Combining Philosophers

Ideas for Anaxarchus, Oliver,A/Smiley,T and Bernard Bolzano

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5 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 / g. Applying mathematics

14246	If mathematics purely concerned mathematical objects, there would be no applied mathematics [Oliver/Smiley]
	Full Idea: If mathematics was purely concerned with mathematical objects, there would be no room for applied mathematics.
	From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.1)
	A reaction: Love it! Of course, they are using 'objects' in the rather Fregean sense of genuine abstract entities. I don't see why fictionalism shouldn't allow maths to be wholly 'pure', although we have invented fictions which actually have application.

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite

10856	A truly infinite quantity does not need to be a variable [Bolzano]
	Full Idea: A truly infinite quantity (for example, the length of a straight line, unbounded in either direction) does not by any means need to be a variable.
	From: Bernard Bolzano (Paradoxes of the Infinite [1846]), quoted by Brian Clegg - Infinity: Quest to Think the Unthinkable §10
	A reaction: This is an important idea, followed up by Cantor, which relegated to the sidelines the view of infinity as simply something that could increase without limit. Personally I like the old view, but there is something mathematically stable about infinity.

6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory

14247	Sets might either represent the numbers, or be the numbers, or replace the numbers [Oliver/Smiley]
	Full Idea: Identifying numbers with sets may mean one of three quite different things: 1) the sets represent the numbers, or ii) they are the numbers, or iii) they replace the numbers.
	From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.2)
	A reaction: Option one sounds the most plausible to me. I will take numbers to be patterns embedded in nature, and sets are one way of presenting them in shorthand form, in order to bring out what is repeated.

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.