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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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4993
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If a bird captures a worm, we could say its behaviour is 'about' the worm [Kirk,R]
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Full Idea:
When a bird pulls a worm from the ground, then swallows it piece by piece, there is a sense in which its behaviour can be said to be about the worm.
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From:
Robert Kirk (Mind and Body [2003], §5.4)
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A reaction:
This is preparing the ground for a possible behaviourist account of intentionality. Reply: you could say rain is about puddles, or you could say we have adopted Dennett's 'intentional stance' to birds, but it tells us nothing about their psychology.
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4982
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Dualism implies some brain events with no physical cause, and others with no physical effect [Kirk,R]
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Full Idea:
If the mind causes brain events, then they are not caused by other brain events, and such causal gaps should be detectable by scientists; there should also be a gap of brain-events which cause no other brain events, because they are causing mind events.
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From:
Robert Kirk (Mind and Body [2003], §2.5)
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A reaction:
This is the double causation problem which Spinoza had spotted (Idea 4862). Expressed this way, it seems a screamingly large problem for dualism. We should be able to discover some VERY strange physical activity in the brain.
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4991
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Behaviourism seems a good theory for intentional states, but bad for phenomenal ones [Kirk,R]
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Full Idea:
For many kinds of mental states, notably intentional ones such as beliefs and desires, behaviourism is appealing, ..but for sensations and experiences such as pain, it seems grossly implausible.
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From:
Robert Kirk (Mind and Body [2003], §5.1)
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A reaction:
The theory does indeed make a bit more sense for intentional states, but it still strikes me as nonsense that there is no more to my belief that 'Whales live in the Atlantic' than a disposition to say something. WHY do I say this something?
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4994
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Behaviourism offers a good alternative to simplistic unitary accounts of mental relationships [Kirk,R]
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Full Idea:
There is a temptation to think that 'aboutness', and the 'contents' of thoughts, and the relation of 'reference', are single and unitary relationships, but behaviourism offers an alternative approach.
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From:
Robert Kirk (Mind and Body [2003], §5.5)
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A reaction:
Personally I wouldn't touch behaviourism with a barge-pole (as it ducks the question of WHY certain behaviour occurs), but a warning against simplistic accounts of intentional states is good. I am sure there cannot be a single neat theory of refererence.
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4984
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All meaningful psychological statements can be translated into physics [Kirk,R]
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Full Idea:
All psychological statements which are meaningful, that is to say, which are in principle verifiable, are translatable into propositions which do not involve psychological concepts, but only the concepts of physics.
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From:
Robert Kirk (Mind and Body [2003], §3.8)
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A reaction:
This shows how eliminativist behaviourism arises out of logical positivism (by only allowing what is verifiable). The simplest objection: we can't verify the mental states of others, because they are private, but they are still the best explanation.
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4997
|
It seems unlikely that most concepts are innate, if a theory must be understood to grasp them [Kirk,R]
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Full Idea:
It is widely accepted that for many concepts, if not all, grasping the concept requires grasping some theory, ...which makes difficulties for the view that concepts are not learned: for 'radical concept nativism', as Fodor calls it.
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From:
Robert Kirk (Mind and Body [2003], §7.3)
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A reaction:
Not a problem for traditional rationalist theories, where the whole theory can be innate along with the concept, but a big objection to modern more cautious non-holistic views (such as Fodor's). Does a bird have a concept AND theory of a nest?
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4995
|
Behaviourists doubt whether reference is a single type of relation [Kirk,R]
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Full Idea:
To most behaviourists it seems misguided to expect there to be a single relation that connects referring expressions with their referents.
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From:
Robert Kirk (Mind and Body [2003], §5.5)
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A reaction:
You don't need to be a behaviourist to feel this doubt. Think about names of real people, names of fictional people, reference to misunderstood items, or imagined items, or reference in dreams, or to mathematical objects, or negations etc.
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19673
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Galileo mathematised movement, and revealed its invariable component - acceleration [Galileo, by Meillassoux]
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Full Idea:
Galileo conceives of movement in mathematical terms. ...In doing so, he uncovered, beyond the variations of position and speed, the mathematical invariant of movement - that is to say, acceleration.
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From:
report of Galileo Galilei (Two Chief World Systems [1632]) by Quentin Meillassoux - After Finitude; the necessity of contingency 5
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A reaction:
That is a very nice advert for the mathematical physics which replaced the Aristotelian substantial forms. ...And yet, is acceleration some deep fact about nature, or a concept which is only needed if you insist on being mathematical?
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