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All the ideas for 'Logicism, Some Considerations (PhD)', 'Mathematical Intuition' and 'Logic [1897]'

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6 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers

13412	Obtaining numbers by abstraction is impossible - there are too many; only a rule could give them, in order [Benacerraf]
	Full Idea: Not all numbers could possibly have been learned à la Frege-Russell, because we could not have performed that many distinct acts of abstraction. Somewhere along the line a rule had to come in to enable us to obtain more numbers, in the natural order.
	From: Paul Benacerraf (Logicism, Some Considerations (PhD) [1960], p.165)
	A reaction: Follows on from Idea 13411. I'm not sure how Russell would deal with this, though I am sure his account cannot be swept aside this easily. Nevertheless this seems powerful and convincing, approaching the problem through the epistemology.

13413	We must explain how we know so many numbers, and recognise ones we haven't met before [Benacerraf]
	Full Idea: Both ordinalists and cardinalists, to account for our number words, have to account for the fact that we know so many of them, and that we can 'recognize' numbers which we've neither seen nor heard.
	From: Paul Benacerraf (Logicism, Some Considerations (PhD) [1960], p.166)
	A reaction: This seems an important contraint on any attempt to explain numbers. Benacerraf is an incipient structuralist, and here presses the importance of rules in our grasp of number. Faced with 42,578,645, we perform an act of deconstruction to grasp it.

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers

13411	If numbers are basically the cardinals (Frege-Russell view) you could know some numbers in isolation [Benacerraf]
	Full Idea: If we accept the Frege-Russell analysis of number (the natural numbers are the cardinals) as basic and correct, one thing which seems to follow is that one could know, say, three, seventeen, and eight, but no other numbers.
	From: Paul Benacerraf (Logicism, Some Considerations (PhD) [1960], p.164)
	A reaction: It seems possible that someone might only know those numbers, as the patterns of members of three neighbouring families (the only place where they apply number). That said, this is good support for the priority of ordinals. See Idea 13412.

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism

13415	An adequate account of a number must relate it to its series [Benacerraf]
	Full Idea: No account of an individual number is adequate unless it relates that number to the series of which it is a member.
	From: Paul Benacerraf (Logicism, Some Considerations (PhD) [1960], p.169)
	A reaction: Thus it is not totally implausible to say that 2 is several different numbers or concepts, depending on whether you see it as a natural number, an integer, a rational, or a real. This idea is the beginning of modern structuralism.

6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism

18201	General principles can be obvious in mathematics, but bold speculations in empirical science [Parsons,C]
	Full Idea: The existence of very general principles in mathematics are universally regarded as obvious, where on an empiricist view one would expect them to be bold hypotheses, about which a prudent scientist would maintain reserve.
	From: Charles Parsons (Mathematical Intuition [1980], p.152), quoted by Penelope Maddy - Naturalism in Mathematics
	A reaction: This is mainly aimed at Quine's and Putnam's indispensability (to science) argument about mathematics.

13. Knowledge Criteria / C. External Justification / 2. Causal Justification

11052	Psychological logic can't distinguish justification from causes of a belief [Frege]
	Full Idea: With the psychological conception of logic we lose the distinction between the grounds that justify a conviction and the causes that actually produce it.
	From: Gottlob Frege (Logic [1897] [1897])
	A reaction: Thus Frege kicked the causal theory of justification well into touch long before it had even been properly formulated. That is not to say that there is no psychological aspect to logic, because there is.