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9978
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Analytic philosophy focuses too much on forms of expression, instead of what is actually said [Tait]
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Full Idea:
The tendency to attack forms of expression rather than attempting to appreciate what is actually being said is one of the more unfortunate habits that analytic philosophy inherited from Frege.
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From:
William W. Tait (Frege versus Cantor and Dedekind [1996], IV)
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A reaction:
The key to this, I say, is to acknowledge the existence of propositions (in brains). For example, this belief will make teachers more sympathetic to pupils who are struggling to express an idea, and verbal nit-picking becomes totally irrelevant.
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9986
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The null set was doubted, because numbering seemed to require 'units' [Tait]
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Full Idea:
The conception that what can be numbered is some object (including flocks of sheep) relative to a partition - a choice of unit - survived even in the late nineteenth century in the form of the rejection of the null set (and difficulties with unit sets).
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From:
William W. Tait (Frege versus Cantor and Dedekind [1996], IX)
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A reaction:
This old view can't be entirely wrong! Frege makes the point that if asked to count a pack of cards, you must decide whether to count cards, or suits, or pips. You may not need a 'unit', but you need a concept. 'Units' name concept-extensions nicely!
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9984
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We can have a series with identical members [Tait]
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Full Idea:
Why can't we have a series (as opposed to a linearly ordered set) all of whose members are identical, such as (a, a, a...,a)?
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From:
William W. Tait (Frege versus Cantor and Dedekind [1996], VII)
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A reaction:
The question is whether the items order themselves, which presumably the natural numbers are supposed to do, or whether we impose the order (and length) of the series. What decides how many a's there are? Do we order, or does nature?
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13416
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Mathematics must be based on axioms, which are true because they are axioms, not vice versa [Tait, by Parsons,C]
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Full Idea:
The axiomatic conception of mathematics is the only viable one. ...But they are true because they are axioms, in contrast to the view advanced by Frege (to Hilbert) that to be a candidate for axiomhood a statement must be true.
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From:
report of William W. Tait (Intro to 'Provenance of Pure Reason' [2005], p.4) by Charles Parsons - Review of Tait 'Provenance of Pure Reason' §2
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A reaction:
This looks like the classic twentieth century shift in the attitude to axioms. The Greek idea is that they must be self-evident truths, but the Tait-style view is that they are just the first steps in establishing a logical structure. I prefer the Greeks.
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9982
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Cantor and Dedekind use abstraction to fix grammar and objects, not to carry out proofs [Tait]
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Full Idea:
Although (in Cantor and Dedekind) abstraction does not (as has often been observed) play any role in their proofs, but it does play a role, in that it fixes the grammar, the domain of meaningful propositions, and so determining the objects in the proofs.
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From:
William W. Tait (Frege versus Cantor and Dedekind [1996], V)
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A reaction:
[compressed] This is part of a defence of abstractionism in Cantor and Dedekind (see K.Fine also on the subject). To know the members of a set, or size of a domain, you need to know the process or function which created the set.
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9985
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Abstraction may concern the individuation of the set itself, not its elements [Tait]
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Full Idea:
A different reading of abstraction is that it concerns, not the individuating properties of the elements relative to one another, but rather the individuating properties of the set itself, for example the concept of what is its extension.
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From:
William W. Tait (Frege versus Cantor and Dedekind [1996], VIII)
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A reaction:
If the set was 'objects in the room next door', we would not be able to abstract from the objects, but we might get to the idea of things being contain in things, or the concept of an object, or a room. Wrong. That's because they are objects... Hm.
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9980
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If abstraction produces power sets, their identity should imply identity of the originals [Tait]
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Full Idea:
If the power |A| is obtained by abstraction from set A, then if A is equipollent to set B, then |A| = |B|. But this does not imply that A = B. So |A| cannot just be A, taken in abstraction, unless that can identify distinct sets, ..or create new objects.
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From:
William W. Tait (Frege versus Cantor and Dedekind [1996], V)
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A reaction:
An elegant piece of argument, which shows rather crucial facts about abstraction. We are then obliged to ask how abstraction can create an object or a set, if the central activity of abstraction is just ignoring certain features.
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8364
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We can imagine controlling floods by controlling rain, but not vice versa [Wright,GHv]
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Full Idea:
Given our present knowledge of the laws of nature, we can imagine ways of controlling floods by controlling rainfall, but not the other way round. That is should be so, however, is contingent.
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From:
G.H. von Wright (Logic and Epistemology of Causal Relations [1973], §8)
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A reaction:
Despite my objections to Idea 8363, this is a good example. It won't establish the metaphysics of the direction of causation, though, because God might control rainfall by controlling floods. Maybe causation is more like a motorway pile-up than dominoes.
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8360
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We must further analyse conditions for causation, into quantifiers or modal concepts [Wright,GHv]
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Full Idea:
We may be able to analyse causation into conditionship relations between events or states of affairs, ...but conditions cannot be regarded as logical primitives, ... and must be analysed into quantifiers, or modal concepts.
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From:
G.H. von Wright (Logic and Epistemology of Causal Relations [1973], §2)
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A reaction:
[very compressed] A nice illustration of the aim of analytical philosophy - to analyse the elements of reality down to logical primitives. This is the dream of Descartes and Leibniz, continued by Russell and co. Do we still have this aspiration?
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8365
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Some laws are causal (Ohm's Law), but others are conceptual principles (conservation of energy) [Wright,GHv]
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Full Idea:
Not all laws are causal 'experimentalist' laws, such as those for falling bodies, or the Gas Law, or Ohm's Law. Some are more like conceptual principles, giving a frame of reference, such as inertia, or conservation of energy, or the law of entropy.
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From:
G.H. von Wright (Logic and Epistemology of Causal Relations [1973], §9)
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A reaction:
An interesting and important distinction, whenever one is exploring the links between theories of causation and of laws of nature. If one wished to attack the whole concept of 'laws of nature', this might be a good place to start.
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