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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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14508
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A 'thisness' is a thing's property of being identical with itself (not the possession of self-identity) [Adams,RM]
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Full Idea:
A thisness is the property of being identical with a certain particular individual - not the property that we all share, of being identical with some individual, but my property of being identical with me, your property of being identical with you etc.
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From:
Robert Merrihew Adams (Primitive Thisness and Primitive Identity [1979], 1)
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A reaction:
These philosophers tell you that a thisness 'is' so-and-so, and don't admit that he (and Plantinga) are putting forward a new theory about haecceities, and one I find implausible. I just don't believe in the property of 'being-identical-to-me'.
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12034
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If the universe was cyclical, totally indiscernible events might occur from time to time [Adams,RM]
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Full Idea:
There is a temporal argument for the possibility of non-identical indiscernibles, if there could be a cyclical universe, in which each event was preceded and followed by infinitely many other events qualitatively indiscernible from itself.
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From:
Robert Merrihew Adams (Primitive Thisness and Primitive Identity [1979], 3)
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A reaction:
The argument is a parallel to Max Black's indiscernible spheres in space. Adams offers the reply that time might be tightly 'curved', so that the repetition was indeed the same event again.
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14510
|
Two events might be indiscernible yet distinct, if there was a universe cyclical in time [Adams,RM]
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Full Idea:
Similar to the argument from spatial dispersal, we can argue against the Identity of Indiscernibles from temporal dispersal. It seems there could be a cyclic universe, ..and thus there could be distinct but indiscernible events, separated temporally.
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From:
Robert Merrihew Adams (Primitive Thisness and Primitive Identity [1979], 3)
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A reaction:
See Idea 14509 for spatial dispersal. If cosmologists decided that a cyclical universe was incoherent, would that ruin the argument? Presumably there might even be indistinguishable events in the one universe (in principle!).
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16455
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Black's two globes might be one globe in highly curved space [Adams,RM]
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Full Idea:
If God creates a globe reached by travelling two diameters in a straight line from another globe, this can be described as two globes in Euclidean space, or a single globe in a tightly curved non-Euclidean space.
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From:
Robert Merrihew Adams (Primitive Thisness and Primitive Identity [1979], 3)
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A reaction:
[my compression of Adams's version of Hacking's response to Black, as spotted by Stalnaker] Hence we save the identity of indiscernibles, by saying we can't be sure that two indiscernibles are not one thing, unusually described.
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11901
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Haecceitism may or may not involve some logical connection to essence [Adams,RM, by Mackie,P]
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Full Idea:
Moderate Haecceitism says that thisnesses and transworld identities are primitive, but logically connected with suchnesses. ..Extreme Haecceitism involves the rejection of all logical connections between suchness and thisness, for persons.
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From:
report of Robert Merrihew Adams (Primitive Thisness and Primitive Identity [1979]) by Penelope Mackie - How Things Might Have Been
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A reaction:
I am coming to the conclusion that they are not linked. That thisness is a feature of our conceptual thinking, and is utterly atomistic and content-free, while suchness is rich and a feature of reality.
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22449
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When we say 'is red' we don't mean 'seems red to most people' [Foot]
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Full Idea:
One might think that 'is red' means the same as 'seems red to most people', forgetting that when asked if an object is red we look at it to see if it is red, and not in order to estimate the reaction that others will have to it.
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From:
Philippa Foot (Moral Relativism [1979], p.23)
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A reaction:
True, but we are conscious of our own reliability as observers (e.g. if colourblind, or with poor hearing or eyesight). I don't take my glasses off, have a look, and pronounce that the object is blurred. Ordinary language philosophy in action.
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12032
|
Direct reference is by proper names, or indexicals, or referential uses of descriptions [Adams,RM]
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Full Idea:
Direct reference is commonly effected by the use of proper names and indexical expressions, and sometimes by what has been called (by Donnellan) the 'referential' use of descriptions.
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From:
Robert Merrihew Adams (Primitive Thisness and Primitive Identity [1979], 2)
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A reaction:
One might enquire whether the third usage should be described as 'direct', but then I am not sure that there is much of a distinction between references which are or are not 'direct'. Either you (or a sentence) refer or you (or it) don't.
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22451
|
All people need affection, cooperation, community and help in trouble [Foot]
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Full Idea:
There is a great deal that all men have in common; all need affection, the cooperation of others, a place in a community, and help in trouble.
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From:
Philippa Foot (Moral Relativism [1979], p.33)
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A reaction:
There seem to be some people who don't need affection or a place in a community, though it is hard to imagine them being happy. These kind of facts are the basis for any sensible cognitivist view of ethics. They are basic to Foot's view.
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22452
|
Do we have a concept of value, other than wanting something, or making an effort to get it? [Foot]
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Full Idea:
Do we know what we mean by saying that anything has value, or even that we value it, as opposed to wanting it or being prepared to go to trouble to get it?
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From:
Philippa Foot (Moral Relativism [1979], p.35)
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A reaction:
Well, I value Rembrandt paintings, but have no aspiration to own one (and would refuse it if offered, because I couldn't look after it properly). And 'we' don't want to move the Taj Mahal to London. She has not expressed this good point very well.
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