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4742
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Correspondence may be one-many or many one, as when either p or q make 'p or q' true [Armstrong]
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Full Idea:
In Armstrong's version of the correspondence theory, the truth-making relation is not one-one, but one-many or many-one. Thus 'p or q' has two truth makers, p and q.
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From:
David M. Armstrong (A World of States of Affairs [1997], p.129), quoted by Pascal Engel - Truth Ch.1
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A reaction:
Interesting. Armstrong deals in universals. He also cites many swans as truth-makers for 'there is a least one black swan'. Not correspondence as we know it, Jim.
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8729
|
Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro]
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Full Idea:
Intuitionists in mathematics deny excluded middle, because it is symptomatic of faith in the transcendent existence of mathematical objects and/or the truth of mathematical statements.
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From:
Stewart Shapiro (Thinking About Mathematics [2000], 1.2)
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A reaction:
There are other problems with excluded middle, such as vagueness, but on the whole I, as a card-carrying 'realist', am committed to the law of excluded middle.
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8763
|
The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro]
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Full Idea:
It is surely wise to identify the positions in the natural numbers structure with their counterparts in the integer, rational, real and complex number structures.
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From:
Stewart Shapiro (Thinking About Mathematics [2000], 10.2)
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A reaction:
The point is that this might be denied, since 3, 3/1, 3.00.., and -3*i^2 are all arrived at by different methods of construction. Natural 3 has a predecessor, but real 3 doesn't. I agree, intuitively, with Shapiro. Russell (1919) disagreed.
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8762
|
Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro]
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Full Idea:
Zermelo said that for each number n, its successor is the singleton of n, so 3 is {{{null}}}, and 1 is not a member of 3. Von Neumann said each number n is the set of numbers less than n, so 3 is {null,{null},{null,{null}}}, and 1 is a member of 3.
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From:
Stewart Shapiro (Thinking About Mathematics [2000], 10.2)
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A reaction:
See Idea 645 - Zermelo could save Plato from the criticisms of Aristotle! These two accounts are cited by opponents of the set-theoretical account of numbers, because it seems impossible to arbitrate between them.
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8749
|
Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro]
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Full Idea:
Term Formalism is the view that mathematics is just about characters or symbols - the systems of numerals and other linguistic forms. ...This will cover integers and rational numbers, but what are real numbers supposed to be, if they lack names?
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From:
Stewart Shapiro (Thinking About Mathematics [2000], 6.1.1)
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A reaction:
Real numbers (such as pi and root-2) have infinite decimal expansions, so we can start naming those. We could also start giving names like 'Harry' to other reals, though it might take a while. OK, I give up.
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8750
|
Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro]
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Full Idea:
Game Formalism likens mathematics to chess, where the 'content' of mathematics is exhausted by the rules of operating with its language. ...This, however, leaves the problem of why the mathematical games are so useful to the sciences.
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From:
Stewart Shapiro (Thinking About Mathematics [2000], 6.1.2)
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A reaction:
This thought pushes us towards structuralism. It could still be a game, but one we learned from observing nature, which plays its own games. Chess is, after all, modelled on warfare.
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8753
|
Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro]
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Full Idea:
Critics commonly complain that the intuitionist restrictions cripple the mathematician. On the other hand, intuitionist mathematics allows for many potentially important distinctions not available in classical mathematics, and is often more subtle.
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From:
Stewart Shapiro (Thinking About Mathematics [2000], 7.1)
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|
A reaction:
The main way in which it cripples is its restriction on talk of infinity ('Cantor's heaven'), which was resented by Hilbert. Since high-level infinities are interesting, it would be odd if we were not allowed to discuss them.
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8731
|
Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro]
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Full Idea:
I classify conceptualists according to what they say about properties or concepts. If someone classified properties as existing independent of language I would classify her as a realist in ontology of mathematics. Or they may be idealists or nominalists.
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|
From:
Stewart Shapiro (Thinking About Mathematics [2000], 2.2.1)
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|
A reaction:
In other words, Shapiro wants to eliminate 'conceptualist' as a useful label in philosophy of mathematics. He's probably right. All thought involves concepts, but that doesn't produce a conceptualist theory of, say, football.
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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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