8251
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The logical space of reasons is a natural phenomenon, and it is the realm of freedom [McDowell]
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Full Idea:
The logical space of reasons is just part of the logical space of nature. ...And, in a Kantian slogan, the space of reasons is the realm of freedom.
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From:
John McDowell (Mind and World [1994], Intro 7)
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A reaction:
[second half on p.5] This is a modern have-your-cake-and-eat-it view of which I am becoming very suspicious. The modern Kantians (Davidson, Nagel, McDowell) are struggling to naturalise free will, but it won't work. Just dump it!
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8729
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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
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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
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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
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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
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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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19092
|
There is no pure Given, but it is cultured, rather than entirely relative [McDowell, by Macbeth]
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Full Idea:
McDowell argues that the Myth of the Given shows not that there is no content to a concept that is not a matter of its inferential relations to other concepts but only that awareness of the sort that we enjoy ...is acquired in the course of acculturation.
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From:
report of John McDowell (Mind and World [1994]) by Danielle Macbeth - Pragmatism and Objective Truth p.185
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A reaction:
The first view is of Wilfred Sellars, who derives pragmatic relativism from his rejection of the Myth. This idea is helpful is seeing why McDowell has a good proposal. As I look out of my window, my immediate experience seems 'cultured'.
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20759
|
Feelings are prior to intelligence; we should be content to live with our simplest feelings [Rousseau]
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Full Idea:
To exist is to feel, our feeling is undoubtedly earlier than our intelligence, and we had feelings before we had ideas. …Let us be simpler and less pretentious; let us be content with the first feelings we experience in ourselves.
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From:
Jean-Jacques Rousseau (Emile: treatise on education [1762], p.210), quoted by Kevin Aho - Existentialism: an introduction 6 'Moods'
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A reaction:
Beginning of the sentimental movement, and precursor to romanticism, but here seen as the basis for the existentialist concept of authenticity. Wallowing in emotions doesn't appeal to me, but you have to make space for them. Rousseau loved walking.
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19854
|
We all owe labour in return for our keep, and every idle citizen is a thief [Rousseau]
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Full Idea:
In a society where a man has to live at others' expense, he owes in labour the price of his keep; and that is without exception. Work, then, is an indispensable social obligation. Rich or poor, powerful or weak, every idle citizen is a thief.
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From:
Jean-Jacques Rousseau (Emile: treatise on education [1762], Bk III)
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A reaction:
Presumably rich landowners who live on rents can justify their position by good husbandry, at a higher level than tilling the soil. But Bertie Wooster won't last long in Rousseau's new world. This is a big challenge to the welfare state.
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