|
4901
|
Truth has to be correspondence to facts, and a match between relations of ideas and relations in the world [Perry]
|
|
Full Idea:
I think knowledge and truth are a matter of correspondence to facts, despite all the energy spent showing the naïveté of this view. The connections of our ideas in our heads correspond to relations in the outside world.
|
|
From:
John Perry (Knowledge, Possibility and Consciousness [2001], §8.1)
|
|
A reaction:
Yes. Modern books offer the difficulties of defining 'correspondence', and finding an independent account of 'facts', as conclusive objections, but I say a brain is a truth machine, and it had better be useful. Indefinability doesn't nullify concepts.
|
|
8729
|
Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro]
|
|
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.
|
|
From:
Stewart Shapiro (Thinking About Mathematics [2000], 1.2)
|
|
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.
|
|
8763
|
The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro]
|
|
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.
|
|
From:
Stewart Shapiro (Thinking About Mathematics [2000], 10.2)
|
|
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.
|
|
8762
|
Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro]
|
|
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.
|
|
From:
Stewart Shapiro (Thinking About Mathematics [2000], 10.2)
|
|
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.
|
|
8749
|
Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro]
|
|
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?
|
|
From:
Stewart Shapiro (Thinking About Mathematics [2000], 6.1.1)
|
|
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.
|
|
8750
|
Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro]
|
|
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.
|
|
From:
Stewart Shapiro (Thinking About Mathematics [2000], 6.1.2)
|
|
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.
|
|
8753
|
Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro]
|
|
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.
|
|
From:
Stewart Shapiro (Thinking About Mathematics [2000], 7.1)
|
|
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.
|
|
8731
|
Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro]
|
|
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.
|
|
From:
Stewart Shapiro (Thinking About Mathematics [2000], 2.2.1)
|
|
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.
|
|
4885
|
Identity is a very weak relation, which doesn't require interdefinability, or shared properties [Perry]
|
|
Full Idea:
The truth of "a=b" doesn't require much of 'a' and 'b' other than that there is a single thing to which they both refer. They needn't be interdefinable, or have supervenient properties. In this sense, identity is a very weak relation.
|
|
From:
John Perry (Knowledge, Possibility and Consciousness [2001], §1.2)
|
|
A reaction:
Interesting. This is seeing the epistemological aspects of identity. Ontologically, identity must invoke Leibniz's Law, and is the ultimately powerful 'relation'. A given student, and the cause of a crop circle, may APPEAR to be quite different.
|
|
4899
|
Possible worlds thinking has clarified the logic of modality, but is problematic in epistemology [Perry]
|
|
Full Idea:
Using possible worlds to model truth-conditions of statements has led to considerable clarity about the logic of modality. Attempts to use the system for epistemic purposes, however, have been plagued by problems.
|
|
From:
John Perry (Knowledge, Possibility and Consciousness [2001], §8.1)
|
|
A reaction:
Presumably what lurks behind this is a distinction between what is logically or naturally possible, and what appears to be possible from the perspective of a conscious mind. Is there a possible world in which I can fly?
|
|
4898
|
Possible worlds are indices for a language, or concrete realities, or abstract possibilities [Perry]
|
|
Full Idea:
Possible worlds can be thought of as indices for models of the language in question, or as concrete realities (David Lewis), or as abstract ways the world might be (Robert Stalnaker), or in various other ways.
|
|
From:
John Perry (Knowledge, Possibility and Consciousness [2001], §8.1)
|
|
A reaction:
I strongly favour the Stalnaker route here. Reducing great metaphysics to mere language I find abhorrent, and I suspect that Lewis was trapped by his commitment to strong empiricism. We must embrace abstractions into our ontology.
|
|
12737
|
Nature can be fully explained by final causes alone, or by efficient causes alone [Leibniz]
|
|
Full Idea:
All the phenomena of nature can be explained solely by final causes, exactly as if there were no efficient causes; and all the phenomena of nature can be explained solely by efficient causes, as if there were no final causes.
|
|
From:
Gottfried Leibniz (Definitiones cogitationesque metaphysicae [1678], A6.4.1403), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 6
|
|
A reaction:
Somewhat speculative (a virtue!), but it is interesting to see him suggesting that there might be two complete and satisfactory explanations, which never touched one another. I can't see Aristotle agreeing with that.
|
|
4891
|
If epiphenomenalism just says mental events are effects but not causes, it is consistent with physicalism [Perry]
|
|
Full Idea:
Epiphenomenalism is usually considered to be a form of dualism, but if we define it as the doctrine that conscious events are effects but not causes, it appears to be consistent with physicalism.
|
|
From:
John Perry (Knowledge, Possibility and Consciousness [2001], §4.2)
|
|
A reaction:
Interesting. The theory was invented to put mind outside physics, and make the closure of physics possible. However, being capable of causing things seems to be a necessary condition for physical objects. An effect in one domain is a cause in another.
|
|
4889
|
Although we may classify ideas by content, we individuate them differently, as their content can change [Perry]
|
|
Full Idea:
Although we classify ideas by content for many purposes, we do not individuate them by content. The content of an idea can change.
|
|
From:
John Perry (Knowledge, Possibility and Consciousness [2001], §3.2)
|
|
A reaction:
As the compiler of this database, I find this very appealing. The mind works exactly like a database. I have a 'file' (Perry's word) marked "London", the content of which undergoes continual change. I am a database management system.
|
|
4896
|
The intension of an expression is a function from possible worlds to an appropriate extension [Perry]
|
|
Full Idea:
In possible-worlds semantics, expressions have intensions, which are functions from possible worlds to appropriate extensions (names to individuals, n-place predicates to n-tuples, and sentences to truth values, built from parts).
|
|
From:
John Perry (Knowledge, Possibility and Consciousness [2001], §8.1)
|
|
A reaction:
Interesting. Perry distinguishes 'referential' (or 'subject matter') content, which is prior to the link to extensions - a link which creates 'reflexive' content. He is keen that they should not become confused. True knowledge is 'situated'.
|