|
7807
|
The laws of thought are true, but they are not the axioms of logic [Bolzano, by George/Van Evra]
|
|
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.
|
|
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.
|
|
17265
|
Philosophical proofs in mathematics establish truths, and also show their grounds [Bolzano, by Correia/Schnieder]
|
|
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.
|
|
12312
|
The real essence of a thing is its powers, or 'dispositional properties' [Copi]
|
|
Full Idea:
With respect to scientific usage, we can say that the real essence of a thing will consist very largely of powers or, in modern terms, dispositional properties.
|
|
From:
Irving M. Copi (Essence and Accident [1954], p.718)
|
|
A reaction:
Once again, Copi is a hero. I personally love the word 'powers' in metaphysics (and dislike the word 'properties', which is lost in a fog of confusion). See Molnar on 'powers' and Mumford on 'dispositions'.
|
|
12303
|
Within the four types of change, essential attributes are those whose loss means destruction [Copi]
|
|
Full Idea:
If we can distinguish the different kinds of change (alteration, locomotion, growth, diminution), then we can say that a given attribute is essential to an object if its loss would result in the destruction of that object.
|
|
From:
Irving M. Copi (Essence and Accident [1954], p.707-8)
|
|
A reaction:
As Copi is aware, this is a necessary condition for a property for essence, but not sufficient. If an attribute were necessary but non-essential, its loss would also be destruction. We say the essential attributes must also have some explanatory role.
|
|
9185
|
Bolzano wanted to avoid Kantian intuitions, and prove everything that could be proved [Bolzano, by Dummett]
|
|
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.
|
|
17264
|
Propositions are abstract structures of concepts, ready for judgement or assertion [Bolzano, by Correia/Schnieder]
|
|
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'.
|
|
12232
|
A 'proposition' is the sense of a linguistic expression, and can be true or false [Bolzano]
|
|
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).
|