17 ideas
| 16489 | Is it possible to state every possible truth about the whole course of nature without using 'not'? [Russell] |
| Full Idea: Imagine a person who knew everything that can be stated without using the word 'not' or some equivalent; would such a person know the whole course of nature, or would he not? | |
| From: Bertrand Russell (Human Knowledge: its scope and limits [1948], 9) | |
| A reaction: Nowadays we might express Russell's thought as 'Does God need the word 'not'?'. Russell's thesis is that such words concern psychology, and not physics. God would need 'not' to describe how human minds work. |
| 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. |
| 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? |
| 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. |
| 16490 | Some facts about experience feel like logical necessities [Russell] |
| Full Idea: The impossibility of seeing two colours simultaneously in a given direction feels like a logical impossibility. | |
| From: Bertrand Russell (Human Knowledge: its scope and limits [1948], 9) | |
| A reaction: I presume all necessities feel equally necessary. If we distinguish necessities by what gives rise to them (a view I favour) then how strong they 'feel' will be irrelevant. We can see why Russell is puzzled by the phenomenon, though. |
| 16488 | It is hard to explain how a sentence like 'it is not raining' can be found true by observation [Russell] |
| Full Idea: If 'it is not raining' means 'the sentence "it is raining" is false', that makes it almost impossible to understand how a sentence containing the word 'not' can be found true by observation. | |
| From: Bertrand Russell (Human Knowledge: its scope and limits [1948], 9) | |
| A reaction: Russell goes on to explore the general difficulty of deciding negative truths by observation. The same problem arises for truthmaker theory. Obviously I can observe that it isn't raining, but it seems parasitic on observing when it is raining. |
| 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'). |
| 16491 | If we define 'this is not blue' as disbelief in 'this is blue', we eliminate 'not' as an ingredient of facts [Russell] |
| Full Idea: We can reintroduce 'not' by a definition: the words 'this is not blue' are defined as expressing disbelief in what is expressed by the words 'this is blue'. In this way the need of 'not' as an indefinable constituent of facts is avoided. | |
| From: Bertrand Russell (Human Knowledge: its scope and limits [1948], 9) | |
| A reaction: This is part of Russell's programme of giving a psychological account of logical connectives. See other ideas from his 1940 and 1948 works. He observes that disbelief is a state just as positive as belief. I love it. |
| 13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
| Full Idea: In a single day there lies open to men of learning more than there ever does to the unenlightened in the longest of lifetimes. | |
| From: Posidonius (fragments/reports [c.95 BCE]), quoted by Seneca the Younger - Letters from a Stoic 078 | |
| A reaction: These remarks endorsing the infinite superiority of the educated to the uneducated seem to have been popular in late antiquity. It tends to be the religions which discourage great learning, especially in their emphasis on a single book. |
| 4786 | Russell's 'at-at' theory says motion is to be at the intervening points at the intervening instants [Russell, by Psillos] |
| Full Idea: To reply to Zeno's Arrow Paradox, Russell developed his 'at-at' theory of motion, which says that to move from A to B is to be at the intervening points at the intervening instants. | |
| From: report of Bertrand Russell (Human Knowledge: its scope and limits [1948]) by Stathis Psillos - Causation and Explanation §4.2 | |
| A reaction: I wonder whether Russell's target was actually Zeno, or was it a simplified ontology of points and instants? The ontology will also need identity, to ensure it is the same thing which arrives at each point. |
| 20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
| Full Idea: Posidonius defined time thus: it is an interval of motion, or the measure of speed and slowness. | |
| From: report of Posidonius (fragments/reports [c.95 BCE]) by John Stobaeus - Anthology 1.08.42 | |
| A reaction: Hm. Can we define motion or speed without alluding to time? Looks like we have to define them as a conjoined pair, which means we cannot fully understand either of them. |