21 ideas
| 10001 | 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. |
| 10007 | 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. |
| 12456 | I aim to establish certainty for mathematical methods [Hilbert] |
| Full Idea: The goal of my theory is to establish once and for all the certitude of mathematical methods. | |
| From: David Hilbert (On the Infinite [1925], p.184) | |
| A reaction: This is the clearest statement of the famous Hilbert Programme, which is said to have been brought to an abrupt end by Gödel's Incompleteness Theorems. |
| 12461 | We believe all mathematical problems are solvable [Hilbert] |
| Full Idea: The thesis that every mathematical problem is solvable - we are all convinced that it really is so. | |
| From: David Hilbert (On the Infinite [1925], p.200) | |
| A reaction: This will include, for example, Goldbach's Conjecture (every even is the sum of two primes), which is utterly simple but with no proof anywhere in sight. |
| 10002 | '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. |
| 9998 | 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? |
| 9633 | No one shall drive us out of the paradise the Cantor has created for us [Hilbert] |
| Full Idea: No one shall drive us out of the paradise the Cantor has created for us. | |
| From: David Hilbert (On the Infinite [1925], p.191), quoted by James Robert Brown - Philosophy of Mathematics | |
| A reaction: This is Hilbert's famous refusal to accept any account of mathematics, such as Kant's, which excludes actual infinities. Cantor had laid out a whole glorious hierarchy of different infinities. |
| 12460 | We extend finite statements with ideal ones, in order to preserve our logic [Hilbert] |
| Full Idea: To preserve the simple formal rules of ordinary Aristotelian logic, we must supplement the finitary statements with ideal statements. | |
| From: David Hilbert (On the Infinite [1925], p.195) | |
| A reaction: I find very appealing the picture of mathematics as rooted in the physical world, and then gradually extended by a series of 'idealisations', which should perhaps be thought of as fictions. |
| 12462 | Only the finite can bring certainty to the infinite [Hilbert] |
| Full Idea: Operating with the infinite can be made certain only by the finitary. | |
| From: David Hilbert (On the Infinite [1925], p.201) | |
| A reaction: See 'Compactness' for one aspect of this claim. I think Hilbert was fighting a rearguard action, and his idea now has few followers. |
| 12455 | The idea of an infinite totality is an illusion [Hilbert] |
| Full Idea: Just as in the limit processes of the infinitesimal calculus, the infinitely large and small proved to be a mere figure of speech, so too we must realise that the infinite in the sense of an infinite totality, used in deductive methods, is an illusion. | |
| From: David Hilbert (On the Infinite [1925], p.184) | |
| A reaction: This is a very authoritative rearguard action. I no longer think the dispute matters much, it being just a dispute over a proposed new meaning for the word 'number'. |
| 12457 | There is no continuum in reality to realise the infinitely small [Hilbert] |
| Full Idea: A homogeneous continuum which admits of the sort of divisibility needed to realise the infinitely small is nowhere to be found in reality. | |
| From: David Hilbert (On the Infinite [1925], p.186) | |
| A reaction: He makes this remark as a response to Planck's new quantum theory (the year before the big works of Heisenberg and Schrödinger). Personally I don't see why infinities should depend on the physical world, since they are imaginary. |
| 10003 | 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. |
| 10008 | 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. |
| 10005 | 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. |
| 10000 | 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. |
| 10006 | 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. |
| 12459 | The subject matter of mathematics is immediate and clear concrete symbols [Hilbert] |
| Full Idea: The subject matter of mathematics is the concrete symbols themselves whose structure is immediately clear and recognisable. | |
| From: David Hilbert (On the Infinite [1925], p.192) | |
| A reaction: I don't think many people will agree with Hilbert here. Does he mean token-symbols or type-symbols? You can do maths in your head, or with different symbols. If type-symbols, you have to explain what a type is. |
| 18112 | Mathematics divides in two: meaningful finitary statements, and empty idealised statements [Hilbert] |
| Full Idea: We can conceive mathematics to be a stock of two kinds of formulas: first, those to which the meaningful communications of finitary statements correspond; and secondly, other formulas which signify nothing and which are ideal structures of our theory. | |
| From: David Hilbert (On the Infinite [1925], p.196), quoted by David Bostock - Philosophy of Mathematics 6.1 |
| 9636 | My theory aims at the certitude of mathematical methods [Hilbert] |
| Full Idea: The goal of my theory is to establish once and for all the certitude of mathematical methods. | |
| From: David Hilbert (On the Infinite [1925], p.184), quoted by James Robert Brown - Philosophy of Mathematics Ch.5 | |
| A reaction: This dream is famous for being shattered by Gödel's Incompleteness Theorem a mere six years later. Neverless there seem to be more limited certainties which are accepted in mathematics. The certainty of the whole of arithmetic is beyond us. |
| 10004 | 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'). |
| 7903 | 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'). |