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19076
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Coherence theories differ over the coherence relation, and over the set of proposition with which to cohere [Young,JO]
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Full Idea:
Coherence theories of truth differ on their accounts of the coherence relation, and on their accounts of the set (or sets) of propositions with which true propositions occur (the 'specified set').
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From:
James O. Young (The Coherence Theory of Truth [2013], §1)
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A reaction:
Coherence is clearly more than consistency or mutual entailment, and I like to invoke explanation. The set has to be large, or the theory is absurd (as two absurdities can 'cohere'). So very large, or very very large, or maximally large?
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19077
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Two propositions could be consistent with your set, but inconsistent with one another [Young,JO]
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Full Idea:
It is unsatisfactory for the coherence relation to be consistency, because two propositions could be consistent with a 'specified set', and yet be inconsistent with each other. That would imply they are both true, which is impossible.
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From:
James O. Young (The Coherence Theory of Truth [2013], §1)
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A reaction:
I'm not convinced by this. You first accept P because it is consistent with the set; then Q turns up, which is consistent with everything in the set except P. So you have to choose between them, and might eject P. Your set was too small.
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19078
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Coherence with actual beliefs, or our best beliefs, or ultimate ideal beliefs? [Young,JO]
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Full Idea:
One extreme for the specified set is the largest consistent set of propositions currently believed by actual people. A moderate position makes it the limit of people's enquiries. The other extreme is what would be believed by an omniscient being.
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From:
James O. Young (The Coherence Theory of Truth [2013], §1)
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A reaction:
One not considered is the set of propositions believed by each individual person. Thoroughgoing relativists might well embrace that one. Peirce and Putnam liked the moderate one. I'm taken with the last one, since truth is an ideal, not a phenomenon.
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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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19074
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Are truth-condtions other propositions (coherence) or features of the world (correspondence)? [Young,JO]
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Full Idea:
For the coherence theory of truth, the truth conditions of propositions consist in other propositions. The correspondence theory, in contrast, states that the truth conditions of propositions are ... objective features of the world.
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From:
James O. Young (The Coherence Theory of Truth [2013], Intro)
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A reaction:
It is obviously rather important for your truth-conditions theory of meaning that you are clear about your theory of truth. A correspondence theory is evidently taken for granted, even in possible worlds versions.
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19082
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Coherence truth suggests truth-condtions are assertion-conditions, which need knowledge of justification [Young,JO]
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Full Idea:
Coherence theorists can argue that the truth conditions of a proposition are those under which speakers tend to assert it, ...and that speakers can only make a practice of asserting a proposition under conditions they can recognise as justifying it.
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From:
James O. Young (The Coherence Theory of Truth [2013], §2.2)
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A reaction:
[compressed] This sounds rather verificationist, and hence wrong, since if you then asserted anything for which you didn't know the justification, that would remove its truth, and thus make it meaningless.
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21991
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The middle class gain freedom through property, but workers can only free all of humanity [Marx, by Singer]
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Full Idea:
Where the middle class can win freedom for themselves on the basis of rights to property - thus excluding others from their freedom - the working class have nothing but their title as human beings. They only liberate themselves by liberating humanity.
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From:
report of Karl Marx (Contrib to Critique of Hegel's Phil of Right [1844]) by Peter Singer - Marx 4
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A reaction:
Individual workers might gain freedom via education, marriage, or entrepreneurship, or by opting for total simplicity of life, but in general Marx seems to be right about this. But we must ask what sort of 'freedom' is needed.
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