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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).
Gist of Idea
Bolzano wanted to reduce all of geometry to arithmetic
Source
report of Bernard Bolzano (Theory of Science (Wissenschaftslehre, 4 vols) [1837]) by James Robert Brown - Philosophy of Mathematics Ch. 3
Book Ref
Brown,James Robert: 'Philosophy of Mathematics' [Routledge 2002], p.28
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.
| 10856 | A truly infinite quantity does not need to be a variable [Bolzano] |
| 9987 | An aggregate in which order does not matter I call a 'set' [Bolzano] |
| 9618 | Bolzano wanted to reduce all of geometry to arithmetic [Bolzano, by Brown,JR] |
| 9830 | Bolzano began the elimination of intuition, by proving something which seemed obvious [Bolzano, by Dummett] |
| 17265 | Philosophical proofs in mathematics establish truths, and also show their grounds [Bolzano, by Correia/Schnieder] |
| 9185 | Bolzano wanted to avoid Kantian intuitions, and prove everything that could be proved [Bolzano, by Dummett] |
| 22276 | Bolzano saw propositions as objective entities, existing independently of us [Bolzano, by Potter] |
| 17264 | Propositions are abstract structures of concepts, ready for judgement or assertion [Bolzano, by Correia/Schnieder] |
| 12233 | The ground of a pure conceptual truth is only in other conceptual truths [Bolzano] |
| 7807 | The laws of thought are true, but they are not the axioms of logic [Bolzano, by George/Van Evra] |
| 12232 | A 'proposition' is the sense of a linguistic expression, and can be true or false [Bolzano] |