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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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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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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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22064
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Fichte's logic is much too narrow, and doesn't deduce ethics, art, society or life [Schlegel,F on Fichte]
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Full Idea:
Only Fichte's principles are deduced in his book, that is, the logical ones, and not even these completely. And what about the practical, the moral and ethical ones. Society, learning, wit, art, and so on are also entitled to be deduced here.
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From:
comment on Johann Fichte (The Science of Knowing (Wissenschaftslehre) [1st ed] [1794]) by Friedrich Schlegel - works Vol 18 p.34
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A reaction:
This is the beginnings of the romantic rebellion against a rather narrowly rationalist approach to philosophy. Schlegel also objects to the fact that Fichte only had one axiom (presumably the idea of the not-Self).
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22060
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The Self is the spontaneity, self-relatedness and unity needed for knowledge [Fichte, by Siep]
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Full Idea:
According to Fichte, spontaneity, self-relatedness, and unity are the basic traits of knowledge (which includes conscience). ...This principle of all knowledge is what he calls the 'I' or the Self.
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From:
report of Johann Fichte (The Science of Knowing (Wissenschaftslehre) [1st ed] [1794]) by Ludwig Siep - Fichte p.58
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A reaction:
This is the idealist view. He gets 'spontaneity' from Kant, which is the mind's contribution to experience. Self-relatedness is the distinctive Fichte idea. Unity presumably means total coherence, which is typical of idealists.
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22016
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The self is not a 'thing', but what emerges from an assertion of normativity [Fichte, by Pinkard]
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Full Idea:
Fichte said the self is not a natural 'thing' but is itself a normative status, and 'it' can obtain this status, so it seems, only by an act of attributing it to itself. ...He continually identified the 'I' with 'reason' itself.
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From:
report of Johann Fichte (The Science of Knowing (Wissenschaftslehre) [1st ed] [1794]) by Terry Pinkard - German Philosophy 1760-1860 05
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A reaction:
Pinkard says Fichte gradually qualified this claim. Fichte struggled to state his view in a way that avoided obvious paradoxes. 'My mind produces decisions, so there must be someone in charge of them'? Is this transcendental?
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22065
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Fichte reduces nature to a lifeless immobility [Schlegel,F on Fichte]
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Full Idea:
Fichte reduces the non-Ego or nature to a state of constant calm, standstill, immobility, lack of all change, movement and life, that is death.
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From:
comment on Johann Fichte (The Science of Knowing (Wissenschaftslehre) [1st ed] [1794]) by Friedrich Schlegel - works vol 12 p.190
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A reaction:
The point is that Fichte's nature is a merely logical or conceptual deduction from the spontaneous reason of the self, so it can't have the lively diversity we find in nature.
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