40 ideas
| 10017 | Truth in a model is more tractable than the general notion of truth [Hodes] |
| Full Idea: Truth in a model is interesting because it provides a transparent and mathematically tractable model - in the 'ordinary' rather than formal sense of the term 'model' - of the less tractable notion of truth. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.131) | |
| A reaction: This is an important warning to those who wish to build their entire account of truth on Tarski's rigorously formal account of the term. Personally I think we should start by deciding whether 'true' can refer to the mental state of a dog. I say it can. |
| 10018 | Truth is quite different in interpreted set theory and in the skeleton of its language [Hodes] |
| Full Idea: There is an enormous difference between the truth of sentences in the interpreted language of set theory and truth in some model for the disinterpreted skeleton of that language. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.132) | |
| A reaction: This is a warning to me, because I thought truth and semantics only entered theories at the stage of 'interpretation'. I must go back and get the hang of 'skeletal' truth, which sounds rather charming. [He refers to set theory, not to logic.] |
| 15845 | It seems absurd that seeing a person's limbs, the one is many, and yet the many are one [Plato] |
| Full Idea: Someone first distinguishes a person's limbs and parts and asks your agreement that all the parts are identical with that unity, then ridicules you that you have to admit one is many, and indefinitely many, and again that the many are only only one thing. | |
| From: Plato (Philebus [c.353 BCE], 14e) | |
| A reaction: This is a passing aporia, but actually seems to approach the central mystery of the metaphysics of identity. A thing can't be a 'unity' if there are not things to unify? So what sorts of 'unification' are there? |
| 10015 | Higher-order logic may be unintelligible, but it isn't set theory [Hodes] |
| Full Idea: Brand higher-order logic as unintelligible if you will, but don't conflate it with set theory. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.131) | |
| A reaction: [he gives Boolos 1975 as a further reference] This is simply a corrective, because the conflation of second-order logic with set theory is an idea floating around in the literature. |
| 10011 | Identity is a level one relation with a second-order definition [Hodes] |
| Full Idea: Identity should he considered a logical notion only because it is the tip of a second-order iceberg - a level 1 relation with a pure second-order definition. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984]) |
| 9148 | I think of variables as objects rather than as signs [Fine,K] |
| Full Idea: It is natural nowadays to think of variables as a certain kind of sign, but I wish to think of them as a certain kind of object. | |
| From: Kit Fine (Cantorian Abstraction: Recon. and Defence [1998], §2) | |
| A reaction: Fine has a theory based on 'arbitrary objects', which is a rather charming idea. The cell of a spreadsheet is a kind of object, I suppose. A variable might be analogous to a point in space, where objects can locate themselves. |
| 10016 | When an 'interpretation' creates a model based on truth, this doesn't include Fregean 'sense' [Hodes] |
| Full Idea: A model is created when a language is 'interpreted', by assigning non-logical terms to objects in a set, according to a 'true-in' relation, but we must bear in mind that this 'interpretation' does not associate anything like Fregean senses with terms. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.131) | |
| A reaction: This seems like a key point (also made by Hofweber) that formal accounts of numbers, as required by logic, will not give an adequate account of the semantics of number-terms in natural languages. |
| 9867 | It is absurd to define a circle, but not be able to recognise a real one [Plato] |
| Full Idea: It will be ridiculous if our student knows the definition of the circle and of the divine sphere itself, but cannot recognize the human sphere and these our circles, used in housebuilding. | |
| From: Plato (Philebus [c.353 BCE], 62a) | |
| A reaction: This is the equivalent of being able to recite numbers, but not to count objects. It also resembles Molyneux's question (to Locke), of whether recognition by one sense entails recognition by others. Nice (and a bit anti-platonist!). |
| 10027 | Mathematics is higher-order modal logic [Hodes] |
| Full Idea: I take the view that (agreeing with Aristotle) mathematics only requires the notion of a potential infinity, ...and that mathematics is higher-order modal logic. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984]) | |
| A reaction: Modern 'modal' accounts of mathematics I take to be heirs of 'if-thenism', which seems to have been Russell's development of Frege's original logicism. I'm beginning to think it is right. But what is the subject-matter of arithmetic? |
| 9865 | Daily arithmetic counts unequal things, but pure arithmetic equalises them [Plato] |
| Full Idea: The arithmetic of the many computes sums of unequal units, such as two armies, or two herds, ..but philosopher's arithmetic computes when it is guaranteed that none of those infinitely many units differed in the least from any of the others. | |
| From: Plato (Philebus [c.353 BCE], 56d) | |
| A reaction: But of course 'the many' are ironing out the differences too, when they say there are 'three armies'. Shocking snob, Plato. Even philosophers are interested in the difference between three armies and three platoons. |
| 10026 | Arithmetic must allow for the possibility of only a finite total of objects [Hodes] |
| Full Idea: Arithmetic should be able to face boldly the dreadful chance that in the actual world there are only finitely many objects. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.148) | |
| A reaction: This seems to be a basic requirement for any account of arithmetic, but it was famously a difficulty for early logicism, evaded by making the existence of an infinity of objects into an axiom of the system. |
| 10021 | It is claimed that numbers are objects which essentially represent cardinality quantifiers [Hodes] |
| Full Idea: The mathematical object-theorist says a number is an object that represents a cardinality quantifier, with the representation relation as the entire essence of the nature of such objects as cardinal numbers like 4. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984]) | |
| A reaction: [compressed] This a classic case of a theory beginning to look dubious once you spell it our precisely. The obvious thought is to make do with the numerical quantifiers, and dispense with the objects. Do other quantifiers need objects to support them? |
| 10022 | Numerical terms can't really stand for quantifiers, because that would make them first-level [Hodes] |
| Full Idea: The dogmatic Frege is more right than wrong in denying that numerical terms can stand for numerical quantifiers, for there cannot be a language in which object-quantifiers and objects are simultaneously viewed as level zero. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.142) | |
| A reaction: Subtle. We see why Frege goes on to say that numbers are level zero (i.e. they are objects). We are free, it seems, to rewrite sentences containing number terms to suit whatever logical form appeals. Numbers are just quantifiers? |
| 14503 | If a mixture does not contain measure and proportion, it is corrupted and destroyed [Plato] |
| Full Idea: Any kind of mixture that does not ...possess measure or the nature of proportion will necessarily corrupt its ingredients and most of all itself. For there would be no blending in such cases but really an unconnected medley, and ruin what contains it. | |
| From: Plato (Philebus [c.353 BCE], 64d) | |
| A reaction: My guess is that Plato is thinking of the decay of living things when they die, losing the proportions of psuché, and then applying this to the unity of inanimate objects as well. One might compare Leibniz's monads. |
| 15857 | Any mixture which lacks measure and proportion doesn't even count as a mixture at all [Plato] |
| Full Idea: Any blend [mixture] which does not have measure or the nature of proportion in any way whatsoever, of necessity destroys both its ingredients and, primarily, itself. It is truly no blend at all, but a kind of unblended disaster. | |
| From: Plato (Philebus [c.353 BCE], 64e) | |
| A reaction: Obviously there can be chaotic mixtures, but I guess Plato is picking out mixtures about which we can say something |
| 10023 | Talk of mirror images is 'encoded fictions' about real facts [Hodes] |
| Full Idea: Talk about mirror images is a sort of fictional discourse. Statements 'about' such fictions are not made true or false by our whims; rather they 'encode' facts about the things reflected in mirrors. | |
| From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.146) | |
| A reaction: Hodes's proposal for how we should view abstract objects (c.f. Frege and Dummett on 'the equator'). The facts involved are concrete, but Hodes is offering 'encoding fictionalism' as a linguistic account of such abstractions. He applies it to numbers. |
| 4447 | If the good is one, is it unchanged when it is in particulars, and is it then separated from itself? [Plato] |
| Full Idea: If man is one, and the good is one, how are they supposed to exist? Do they stay the same even though they are found in many things at the same time, and are they then entirely separated from themselves, which seems most impossible of all? | |
| From: Plato (Philebus [c.353 BCE], 15a) | |
| A reaction: Presumably Plato anguishes over this because he thinks Forms are self-predicating (the Good is good). Big mistake. The Good fathers good particulars which resemble itself, but are diluted? |
| 15856 | A thing can become one or many, depending on how we talk about it [Plato] |
| Full Idea: It is through discourse that the same thing flits around, becoming one and many in all sorts of ways. | |
| From: Plato (Philebus [c.353 BCE], 15d) | |
| A reaction: This is not scepticism about wholes on Plato's part, but a reminder of an obvious fact, that in thought we can break the world up and put it back together again. It is a touchstone of the debate, though. |
| 374 | If one object is divided into its parts, someone can then say that one are many and many is one [Plato] |
| Full Idea: Someone can theoretically divide an object into constituent parts, concede that they are one object, and then claim that therefore the one is many and the many are one. | |
| From: Plato (Philebus [c.353 BCE], 14e) |
| 389 | How can you be certain about aspects of the world if they aren't constant? [Plato] |
| Full Idea: Could we attribute certainty to studying aspects of the world, such as how it arose, or acts, or is acted upon, when none of them ever was or will be constant? Of course not. | |
| From: Plato (Philebus [c.353 BCE], 59b) |
| 9152 | If green is abstracted from a thing, it is only seen as a type if it is common to many things [Fine,K] |
| Full Idea: In traditional abstraction, the colour green merely has the intrinsic property of being green, other properties of things being abstracted away. But why should that be regarded as a type? It must be because the property is common to the instances. | |
| From: Kit Fine (Cantorian Abstraction: Recon. and Defence [1998], §5) | |
| A reaction: A nice question which shows that the much-derided single act of abstraction is not sufficient to arrive at a concept, so that abstraction is a more complex matter (perhaps even a rational one) than simple empiricists believe. |
| 9149 | To obtain the number 2 by abstraction, we only want to abstract the distinctness of a pair of objects [Fine,K] |
| Full Idea: In abstracting from the elements of a doubleton to obtain 2, we do not wish to abstract away from all features of the objects. We wish to take account of the fact that the two objects are distinct; this alone should be preserved under abstraction. | |
| From: Kit Fine (Cantorian Abstraction: Recon. and Defence [1998], §3) | |
| A reaction: This is Fine's strategy for meeting Frege's objection to abstraction, summarised in Idea 9146. It seems to use the common sense idea that abstraction is not all-or-nothing. Abstraction has degrees (and levels). |
| 9150 | We should define abstraction in general, with number abstraction taken as a special case [Fine,K] |
| Full Idea: Number abstraction can be taken to be a special case of abstraction in general, which can then be defined without recourse to the concept of number. | |
| From: Kit Fine (Cantorian Abstraction: Recon. and Defence [1998], §3) | |
| A reaction: At last, a mathematical logician recognising that they don't have a monopoly on abstraction. It is perfectly obvious that abstractions of simple daily concepts must be chronologically and logically prior to number abstraction. Number of what? |
| 9146 | After abstraction all numbers seem identical, so only 0 and 1 will exist! [Fine,K] |
| Full Idea: In Cantor's abstractionist account there can only be two numbers, 0 and 1. For abs(Socrates) = abs(Plato), since their numbers are the same. So the number of {Socrates,Plato} is {abs(Soc),abs(Plato)}, which is the same number as {Socrates}! | |
| From: Kit Fine (Cantorian Abstraction: Recon. and Defence [1998], §1) | |
| A reaction: Fine tries to answer this objection, which arises from §45 of Frege's Grundlagen. Fine summarises that "indistinguishability without identity appears to be impossible". Maybe we should drop talk of numbers in terms of sets. |
| 390 | If goodness involves moderation and proportion, then it seems to be found in beauty [Plato] |
| Full Idea: Moderation and proportion seem, in effect, to be beauty and excellence. So now this property we're looking for, goodness, has taken refuge in beauty. | |
| From: Plato (Philebus [c.353 BCE], 64e) |
| 391 | The good involves beauty, proportion and truth [Plato] |
| Full Idea: If we are unable to net the good in a single concept, three must capture it: namely, beauty, proportion and truth. | |
| From: Plato (Philebus [c.353 BCE], 65a) | |
| A reaction: Very interesting. More illuminating than the discussion of the Good in 'Republic'. Is a handsome and honest murderer good? Is beauty part of the nature of the good, or a hallmark of it? |
| 392 | Neither intellect nor pleasure are the good, because they are not perfect and self-sufficient [Plato] |
| Full Idea: Both intellect and pleasure are completely absolved of being the good itself, since they both lack independence, that is, sufficiency and perfection. | |
| From: Plato (Philebus [c.353 BCE], 67a) | |
| A reaction: This seems to be Plato disagreeing with Socrates, who sees reason and intellect as central to morality. Presumable he means that the good should be a primitive. Why is pleasure not sufficient? |
| 393 | Good first, then beauty, then reason, then knowledge, then pleasure [Plato, by PG] |
| Full Idea: Good is supreme, followed by beauty, then reason, then knowledge, then pure pleasure, then mixed pleasure. | |
| From: report of Plato (Philebus [c.353 BCE], 67a) by PG - Db (ideas) | |
| A reaction: He tells us that pure pleasures are simple pleasures. Epicurus presumably read this. No mention of truth, unless that is part of reason. Why does he value beauty so highly? |
| 385 | Some of the pleasures and pains we feel are false [Plato] |
| Full Idea: Living beings experience pleasures and pains which seem, and indeed are, false. | |
| From: Plato (Philebus [c.353 BCE], 42c) | |
| A reaction: The idea that there are 'authentic' pleasures and pains needs some investigation. Misguided anger is a false pain? Vanity is a false pleasure? |
| 387 | A small pure pleasure is much finer than a large one contaminated with pain [Plato] |
| Full Idea: A tiny little pleasure is, if uncontaminated by pain, always more pleasant, truer and finer than a large amount. | |
| From: Plato (Philebus [c.353 BCE], 53b) | |
| A reaction: More Platonic puritanism. Is a complete absence of pleasure the highest pleasure of all? I don't think I understand 'truer'. Why would a pleasure be false because it is intense? |
| 373 | Pleasure is certainly very pleasant, but it doesn't follow that all pleasures are good [Plato] |
| Full Idea: The pleasantness of pleasure is not in dispute, but where we say the majority of pleasures are bad, though some are good, you are attributing goodness to all of them. | |
| From: Plato (Philebus [c.353 BCE], 13b) | |
| A reaction: Bentham's plausible view is that the feeling of pleasure is always good, and the badness is in some other aspect of the event. Compare sadistic fantasy with sadistic action. |
| 379 | The good must be sufficient and perfect, and neither intellect nor pleasure are that [Plato] |
| Full Idea: Neither pleasure nor intellect comprises the good. If it did it would have to be sufficient and perfect. | |
| From: Plato (Philebus [c.353 BCE], 22b) | |
| A reaction: Seems sensible. I can't make sense of any vision of the good which consists of suppressing some aspect of human nature. (Hm. Our capacity for violence?) |
| 371 | Reason, memory, truth and wisdom are far better than pleasure, for those who can attain them [Plato] |
| Full Idea: My contention is that reason, intellect, memory - along with correct belief and true calculation - are far better than pleasure for all creatures capable of attaining them. | |
| From: Plato (Philebus [c.353 BCE], 11b) | |
| A reaction: Why? Is it better to understand deeply, or to act well? Can we just say there is objective good and subjective good, and they have little in common? Depressed heroes. |
| 376 | Would you prefer a life of pleasure without reason, or one of reason without pleasure? [Plato] |
| Full Idea: Try thinking about the life of pleasure without reason, and the life of reason without pleasure. | |
| From: Plato (Philebus [c.353 BCE], 20e) | |
| A reaction: I suspect that we see the two as more deeply entangled that Plato did. It would be hard to motivate reasoning if we didn't enjoy it. Pleasure without reason sound dire. |
| 382 | It is unlikely that the gods feel either pleasure or pain [Plato] |
| Full Idea: It is unlikely that the gods feel pleasure or the opposite. | |
| From: Plato (Philebus [c.353 BCE], 33b) | |
| A reaction: Compare Idea 383. |
| 381 | We feel pleasure when we approach our natural state of harmony [Plato] |
| Full Idea: When harmony is being restored, and the natural state of harmony is approached, then pleasure arises. | |
| From: Plato (Philebus [c.353 BCE], 31d) | |
| A reaction: The supreme value of harmony was important to Plato, but most of us are less convinced, I suspect. The way to achieve harmony is to avoid anything stressful. |
| 386 | Intense pleasure and pain are not felt in a good body, but in a worthless one [Plato] |
| Full Idea: Intensity of pleasure and pain is to be found not in a good state of body and soul, but in a worthless one. | |
| From: Plato (Philebus [c.353 BCE], 45e) | |
| A reaction: This just seems to be Plato's puritanism. How can you criticise someone for experience genuine intense pain? Experiencing intense pleasure is no crime, but pursuit of it might be. |
| 388 | Hedonists must say that someone in pain is bad, even if they are virtuous [Plato] |
| Full Idea: A hedonist must say that someone who happens to be feeling pain rather than pleasure is, as long as the pain lasts, a bad man, even if he is the most virtuous man in the world. | |
| From: Plato (Philebus [c.353 BCE], 55b) |
| 377 | If you lived a life of maximum pleasure, would you still be lacking anything? [Plato] |
| Full Idea: Would you, Protarchus, gladly live your whole life experiencing only the greatest pleasure? Would you think you were still lacking anything? | |
| From: Plato (Philebus [c.353 BCE], 21a) | |
| A reaction: the pleasure machine problem |
| 378 | A life of pure pleasure with no intellect is the life of a jellyfish [Plato] |
| Full Idea: A life of pure pleasure with no intellect is not the life of a human being, but the life of a jellyfish. | |
| From: Plato (Philebus [c.353 BCE], 21c) |