Combining Texts

Ideas for 'The Fixation of Belief', 'Theory of Science (Wissenschaftslehre, 4 vols)' and 'Writings from Late Notebooks'

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3 ideas

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

9618	Bolzano wanted to reduce all of geometry to arithmetic [Bolzano, by Brown,JR]
	Full Idea: Bolzano if the father of 'arithmetization', which sought to found all of analysis on the concepts of arithmetic and to eliminate geometrical notions entirely (with logicism taking it a step further, by reducing arithmetic to logic).
	From: report of Bernard Bolzano (Theory of Science (Wissenschaftslehre, 4 vols) [1837]) by James Robert Brown - Philosophy of Mathematics Ch. 3
	A reaction: Brown's book is a defence of geometrical diagrams against Bolzano's approach. Bolzano sounds like the modern heir of Pythagoras, if he thinks that space is essentially numerical.

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units

7207	Counting needs unities, but that doesn't mean they exist; we borrowed it from the concept of 'I' [Nietzsche]
	Full Idea: We need unities in order to be able to count: we should not therefore assume that such unities exist. We have borrowed the concept of unity from our concept of 'I' - our oldest article of faith.
	From: Friedrich Nietzsche (Writings from Late Notebooks [1887], 14[79])
	A reaction: Personally I think that counting derives from patterns, and that all creatures can discern patterns in their environment, which means discriminating the parts of the pattern, which are therefore real and existing entities.

6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics

9830	Bolzano began the elimination of intuition, by proving something which seemed obvious [Bolzano, by Dummett]
	Full Idea: Bolzano began the process of eliminating intuition from analysis, by proving something apparently obvious (that as continuous function must be zero at some point). Proof reveals on what a theorem rests, and that it is not intuition.
	From: report of Bernard Bolzano (Theory of Science (Wissenschaftslehre, 4 vols) [1837]) by Michael Dummett - Frege philosophy of mathematics Ch.6
	A reaction: Kant was the target of Bolzano's attack. Two responses might be to say that many other basic ideas are intuited but impossible to prove, or to say that proof itself depends on intuition, if you dig deep enough.