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10007
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Quantifiers for domains and for inference come apart if there are no entities [Hofweber]
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Full Idea:
Quantifiers have two functions in communication - to range over a domain of entities, and to have an inferential role (e.g. F(t)→'something is F'). In ordinary language these two come apart for singular terms not standing for any entities.
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From:
Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §6.3)
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A reaction:
This simple observations seems to me to be wonderfully illuminating of a whole raft of problems, the sort which logicians get steamed up about, and ordinary speakers don't. Context is the key to 90% of philosophical difficulties (?). See Idea 10008.
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13412
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Obtaining numbers by abstraction is impossible - there are too many; only a rule could give them, in order [Benacerraf]
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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.
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From:
Paul Benacerraf (Logicism, Some Considerations (PhD) [1960], p.165)
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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.
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13413
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We must explain how we know so many numbers, and recognise ones we haven't met before [Benacerraf]
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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.
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From:
Paul Benacerraf (Logicism, Some Considerations (PhD) [1960], p.166)
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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.
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10002
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'2 + 2 = 4' can be read as either singular or plural [Hofweber]
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Full Idea:
There are two ways to read to read '2 + 2 = 4', as singular ('two and two is four'), and as plural ('two and two are four').
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From:
Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §4.1)
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A reaction:
Hofweber doesn't notice that this phenomenon occurs elsewhere in English. 'The team is playing well', or 'the team are splitting up'; it simply depends whether you are holding the group in though as an entity, or as individuals. Important for numbers.
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9998
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What is the relation of number words as singular-terms, adjectives/determiners, and symbols? [Hofweber]
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Full Idea:
There are three different uses of the number words: the singular-term use (as in 'the number of moons of Jupiter is four'), the adjectival (or determiner) use (as in 'Jupiter has four moons'), and the symbolic use (as in '4'). How are they related?
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From:
Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §1)
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A reaction:
A classic philosophy of language approach to the problem - try to give the truth-conditions for all three types. The main problem is that the first one implies that numbers are objects, whereas the others do not. Why did Frege give priority to the first?
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13411
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If numbers are basically the cardinals (Frege-Russell view) you could know some numbers in isolation [Benacerraf]
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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.
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From:
Paul Benacerraf (Logicism, Some Considerations (PhD) [1960], p.164)
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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.
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10003
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Why is arithmetic hard to learn, but then becomes easy? [Hofweber]
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Full Idea:
Why is arithmetic so hard to learn, and why does it seem so easy to us now? For example, subtracting 789 from 26,789.
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From:
Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §4.2)
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A reaction:
His answer that we find thinking about objects very easy, but as children we have to learn with difficulty the conversion of the determiner/adjectival number words, so that we come to think of them as objects.
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10008
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Arithmetic is not about a domain of entities, as the quantifiers are purely inferential [Hofweber]
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Full Idea:
I argue for an internalist conception of arithmetic. Arithmetic is not about a domain of entities, not even quantified entities. Quantifiers over natural numbers occur in their inferential-role reading in which they merely generalize over the instances.
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From:
Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §6.3)
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A reaction:
Hofweber offers the hope that modern semantics can disentangle the confusions in platonist arithmetic. Very interesting. The fear is that after digging into the semantics for twenty years, you find the same old problems re-emerging at a lower level.
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10005
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Arithmetic doesn’t simply depend on objects, since it is true of fictional objects [Hofweber]
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Full Idea:
That 'two dogs are more than one' is clearly true, but its truth doesn't depend on the existence of dogs, as is seen if we consider 'two unicorns are more than one', which is true even though there are no unicorns.
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From:
Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §6.2)
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A reaction:
This is an objection to crude empirical accounts of arithmetic, but the idea would be that there is a generalisation drawn from objects (dogs will do nicely), which then apply to any entities. If unicorns are entities, it will be true of them.
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10000
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We might eliminate adjectival numbers by analysing them into blocks of quantifiers [Hofweber]
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Full Idea:
Determiner uses of number words may disappear on analysis. This is inspired by Russell's elimination of the word 'the'. The number becomes blocks of first-order quantifiers at the level of semantic representation.
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From:
Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §2)
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A reaction:
[compressed] The proposal comes from platonists, who argue that numbers cannot be analysed away if they are objects. Hofweber says the analogy with Russell is wrong, as 'the' can't occur in different syntactic positions, the way number words can.
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10006
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First-order logic captures the inferential relations of numbers, but not the semantics [Hofweber]
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Full Idea:
Representing arithmetic formally we do not primarily care about semantic features of number words. We are interested in capturing the inferential relations of arithmetical statements to one another, which can be done elegantly in first-order logic.
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From:
Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §6.3)
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A reaction:
This begins to pinpoint the difference between the approach of logicists like Frege, and those who are interested in the psychology of numbers, and the empirical roots of numbers in the process of counting.
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19378
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Early modern possibility is what occurs sometime; for Leibniz, it is what is not contradictory [Arthur,R]
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Full Idea:
For Descartes, Hobbes and Spinoza, if a state of things is possible, it must occur at some time, whether past, present or future. For Leibniz possibility makes no reference to time; an individual is possible if its concept contains no contradiction.
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From:
Richard T.W. Arthur (Leibniz [2014], 4 'Contingent')
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A reaction:
It has always struck me as fallacious to say that anything that is possible must at some time occur. If '6' is possible on the die, what will constrain it to eventually come up when thrown? Mere non-contradiction doesn't imply possibility either.
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