15054
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'Quietist' says abandon metaphysics because answers are unattainable (as in Kant's noumenon) [Fine,K]
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Full Idea:
The 'quietist' view of metaphysics says that realist metaphysics should be abandoned, not because its questions cannot be framed, but because their answers cannot be found. The real world of metaphysics is akin to Kant's noumenal world.
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From:
Kit Fine (The Question of Realism [2001], 4)
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A reaction:
[He cites Blackburn, Dworkin, A.Fine, and Putnam-1987 as quietists] Fine aims to clarify the concepts of factuality and of ground, in order to show that metaphysics is possible.
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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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15006
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Something is grounded when it holds, and is explained, and necessitated by something else [Fine,K, by Sider]
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Full Idea:
When p 'grounds' q then q holds in virtue of p's holding; q's holding is nothing beyond p's holding; the truth of p explains the truth of q in a particularly tight sense (explanation of q by p in this sense requires that p necessitates q).
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From:
report of Kit Fine (The Question of Realism [2001], 15-16) by Theodore Sider - Writing the Book of the World 08.1
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A reaction:
This proposal has become a hot topic in current metaphysics, as attempts are made to employ 'grounding' in various logical, epistemological and ontological contexts. I'm a fan - it is at the heart of metaphysics as structure of reality.
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15055
|
Grounding relations are best expressed as relations between sentences [Fine,K]
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Full Idea:
I recommend that a statement of ground be cast in the following 'canonical' form: Its being the case that S consists in nothing more than its being the case that T, U... (where S, T, U... are particular sentences).
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From:
Kit Fine (The Question of Realism [2001], 5)
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A reaction:
The point here is that grounding is to be undestood in terms of sentences (and 'its being the case that...'), rather than in terms of objects, properties or relations. Fine thus makes grounding a human activity, rather than a natural activity.
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15052
|
Reduction is modal, if the reductions necessarily entail the truth of the target sentence [Fine,K]
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Full Idea:
The third, more recent, approach to reduction is a modal matter. A class of propositions will reduce to - or supervene upon - another if, necessarily, any truth from the one is entailed by truths from the other.
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From:
Kit Fine (The Question of Realism [2001], 3)
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A reaction:
[He cites Armstrong, Chalmers and Jackson for this approach] Fine notes that some people reject supervenience as a sort of reduction. He objects that this reduction doesn't necessarily lead to something more basic.
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15048
|
In metaphysics, reality is regarded as either 'factual', or as 'fundamental' [Fine,K]
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Full Idea:
The first main approach says metaphysical reality is to be identified with what is 'objective' or 'factual'. ...According to the second conception, metaphysical reality is to be identified with what is 'irreducible' or 'fundamental'.
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From:
Kit Fine (The Question of Realism [2001], 1)
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A reaction:
Fine is defending the 'fundamental' approach, via the 'grounding' relation. The whole structure, though, seems to be reality. In particular, a complete story must include the relations which facilitate more than mere fundamentals.
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15059
|
Grounding is an explanation of truth, and needs all the virtues of good explanations [Fine,K]
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Full Idea:
The main sources of evidence for judgments of ground are intuitive and explanatory. The relationship of ground is a form of explanation, ..explaining what makes a proposition true, which needs simplicity, breadth, coherence, non-circularity and strength.
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From:
Kit Fine (The Question of Realism [2001], 7)
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A reaction:
My thought is that not only must grounding explain, and therefore be a good explanation, but that the needs of explanation drive our decisions about what are the grounds. It is a bit indeterminate which is tail and which is dog.
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15057
|
Ultimate explanations are in 'grounds', which account for other truths, which hold in virtue of the grounding [Fine,K]
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Full Idea:
We take ground to be an explanatory relation: if the truth that P is grounded in other truths, then they account for its truth; P's being the case holds in virtue of the other truths' being the case. ...It is the ultimate form of explanation.
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From:
Kit Fine (The Question of Realism [2001], 5)
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A reaction:
To be 'ultimate' that which grounds would have to be something which thwarted all further explanation. Popper, for example, got quite angry at the suggestion that we should put a block on further investigation in this way.
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6616
|
Least action is not a causal law, but a 'global law', describing a global essence [Ellis]
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Full Idea:
The principle of least action is not a causal law, but is what I call a 'global law', which describes the essence of the global kind, which every object in the universe necessarily instantiates.
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From:
Brian Ellis (Katzav on limitations of dispositions [2005])
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A reaction:
As a fan of essentialism I find this persuasive. If I inherit part of my essence from being a mammal, I inherit other parts of my essence from being an object, and all objects would share that essence, so it would look like a 'law' for all objects.
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6615
|
A species requires a genus, and its essence includes the essence of the genus [Ellis]
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Full Idea:
A specific universal can exist only if the generic universal of which it is a species exists, but generic universals don't depend on species; …the essence of any genus is included in its species, but not conversely.
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From:
Brian Ellis (Katzav on limitations of dispositions [2005], 91)
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A reaction:
Thus the species 'electron' would be part of the genus 'lepton', or 'human' part of 'mammal'. The point of all this is to show how individual items connect up with the rest of the universe, giving rise to universal laws, such as Least Action.
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6614
|
A hierarchy of natural kinds is elaborate ontology, but needed to explain natural laws [Ellis]
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Full Idea:
The hierarchy of natural kinds proposed by essentialism may be more elaborate than is strictly required for purposes of ontology, but it is necessary to explain the necessity of the laws of nature, and the universal applicability of global principles.
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From:
Brian Ellis (Katzav on limitations of dispositions [2005], 91)
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A reaction:
I am all in favour of elaborating ontology in the name of best explanation. There seem, though, to be some remaining ontological questions at the point where the explanations of essentialism run out.
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6612
|
Without general principles, we couldn't predict the behaviour of dispositional properties [Ellis]
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Full Idea:
It is objected to dispositionalism that without the principle of least action, or some general principle of equal power, the specific dispositional properties of things could tell us very little about how these things would be disposed to behave.
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From:
Brian Ellis (Katzav on limitations of dispositions [2005], 90)
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A reaction:
Ellis attempts to meet this criticism, by placing dispositional properties within a hierarchy of broader properties. There remains a nagging doubt about how essentialism can account for space, time, order, and the existence of essences.
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