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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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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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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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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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15643
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Nominal essence mistakenly gives equal weight to all underlying properties that produce appearances [Eagle]
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Full Idea:
Nominal essence does not allow for gradations in significance for the underlying properties. Those are all essential for the object behaving as it observably does, and they must all be given equal weight when deciding what the object does.
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From:
Antony Eagle (Locke on Essences and Kinds [2005], IV)
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A reaction:
This is where 'scientific' essentialism comes in. If we take one object, or one kind of object, in isolation, Eagle is right. When we start to compare, and to set up controlled conditions tests, we can dig into the 'gradations' he cares about.
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20322
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I am aware that freedom is possible, and the freedom is not in theory, but in seeking freedom [Jaspers]
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Full Idea:
Either there is no freedom or it is in asking about it. But what makes me ask is an original will to be free, so my freedom is anticipated in the fact of asking. I cannot prove it first, then will it. I will it because I am conscious of its possibility.
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From:
Karl Jaspers (Philosophy [1932], vol.2)
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A reaction:
This presents the subjective claims for free will rather more persuasively than usual. I am conscious of a possibility that I might flap my arms and fly, so that doesn't establish anything. But yearning to be free is a sort of freedom.
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20318
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My helplessness in philosophising reveals my being, and begins its upsurge [Jaspers]
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Full Idea:
Philosophising, not knowing, brings me to myself. The helplessness to which philosophising reduces me when I doubt its origin is an expressions of the helplessness of my self-being, and the reality of philosophising is the incipient upsurge of that being.
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From:
Karl Jaspers (Philosophy [1932], vol.2)
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A reaction:
I like the sound of 'philosophy as a way of life', and loosely aspire to it, but I'm still not sure what it means, other than a good way to pass the time. The idea that it leads to higher modes of being sounds a bit arrogant. But it is a good thing!
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20317
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Mundane existence is general, falling under universals, but Existens is unique to individuals [Jaspers]
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Full Idea:
Mundane being, the being we know, is general because it is generally valid for everyone. ...Existenz is never general, and thus not a case that might be subsumed as particular under a universal.
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From:
Karl Jaspers (Philosophy [1932], vol.2)
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A reaction:
I'm trying to visualise a mode of existence which would fulfil only me, answering to my unique nature, but it looks like a vain delusion. I may be a one-off combination, but I see all of my ingredients in various other people.
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20316
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Every decision I make moves towards or away from fulfilled Existenz [Jaspers]
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Full Idea:
My Existenz, as a possibility, takes a step toward being or away from being, toward nothingness, in every choice or decision I make.
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From:
Karl Jaspers (Philosophy [1932], vol.2)
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A reaction:
The existential idea of action involves what you are, as well as what you do. There seems to be a paradox. My being is plastic, and can change enormously, so I should take responsibility for the change. But who is in charge of the changes?
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