1837 | Theory of Science (Wissenschaftslehre, 4 vols) |
p.6 | 17265 | 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 (Wissenschaftslehre, 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. |
p.6 | 17264 | 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 (Wissenschaftslehre, 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'. |
p.17 | 22276 | Bolzano saw propositions as objective entities, existing independently of us | |
Full Idea: Bolzano took the entities of which truth is predicated to be not propositions in the subjective sense but 'propositions-in-themselves' - objective entities existing independent of our apprehension. | |||
From: report of Bernard Bolzano (Theory of Science (Wissenschaftslehre, 4 vols) [1837]) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 02 'Emp' | |||
A reaction: A serious mistake. Presumably the objective propositions are all true (or there would be endless infinities of them). So what is assessed in the case of error? Something other than the objective propositions! We assess these other things! |
p.28 | 9618 | 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 (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. |
p.67 | 9830 | 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 (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. |
p.129 | 9185 | 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 (Wissenschaftslehre, 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. |
Pref | p. | 12233 | 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 (Wissenschaftslehre, 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. |
§3 | p.39 | 7807 | 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 (Wissenschaftslehre, 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. |
Pref? | p. | 12232 | 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 (Wissenschaftslehre, 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). |
1846 | Paradoxes of the Infinite |
p.131 | 10856 | A truly infinite quantity does not need to be a variable | |
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. |
§4 | p.56 | 9987 | An aggregate in which order does not matter I call a 'set' |
Full Idea: An aggregate whose basic conception renders the arrangement of its members a matter of indifference, and whose permutation therefore produces no essential difference, I call a 'set'. | |||
From: Bernard Bolzano (Paradoxes of the Infinite [1846], §4), quoted by William W. Tait - Frege versus Cantor and Dedekind IX | |||
A reaction: The idea of 'sets' was emerging before Cantor formalised it, and clarified it by thinking about infinite sets. Nowadays we also have 'ordered' sets, which rather contradicts Bolzano, and we also expect the cardinality to be determinate. |