72 ideas
| 14456 | 'Socrates is human' expresses predication, and 'Socrates is a man' expresses identity [Russell] |
| Full Idea: The is of 'Socrates is human' expresses the relation of subject and predicate; the is of 'Socrates is a man' expresses identity. It is a disgrace to the human race that it employs the same word 'is' for these entirely different ideas. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVI) | |
| A reaction: Does the second one express identity? It sounds more like membership to me. 'Socrates is the guy with the hemlock' is more like identity. |
| 14426 | A definition by 'extension' enumerates items, and one by 'intension' gives a defining property [Russell] |
| Full Idea: The definition of a class or collection which enumerates is called a definition by 'extension', and one which mentions a defining property is called a definition by 'intension'. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], II) | |
| A reaction: In ordinary usage we take intensional definitions for granted, so it is interesting to realise that you might define 'tiger' by just enumerating all the tigers. But all past tigers? All future tigers? All possible tigers which never exist? |
| 8468 | The sentence 'procrastination drinks quadruplicity' is meaningless, rather than false [Russell, by Orenstein] |
| Full Idea: Russell proposed (in his theory of types) that sentences like 'The number two is fond of cream cheese' or 'Procrastination drinks quadruplicity' should be regarded as not false but meaningless. | |
| From: report of Bertrand Russell (Introduction to Mathematical Philosophy [1919]) by Alex Orenstein - W.V. Quine Ch.3 | |
| A reaction: This seems to be the origin of the notion of a 'category mistake', which Ryle made famous. The problem is always poetry, where abstractions can be reified, or personified, and meaning can be squeezed out of almost anything. |
| 14454 | An argument 'satisfies' a function φx if φa is true [Russell] |
| Full Idea: We say that an argument a 'satisfies' a function φx if φa is true. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XV) | |
| A reaction: We end up with Tarski defining truth in terms of satisfaction, so we shouldn't get too excited about what he achieved (any more than he got excited). |
| 14453 | The Darapti syllogism is fallacious: All M is S, all M is P, so some S is P' - but if there is no M? [Russell] |
| Full Idea: Some moods of the syllogism are fallacious, e.g. 'Darapti': 'All M is S, all M is P, therefore some S is P', which fails if there is no M. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XV) | |
| A reaction: This critique rests on the fact that the existential quantifier entails some existence, but the universal quantifier does not. |
| 9572 | Realists about sets say there exists a null set in the real world, with no members [Chihara] |
| Full Idea: In the Gödelian realistic view of set theory the statement that there is a null set as the assertion of the existence in the real world of a set that has no members. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 11.6) | |
| A reaction: It seems to me obvious that such a claim is nonsense on stilts. 'In the beginning there was the null set'? |
| 9550 | We only know relational facts about the empty set, but nothing intrinsic [Chihara] |
| Full Idea: Everything we know about the empty set is relational; we know that nothing is the membership relation to it. But what do we know about its 'intrinsic properties'? | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 01.5) | |
| A reaction: Set theory seems to depend on the concept of the empty set. Modern theorists seem over-influenced by the Quine-Putnam view, that if science needs it, we must commit ourselves to its existence. |
| 9562 | In simple type theory there is a hierarchy of null sets [Chihara] |
| Full Idea: In simple type theory, there is a null set of type 1, a null set of type 2, a null set of type 3..... (Quine has expressed his distaste for this). | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 07.4) | |
| A reaction: It is bad enough trying to individuate the unique null set, without whole gangs of them drifting indistinguishably through the logical fog. All rational beings should share Quine's distaste, even if Quine is wrong. |
| 9573 | The null set is a structural position which has no other position in membership relation [Chihara] |
| Full Idea: In the structuralist view of sets, in structures of a certain sort the null set is taken to be a position (or point) that will be such that no other position (or point) will be in the membership relation to it. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 11.6) | |
| A reaction: It would be hard to conceive of something having a place in a structure if nothing had a relation to it, so is the null set related to singeton sets but not there members. It will be hard to avoid Platonism here. Set theory needs the null set. |
| 9551 | What is special about Bill Clinton's unit set, in comparison with all the others? [Chihara] |
| Full Idea: What is it about the intrinsic properties of just that one unit set in virtue of which Bill Clinton is related to just it and not to any other unit sets in the set-theoretical universe? | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 01.5) | |
| A reaction: If we all kept pet woodlice, we had better not hold a wood louse rally, or we might go home with the wrong one. My singleton seems seems remarkably like yours. Could we, perhaps, swap, just for a change? |
| 14427 | We can enumerate finite classes, but an intensional definition is needed for infinite classes [Russell] |
| Full Idea: We know a great deal about a class without enumerating its members …so definition by extension is not necessary to knowledge about a class ..but enumeration of infinite classes is impossible for finite beings, so definition must be by intension. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], II) | |
| A reaction: Presumably mathematical induction (which keeps apply the rule to extend the class) will count as an intension here. |
| 14428 | Members define a unique class, whereas defining characteristics are numerous [Russell] |
| Full Idea: There is only one class having a given set of members, whereas there are always many different characteristics by which a given class may be defined. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], II) |
| 14447 | Infinity says 'for any inductive cardinal, there is a class having that many terms' [Russell] |
| Full Idea: The Axiom of Infinity may be enunciated as 'If n be any inductive cardinal number, there is at least one class of individuals having n terms'. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XIII) | |
| A reaction: So for every possible there exists a set of terms for it. Notice that they are 'terms', not 'objects'. We must decide whether we are allowed terms which don't refer to real objects. |
| 14440 | We may assume that there are infinite collections, as there is no logical reason against them [Russell] |
| Full Idea: There is no logical reason against infinite collections, and we are therefore justified, in logic, in investigating the hypothesis that there are such collections. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], VIII) |
| 14443 | The British parliament has one representative selected from each constituency [Russell] |
| Full Idea: We have a class of representatives, who make up our Parliament, one being selected out of each constituency. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XII) | |
| A reaction: You can rely on Russell for the clearest illustrations of these abstract ideas. He calls the Axiom of Choice the 'Multiplicative' Axiom. |
| 14445 | Choice shows that if any two cardinals are not equal, one must be the greater [Russell] |
| Full Idea: The [Axiom of Choice] is also equivalent to the assumption that of any two cardinals which are not equal, one must be the greater. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XII) | |
| A reaction: It is illuminating for the uninitiated to learn that this result can't be taken for granted (with infinite cardinals). |
| 14444 | Choice is equivalent to the proposition that every class is well-ordered [Russell] |
| Full Idea: Zermelo has shown that [the Axiom of Choice] is equivalent to the proposition that every class is well-ordered, i.e. can be arranged in a series in which every sub-class has a first term (except, of course, the null class). | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XII) | |
| A reaction: Russell calls Choice the 'Multiplicative' Axiom. |
| 14446 | We can pick all the right or left boots, but socks need Choice to insure the representative class [Russell] |
| Full Idea: Among boots we distinguish left and right, so we can choose all the right or left boots; with socks no such principle suggests itself, and we cannot be sure, without the [Axiom of Choice], that there is a class consisting of one sock from each pair. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XII) | |
| A reaction: A deservedly famous illustration of a rather tricky part of set theory. |
| 14459 | Reducibility: a family of functions is equivalent to a single type of function [Russell] |
| Full Idea: The Axiom of Reducibility says 'There is a type of a-functions such that, given any a-function, it is formally equivalent to some function of the type in question'. ..It involves all that is really essential in the theory of classes. But is it true? | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVII) | |
| A reaction: I take this to say that in the theory of types, it is possible to reduce each level of type down to one type. |
| 9549 | The set theorist cannot tell us what 'membership' is [Chihara] |
| Full Idea: The set theorist cannot tell us anything about the true relationship of membership. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 01.5) | |
| A reaction: If three unrelated objects suddenly became members of a set, it is hard to see how the world would have changed, except in the minds of those thinking about it. |
| 14461 | Propositions about classes can be reduced to propositions about their defining functions [Russell] |
| Full Idea: It is right (in its main lines) to say that there is a reduction of propositions nominally about classes to propositions about their defining functions. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVII) | |
| A reaction: The defining functions will involve the theory of types, in order to avoid the paradoxes of naďve set theory. This is Russell's strategy for rejecting the existence of sets. |
| 9571 | ZFU refers to the physical world, when it talks of 'urelements' [Chihara] |
| Full Idea: ZFU set theory talks about physical objects (the urelements), and hence is in some way about the physical world. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 11.5) | |
| A reaction: This sounds a bit surprising, given that the whole theory would appear to be quite unaffected if God announced that idealism is true and there are no physical objects. |
| 8469 | Russell's proposal was that only meaningful predicates have sets as their extensions [Russell, by Orenstein] |
| Full Idea: Russell's solution (in the theory of types) consists of restricting the principle that every predicate has a set as its extension so that only meaningful predicates have sets as their extensions. | |
| From: report of Bertrand Russell (Introduction to Mathematical Philosophy [1919]) by Alex Orenstein - W.V. Quine Ch.3 | |
| A reaction: There might be a chicken-and-egg problem here. How do you decide the members of a set (apart from ostensively) without deciding the predicate(s) that combine them? |
| 8745 | Classes are logical fictions, and are not part of the ultimate furniture of the world [Russell] |
| Full Idea: The symbols for classes are mere conveniences, not representing objects called 'classes'. Classes are in fact logical fictions; they cannot be regarded as part of the ultimate furniture of the world. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], Ch.18), quoted by Stewart Shapiro - Thinking About Mathematics 5.2 | |
| A reaction: I agree. For 'logical fictions' read 'abstractions'. To equate abstractions with fictions is to underline the fact that they are a human creation. They are either that or platonic objects - there is no middle way. |
| 9563 | A pack of wolves doesn't cease when one member dies [Chihara] |
| Full Idea: A pack of wolves is not thought to go out of existence just because some member of the pack is killed. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 07.5) | |
| A reaction: The point is that the formal extensional notion of a set doesn't correspond to our common sense notion of a group or class. Even a highly scientific theory about wolves needs a loose notion of a wolf pack. |
| 14452 | All the propositions of logic are completely general [Russell] |
| Full Idea: It is part of the definition of logic that all its propositions are completely general. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XV) |
| 14462 | In modern times, logic has become mathematical, and mathematics has become logical [Russell] |
| Full Idea: Logic has become more mathematical, and mathematics has become more logical. The consequence is that it has now become wholly impossible to draw a line between the two; in fact, the two are one. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVIII) | |
| A reaction: This appears to be true even if you reject logicism about mathematics. Logicism is sometimes rejected because it always ends up with a sneaky ontological commitment, but maybe mathematics shares exactly the same commitment. |
| 10057 | Logic can only assert hypothetical existence [Russell] |
| Full Idea: No proposition of logic can assert 'existence' except under a hypothesis. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVIII) | |
| A reaction: I am prepared to accept this view fairly dogmatically, though Musgrave shows some of the difficulties of the if-thenist view (depending on which 'order' of logic is being used). |
| 12444 | Logic is concerned with the real world just as truly as zoology [Russell] |
| Full Idea: Logic is concerned with the real world just as truly as zoology, though with its more abstract and general features. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVI) | |
| A reaction: I love this idea and am very sympathetic to it. The rival view seems to be that logic is purely conventional, perhaps defined by truth tables etc. It is hard to see how a connective like 'tonk' could be self-evidently silly if it wasn't 'unnatural'. |
| 14464 | Logic can be known a priori, without study of the actual world [Russell] |
| Full Idea: Logical propositions are such as can be known a priori, without study of the actual world. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVIII) | |
| A reaction: This remark constrasts strikingly with Idea 12444, which connects logic to the actual world. Is it therefore a priori synthetic? |
| 9561 | The mathematics of relations is entirely covered by ordered pairs [Chihara] |
| Full Idea: Everything one needs to do with relations in mathematics can be done by taking a relation to be a set of ordered pairs. (Ordered triples etc. can be defined as order pairs, so that <x,y,z> is <x,<y,z>>). | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 07.2) | |
| A reaction: How do we distinguish 'I own my cat' from 'I love my cat'? Or 'I quite like my cat' from 'I adore my cat'? Nevertheless, this is an interesting starting point for a discussion of relations. |
| 14458 | Asking 'Did Homer exist?' is employing an abbreviated description [Russell] |
| Full Idea: When we ask whether Homer existed, we are using the word 'Homer' as an abbreviated description. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVI) | |
| A reaction: It is hard to disagree with Russell over this rather unusual example. It doesn't seem so plausible when Ottiline refers to 'Bertie'. |
| 10450 | Russell admitted that even names could also be used as descriptions [Russell, by Bach] |
| Full Idea: Russell clearly anticipated Donnellan when he said proper names can also be used as descriptions, adding that 'there is nothing in the phraseology to show whether they are being used in this way or as names'. | |
| From: report of Bertrand Russell (Introduction to Mathematical Philosophy [1919], p.175) by Kent Bach - What Does It Take to Refer? 22.2 L1 | |
| A reaction: This seems also to anticipate Strawson's flexible and pragmatic approach to these things, which I am beginning to think is correct. |
| 14457 | Names are really descriptions, except for a few words like 'this' and 'that' [Russell] |
| Full Idea: We can even say that, in all such knowledge as can be expressed in words, with the exception of 'this' and 'that' and a few other words of which the meaning varies on different occasions - no names occur, but what seem like names are really descriptions. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVI) | |
| A reaction: I like the caveat about what is expressed in words. Russell is very good at keeping non-verbal thought in the picture. This is his famous final reduction of names to simple demonstratives. |
| 7311 | The only genuine proper names are 'this' and 'that' [Russell] |
| Full Idea: In all knowledge that can be expressed in words - with the exception of "this" and "that", and a few other such words - no genuine proper names occur, but what seem like genuine proper names are really descriptions | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVI) | |
| A reaction: This is the terminus of Russell's train of thought about descriptions. Suppose you point to something non-existent, like a ghost in a misty churchyard? You'd be back to the original problem of naming a non-existent! |
| 14455 | 'I met a unicorn' is meaningful, and so is 'unicorn', but 'a unicorn' is not [Russell] |
| Full Idea: In 'I met a unicorn' the four words together make a significant proposition, and the word 'unicorn' is significant, …but the two words 'a unicorn' do not form a group having a meaning of its own. It is an indefinite description describing nothing. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVI) |
| 9552 | Sentences are consistent if they can all be true; for Frege it is that no contradiction can be deduced [Chihara] |
| Full Idea: In first-order logic a set of sentences is 'consistent' iff there is an interpretation (or structure) in which the set of sentences is true. ..For Frege, though, a set of sentences is consistent if it is not possible to deduce a contradiction from it. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 02.1) | |
| A reaction: The first approach seems positive, the second negative. Frege seems to have a higher standard, which is appealing, but the first one seems intuitively right. There is a possible world where this could work. |
| 14442 | If straight lines were like ratios they might intersect at a 'gap', and have no point in common [Russell] |
| Full Idea: We wish to say that when two straight lines cross each other they have a point in common, but if the series of points on a line were similar to the series of ratios, the two lines might cross in a 'gap' and have no point in common. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], X) | |
| A reaction: You can make a Dedekind Cut in the line of ratios (the rationals), so there must be gaps. I love this idea. We take for granted intersection at a point, but physical lines may not coincide. That abstract lines might fail also is lovely! |
| 14438 | New numbers solve problems: negatives for subtraction, fractions for division, complex for equations [Russell] |
| Full Idea: Every generalisation of number has presented itself as needed for some simple problem. Negative numbers are needed to make subtraction always possible; fractions to make division always possible; complex numbers to make solutions of equations possible. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], VII) | |
| A reaction: Doesn't this rather suggest that we made them up? If new problems turn up, we'll invent another lot. We already have added 'surreal' numbers. |
| 13510 | Could a number just be something which occurs in a progression? [Russell, by Hart,WD] |
| Full Idea: Russell toyed with the idea that there is nothing to being a natural number beyond occurring in a progression | |
| From: report of Bertrand Russell (Introduction to Mathematical Philosophy [1919], p.8) by William D. Hart - The Evolution of Logic 5 | |
| A reaction: How could you define a progression, without a prior access to numbers? - Arrange all the objects in the universe in ascending order of mass. Use scales to make the selection. Hence a finite progression, with no numbers! |
| 14436 | A series can be 'Cut' in two, where the lower class has no maximum, the upper no minimum [Russell] |
| Full Idea: There is no maximum to the ratios whose square is less than 2, and no minimum to those whose square is greater than 2. This division of a series into two classes is called a 'Dedekind Cut'. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], VII) |
| 14439 | A complex number is simply an ordered couple of real numbers [Russell] |
| Full Idea: A complex number may be regarded and defined as simply an ordered couple of real numbers | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], VII) |
| 14421 | Discovering that 1 is a number was difficult [Russell] |
| Full Idea: The discovery that 1 is a number must have been difficult. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], I) | |
| A reaction: Interesting that he calls it a 'discovery'. I am tempted to call it a 'decision'. |
| 14424 | Numbers are needed for counting, so they need a meaning, and not just formal properties [Russell] |
| Full Idea: We want our numbers to be such as can be used for counting common objects, and this requires that our numbers should have a definite meaning, not merely that they should have certain formal properties. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], I) | |
| A reaction: Why would just having certain formal properties be insufficient for counting? You just need an ordered series of unique items. It isn't just that we 'want' this. If you define something that we can't count with, you haven't defined numbers. |
| 14441 | The formal laws of arithmetic are the Commutative, the Associative and the Distributive [Russell] |
| Full Idea: The usual formal laws of arithmetic are the Commutative Law [a+b=b+a and axb=bxa], the Associative Law [(a+b)+c=a+(b+c) and (axb)xc=ax(bxc)], and the Distributive Law [a(b+c)=ab+ac)]. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], IX) |
| 14420 | Infinity and continuity used to be philosophy, but are now mathematics [Russell] |
| Full Idea: The nature of infinity and continuity belonged in former days to philosophy, but belongs now to mathematics. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], Pref) | |
| A reaction: It is hard to disagree, since mathematicians since Cantor have revealed so much about infinite numbers (through set theory), but I think it remains an open question whether philosophers have anything distinctive to contribute. |
| 9553 | Analytic geometry gave space a mathematical structure, which could then have axioms [Chihara] |
| Full Idea: With the invention of analytic geometry (by Fermat and then Descartes) physical space could be represented as having a mathematical structure, which could eventually lead to its axiomatization (by Hilbert). | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 02.3) | |
| A reaction: The idea that space might have axioms seems to be pythagoreanism run riot. I wonder if there is some flaw at the heart of Einstein's General Theory because of this? |
| 14431 | The definition of order needs a transitive relation, to leap over infinite intermediate terms [Russell] |
| Full Idea: Order must be defined by means of a transitive relation, since only such a relation is able to leap over an infinite number of intermediate terms. ...Without it we would not be able to define the order of magnitude among fractions. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], IV) |
| 14422 | Any founded, non-repeating series all reachable in steps will satisfy Peano's axioms [Russell] |
| Full Idea: Given any series which is endless, contains no repetitions, has a beginning, and has no terms that cannot be reached from the beginning in a finite number of steps, we have a set of terms verifying Peano's axioms. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], I) |
| 14423 | '0', 'number' and 'successor' cannot be defined by Peano's axioms [Russell] |
| Full Idea: That '0', 'number' and 'successor' cannot be defined by means of Peano's five axioms, but must be independently understood. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], I) |
| 14425 | A number is something which characterises collections of the same size [Russell] |
| Full Idea: The number 3 is something which all trios have in common, and which distinguishes them from other collections. A number is something that characterises certain collections, namely, those that have that number. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], II) | |
| A reaction: This is a verbal summary of the Fregean view of numbers, which marks the arrival of set theory as the way arithmetic will in future be characterised. The question is whether set theory captures all aspects of numbers. Does it give a tool for counting? |
| 14434 | What matters is the logical interrelation of mathematical terms, not their intrinsic nature [Russell] |
| Full Idea: What matters in mathematics is not the intrinsic nature of our terms, but the logical nature of their interrelations. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], VI) | |
| A reaction: If they have an instrinsic nature, that would matter far more, because that would dictate the interrelations. Structuralism seems to require that they don't actually have any intrinsic nature. |
| 10192 | We can replace existence of sets with possibility of constructing token sentences [Chihara, by MacBride] |
| Full Idea: Chihara's 'constructability theory' is nominalist - mathematics is reducible to a simple theory of types. Instead of talk of sets {x:x is F}, we talk of open sentences Fx defining them. Existence claims become constructability of sentence tokens. | |
| From: report of Charles Chihara (A Structural Account of Mathematics [2004]) by Fraser MacBride - Review of Chihara's 'Structural Acc of Maths' p.81 | |
| A reaction: This seems to be approaching the problem in a Fregean way, by giving an account of the semantics. Chihara is trying to evade the Quinean idea that assertion is ontological commitment. But has Chihara retreated too far? How does he assert existence? |
| 14465 | Maybe numbers are adjectives, since 'ten men' grammatically resembles 'white men' [Russell] |
| Full Idea: 'Ten men' is grammatically the same form as 'white men', so that 10 might be thought to be an adjective qualifying 'men'. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVIII) | |
| A reaction: The immediate problem, as Frege spotted, is that such expressions can be rephrased to remove the adjective (by saying 'the number of men is ten'). |
| 13414 | For Russell, numbers are sets of equivalent sets [Russell, by Benacerraf] |
| Full Idea: Russell's own stand was that numbers are really only sets of equivalent sets. | |
| From: report of Bertrand Russell (Introduction to Mathematical Philosophy [1919]) by Paul Benacerraf - Logicism, Some Considerations (PhD) p.168 | |
| A reaction: Benacerraf is launching a nice attack on this view, based on our inability to grasp huge numbers on this basis, or to see their natural order. |
| 14449 | There is always something psychological about inference [Russell] |
| Full Idea: There is always unavoidably something psychological about inference. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XIV) | |
| A reaction: Glad to find Russell saying that. Only pure Fregeans dream of a logic that rises totally above the minds that think it. See Robert Hanna on the subject. |
| 14463 | Existence can only be asserted of something described, not of something named [Russell] |
| Full Idea: Existence can only be asserted of something described, not of something named. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVIII) | |
| A reaction: This is the motivation behind Russell's theory of definite descriptions, and epitomises the approach to ontology through language. Sounds wrong to me! |
| 14429 | Classes are logical fictions, made from defining characteristics [Russell] |
| Full Idea: Classes may be regarded as logical fictions, manufactured out of defining characteristics. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], II n1) | |
| A reaction: I agree with this. The idea that in addition to the members there is a further object, the set containing them, is absurd. Sets are a tool for thinking about the world. |
| 9559 | If a successful theory confirms mathematics, presumably a failed theory disconfirms it? [Chihara] |
| Full Idea: If mathematics shares whatever confirmation accrues to the theories using it, would it not be reasonable to suppose that mathematics shares whatever disconfirmation accrues to the theories using it? | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 05.8) | |
| A reaction: Presumably Quine would bite the bullet here, although maths is much closer to the centre of his web of belief, and so far less likely to require adjustment. In practice, though, mathematics is not challenged whenever an experiment fails. |
| 9566 | No scientific explanation would collapse if mathematical objects were shown not to exist [Chihara] |
| Full Idea: Evidently, no scientific explanations of specific phenomena would collapse as a result of any hypothetical discovery that no mathematical objects exist. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 09.1) | |
| A reaction: It is inconceivable that anyone would challenge this claim. A good model seems to be drama; a play needs commitment from actors and audience, even when we know it is fiction. The point is that mathematics doesn't collapse either. |
| 14430 | If a relation is symmetrical and transitive, it has to be reflexive [Russell] |
| Full Idea: It is obvious that a relation which is symmetrical and transitive must be reflexive throughout its domain. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], II) | |
| A reaction: Compare Idea 13543! The relation will return to its originator via its neighbours, rather than being directly reflexive? |
| 14432 | 'Asymmetry' is incompatible with its converse; a is husband of b, so b can't be husband of a [Russell] |
| Full Idea: The relation of 'asymmetry' is incompatible with the converse. …The relation 'husband' is asymmetrical, so that if a is the husband of b, b cannot be the husband of a. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], V) | |
| A reaction: This is to be contrasted with 'non-symmetrical', where there just happens to be no symmetry. |
| 14435 | The essence of individuality is beyond description, and hence irrelevant to science [Russell] |
| Full Idea: The essence of individuality always eludes words and baffles description, and is for that very reason irrelevant to science. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], VI) | |
| A reaction: [context needed for a full grasp of this idea] Russell seems to refer to essence as much as to individuality. The modern essentialist view is that essences are not beyond description after all. Fundamental physics is clearer now than in 1919. |
| 12197 | Inferring q from p only needs p to be true, and 'not-p or q' to be true [Russell] |
| Full Idea: In order that it be valid to infer q from p, it is only necessary that p should be true and that the proposition 'not-p or q' should be true. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XIV) | |
| A reaction: Rumfitt points out that this approach to logical consequences is a denial of any modal aspect, such as 'logical necessity'. Russell observes that for a good inference you must know the disjunction as a whole. Could disjunction be modal?... |
| 14450 | All forms of implication are expressible as truth-functions [Russell] |
| Full Idea: There is no need to admit as a fundamental notion any form of implication not expressible as a truth-function. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XIV) | |
| A reaction: Note that this is from a book about 'mathematical' philosophy. Nevertheless, it seems to have the form of a universal credo for Russell. He wasn't talking about conditionals here. Maybe conditionals are not implications (in isolation, that is). |
| 14460 | If something is true in all possible worlds then it is logically necessary [Russell] |
| Full Idea: Saying that the axiom of reducibility is logically necessary is what would be meant by saying that it is true in all possible worlds. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVII) | |
| A reaction: This striking remark is a nice bridge between Leibniz (about whom Russell wrote a book) and Kripke. |
| 14433 | Mathematically expressed propositions are true of the world, but how to interpret them? [Russell] |
| Full Idea: We know that certain scientific propositions - often expressed in mathematical symbols - are more or less true of the world, but we are very much at sea as to the interpretation to be put upon the terms which occur in these propositions. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], VI) | |
| A reaction: Enter essentialism, say I! Russell's remark is pretty understandable in 1919, but I don't think the situation has changed much. The problem of interpretation may be of more interest to philosophers than to physicists. |
| 9568 | I prefer the open sentences of a Constructibility Theory, to Platonist ideas of 'equivalence classes' [Chihara] |
| Full Idea: What I refer to as an 'equivalence class' (of line segments of a particular length) is an open sentence in my Constructibility Theory. I just use this terminology of the Platonist for didactic purposes. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 09.10) | |
| A reaction: This is because 'equivalence classes' is committed to the existence of classes, which is Quinean Platonism. I am with Chihara in wanting a story that avoids such things. Kit Fine is investigating similar notions of rules of construction. |
| 9547 | Mathematical entities are causally inert, so the causal theory of reference won't work for them [Chihara] |
| Full Idea: Causal theories of reference seem doomed to failure for the case of reference to mathematical entities, since such entities are evidently causally inert. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 01.3) | |
| A reaction: Presumably you could baptise a fictional entity such as 'Polonius', and initiate a social causal chain, with a tradition of reference. You could baptise a baby in absentia. |
| 14451 | Propositions are mainly verbal expressions of true or false, and perhaps also symbolic thoughts [Russell] |
| Full Idea: We mean by 'proposition' primarily a form of words which expresses what is either true or false. I say 'primarily' because I do not wish to exclude other than verbal symbols, or even mere thoughts if they have a symbolic character. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XV) | |
| A reaction: I like the last bit, as I think of propositions as pre-verbal thoughts, and I am sympathetic to Fodor's 'language of thought' thesis, that there is a system of representations within the brain. |
| 1590 | The just man does not harm his enemies, but benefits everyone [Plato] |
| Full Idea: First, Socrates, you told me justice is harming your enemies and helping your friends. But later it seemed that the just man, since everything he does is for someone's benefit, never harms anyone. | |
| From: Plato (Clitophon [c.372 BCE], 410b) | |
| A reaction: Socrates certainly didn't subscribe to the first view, which is the traditional consensus in Greek culture. In general Socrates agreed with the views later promoted by Jesus. |
| 9574 | 'Gunk' is an individual possessing no parts that are atoms [Chihara] |
| Full Idea: An 'atomless gunk' is defined to be an individual possessing no parts that are atoms. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], App A) | |
| A reaction: [Lewis coined it] If you ask what are a-toms made of and what are ideas made of, the only answer we can offer is that the a-toms are made of gunk, and the ideas aren't made of anything, which is still bad news for the existence of ideas. |