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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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14330
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To be realists about dispositions, we can only discuss them through their categorical basis [Armstrong]
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Full Idea:
It is only to the extent that we relate disposition to 'categorical basis', and difference of disposition to difference of 'categorical basis', that we can speak of dispositions. We must be Realists, not Phenomenalists, about dispositions.
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From:
David M. Armstrong (A Materialist Theory of Mind (Rev) [1968], 6.VI)
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A reaction:
It is Armstrong's realism which motivates this claim, because he thinks only categorical properties are real. But categorical properties seem to be passive, and the world is active.
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6498
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Armstrong suggests secondary qualities are blurred primary qualities [Armstrong, by Robinson,H]
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Full Idea:
According to D.M. Armstrong and others, when we perceive secondary qualities we are in fact perceiving primary qualities in a confused, indistinct or blurred way.
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From:
report of David M. Armstrong (A Materialist Theory of Mind (Rev) [1968], 270-90) by Howard Robinson - Perception III.1
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A reaction:
This is obviously an attempt to fit secondary qualities into a reductive physicalist account of the mind. Personally I favour Armstrong's project, but doubt whether this strategy is necessary. I just don't think there is anything 'primary' about redness.
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5690
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A mental state without belief refutes self-intimation; a belief with no state refutes infallibility [Armstrong, by Shoemaker]
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Full Idea:
For Armstrong, introspection involves a belief, and mental states and their accompanying beliefs are 'distinct existences', so a state without belief shows states are not self-intimating, and the belief without the state shows beliefs aren't infallible.
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From:
report of David M. Armstrong (A Materialist Theory of Mind (Rev) [1968]) by Sydney Shoemaker - Introspection
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A reaction:
I agree with Armstrong. Introspection is a two-level activity, which animals probably can't do, and there is always the possibility of a mismatch between the two levels, so introspection is neither self-intimating nor infallibe (though incorrigible).
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5493
|
If pains are defined causally, and research shows that the causal role is physical, then pains are physical [Armstrong, by Lycan]
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Full Idea:
Armstrong and Lewis said that mental items were defined in terms of typical causes and effects; if, as seems likely, research reveals that a particular causal niche is occupied by a physical state, it follows that pain is a physical state.
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From:
report of David M. Armstrong (A Materialist Theory of Mind (Rev) [1968]) by William Lycan - Introduction - Ontology p.5
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A reaction:
I am not fully convinced of the first step in the argument. It sounds like the epistemology and the ontology have got muddled (as usual). We define mental states as we define electrons, in terms of observed behaviour, but what are they?
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