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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
|
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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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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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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7339
|
Because human life is what is sacred, Mosaic law has no death penalty for property violations [Johnson,P]
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Full Idea:
Where other codes provided the death penalty for offences against property, in Mosaic law no property offence is capital; human life is too sacred, where the rights of property alone are violated.
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From:
Paul Johnson (The History of the Jews [1987], Pt I)
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A reaction:
We still preserve this idea in our law, and also in our culture, where we are keen to insist that catastrophes like earthquakes or major fires are measured almost entirely by the loss of life, not the loss of property. I approve.
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7353
|
The Pharisees undermined slavery, by giving slaves responsibility and status in law courts [Johnson,P]
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Full Idea:
It is no accident that slavery among Jews disappeared with the rise of the Pharisees, as they insisted that all were equal before God in a court. Masters were no longer responsible for actions of slaves, so a slave had status, and slavery could not work.
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From:
Paul Johnson (The History of the Jews [1987], Pt II)
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A reaction:
As in seventeenth century England, the rise of social freedom comes from religious sources, not social sources. A slave has status in the transcendent world of souls, despite being a nobody in the physical world.
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7340
|
Mosaic law was the first to embody the rule of law, and equality before the law [Johnson,P]
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Full Idea:
Mosaic law meant that God ruled through his laws, and since all were equally subject to the law, the system was the first to embody the double merits of the rule of law and equality before the law.
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From:
Paul Johnson (The History of the Jews [1987], Pt I)
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A reaction:
If this is correct, it seems to be a hugely important step, combined with Idea 1659, that revenge should be the action of a the state, not of the individual. They are the few simple and essential keys to civilization.
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7338
|
Man's life is sacred, because it is made in God's image [Johnson,P]
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Full Idea:
In Mosaic theology, man is made in God's image, and so his life is not just valuable, it is sacred.
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From:
Paul Johnson (The History of the Jews [1987], Pt I)
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A reaction:
The obvious question is what exactly is meant by "in God's image". Physically, spiritually, intellectually, morally? I am guessing that the original idea was intellectual, because we are the only rational animal. The others seem unlikely, or arrogant.
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7336
|
A key moment is the idea of a single moral God, who imposes his morality on humanity [Johnson,P]
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Full Idea:
The discovery of monotheism, and not just of monotheism but of a sole, omnipotent God actuated by ethical principles and seeking methodically to impose them on human beings, is one of the greatest turning-points in history, perhaps the greatest of all.
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From:
Paul Johnson (The History of the Jews [1987], Pt I)
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A reaction:
'Discovery' begs some questions, but when put like this you realise what a remarkable event it was. It is a good candidate for the most influential idea ever, even if large chunks of humanity, especially in the orient, never took to monotheism.
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7341
|
Sampson illustrates the idea that religious heroes often begin as outlaws and semi-criminals [Johnson,P]
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Full Idea:
Sampson is the outstanding example of the point which the Book of Judges makes again and again, that the Lord and society are often served by semi-criminal types, outlaws and misfits, who become folk-heroes and then religious heroes.
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From:
Paul Johnson (The History of the Jews [1987], Pt I)
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A reaction:
This illustrates nicely Nietzsche's claim, that the jews were responsible for his 'inversion of values', in which aristocratic virtues are downgraded, and the virtues of a good slave are elevated (though Sampson may not show that point so well!).
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7355
|
The Torah pre-existed creation, and was its blueprint [Johnson,P]
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Full Idea:
The Torah was not just a book about God. It pre-existed creation, in the same way as God did. In fact, it was the blueprint of creation.
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From:
Paul Johnson (The History of the Jews [1987], Pt III)
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A reaction:
You can only become a 'people of the book' (which Moslems resented in Judaism, and then emulated) if you give this stupendously high status to your book. Hence Christian fundamentalism makes sense, with its emphasis on the divinity of the Bible.
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7344
|
Judaism involves circumcision, Sabbath, Passover, Pentecost, Tabernacles, New Year, and Atonement [Johnson,P]
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Full Idea:
The practices of Judaism developed during their Exile: circumcision, the Sabbath, the Passover (founding of the nation), Pentecost (giving of the laws), the Tabernacles, the New Year, and the Day of Atonement.
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From:
Paul Johnson (The History of the Jews [1987], Pt II)
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A reaction:
These were the elements of ritual created to replace the existence of a physically located state. An astonishing achievement, not even remotely achieved by any other state that was driven off its lands. A culture is an idea, not a country.
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7347
|
Zoroastrians believed in one eternal beneficent being, Creator through the holy spirit [Johnson,P]
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Full Idea:
Cyrus the Great was a Zoroastrian, believing in one, eternal, beneficent being, 'Creator of all things through the holy spirit'.
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From:
Paul Johnson (The History of the Jews [1987], Pt II)
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|
A reaction:
Is this the actual origin of monotheism, or did they absorb this idea from the Jews? The interesting bit is the fact that the supreme being (called Marduk) is 'beneficent', which one doesn't associate with these remote and supposed pagans.
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7349
|
Immortality based on judgement of merit was developed by the Egyptians (not the Jews) [Johnson,P]
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Full Idea:
The idea of judgement at death and immortality on the basis of merit were developed in Egypt before 1000 BCE. It is not Jewish because it was not in the Torah, and the Sadducees, who stuck to their texts, seemed to have denied the afterlife completely.
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From:
Paul Johnson (The History of the Jews [1987], Pt II)
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A reaction:
This is the idea considered crucial to religion by Immanuel Kant (Idea 1455), who should be declared an honorary Egyptian. To me the idea that only the good go to heaven sounds like sadly wishful thinking - a fictional consolation for an unhappy life.
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