Combining Texts

All the ideas for 'Mahaprajnaparamitashastra', 'A Theory of Universals' and 'Number Determiners, Numbers, Arithmetic'

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18 ideas

4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / g. System S4
If what is actual might have been impossible, we need S4 modal logic [Armstrong, by Lewis]
     Full Idea: Armstrong says what is actual (namely a certain roster of universals) might have been impossible. Hence his modal logic is S4, without the 'Brouwersche Axiom'.
     From: report of David M. Armstrong (A Theory of Universals [1978]) by David Lewis - Armstrong on combinatorial possibility 'The demand'
     A reaction: So p would imply possibly-not-possibly-p.
5. Theory of Logic / F. Referring in Logic / 1. Naming / d. Singular terms
An adjective contributes semantically to a noun phrase [Hofweber]
     Full Idea: The semantic value of a determiner (an adjective) is a function from semantic values to nouns to semantic values of full noun phrases.
     From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §3.1)
     A reaction: This kind of states the obvious (assuming one has a compositional view of sentences), but his point is that you can't just eliminate adjectival uses of numbers by analysing them away, as if they didn't do anything.
5. Theory of Logic / G. Quantification / 2. Domain of Quantification
Quantifiers for domains and for inference come apart if there are no entities [Hofweber]
     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.
     From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §6.3)
     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.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
'2 + 2 = 4' can be read as either singular or plural [Hofweber]
     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').
     From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §4.1)
     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.
What is the relation of number words as singular-terms, adjectives/determiners, and symbols? [Hofweber]
     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?
     From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §1)
     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?
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
Why is arithmetic hard to learn, but then becomes easy? [Hofweber]
     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.
     From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §4.2)
     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.
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
Arithmetic is not about a domain of entities, as the quantifiers are purely inferential [Hofweber]
     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.
     From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §6.3)
     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.
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
Arithmetic doesn’t simply depend on objects, since it is true of fictional objects [Hofweber]
     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.
     From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §6.2)
     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.
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
We might eliminate adjectival numbers by analysing them into blocks of quantifiers [Hofweber]
     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.
     From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §2)
     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.
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
First-order logic captures the inferential relations of numbers, but not the semantics [Hofweber]
     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.
     From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §6.3)
     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.
8. Modes of Existence / B. Properties / 1. Nature of Properties
Properties are universals, which are always instantiated [Armstrong, by Heil]
     Full Idea: Armstrong takes properties to be universals, and believes there are no 'uninstantiated' universals.
     From: report of David M. Armstrong (A Theory of Universals [1978]) by John Heil - From an Ontological Point of View §9.3
     A reaction: At first glance this, like many theories of universals, seems to invite Ockham's Razor. If they are always instantiated, perhaps we should perhaps just try to talk about the instantiations (i.e. tropes), and skip the universal?
8. Modes of Existence / B. Properties / 6. Categorical Properties
Even if all properties are categorical, they may be denoted by dispositional predicates [Armstrong, by Bird]
     Full Idea: Armstrong says all properties are categorical, but a dispositional predicate may denote such a property; the dispositional predicate denotes the categorical property in virtue of the dispositional role it happens, contingently, to play in this world.
     From: report of David M. Armstrong (A Theory of Universals [1978]) by Alexander Bird - Nature's Metaphysics 3.1
     A reaction: I favour the fundamentality of the dispositional rather than the categorical. The world consists of powers, and we find ourselves amidst their categorical expressions. I could be persuaded otherwise, though!
8. Modes of Existence / D. Universals / 2. Need for Universals
Universals explain resemblance and causal power [Armstrong, by Oliver]
     Full Idea: Armstrong thinks universals play two roles, namely grounding objective resemblances and grounding causal powers.
     From: report of David M. Armstrong (A Theory of Universals [1978]) by Alex Oliver - The Metaphysics of Properties 11
     A reaction: Personally I don't think universals explain anything at all. They just add another layer of confusion to a difficult problem. Oliver objects that this seems a priori, contrary to Armstrong's principle in Idea 10728.
8. Modes of Existence / E. Nominalism / 3. Predicate Nominalism
It doesn't follow that because there is a predicate there must therefore exist a property [Armstrong]
     Full Idea: I suggest that we reject the notion that just because the predicate 'red' applies to an open class of particulars, therefore there must be a property, redness.
     From: David M. Armstrong (A Theory of Universals [1978], p.8), quoted by DH Mellor / A Oliver - Introduction to 'Properties' §6
     A reaction: At last someone sensible (an Australian) rebuts that absurd idea that our ontology is entirely a feature of our language
9. Objects / F. Identity among Objects / 4. Type Identity
The type-token distinction is the universal-particular distinction [Armstrong, by Hodes]
     Full Idea: Armstrong conflates the type-token distinction with that between universals and particulars.
     From: report of David M. Armstrong (A Theory of Universals [1978], xiii,16/17) by Harold Hodes - Logicism and Ontological Commits. of Arithmetic 147 n23
     A reaction: This seems quite reasonable, even if you don’t believe in the reality of universals. I'm beginning to think we should just use the term 'general' instead of 'universal'. 'Type' also seems to correspond to 'set', with the 'token' as the 'member'.
9. Objects / F. Identity among Objects / 5. Self-Identity
A thing's self-identity can't be a universal, since we can know it a priori [Armstrong, by Oliver]
     Full Idea: Armstrong says that if it can be proved a priori that a thing falls under a certain universal, then there is no such universal - and hence there is no universal of a thing being identical with itself.
     From: report of David M. Armstrong (A Theory of Universals [1978], II p.11) by Alex Oliver - The Metaphysics of Properties 11
     A reaction: This is a distinctively Armstrongian view, based on his belief that universals must be instantiated, and must be discoverable a posteriori, as part of science. I'm baffled by self-identity, but I don't think this argument does the job.
15. Nature of Minds / C. Capacities of Minds / 4. Objectification
Our minds are at their best when reasoning about objects [Hofweber]
     Full Idea: Our minds mainly reason about objects. Most cognitive problems we are faced with deal with particular objects, whether they are people or material things. Reasoning about them is what our minds are good at.
     From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §4.3)
     A reaction: Hofweber is suggesting this as an explanation of why we continually reify various concepts, especially numbers. Very plausible. It works for qualities of character, and explains our tendency to talk about universals as objects ('redness').
23. Ethics / C. Virtue Theory / 3. Virtues / a. Virtues
The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna]
     Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom.
     From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88)
     A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate').