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9143
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Implicit definitions must be satisfiable, creative definitions introduce things, contextual definitions build on things [Fine,K, by Cook/Ebert]
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Full Idea:
Fine distinguishes 'implicit definitions', where we must know it is satisfiable before it is deployed, 'creative definitions', where objects are introduced in virtue of the definition, ..and 'contextual definitions', based on established vocabulary.
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From:
report of Kit Fine (The Limits of Abstraction [2002], 060) by R Cook / P Ebert - Notice of Fine's 'Limits of Abstraction' 3
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A reaction:
Fine is a fan of creative definition. This sounds something like the distinction between cutting nature at the perceived joints, and speculating about where new joints might be inserted. Quite a helpful thought.
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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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9144
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Fine's 'procedural postulationism' uses creative definitions, but avoids abstract ontology [Fine,K, by Cook/Ebert]
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Full Idea:
Fine says creative definitions can found mathematics. His 'procedural postulationism' says one stipulates not truths, but certain procedures for extending a domain. The procedures can be stated without invoking an abstract ontology.
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From:
report of Kit Fine (The Limits of Abstraction [2002], 100) by R Cook / P Ebert - Notice of Fine's 'Limits of Abstraction' 4
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A reaction:
(For creative definitions, see Idea 9143) This sounds close in spirit to fictionalism, but with the emphasis on the procedure (which can presumably be formalized) rather than a pure act of imaginative creation.
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10141
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Many different kinds of mathematical objects can be regarded as forms of abstraction [Fine,K]
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Full Idea:
Many different kinds of mathematical objects (natural numbers, the reals, points, lines, figures, groups) can be regarded as forms of abstraction, with special theories having their basis in a general theory of abstraction.
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From:
Kit Fine (The Limits of Abstraction [2002], I.4)
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A reaction:
This result, if persuasive, would be just the sort of unified account which the whole problem of abstact ideas requires.
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9142
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Fine considers abstraction as reconceptualization, to produce new senses by analysing given senses [Fine,K, by Cook/Ebert]
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Full Idea:
Fine considers abstraction principles as instances of reconceptualization (rather than implicit definition, or using the Context Principle). This centres not on reference, but on new senses emerging from analysis of a given sense.
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From:
report of Kit Fine (The Limits of Abstraction [2002], 035) by R Cook / P Ebert - Notice of Fine's 'Limits of Abstraction' 2
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A reaction:
Fine develops an argument against this view, because (roughly) the procedure does not end in a unique result. Intuitively, the idea that abstraction is 'reconceptualization' sounds quite promising to me.
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5078
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Kant and Mill both try to explain right and wrong, without a divine lawgiver [Taylor,R]
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Full Idea:
Kant and Mill were in total agreement in trying to give content to the distinction between moral right and wrong, without recourse to any divine lawgiver.
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From:
Richard Taylor (Virtue Ethics: an Introduction [2002], Ch.14)
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A reaction:
A nice analysis, in tune with MacIntyre and others, who see such attempts as failures. It is hard, however, to deny the claims of rational principles, or of suffering, in our moral framework. I agree with Taylor's move back to virtue, but it ain't simple.
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5067
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Morality based on 'forbid', 'permit' and 'require' implies someone who does these things [Taylor,R]
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Full Idea:
If morality is based on wrong (meaning 'forbidden'), right ('permitted'), and obligatory ('required'), we are led to ask 'Who is it that thus permits, forbids or requires that certain things be done or not done?'
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From:
Richard Taylor (Virtue Ethics: an Introduction [2002], Ch.2)
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A reaction:
Clear reinforcement for Nietzsche's attack on conventional morals, which Taylor sees as a relic of medieval religious attitudes. Taylor says Kant offered a non-religious version of the same authority. I agree. Back to the Greek pursuit of excellence!
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5079
|
Pleasure can have a location, and be momentary, and come and go - but happiness can't [Taylor,R]
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Full Idea:
Pleasures can be located in a particular part of the body, and can be momentary, and come and go, but this is not the case with happiness.
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From:
Richard Taylor (Virtue Ethics: an Introduction [2002], Ch.16)
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A reaction:
Probably no one ever thought that pleasure and happiness were actually identical - merely that pleasure is the only cause and source of happiness. These are good objections to that hypothesis. Pleasure simply isn't 'the good'.
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5068
|
'Eudaimonia' means 'having a good demon', implying supreme good fortune [Taylor,R]
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Full Idea:
The word 'eudaimonia' means literally 'having a good demon', which is apt, because it suggests some kind of supreme good fortune, of the sort which might be thought of as a bestowal.
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From:
Richard Taylor (Virtue Ethics: an Introduction [2002], Ch.5)
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A reaction:
Beware of etymology. This implies that eudaimonia is almost entirely beyond a person's control, but Aristotle doesn't think that. A combination of education and effort can build on some natural gifts to create a fully successful life.
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5077
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The modern idea of obligation seems to have lost the idea of an obligation 'to' something [Taylor,R]
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Full Idea:
In modern moral thinking, obligation is something every responsible person is supposed to have, but it is not an obligation to the state, or society, or humanity, or even to God. It is an obligation standing by itself.
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From:
Richard Taylor (Virtue Ethics: an Introduction [2002], Ch.12)
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A reaction:
This nicely pinpoints how some our moral attitudes are relics of religion. Taylor wants a return to virtue, but one could respond by opting for the social contract (with very clear obligations) or Kantian 'contractualism' (answering to rational beings).
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5066
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If we are made in God's image, pursuit of excellence is replaced by duty to obey God [Taylor,R]
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Full Idea:
Once people are declared to be images of God, just by virtue of minimal humanity, they have, therefore, no greater individual excellence to aspire to, and their purpose became one of obligation, that is, obedience to God's will.
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From:
Richard Taylor (Virtue Ethics: an Introduction [2002], Ch.2)
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A reaction:
An interesting and plausible historical analysis. There is a second motivation for the change, though, in Grotius's desire to develop a more legalistic morality, focusing on actions rather than character. Taylor's point is more interesting, though.
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