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7807
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The laws of thought are true, but they are not the axioms of logic [Bolzano, by George/Van Evra]
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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.
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From:
report of Bernard Bolzano (Theory of Science (Wissenschaftslehre, 4 vols) [1837], §3) by George / Van Evra - The Rise of Modern Logic
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A reaction:
An interesting and crucial distinction. For samples of proposed axioms of logic, see Ideas 6408, 7798 and 7797.
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9987
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An aggregate in which order does not matter I call a 'set' [Bolzano]
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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'.
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From:
Bernard Bolzano (Paradoxes of the Infinite [1846], §4), quoted by William W. Tait - Frege versus Cantor and Dedekind IX
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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.
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9618
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Bolzano wanted to reduce all of geometry to arithmetic [Bolzano, by Brown,JR]
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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).
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From:
report of Bernard Bolzano (Theory of Science (Wissenschaftslehre, 4 vols) [1837]) by James Robert Brown - Philosophy of Mathematics Ch. 3
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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.
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9830
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Bolzano began the elimination of intuition, by proving something which seemed obvious [Bolzano, by Dummett]
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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.
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From:
report of Bernard Bolzano (Theory of Science (Wissenschaftslehre, 4 vols) [1837]) by Michael Dummett - Frege philosophy of mathematics Ch.6
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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.
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17302
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Ground is irreflexive, asymmetric, transitive, non-monotonic etc. [Audi,P]
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Full Idea:
The logical principles about grounding include irreflexivity, asymmetry, transitivity, non-monotonicity, and so forth.
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From:
Paul Audi (Clarification and Defense of Grounding [2012], 3.8)
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A reaction:
[It can't ground itself, there is no mutual grounding, grounds of grounds ground, and grounding judgements are not fixed]
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17292
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Avoid 'in virtue of' for grounding, since it might imply a reflexive relation such as identity [Audi,P]
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Full Idea:
We should not use 'in virtue of' where it might express a reflexive relation, such as identity. Since grounding is a relation of determination, and closely linked to the concept of explanation, it is irreflexive and asymmetric.
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From:
Paul Audi (Clarification and Defense of Grounding [2012], 3.2)
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A reaction:
E.g. he says someone isn't a bachelor in virtue of being an unmarried man, since a bachelor just is an unmarried man. I can't disagree. 'Determination' looks like the magic word, even if we don't know how it cashes out.
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17294
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Grounding is a singular relation between worldly facts [Audi,P]
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Full Idea:
On my view, grounding is a singular relation between facts. ...Facts, on this view, are obtaining states of affairs.
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From:
Paul Audi (Clarification and Defense of Grounding [2012], 3.2)
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A reaction:
He rest this claim on his 'worldly' view of facts, Idea 17293. I seem to be agreeing with him. Note that it is not between types of fact, even if there are such general truths, such as in chemistry.
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17300
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If grounding relates facts, properties must be included, as well as objects [Audi,P]
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Full Idea:
Taking facts to be the relata of grounding has the interesting consequence that it does not relate ordinary particulars, objects, considered apart from their properties.
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From:
Paul Audi (Clarification and Defense of Grounding [2012], 3.4)
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A reaction:
It will depend on what you mean by properties, and it seems to me that something like 'powers' must be invoked, to get the active character that seems to be involved in grounding.
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17265
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Philosophical proofs in mathematics establish truths, and also show their grounds [Bolzano, by Correia/Schnieder]
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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.
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From:
report of Bernard Bolzano (Theory of Science (Wissenschaftslehre, 4 vols) [1837]) by Correia,F/Schnieder,B - Grounding: an opinionated introduction 2.3
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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.
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17301
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Reduction is just identity, so the two things are the same fact, so reduction isn't grounding [Audi,P]
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Full Idea:
I deny that when p grounds q, q thereby reduces to p, and I deny that if q reduces to p, then p grounds q. ...On my view, reduction is nothing other than identity, so p is the same fact as q.
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From:
Paul Audi (Clarification and Defense of Grounding [2012], 3.5)
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A reaction:
Very good. I can't disagree with any of it, and it is crystal clear. Philosophical heaven.
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9185
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Bolzano wanted to avoid Kantian intuitions, and prove everything that could be proved [Bolzano, by Dummett]
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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.
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From:
report of Bernard Bolzano (Theory of Science (Wissenschaftslehre, 4 vols) [1837]) by Michael Dummett - The Philosophy of Mathematics 2.3
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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.
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17299
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There are plenty of examples of non-causal explanation [Audi,P]
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Full Idea:
There are a number of explanations where it seems clear that causation is not involved at all: normative grounded in non-normative, disposition grounded in categorical, aesthetic grounded in non-aesthetic, semantic in social and psychological.
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From:
Paul Audi (Clarification and Defense of Grounding [2012], 3.3)
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A reaction:
Apart from dispositions, perhaps, these all seem to be experienced phenomena grounded in the physical world. 'Determination' is the preferred term for non-causal grounding.
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17264
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Propositions are abstract structures of concepts, ready for judgement or assertion [Bolzano, by Correia/Schnieder]
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Full Idea:
Bolzano conceived of propositions as abstract objects which are structured compounds of concepts and potential contents of judgements and assertions.
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From:
report of Bernard Bolzano (Theory of Science (Wissenschaftslehre, 4 vols) [1837]) by Correia,F/Schnieder,B - Grounding: an opinionated introduction 2.3
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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'.
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12232
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A 'proposition' is the sense of a linguistic expression, and can be true or false [Bolzano]
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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.
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From:
Bernard Bolzano (Theory of Science (Wissenschaftslehre, 4 vols) [1837], Pref?)
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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).
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