Combining Texts

All the ideas for 'Bayesianism', 'Continuity and Irrational Numbers' and 'On the Soul (frags)'

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7 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
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'].
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
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.
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic
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?
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / l. Limits
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]
14. Science / C. Induction / 6. Bayes's Theorem
Probability of H, given evidence E, is prob(H) x prob(E given H) / prob(E) [Horwich]
     Full Idea: Bayesianism says ideally rational people should have degrees of belief (not all-or-nothing beliefs), corresponding with probability theory. Probability of H, given evidence E, is prob(H) X prob(E given H) / prob(E).
     From: Paul Horwich (Bayesianism [1992], p.41)
Bayes' theorem explains why very surprising predictions have a higher value as evidence [Horwich]
     Full Idea: Bayesianism can explain the fact that in science surprising predictions have greater evidential value, as the equation produces a higher degree of confirmation.
     From: Paul Horwich (Bayesianism [1992], p.42)
17. Mind and Body / E. Mind as Physical / 3. Eliminativism
Dicaearchus said soul does not exist, but is just a configuration of the body [Dicaearchus, by Fortenbaugh]
     Full Idea: Dicaearchus advanced the view that mind and soul do not exist; there is only body configured in a certain way.
     From: report of Dicaearchus (On the Soul (frags) [c.320 BCE]) by William W. Fortenbaugh - Dicaearchus
     A reaction: Pure eliminativism! It is hard to find even ruthless modern physicalists taking such a bold view. Note that he is a pupil of Aristotle, and this does not sound like a major disagreement with his teacher's views.