42 ideas
| 9901 | Numbers can't be sets if there is no agreement on which sets they are [Benacerraf] |
| Full Idea: The fact that Zermelo and Von Neumann disagree on which particular sets the numbers are is fatal to the view that each number is some particular set. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], II) | |
| A reaction: I agree. A brilliantly simple argument. There is the possibility that one of the two accounts is correct (I would vote for Zermelo), but it is not actually possible to prove it. |
| 9912 | There are no such things as numbers [Benacerraf] |
| Full Idea: There are no such things as numbers. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], IIIC) | |
| A reaction: Mill said precisely the same (Idea 9794). I think I agree. There has been a classic error of reification. An abstract pattern is not an object. If I coin a word for all the three-digit numbers in our system, I haven't created a new 'object'. |
| 9151 | Benacerraf says numbers are defined by their natural ordering [Benacerraf, by Fine,K] |
| Full Idea: Benacerraf thinks of numbers as being defined by their natural ordering. | |
| From: report of Paul Benacerraf (What Numbers Could Not Be [1965]) by Kit Fine - Cantorian Abstraction: Recon. and Defence §5 | |
| A reaction: My intuition is that cardinality is logically prior to ordinality, since that connects better with the experienced physical world of objects. Just as the fact that people have different heights must precede them being arranged in height order. |
| 13891 | To understand finite cardinals, it is necessary and sufficient to understand progressions [Benacerraf, by Wright,C] |
| Full Idea: Benacerraf claims that the concept of a progression is in some way the fundamental arithmetical notion, essential to understanding the idea of a finite cardinal, with a grasp of progressions sufficing for grasping finite cardinals. | |
| From: report of Paul Benacerraf (What Numbers Could Not Be [1965]) by Crispin Wright - Frege's Concept of Numbers as Objects 3.xv | |
| A reaction: He cites Dedekind (and hence the Peano Axioms) as the source of this. The interest is that progression seems to be fundamental to ordianls, but this claims it is also fundamental to cardinals. Note that in the first instance they are finite. |
| 17904 | A set has k members if it one-one corresponds with the numbers less than or equal to k [Benacerraf] |
| Full Idea: Any set has k members if and only if it can be put into one-to-one correspondence with the set of numbers less than or equal to k. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], I) | |
| A reaction: This is 'Ernie's' view of things in the paper. This defines the finite cardinal numbers in terms of the finite ordinal numbers. He has already said that the set of numbers is well-ordered. |
| 17906 | To explain numbers you must also explain cardinality, the counting of things [Benacerraf] |
| Full Idea: I would disagree with Quine. The explanation of cardinality - i.e. of the use of numbers for 'transitive counting', as I have called it - is part and parcel of the explication of number. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], I n2) | |
| A reaction: Quine says numbers are just a progression, with transitive counting as a bonus. Interesting that Benacerraf identifies cardinality with transitive counting. I would have thought it was the possession of numerical quantity, not ascertaining it. |
| 9898 | We can count intransitively (reciting numbers) without understanding transitive counting of items [Benacerraf] |
| Full Idea: Learning number words in the right order is counting 'intransitively'; using them as measures of sets is counting 'transitively'. ..It seems possible for someone to learn the former without learning the latter. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], I) | |
| A reaction: Scruton's nice question (Idea 3907) is whether you could be said to understand numbers if you could only count intransitively. I would have thought such a state contained no understanding at all of numbers. Benacerraf agrees. |
| 17903 | Someone can recite numbers but not know how to count things; but not vice versa [Benacerraf] |
| Full Idea: It seems that it is possible for someone to learn to count intransitively without learning to count transitively. But not vice versa. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], I) | |
| A reaction: Benacerraf favours the priority of the ordinals. It is doubtful whether you have grasped cardinality properly if you don't know how to count things. Could I understand 'he has 27 sheep', without understanding the system of natural numbers? |
| 9897 | The application of a system of numbers is counting and measurement [Benacerraf] |
| Full Idea: The application of a system of numbers is counting and measurement. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], I) | |
| A reaction: A simple point, but it needs spelling out. Counting seems prior, in experience if not in logic. Measuring is a luxury you find you can indulge in (by imagining your quantity) split into parts, once you have mastered counting. |
| 9900 | For Zermelo 3 belongs to 17, but for Von Neumann it does not [Benacerraf] |
| Full Idea: Ernie's number progression is [φ],[φ,[φ]],[φ,[φ],[φ,[φ,[φ]]],..., whereas Johnny's is [φ],[[φ]],[[[φ]]],... For Ernie 3 belongs to 17, not for Johnny. For Ernie 17 has 17 members; for Johnny it has one. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], II) | |
| A reaction: Benacerraf's point is that there is no proof-theoretic way to choose between them, though I am willing to offer my intuition that Ernie (Zermelo) gives the right account. Seventeen pebbles 'contains' three pebbles; you must pass 3 to count to 17. |
| 9899 | The successor of x is either x and all its members, or just the unit set of x [Benacerraf] |
| Full Idea: For Ernie, the successor of a number x was the set consisting of x and all the members of x, while for Johnny the successor of x was simply [x], the unit set of x - the set whose only member is x. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], II) | |
| A reaction: See also Idea 9900. Benacerraf's famous point is that it doesn't seem to make any difference to arithmetic which version of set theory you choose as its basis. I take this to conclusively refute the idea that numbers ARE sets. |
| 8697 | Disputes about mathematical objects seem irrelevant, and mathematicians cannot resolve them [Benacerraf, by Friend] |
| Full Idea: If two children were brought up knowing two different set theories, they could entirely agree on how to do arithmetic, up to the point where they discuss ontology. There is no mathematical way to tell which is the true representation of numbers. | |
| From: report of Paul Benacerraf (What Numbers Could Not Be [1965]) by Michèle Friend - Introducing the Philosophy of Mathematics | |
| A reaction: Benacerraf ends by proposing a structuralist approach. If mathematics is consistent with conflicting set theories, then those theories are not shedding light on mathematics. |
| 8304 | No particular pair of sets can tell us what 'two' is, just by one-to-one correlation [Benacerraf, by Lowe] |
| Full Idea: Hume's Principle can't tell us what a cardinal number is (this is one lesson of Benacerraf's well-known problem). An infinity of pairs of sets could actually be the number two (not just the simplest sets). | |
| From: report of Paul Benacerraf (What Numbers Could Not Be [1965]) by E.J. Lowe - The Possibility of Metaphysics 10.3 | |
| A reaction: The drift here is for numbers to end up as being basic, axiomatic, indefinable, universal entities. Since I favour patterns as the basis of numbers, I think the basis might be in a pre-verbal experience, which even a bird might have, viewing its eggs. |
| 9906 | If ordinal numbers are 'reducible to' some set-theory, then which is which? [Benacerraf] |
| Full Idea: If a particular set-theory is in a strong sense 'reducible to' the theory of ordinal numbers... then we can still ask, but which is really which? | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], IIIB) | |
| A reaction: A nice question about all reductions. If we reduce mind to brain, does that mean that brain is really just mind. To have a direction (up/down?), reduction must lead to explanation in a single direction only. Do numbers explain sets? |
| 9907 | If any recursive sequence will explain ordinals, then it seems to be the structure which matters [Benacerraf] |
| Full Idea: If any recursive sequence whatever would do to explain ordinal numbers suggests that what is important is not the individuality of each element, but the structure which they jointly exhibit. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], IIIC) | |
| A reaction: This sentence launched the whole modern theory of Structuralism in mathematics. It is hard to see what properties a number-as-object could have which would entail its place in an ordinal sequence. |
| 9908 | The job is done by the whole system of numbers, so numbers are not objects [Benacerraf] |
| Full Idea: 'Objects' do not do the job of numbers singly; the whole system performs the job or nothing does. I therefore argue that numbers could not be objects at all. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], IIIC) | |
| A reaction: This thought is explored by structuralism - though it is a moot point where mere 'nodes' in a system (perhaps filled with old bits of furniture) will do the job either. No one ever explains the 'power' of numbers (felt when you do a sudoku). Causal? |
| 9909 | The number 3 defines the role of being third in a progression [Benacerraf] |
| Full Idea: Any object can play the role of 3; that is, any object can be the third element in some progression. What is peculiar to 3 is that it defines that role, not by being a paradigm, but by representing the relation of any third member of a progression. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], IIIC) | |
| A reaction: An interesting early attempt to spell out the structuralist idea. I'm thinking that the role is spelled out by the intersection of patterns which involve threes. |
| 9911 | Number words no more have referents than do the parts of a ruler [Benacerraf] |
| Full Idea: Questions of the identification of the referents of number words should be dismissed as misguided in just the way that a question about the referents of the parts of a ruler would be seen as misguided. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], IIIC) | |
| A reaction: What a very nice simple point. It would be very strange to insist that every single part of the continuum of a ruler should be regarded as an 'object'. |
| 8925 | Mathematical objects only have properties relating them to other 'elements' of the same structure [Benacerraf] |
| Full Idea: Mathematical objects have no properties other than those relating them to other 'elements' of the same structure. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], p.285), quoted by Fraser MacBride - Structuralism Reconsidered §3 n13 | |
| A reaction: Suppose we only had one number - 13 - and we all cried with joy when we recognised it in a group of objects. Would that be a number, or just a pattern, or something hovering between the two? |
| 9938 | How can numbers be objects if order is their only property? [Benacerraf, by Putnam] |
| Full Idea: Benacerraf raises the question how numbers can be 'objects' if they have no properties except order in a particular ω-sequence. | |
| From: report of Paul Benacerraf (What Numbers Could Not Be [1965], p.301) by Hilary Putnam - Mathematics without Foundations | |
| A reaction: Frege certainly didn't think that order was their only property (see his 'borehole' metaphor in Grundlagen). It might be better to say that they are objects which only have relational properties. |
| 9910 | Number-as-objects works wholesale, but fails utterly object by object [Benacerraf] |
| Full Idea: The identification of numbers with objects works wholesale but fails utterly object by object. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], IIIC) | |
| A reaction: This seems to be a glaring problem for platonists. You can stare at 1728 till you are blue in the face, but it only begins to have any properties at all once you examine its place in the system. This is unusual behaviour for an object. |
| 9903 | Number words are not predicates, as they function very differently from adjectives [Benacerraf] |
| Full Idea: The unpredicative nature of number words can be seen by noting how different they are from, say, ordinary adjectives, which do function as predicates. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], II) | |
| A reaction: He points out that 'x is seventeen' is a rare construction in English, unlike 'x is happy/green/interesting', and that numbers outrank all other adjectives (having to appear first in any string of them). |
| 9904 | The set-theory paradoxes mean that 17 can't be the class of all classes with 17 members [Benacerraf] |
| Full Idea: In no consistent theory is there a class of all classes with seventeen members. The existence of the paradoxes is a good reason to deny to 'seventeen' this univocal role of designating the class of all classes with seventeen members. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], II) | |
| A reaction: This was Frege's disaster, and seems to block any attempt to achieve logicism by translating numbers into sets. It now seems unclear whether set theory is logic, or mathematics, or sui generis. |
| 9905 | Identity statements make sense only if there are possible individuating conditions [Benacerraf] |
| Full Idea: Identity statements make sense only in contexts where there exist possible individuating conditions. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], III) | |
| A reaction: He is objecting to bizarre identifications involving numbers. An identity statement may be bizarre even if we can clearly individuate the two candidates. Winston Churchill is a Mars Bar. Identifying George Orwell with Eric Blair doesn't need a 'respect'. |
| 7337 | In Mosaic legal theory, crimes are sins and sins are crimes [Johnson,P] |
| Full Idea: In Mosaic legal theory, all breaches of the law offend God. All crimes are sins, just as all sins are crimes. | |
| From: Paul Johnson (The History of the Jews [1987], Pt I) | |
| A reaction: This would seem to define Josephus called a 'theocracy'. Not just rule by a priesthood, but also an attempt to make civil law coincide with the teachings of sacred texts. But doing 80 m.p.h. on a motorway at 2 a.m. hardly seems like a sin. |
| 7339 | Because human life is what is sacred, Mosaic law has no death penalty for property violations [Johnson,P] |
| Full Idea: Where other codes provided the death penalty for offences against property, in Mosaic law no property offence is capital; human life is too sacred, where the rights of property alone are violated. | |
| From: Paul Johnson (The History of the Jews [1987], Pt I) | |
| A reaction: We still preserve this idea in our law, and also in our culture, where we are keen to insist that catastrophes like earthquakes or major fires are measured almost entirely by the loss of life, not the loss of property. I approve. |
| 7353 | The Pharisees undermined slavery, by giving slaves responsibility and status in law courts [Johnson,P] |
| Full Idea: It is no accident that slavery among Jews disappeared with the rise of the Pharisees, as they insisted that all were equal before God in a court. Masters were no longer responsible for actions of slaves, so a slave had status, and slavery could not work. | |
| From: Paul Johnson (The History of the Jews [1987], Pt II) | |
| A reaction: As in seventeenth century England, the rise of social freedom comes from religious sources, not social sources. A slave has status in the transcendent world of souls, despite being a nobody in the physical world. |
| 7340 | Mosaic law was the first to embody the rule of law, and equality before the law [Johnson,P] |
| Full Idea: Mosaic law meant that God ruled through his laws, and since all were equally subject to the law, the system was the first to embody the double merits of the rule of law and equality before the law. | |
| From: Paul Johnson (The History of the Jews [1987], Pt I) | |
| A reaction: If this is correct, it seems to be a hugely important step, combined with Idea 1659, that revenge should be the action of a the state, not of the individual. They are the few simple and essential keys to civilization. |
| 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. |
| 7338 | Man's life is sacred, because it is made in God's image [Johnson,P] |
| Full Idea: In Mosaic theology, man is made in God's image, and so his life is not just valuable, it is sacred. | |
| From: Paul Johnson (The History of the Jews [1987], Pt I) | |
| A reaction: The obvious question is what exactly is meant by "in God's image". Physically, spiritually, intellectually, morally? I am guessing that the original idea was intellectual, because we are the only rational animal. The others seem unlikely, or arrogant. |
| 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. |
| 7348 | The Jews sharply distinguish human and divine, but the Greeks pull them closer together [Johnson,P] |
| Full Idea: The Jews drew an absolute distinction between the human and the divine; the Greeks constantly elevated the human - they were Promethean - and lowered the divine. | |
| From: Paul Johnson (The History of the Jews [1987], Pt II) | |
| A reaction: An intriguing observation. The Greek idea runs right through European culture, surfacing (for example) in 'Faust', or 'Frankenstein', or the films of James Cameron. I'm with the Greeks; I want to see how far humanity can be elevated. |
| 7355 | The Torah pre-existed creation, and was its blueprint [Johnson,P] |
| Full Idea: The Torah was not just a book about God. It pre-existed creation, in the same way as God did. In fact, it was the blueprint of creation. | |
| From: Paul Johnson (The History of the Jews [1987], Pt III) | |
| A reaction: You can only become a 'people of the book' (which Moslems resented in Judaism, and then emulated) if you give this stupendously high status to your book. Hence Christian fundamentalism makes sense, with its emphasis on the divinity of the Bible. |
| 7336 | A key moment is the idea of a single moral God, who imposes his morality on humanity [Johnson,P] |
| Full Idea: The discovery of monotheism, and not just of monotheism but of a sole, omnipotent God actuated by ethical principles and seeking methodically to impose them on human beings, is one of the greatest turning-points in history, perhaps the greatest of all. | |
| From: Paul Johnson (The History of the Jews [1987], Pt I) | |
| A reaction: 'Discovery' begs some questions, but when put like this you realise what a remarkable event it was. It is a good candidate for the most influential idea ever, even if large chunks of humanity, especially in the orient, never took to monotheism. |
| 7341 | Sampson illustrates the idea that religious heroes often begin as outlaws and semi-criminals [Johnson,P] |
| Full Idea: Sampson is the outstanding example of the point which the Book of Judges makes again and again, that the Lord and society are often served by semi-criminal types, outlaws and misfits, who become folk-heroes and then religious heroes. | |
| From: Paul Johnson (The History of the Jews [1987], Pt I) | |
| A reaction: This illustrates nicely Nietzsche's claim, that the jews were responsible for his 'inversion of values', in which aristocratic virtues are downgraded, and the virtues of a good slave are elevated (though Sampson may not show that point so well!). |
| 7342 | Isaiah moved Israelite religion away from the local, onto a more universal plane [Johnson,P] |
| Full Idea: The works of Isaiah (740-700 BCE) mark the point at which the Israelite religion began to spiritualize itself, to move from a specific location in space and time on to the universalist plane. | |
| From: Paul Johnson (The History of the Jews [1987], Pt I) | |
| A reaction: This is necessary if any religion is going to make converts outside the local culture. The crucial step would be to disembody God, so that He cannot be represented by a statue. The difficulty is for him to be universal, but retain a 'chosen people'. |
| 7344 | Judaism involves circumcision, Sabbath, Passover, Pentecost, Tabernacles, New Year, and Atonement [Johnson,P] |
| Full Idea: The practices of Judaism developed during their Exile: circumcision, the Sabbath, the Passover (founding of the nation), Pentecost (giving of the laws), the Tabernacles, the New Year, and the Day of Atonement. | |
| From: Paul Johnson (The History of the Jews [1987], Pt II) | |
| A reaction: These were the elements of ritual created to replace the existence of a physically located state. An astonishing achievement, not even remotely achieved by any other state that was driven off its lands. A culture is an idea, not a country. |
| 7345 | In exile the Jews became a nomocracy [Johnson,P] |
| Full Idea: In exile the Jews, deprived of a state, became a nomocracy - voluntarily submitting to rule by a Law which could only be enforced by consent. Nothing like this had occurred before in history. | |
| From: Paul Johnson (The History of the Jews [1987], Pt II) | |
| A reaction: It is the most remarkable case in history of a people united and strengthened by adversity, and it became an important experiment in the building of human cultures. But what is the point of preserving a culture, with no land? Why not just integrate? |
| 7347 | Zoroastrians believed in one eternal beneficent being, Creator through the holy spirit [Johnson,P] |
| Full Idea: Cyrus the Great was a Zoroastrian, believing in one, eternal, beneficent being, 'Creator of all things through the holy spirit'. | |
| From: Paul Johnson (The History of the Jews [1987], Pt II) | |
| A reaction: Is this the actual origin of monotheism, or did they absorb this idea from the Jews? The interesting bit is the fact that the supreme being (called Marduk) is 'beneficent', which one doesn't associate with these remote and supposed pagans. |
| 7349 | Immortality based on judgement of merit was developed by the Egyptians (not the Jews) [Johnson,P] |
| Full Idea: The idea of judgement at death and immortality on the basis of merit were developed in Egypt before 1000 BCE. It is not Jewish because it was not in the Torah, and the Sadducees, who stuck to their texts, seemed to have denied the afterlife completely. | |
| From: Paul Johnson (The History of the Jews [1987], Pt II) | |
| A reaction: This is the idea considered crucial to religion by Immanuel Kant (Idea 1455), who should be declared an honorary Egyptian. To me the idea that only the good go to heaven sounds like sadly wishful thinking - a fictional consolation for an unhappy life. |
| 7354 | The main doctrine of the Pharisees was belief in resurrection and the afterlife [Johnson,P] |
| Full Idea: Belief in resurrection and the afterlife was the main distinguishing mark of Pharisaism, and thus fundamental of rabbinic Judaism. | |
| From: Paul Johnson (The History of the Jews [1987], Pt II) | |
| A reaction: Belief in an afterlife seems to go back to the Egyptians, but this development in Judaism was obviously very influential, even among early Christians, who initially seem to have only believed in resurrection of the body. |
| 7356 | Pious Jews saw heaven as a vast library [Johnson,P] |
| Full Idea: Pious Jews saw heaven as a vast library, with the Archangel Metatron as the librarian: the books in the shelves there pressed themselves together to make room for a newcomer. | |
| From: Paul Johnson (The History of the Jews [1987], Pt III) | |
| A reaction: I'm tempted to convert to Judaism. |