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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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16554
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Activities have place, rate, duration, entities, properties, modes, direction, polarity, energy and range [Machamer/Darden/Craver]
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Full Idea:
Activities can be identified spatiotemporally, and individuated by rate, duration, and types of entity and property that engage in them. They also have modes of operation, directionality, polarity, energy requirements and a range.
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From:
Machamer,P/Darden,L/Craver,C (Thinking About Mechanisms [2000], 3)
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A reaction:
This is their attempt at making 'activity' one of the two central concepts of ontology, along with 'entity'. A helpful analysis. It just seems to be one way of slicing the cake.
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5998
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From the necessity of the past we can infer the impossibility of what never happens [Diod.Cronus, by White,MJ]
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Full Idea:
Diodorus' Master Argument inferred that since what is past (i.e. true in the past) is necessary, and the impossible cannot follow from the possible, that therefore if something neither is nor ever will be the case, then it is impossible.
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From:
report of Diodorus Cronus (fragments/reports [c.300 BCE]) by Michael J. White - Diodorus Cronus
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A reaction:
The argument is, apparently, no longer fully clear, but it seems to imply determinism, or at least a rejection of the idea that free will and determinism are compatible. (Epictetus 2.19)
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16562
|
We understand something by presenting its low-level entities and activities [Machamer/Darden/Craver]
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Full Idea:
The intelligibility of a phenomenon consists in the mechanisms being portrayed in terms of a field's bottom out entities and activities.
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From:
Machamer,P/Darden,L/Craver,C (Thinking About Mechanisms [2000], 7)
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A reaction:
In other words, we understand complex things by reducing them to things we do understand. It would, though, be illuminating to see a nest of interconnected activities, even if we understood none of them.
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16555
|
Functions are not properties of objects, they are activities contributing to mechanisms [Machamer/Darden/Craver]
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Full Idea:
It is common to speak of functions as properties 'had by' entities, …but they should rather be understood in terms of the activities by virtue of which entities contribute to the workings of a mechanism.
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From:
Machamer,P/Darden,L/Craver,C (Thinking About Mechanisms [2000], 3)
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A reaction:
I'm certainly quite passionately in favour of cutting down on describing the world almost entirely in terms of entities which have properties. An 'activity', though, is a bit of an elusive concept.
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16528
|
Mechanisms are not just push-pull systems [Machamer/Darden/Craver]
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Full Idea:
One should not think of mechanisms as exclusively mechanical (push-pull) systems.
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From:
Machamer,P/Darden,L/Craver,C (Thinking About Mechanisms [2000], 1)
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A reaction:
The difficulty seems to be that you could broaden the concept of 'mechanism' indefinitely, so that it covered history, mathematics, populations, cultural change, and even mathematics. Where to stop?
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16553
|
Our account of mechanism combines both entities and activities [Machamer/Darden/Craver]
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Full Idea:
We emphasise the activities in mechanisms. This is explicitly dualist. Substantivalists speak of entities with dispositions to act. Process ontologists reify activities and try to reduce entities to processes. We try to capture both intuitions.
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From:
Machamer,P/Darden,L/Craver,C (Thinking About Mechanisms [2000], 3)
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A reaction:
[A quotation of selected fragments] The problem here seems to be the raising of an 'activity' to a central role in ontology, when it doesn't seem to be primitive, and will typically be analysed in a variety of ways.
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16559
|
Descriptions of explanatory mechanisms have a bottom level, where going further is irrelevant [Machamer/Darden/Craver]
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Full Idea:
Nested hierachical descriptions of mechanisms typically bottom out in lowest level mechanisms. …Bottoming out is relative …the explanation comes to an end, and description of lower-level mechanisms would be irrelevant.
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From:
Machamer,P/Darden,L/Craver,C (Thinking About Mechanisms [2000], 5.1)
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A reaction:
This seems to me exactly the right story about mechanism, and it is a story I am associating with essentialism. The relevance is ties to understanding. The lower level is either fully understood, or totally baffling.
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16564
|
There are four types of bottom-level activities which will explain phenomena [Machamer/Darden/Craver]
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Full Idea:
There are four bottom-out kinds of activities: geometrico-mechanical, electro-chemical, electro-magnetic and energetic. These are abstract means of production that can be fruitfully applied in particular cases to explain phenomena.
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From:
Machamer,P/Darden,L/Craver,C (Thinking About Mechanisms [2000], 7)
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A reaction:
I like that. It gives a nice core for a metaphysics for physicalists. I suspect that 'mechanical' can be reduced to something else, and that 'energetic' will disappear in the final story.
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16558
|
Laws of nature have very little application in biology [Machamer/Darden/Craver]
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Full Idea:
The traditional notion of a law of nature has few, if any, applications in neurobiology or molecular biology.
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From:
Machamer,P/Darden,L/Craver,C (Thinking About Mechanisms [2000], 3.2)
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A reaction:
This is a simple and self-evident fact, and bad news for anyone who want to build their entire ontology around laws of nature. I take such a notion to be fairly empty, except as a convenient heuristic device.
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