Ideas from 'Theory of Science (4 vols)' by Bernard Bolzano [1837], by Theme Structure

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2. Reason / B. Laws of Thought / 1. Laws of Thought
The laws of thought are true, but they are not the axioms of logic
                        Full Idea: Bolzano said the 'laws of thought' (identity, contradiction, excluded middle) are true, but nothing of interest follows from them. Logic obeys them, but they are not logic's first principles or axioms.
                        From: report of Bernard Bolzano (Theory of Science (4 vols) [1837], 3) by George / Van Evra - The Rise of Modern Logic
                        A reaction: An interesting and crucial distinction. For samples of proposed axioms of logic, see Ideas 6408, 7798 and 7797.
6. Mathematics / A. Nature of Mathematics / 2. Geometry
Bolzano wanted to reduce all of geometry to arithmetic
                        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 (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 / C. Sources of Mathematics / 2. Intuition of Mathematics
Bolzano began the elimination of intuition, by proving something which seemed obvious
                        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 (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.
7. Existence / C. Structure of Existence / 1. Grounding / c. Grounding and explanation
Philosophical proofs in mathematics establish truths, and also show their grounds
                        Full Idea: Mathematical proofs are philosophical in method if they do not only demonstrate that a certain mathematical truth holds but if they also disclose why it holds, that is, if they uncover its grounds.
                        From: report of Bernard Bolzano (Theory of Science (4 vols) [1837]) by Correia,F/Schnieder,B - Grounding: an opinionated introduction 2.3
                        A reaction: I aim to defend the role of explanation in mathematics, but this says that this is only if the proofs are 'philosophical', which may be of no interest to mathematicians. Oh well, that's their loss.
12. Knowledge Sources / E. Direct Knowledge / 2. Intuition
Bolzano wanted to avoid Kantian intuitions, and prove everything that could be proved
                        Full Idea: Bolzano was determined to expel Kantian intuition from analysis, and to prove from first principles anything that could be proved, no matter how obvious it might seem when thought of in geometrical terms.
                        From: report of Bernard Bolzano (Theory of Science (4 vols) [1837]) by Michael Dummett - The Philosophy of Mathematics 2.3
                        A reaction: This is characteristic of the Enlightenment Project, well after the Enlightenment. It is a step towards Frege's attack on 'psychologism' in mathematics. The problem is that it led us into a spurious platonism. We live in troubled times.
19. Language / D. Propositions / 2. Abstract Propositions / a. Propositions as sense
Propositions are abstract structures of concepts, ready for judgement or assertion
                        Full Idea: Bolzano conceived of propositions as abstract objects which are structured compounds of concepts and potential contents of judgements and assertions.
                        From: report of Bernard Bolzano (Theory of Science (4 vols) [1837]) by Correia,F/Schnieder,B - Grounding: an opinionated introduction 2.3
                        A reaction: Personally I think of propositions as brain events, the constituents of thought about the world, but that needn't contradict the view of them as 'abstract'.
A 'proposition' is the sense of a linguistic expression, and can be true or false
                        Full Idea: What I mean by 'propositions' is not what the grammarians call a proposition, namely the linguistic expression, but the mere sense of this expression, is what is meant by proposition in itself or object proposition. This sense can be true or false.
                        From: Bernard Bolzano (Theory of Science (4 vols) [1837], Pref?)
                        A reaction: This seems to be the origin of what we understand by 'proposition'. The disputes are over whether such things exists, and whether they are features of minds or features of the world (resembling facts).
19. Language / E. Analyticity / 2. Analytic Truths
The ground of a pure conceptual truth is only in other conceptual truths
                        Full Idea: We can find the ground of a pure conceptual truth only in other conceptual truths.
                        From: Bernard Bolzano (Theory of Science (4 vols) [1837], Pref)
                        A reaction: Elsewhere he insists that these grounds must be in 'truths', and not just in the attributes of the concepts of involved. This conflicts with Kit Fine's view, that the concepts themselves are the source of conceptual truth and necessity.