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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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4483
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If abstract terms are sets of tropes, 'being a unicorn' and 'being a griffin' turn out identical [Loux]
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Full Idea:
If trope theorists say abstract singular terms name sets of tropes, what is the referent of 'is a unicorn'? The only candidate is the null set (with no members), but there is just one null set, so 'being a unicorn' and 'being a griffin' will be identical.
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From:
Michael J. Loux (Metaphysics: contemporary introduction [1998], p.86)
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A reaction:
Not crucial, I would think, given that a unicorn is just a horse with a horn. Hume explains how we do that, combining ideas which arose from actual tropes.
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6248
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Reason is too slow and doubtful to guide all actions, which need external and moral senses [Hutcheson]
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Full Idea:
We boast of our mighty reason above other animals, but its processes are too slow, too full of doubt, to serve us in every exigency, either for our preservation, without external senses, or to influence our actions for good without the moral sense.
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From:
Francis Hutcheson (Treatise 2: Virtue or Moral Good [1725], §VII.III)
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A reaction:
This idea was taken up by Hume, and it must have influence Hume's general scepticism about the importance of reason. What this idea misses is the enormous influence of prior reasoning on our quick decisions.
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6239
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We dislike a traitor, even if they give us great benefit [Hutcheson]
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Full Idea:
Let us consider if a traitor, who would sell his own country to us, may not often be as advantageous to us, as an hero who defends us: and yet we can love the treason, and hate the traitor.
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From:
Francis Hutcheson (Treatise 2: Virtue or Moral Good [1725], §I.VI)
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A reaction:
A nice example, which certainly refutes any claim that morality is entirely and directly self-interested. High-minded idealism, though, is not the only alternative explanation. We admire loyalty, but not loyalty to, say, Hitler.
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6240
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The moral sense is not an innate idea, but an ability to approve or disapprove in a disinterested way [Hutcheson]
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Full Idea:
The moral sense is not an innate idea or knowledge, but a determination of our minds to receive the simple ideas of approbation or condemnation, from actions observed, antecedent to any opinions of advantage or loss to redound to ourselves.
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From:
Francis Hutcheson (Treatise 2: Virtue or Moral Good [1725], §I.VIII)
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A reaction:
This may claim a pure moral intuition, but it is also close to Kantian universalising of the rules for behaviour. It is also a variation on Descartes' 'natural light' of reason. Of course, if we say the ideas are 'received', where are they received from?
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6242
|
We cannot choose our moral feelings, otherwise bribery could affect them [Hutcheson]
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Full Idea:
Neither benevolence nor any other affection or desire can be directly raised by volition; if they could, then we could be bribed into any affection whatsoever toward any object.
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From:
Francis Hutcheson (Treatise 2: Virtue or Moral Good [1725], §II.IV)
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A reaction:
Of course, notoriously, the vast mass of people have often been bribed to love a politician, by low taxes, or bread and circuses. Still, you cannot choose to love or admire someone, you just do. Not much free will there.
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6244
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Human nature seems incapable of universal malice, except what results from self-love [Hutcheson]
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Full Idea:
Human nature seems scarce capable of malicious disinterested hatred, or an ultimate desire of the misery of others, when we imagine them not pernicious to us, or opposite to our interests; ..that is only the effect of self-love, not disinterested malice.
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From:
Francis Hutcheson (Treatise 2: Virtue or Moral Good [1725], §II.VII)
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A reaction:
I suppose it is true that even the worst criminals brooding in prison don't wish the entire population of some foreign country to die in pain. Only a very freakish person would wish the human race were extinct. A very nice observation.
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6243
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As death approaches, why do we still care about family, friends or country? [Hutcheson]
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Full Idea:
How comes it that we do not lose, at the approach of death, all concern for our families, friends, or country?
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From:
Francis Hutcheson (Treatise 2: Virtue or Moral Good [1725], §II.V)
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A reaction:
A nice question. No doubt some people do cease to care, but on the whole it raises the 'last round' problem in social contract theory, which is why fulfil your part of a bargain if it is too late to receive the repayment afterwards?
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6241
|
Contempt of danger is just madness if it is not in some worthy cause [Hutcheson]
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Full Idea:
Mere courage, or contempt of danger, if we conceive it to have no regard to the defence of the innocent, or repairing of wrongs or self-interest, would only entitle its possessor to bedlam.
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From:
Francis Hutcheson (Treatise 2: Virtue or Moral Good [1725], §II.I)
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A reaction:
If many criminals would love to rob a bank, but only a few have the nerve to attempt it, we can hardly deny that the latter exhibit a sort of courage. The Greeks say that good sense must be involved, but few of them were so moral about courage.
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6245
|
That action is best, which procures the greatest happiness for the greatest number [Hutcheson]
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Full Idea:
That action is best, which procures the greatest happiness for the greatest number; and that worst, which, in like manner, occasions misery.
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From:
Francis Hutcheson (Treatise 2: Virtue or Moral Good [1725], §III.VIII)
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A reaction:
The first use of a phrase taken up by Bentham. This is not just an anticipation of utilitarianism, it is utilitarianism, with all its commitment to consequentialism (but see Idea 6246), and to the maximising of happiness. It is a brilliant idea.
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6251
|
The loss of perfect rights causes misery, but the loss of imperfect rights reduces social good [Hutcheson]
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Full Idea:
Perfect rights are necessary to the public good, and it makes those miserable whose rights are thus violated; …imperfect rights tend to the improvement and increase of good in a society, but are not necessary to prevent universal misery.
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From:
Francis Hutcheson (Treatise 2: Virtue or Moral Good [1725], §VII.VI)
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A reaction:
This is a very utilitarian streak in Hutcheson, converting natural law into its tangible outcome in actual happiness or misery. The distinction here is interesting (taken up by Mill), but there is a very blurred borderline.
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6249
|
If goodness is constituted by God's will, it is a tautology to say God's will is good [Hutcheson]
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Full Idea:
To call the laws of the supreme Deity good or holy or just, if these be constituted by laws, or the will of a superior, must be an insignificant tautology, amounting to no more than 'God wills what he wills' or 'His will is conformable to his will'.
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From:
Francis Hutcheson (Treatise 2: Virtue or Moral Good [1725], §VII.V)
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|
A reaction:
This argues not only against God as the source of morality, but also against any rules, such as those of the Categorical Imperative. Why should I follow the Categorical Imperative? What has value must dictate the rules. Is obedience the highest value?
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