem to have been popular in late antiquity. It tends to be the religions which discourage great learning, especially in their emphasis on a single book. |
20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
Full Idea: Posidonius defined time thus: it is an interval of motion, or the measure of speed and slowness. | |
From: report of Posidonius (fragments/reports [c.95 BCE]) by John Stobaeus - Anthology 1.08.42 | |
A reaction: Hm. Can we define motion or speed without alluding to time? Looks like we have to define them as a conjoined pair, which means we cannot fully understand either of them. |