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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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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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5953
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For the Cyrenaics experience was not enough to give certainty about reality [Aristippus young, by Plutarch]
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Full Idea:
The Cyrenaics, placing all experience within themselves, thought such evidence was insufficient warrant for certainty about reality, and withdrew as in a siege from the world, admitting that objects 'appear', but refusing to pronounce the word 'are'.
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From:
report of Aristippus the younger (fragments/reports [c.335 BCE]) by Plutarch - 74: Reply to Colotes §1120
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A reaction:
This seems to be the most extreme position found in ancient thought. It accompanies their extreme hedonism, based on the reality of experience and lack of interest in anything external. A bit daft, really.
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9185
|
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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9316
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How is self-representation possible, does it produce a regress, and is experience like that? [Kriegel/Williford]
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Full Idea:
The difficulties with a self-representational view of consciousness are how self-representation of mental states could be possible, whether it leads to an infinite regress, and whether it can capture the actual phenomenology of experience.
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From:
U Kriegel / K Williford (Intro to 'Self-Representational Consciousness' [2006], §3)
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A reaction:
[compressed] All of these objections strike me as persuasive, especially the first one. I'm not sure I know what self-representation is. Mirrors externally represent, and they can't represent themselves. Two mirrors together achieve something..
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9314
|
Unfortunately, higher-order representations could involve error [Kriegel/Williford]
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Full Idea:
A problem for explaining consciousness by higher-order representations is that, like their first-order counterparts, they can misrepresent; there could be a subjective impression of being in a conscious state without actually being in any conscious state.
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From:
U Kriegel / K Williford (Intro to 'Self-Representational Consciousness' [2006], §1)
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A reaction:
It sounds plausible that this is a logical possibility, but how do you assess whether it is an actual or natural possibility? Are we saying that higher-order representations are judgments, which could be true or false? Hm.
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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
|
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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3026
|
Actions are influenced by circumstances, so Cyrenaics say felons should be reformed, not hated [Aristippus young, by Diog. Laertius]
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Full Idea:
Cyrenaics say errors should be pardoned, because men do not err intentionally but are influenced by circumstances; one should not hate a person, but only teach him better.
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From:
report of Aristippus the younger (fragments/reports [c.335 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.7.9
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A reaction:
A very appealing suggestion, and rather wonderful for its time. There is still implied agreement about what is 'error', and what counts as 'better'.
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3024
|
Cyrenaics teach that honour, justice and shame are all based on custom and fashion [Aristippus young, by Diog. Laertius]
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Full Idea:
The Cyrenaics taught that there was nothing naturally and intrinsically just, or honourable, or disgraceful; but that things were considered so because of law and fashion.
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From:
report of Aristippus the younger (fragments/reports [c.335 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.7.8
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A reaction:
As we would say now, values and virtues are 'cultural constructs'. This obviously contains a lot of truth, but I don't think our opposition of genocide is just 'fashion'.
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