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3 ideas
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. |
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. |
17738 | Quine blurs the difference between knowledge of arithmetic and of physics [Jenkins on Quine] |
Full Idea: Quine cannot deal with the intuition that there is a difference in kind between our knowledge of arithmetic and our knowledge of physics. | |
From: comment on Willard Quine (Two Dogmas of Empiricism [1953]) by Carrie Jenkins - Grounding Concepts 7.5 | |
A reaction: The endorses this criticism, which she says is widespread. I'm not convinced that there is a clear notion of 'difference in kind' here. Jenkins gets arithmetic from concepts and physics from the world. Is that a sharp distinction? |