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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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19377
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A monad and its body are living, so life is everywhere, and comes in infinite degrees [Leibniz]
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Full Idea:
Each monad, together with a particular body, makes up a living substance. Thus, there is not only life everywhere, joined to limbs or organs, but there are also infinite degrees of life in the monads, some dominating more or less over others.
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From:
Gottfried Leibniz (Principles of Nature and Grace based on Reason [1714], 4)
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A reaction:
Two key ideas: that each monad is linked to a body (which is presumably passive), and the infinite degrees of life in monads. Thus rocks consist of monads, but at an exceedingly low degree of life. They are stubborn and responsive.
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19353
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'Perception' is basic internal representation, and 'apperception' is reflective knowledge of perception [Leibniz]
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Full Idea:
We distinguish between 'perception', the internal state of the monad representing external things, and 'apperception', which is consciousness, or the reflective knowledge of this internal state, not given to all souls, nor at all times to a given soul.
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From:
Gottfried Leibniz (Principles of Nature and Grace based on Reason [1714], §4)
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A reaction:
The word 'apperception' is standard in Kant. I find it surprising that modern analytic philosophers don't seem to use it when they write about perception. It strikes me as useful, but maybe specialists have a reason for avoiding it.
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5061
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Animals are semi-rational because they connect facts, but they don't see causes [Leibniz]
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Full Idea:
There is a connexion between the perceptions of animals, which bears some resemblance to reason: but it is based only on the memory of facts or effects, and not at all on the knowledge of causes.
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From:
Gottfried Leibniz (Principles of Nature and Grace based on Reason [1714], §5)
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A reaction:
This amounts to the view that animals can do Humean induction (where you see regularities), but not Leibnizian induction (where you see necessities). I say all minds perceive patterns, but only humans can think about the patterns they have perceived.
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