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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
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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
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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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12189
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Logical necessity involves a decision about usage, and is non-realist and non-cognitive [Wright,C, by McFetridge]
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Full Idea:
Wright espouses a non-realist, indeed non-cognitive account of logical necessity. Crucial to this is the idea that acceptance of a statement as necessary always involves an element of decision (to use it in a necessary way).
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From:
report of Crispin Wright (Inventing Logical Necessity [1986]) by Ian McFetridge - Logical Necessity: Some Issues §3
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A reaction:
This has little appeal to me, as I take (unfashionably) the view that that logical necessity is rooted in the behaviour of the actual physical world, with which you can't argue. We test simple logic by making up examples.
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16638
|
The qualities of the world are mere appearances; reality is the motions which cause them [Hobbes]
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Full Idea:
Whatsoever accidents or qualities our senses make us think there be in the world, they are not there, but are seemings and apparitions only. The things that really are in the world without us are those motions by which these seemings are caused.
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From:
Thomas Hobbes (The Elements of Law [1640], I.2.10), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 10.2
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A reaction:
This seems to count as a sense-datum theory, rather than a representative theory of perception, since it makes no commitment to the qualities containing any accurate information at all. We just start from the qualities and try to work it out.
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16688
|
Evidence is conception, which is imagination, which proceeds from the senses [Hobbes]
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Full Idea:
All evidence is conception, as it is said, and all conception is imagination and proceeds from sense. And spirits we suppose to be those substances which work not upon the sense, and therefore not conceptible.
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From:
Thomas Hobbes (The Elements of Law [1640], I.11.5), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 16.2
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A reaction:
This is exactly the same as Hume's claim that all ideas are the result of impressions, and is the very essence of empiricism. We see here that such an epistemology can have huge consequences.
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7320
|
Holism cannot give a coherent account of scientific methodology [Wright,C, by Miller,A]
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Full Idea:
Crispin Wright has argued that Quine's holism is implausible because it is actually incoherent: he claims that Quine's holism cannot provide us with a coherent account of scientific methodology.
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From:
report of Crispin Wright (Inventing Logical Necessity [1986]) by Alexander Miller - Philosophy of Language 4.5
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A reaction:
This sounds promising, given my intuitive aversion to linguistic holism, and almost everything to do with Quine. Scientific methodology is not isolated, but spreads into our ordinary (experimental) interactions with the world (e.g. Idea 2461).
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7410
|
Self-preservation is basic, and people judge differently about that, implying ethical relativism [Hobbes, by Tuck]
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Full Idea:
If men are their own judges of what conduces to their preservation, ..all men make different decisions about what counts as a danger, so (for Hobbes) the grimmest version of ethical relativism seems to be the only possible ethical vision.
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From:
report of Thomas Hobbes (The Elements of Law [1640]) by Richard Tuck - Hobbes Ch.2
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A reaction:
This might depend on self-preservation being the only fundamental value. But if self-preservation is not a pressing issue, presumably other values might come into play, some of them less concerned with the individual's own interests.
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