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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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18967
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A 'proposition' is said to be the timeless cognitive part of the meaning of a sentence [Quine]
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Full Idea:
A 'proposition' is the meaning of a sentence. More precisely, since propositions are supposed to be true or false once and for all, it is the meaning of an eternal sentence. More precisely still, it is the 'cognitive' meaning, involving truth, not poetry.
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From:
Willard Quine (Propositional Objects [1965], p.139)
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A reaction:
Quine defines this in order to attack it. I equate a proposition with a thought, and take a sentence to be an attempt to express a proposition. I have no idea why they are supposed to be 'timeless'. Philosophers have some very odd ideas.
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18968
|
The problem with propositions is their individuation. When do two sentences express one proposition? [Quine]
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Full Idea:
The trouble with propositions, as cognitive meanings of eternal sentences, is individuation. Given two eternal sentences, themselves visibly different linguistically, it is not sufficiently clear under when to say that they mean the same proposition.
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From:
Willard Quine (Propositional Objects [1965], p.140)
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A reaction:
If a group of people agree that two sentences mean the same thing, which happens all the time, I don't see what gives Quine the right to have a philosophical moan about some dubious activity called 'individuation'.
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21099
|
People must have agreed to authority, because they are naturally equal, prior to education [Hume]
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Full Idea:
When we consider how nearly equal all men are in their bodily force, and even in their mental powers and faculties, till cultivated by education, ...then nothing but their own consent could at first associate them together, and subject them to authority.
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From:
David Hume (Of the original contract [1741], p.276)
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A reaction:
This doesn't sound very convincing. Some people are much better suited than others to training and education. Men vary enormously in size.
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20495
|
We no more give 'tacit assent' to the state than a passenger carried on board a ship while asleep [Hume]
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Full Idea:
[If we give 'tacit' assent to the state] ...we may as well assert that a man, by remaining in a vessel, freely consents to the dominion of the master, though he was carried aboard while asleep.
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From:
David Hume (Of the original contract [1741], p.283)
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A reaction:
We should probably drop the whole idea that we give assent to the state. We are stuck with a state, and a few of us can escape, if it seems important enough, but most of us have no choice. He hope to assent to the controllers of the state.
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6703
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Poor people lack the knowledge or wealth to move to a different state [Hume]
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Full Idea:
Can we seriously say, that a poor peasant or artisan has a free choice to leave his country, when he knows no foreign language or manners, and lives, from day to day, by the small wages that he acquires?
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From:
David Hume (Of the original contract [1741], p.283)
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A reaction:
Of course, in the nineteenth century the Scottish poor did, going to America, which welcomed the poor, and spoke English. Hume's point is the right reply to anyone who says 'If you don't like it, go elsewhere'. Also 'No! Change it!'
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21102
|
We all know that the history of property is founded on injustices [Hume]
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Full Idea:
Reason tells us that there is no property in durable objects, such as land or houses, when carefully examined in passing from hand to hand, but must, in some period, have been founded on fraud and injustice.
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From:
David Hume (Of the original contract [1741], p.288)
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A reaction:
A prime objection to Nozick, who fantasises about an initial position of just ownership, which can then be the subject of just contracts. In 1866 thousands of white people were granted land in the USA, but not a single black freed slave got anything.
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