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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18430
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We accept properties because of type/tokens, reference, and quantification [Edwards]
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Full Idea:
Three main reasons for thinking properties exist: the one-over-many argument (that a type can have many tokens), the reference argument (to understand predicates and singular terms), and the quantification argument (that we quantify over them).
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From:
Douglas Edwards (Properties [2014], 1.1)
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A reaction:
[Bits in brackets are compressions of his explanations]. I don't find any of these remotely persuasive. Why would we infer how the world is, simply from how we talk about or reason about the world? His first reason is the only interesting one.
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8329
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Either causal relations are given in experience, or they are unobserved and theoretical [Sosa/Tooley]
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Full Idea:
There is a fundamental choice between the realist approach to causation which says that the relation is immediately given in experience, and the view that causation is a theoretical relation, and so not directly observable.
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From:
E Sosa / M Tooley (Introduction to 'Causation' [1993], §1)
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A reaction:
Even if immediate experience is involved, there is a step of abstraction in calling it a cause, and picking out events. A 'theoretical relation' is not of much interest there if no observations are involved. I don't think a choice is required here.
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8324
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The problem is to explain how causal laws and relations connect, and how they link to the world [Sosa/Tooley]
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Full Idea:
Causal states of affairs encompass causal laws, and causal relations between events or states of affairs; two key questions concern the relation between causal laws and causal relations, and the relation between these and non-causal affairs.
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From:
E Sosa / M Tooley (Introduction to 'Causation' [1993], §1)
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A reaction:
This is the agenda for modern analytical philosophy. I'm not quite clear what would count as an answer. When have you 'explained' a relation? Does calling it 'gravity', or finding an equation, explain that relation? Do gravitinos explain it?
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8328
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Causation isn't energy transfer, because an electron is caused by previous temporal parts [Sosa/Tooley]
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Full Idea:
The temporal parts of an electron (for example) are causally related, but this relation does not involve any transfer of energy or momentum. Causation cannot be identified with physical energy relations, and physicalist reductions look unpromising.
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From:
E Sosa / M Tooley (Introduction to 'Causation' [1993], §1)
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A reaction:
This idea, plus Idea 8327, are their grounds for rejecting Fair's proposal (Idea 8326). It feels like a different use of 'cause' when we say 'the existence of x was caused by its existence yesterday'. It is more like inertia. Destruction needs energy.
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8325
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The dominant view is that causal laws are prior; a minority say causes can be explained singly [Sosa/Tooley]
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Full Idea:
The dominant view is that causal laws are more basic than causal relations, with relations being logically supervenient on causal laws, and on properties and event relations; some, though, defend the singularist view, in which events alone can be related.
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From:
E Sosa / M Tooley (Introduction to 'Causation' [1993], §1)
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A reaction:
I am deeply suspicious about laws (see Idea 5470). I suspect that the laws are merely descriptions of the regularities that arise from the single instances of causation. We won't explain the single instances, but then laws don't 'explain' them either.
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