155 ideas
| 14122 | Analysis gives us nothing but the truth - but never the whole truth [Russell] |
| Full Idea: Though analysis gives us the truth, and nothing but the truth, yet it can never give us the whole truth | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §138) |
| 14109 | The study of grammar is underestimated in philosophy [Russell] |
| Full Idea: The study of grammar, in my opinion, is capable of throwing far more light on philosophical questions than is commonly supposed by philosophers. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §046) | |
| A reaction: This is a dangerous tendency, which has led to some daft linguistic philosophy, but Russell himself was never guilty of losing the correct perspective on the matter. |
| 14165 | Analysis falsifies, if when the parts are broken down they are not equivalent to their sum [Russell] |
| Full Idea: It is said that analysis is falsification, that the complex is not equivalent to the sum of its constituents and is changed when analysed into these. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §439) | |
| A reaction: Not quite Moore's Paradox of Analysis, but close. Russell is articulating the view we now call 'holism' - that the whole is more than the sum of its parts - which I can never quite believe. |
| 6779 | Instrumentalists say distinctions between observation and theory vanish with ostensive definition [Bird] |
| Full Idea: Instrumentalists treat the theoretical/non-theoretical and the observational/non-observational distinctions as the same, ..because they think words get their meaning by way of ostensive definition. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.4) | |
| A reaction: To be honest, I'm not sure I quite understand this, but it sounds interesting... Ostensive definition seems to match the pragmatic spirit of instrumentalism (for which, see Idea 6778). Bird explains it all more fully. |
| 14115 | Definition by analysis into constituents is useless, because it neglects the whole [Russell] |
| Full Idea: A definition as an analysis of an idea into its constituents is inconvenient and, I think, useless; it overlooks the fact that wholes are not, as a rule, determinate when their constituents are given. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §108) | |
| A reaction: The influence of Leibniz seems rather strong here, since he was obsessed with explaining what creates true unities. |
| 14159 | In mathematics definitions are superfluous, as they name classes, and it all reduces to primitives [Russell] |
| Full Idea: The statement that a class is to be represented by a symbol is a definition in mathematics, and says nothing about mathematical entities. Any formula can be stated in terms of primitive ideas, so the definitions are superfluous. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §412) | |
| A reaction: [compressed wording] I'm not sure that everyone would agree with this (e.g. Kit Fine), as certain types of numbers seem to be introduced by stipulative definitions. |
| 14148 | Infinite regresses have propositions made of propositions etc, with the key term reappearing [Russell] |
| Full Idea: In the objectionable kind of infinite regress, some propositions join to constitute the meaning of some proposition, but one of them is similarly compounded, and so ad infinitum. This comes from circular definitions, where the term defined reappears. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §329) |
| 18002 | As well as a truth value, propositions have a range of significance for their variables [Russell] |
| Full Idea: Every proposition function …has, in addition to its range of truth, a range of significance, i.e. a range within which x must lie if φ(x) is to be a proposition at all, whether true or false. This is the first point of the theory of types. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], App B:523), quoted by Ofra Magidor - Category Mistakes 1.2 | |
| A reaction: Magidor quotes this as the origin of the idea of a 'category mistake'. It is the basis of the formal theory of types, but is highly influential in philosophy generally, especially as a criterion for ruling many propositions as 'meaningless'. |
| 14102 | What is true or false is not mental, and is best called 'propositions' [Russell] |
| Full Idea: I hold that what is true or false is not in general mental, and requiring a name for the true or false as such, this name can scarcely be other than 'propositions'. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], Pref) | |
| A reaction: This is the Fregean and logicians' dream that that there is some fixed eternal realm of the true and the false. I think true and false concern the mental. We can talk about the 'facts' which are independent of minds, but not the 'truth'. |
| 14176 | "The death of Caesar is true" is not the same proposition as "Caesar died" [Russell] |
| Full Idea: "The death of Caesar is true" is not, I think, the same proposition as "Caesar died". | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §478) | |
| A reaction: I suspect that it was this remark which provoked Ramsey into rebellion, because he couldn't see the difference. Nowadays we must talk first of conversational implicature, and then of language and metalanguage. |
| 14113 | The null class is a fiction [Russell] |
| Full Idea: The null class is a fiction. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §079) | |
| A reaction: This does not commit him to regarding all classes as fictions - though he seems to have eventually come to believe that. The null class seems to have a role something like 'Once upon a time...' in story-telling. You can then tell truth or fiction. |
| 15894 | Russell invented the naïve set theory usually attributed to Cantor [Russell, by Lavine] |
| Full Idea: Russell was the inventor of the naïve set theory so often attributed to Cantor. | |
| From: report of Bertrand Russell (The Principles of Mathematics [1903]) by Shaughan Lavine - Understanding the Infinite I |
| 14126 | Order rests on 'between' and 'separation' [Russell] |
| Full Idea: The two sources of order are 'between' and 'separation'. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §204) |
| 14127 | Order depends on transitive asymmetrical relations [Russell] |
| Full Idea: All order depends upon transitive asymmetrical relations. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §208) |
| 14121 | The part-whole relation is ultimate and indefinable [Russell] |
| Full Idea: The relation of whole and part is, it would seem, an indefinable and ultimate relation, or rather several relations, often confounded, of which one at least is indefinable. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §135) | |
| A reaction: This is before anyone had produced a mathematical account of mereology (qv). |
| 14106 | Implication cannot be defined [Russell] |
| Full Idea: A definition of implication is quite impossible. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §016) |
| 14108 | It would be circular to use 'if' and 'then' to define material implication [Russell] |
| Full Idea: It would be a vicious circle to define material implication as meaning that if one proposition is true, then another is true, for 'if' and 'then' already involve implication. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §037) | |
| A reaction: Hence the preference for defining it by the truth table, or as 'not-p or q'. |
| 14167 | The only classes are things, predicates and relations [Russell] |
| Full Idea: The only classes appear to be things, predicates and relations. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §440) | |
| A reaction: This is the first-order logic view of reality, which has begun to look incredibly impoverished in modern times. Processes certainly demand a hearing, as do modal facts. |
| 14105 | There seem to be eight or nine logical constants [Russell] |
| Full Idea: The number of logical constants is not great: it appears, in fact, to be eight or nine. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §012) | |
| A reaction: There is, of course, lots of scope for interdefinability. No one is going to disagree greatly with his claim, so it is an interesting fact, which invites some sort of (non-platonic) explanation. |
| 18722 | Negations are not just reversals of truth-value, since that can happen without negation [Wittgenstein on Russell] |
| Full Idea: Russell explained ¬p by saying that ¬p is true if p is false and false if p is true. But this is not an explanation of negation, for it might apply to propositions other than the negative. | |
| From: comment on Bertrand Russell (The Principles of Mathematics [1903]) by Ludwig Wittgenstein - Lectures 1930-32 (student notes) B XI.3 | |
| A reaction: Presumably he is thinking of 'the light is on' and 'the light is off'. A very astute criticism, which seems to be correct. What would Russell say? Perhaps we add that negation is an 'operation' which achieves flipping of the truth-value? |
| 14104 | Constants are absolutely definite and unambiguous [Russell] |
| Full Idea: A constant is something absolutely definite, concerning which there is no ambiguity whatever. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §006) |
| 14114 | Variables don't stand alone, but exist as parts of propositional functions [Russell] |
| Full Idea: A variable is not any term simply, but any term as entering into a propositional function. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §093) | |
| A reaction: So we should think of variables entirely by their role, rather than as having a semantics of their own (pace Kit Fine? - though see Russell §106, p.107). |
| 14137 | 'Any' is better than 'all' where infinite classes are concerned [Russell] |
| Full Idea: The word 'any' is preferable to the word 'all' where infinite classes are concerned. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §284) | |
| A reaction: The reason must be that it is hard to quantify over 'all' of the infinite members, but it is easier to say what is true of any one of them. |
| 14149 | The Achilles Paradox concerns the one-one correlation of infinite classes [Russell] |
| Full Idea: When the Achilles Paradox is translated into arithmetical language, it is seen to be concerned with the one-one correlation of two infinite classes. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §321) | |
| A reaction: Dedekind's view of infinity (Idea 9826) shows why this results in a horrible tangle. |
| 15895 | Russell discovered the paradox suggested by Burali-Forti's work [Russell, by Lavine] |
| Full Idea: Burali-Forti didn't discover any paradoxes, though his work suggested a paradox to Russell. | |
| From: report of Bertrand Russell (The Principles of Mathematics [1903]) by Shaughan Lavine - Understanding the Infinite I |
| 14152 | In geometry, Kant and idealists aimed at the certainty of the premisses [Russell] |
| Full Idea: The approach to practical geometry of the idealists, and especially of Kant, was that we must be certain of the premisses on their own account. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §353) |
| 14154 | Geometry throws no light on the nature of actual space [Russell] |
| Full Idea: Geometry no longer throws any direct light on the nature of actual space. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §353) | |
| A reaction: This was 1903. Minkowski then contributed a geometry of space which was used in Einstein's General Theory. It looks to me as if geometry reveals the possibilities for actual space. |
| 14151 | Pure geometry is deductive, and neutral over what exists [Russell] |
| Full Idea: As a branch of pure mathematics, geometry is strictly deductive, indifferent to the choice of its premises, and to the question of whether there strictly exist such entities. It just deals with series of more than one dimension. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §352) | |
| A reaction: This seems to be the culmination of the seventeenth century reduction of geometry to algebra. Russell admits that there is also the 'study of actual space'. |
| 14153 | In geometry, empiricists aimed at premisses consistent with experience [Russell] |
| Full Idea: The approach to practical geometry of the empiricists, notably Mill, was to show that no other set of premisses would give results consistent with experience. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §353) | |
| A reaction: The modern phrase might be that geometry just needs to be 'empirically adequate'. The empiricists are faced with the possibility of more than one successful set of premisses, and the idealist don't know how to demonstrate truth. |
| 14155 | Two points have a line joining them (descriptive), a distance (metrical), and a whole line (projective) [Russell, by PG] |
| Full Idea: Two points will define the line that joins them ('descriptive' geometry), the distance between them ('metrical' geometry), and the whole of the extended line ('projective' geometry). | |
| From: report of Bertrand Russell (The Principles of Mathematics [1903], §362) by PG - Db (ideas) | |
| A reaction: [a summary of Russell's §362] Projective Geometry clearly has the highest generality, and the modern view seems to make it the master subject of geometry. |
| 18254 | Russell's approach had to treat real 5/8 as different from rational 5/8 [Russell, by Dummett] |
| Full Idea: Russell defined the rationals as ratios of integers, and was therefore forced to treat the real number 5/8 as an object distinct from the rational 5/8. | |
| From: report of Bertrand Russell (The Principles of Mathematics [1903]) by Michael Dummett - Frege philosophy of mathematics 21 'Frege's' |
| 14144 | Ordinals result from likeness among relations, as cardinals from similarity among classes [Russell] |
| Full Idea: Ordinal numbers result from likeness among relations, as cardinals from similarity among classes. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §293) |
| 14128 | Some claim priority for the ordinals over cardinals, but there is no logical priority between them [Russell] |
| Full Idea: It is claimed that ordinals are prior to cardinals, because they form the progression which is relevant to mathematics, but they both form progressions and have the same ordinal properties. There is nothing to choose in logical priority between them. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §230) | |
| A reaction: We have an intuitive notion of the size of a set without number, but you can't actually start counting without number, so the ordering seems to be the key to the business, which (I would have thought) points to ordinals as prior. |
| 14129 | Ordinals presuppose two relations, where cardinals only presuppose one [Russell] |
| Full Idea: Ordinals presuppose serial and one-one relations, whereas cardinals only presuppose one-one relations. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §232) | |
| A reaction: This seems to award the palm to the cardinals, for their greater logical simplicity, but I have already given the award to the ordinals in the previous idea, and I am not going back on that. |
| 14132 | Properties of numbers don't rely on progressions, so cardinals may be more basic [Russell] |
| Full Idea: The properties of number must be capable of proof without appeal to the general properties of progressions, since cardinals can be independently defined, and must be seen in a progression before theories of progression are applied to them. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §243) | |
| A reaction: Russell says there is no logical priority between ordinals and cardinals, but it is simpler to start an account with cardinals. |
| 14141 | Ordinals are defined through mathematical induction [Russell] |
| Full Idea: The ordinal numbers are defined by some relation to mathematical induction. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §290) |
| 14142 | Ordinals are types of series of terms in a row, rather than the 'nth' instance [Russell] |
| Full Idea: The finite ordinals may be conceived as types of series; ..the ordinal number may be taken as 'n terms in a row'; this is distinct from the 'nth', and logically prior to it. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §290) | |
| A reaction: Worth nothing, because the popular and traditional use of 'ordinal' (as in learning a foreign language) is to mean the nth instance of something, rather than a whole series. |
| 14139 | Transfinite ordinals don't obey commutativity, so their arithmetic is quite different from basic arithmetic [Russell] |
| Full Idea: Unlike the transfinite cardinals, the transfinite ordinals do not obey the commutative law, and their arithmetic is therefore quite different from elementary arithmetic. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §290) |
| 14145 | For Cantor ordinals are types of order, not numbers [Russell] |
| Full Idea: In his most recent article Cantor speaks of ordinals as types of order, not as numbers. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §298) | |
| A reaction: Russell likes this because it supports his own view of ordinals as classes of serial relations. It has become orthodoxy to refer to heaps of things as 'numbers' when the people who introduced them may not have seen them that way. |
| 14146 | We aren't sure if one cardinal number is always bigger than another [Russell] |
| Full Idea: We do not know that of any two different cardinal numbers one must be the greater. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §300) | |
| A reaction: This was 1903, and I don't know whether the situation has changed. I find this thought extremely mind-boggling, given that cardinals are supposed to answer the question 'how many?' Presumably they can't be identical either. See Burali-Forti. |
| 14135 | Real numbers are a class of rational numbers (and so not really numbers at all) [Russell] |
| Full Idea: Real numbers are not really numbers at all, but something quite different; ...a real number, so I shall contend, is nothing but a certain class of rational numbers. ...A segment of rationals is a real number. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §258) |
| 14123 | Some quantities can't be measured, and some non-quantities are measurable [Russell] |
| Full Idea: Some quantities cannot be measured (such as pain), and some things which are not quantities can be measured (such as certain series). | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §150) |
| 14158 | Quantity is not part of mathematics, where it is replaced by order [Russell] |
| Full Idea: Quantity, though philosophers seem to think it essential to mathematics, does not occur in pure mathematics, and does occur in many cases not amenable to mathematical treatment. The place of quantity is taken by order. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §405) | |
| A reaction: He gives pain as an example of a quantity which cannot be treated mathematically. |
| 14120 | Counting explains none of the real problems about the foundations of arithmetic [Russell] |
| Full Idea: The process of counting gives us no indication as to what the numbers are, as to why they form a series, or as to how it is to be proved that there are n numbers from 1 to n. Hence counting is irrelevant to the foundations of arithmetic. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §129) | |
| A reaction: I take it to be the first truth in the philosophy of mathematics that if there is a system of numbers which won't do the job of counting, then that system is irrelevant. Counting always comes first. |
| 14118 | We can define one-to-one without mentioning unity [Russell] |
| Full Idea: It is possible, without the notion of unity, to define what is meant by one-to-one. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §109) | |
| A reaction: This is the trick which enables the Greek account of numbers, based on units, to be abandoned. But when you have arranged the boys and the girls one-to-one, you have not yet got a concept of number. |
| 14119 | We do not currently know whether, of two infinite numbers, one must be greater than the other [Russell] |
| Full Idea: It is not at present known whether, of two different infinite numbers, one must be greater and the other less. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §118) | |
| A reaction: This must refer to cardinal numbers, as ordinal numbers have an order. The point is that the proper subset is equal to the set (according to Dedekind). |
| 14133 | There are cardinal and ordinal theories of infinity (while continuity is entirely ordinal) [Russell] |
| Full Idea: The theory of infinity has two forms, cardinal and ordinal, of which the former springs from the logical theory of numbers; the theory of continuity is purely ordinal. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §249) |
| 14134 | Infinite numbers are distinguished by disobeying induction, and the part equalling the whole [Russell] |
| Full Idea: There are two differences of infinite numbers from finite: that they do not obey mathematical induction (both cardinals and ordinals), and that the whole contains a part consisting of the same number of terms (applying only to ordinals). | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §250) |
| 14143 | ω names the whole series, or the generating relation of the series of ordinal numbers [Russell] |
| Full Idea: The ordinal representing the whole series must be different from what represents a segment of itself, with no immediate predecessor, since the series has no last term. ω names the class progression, or generating relation of series of this class. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §291) | |
| A reaction: He is paraphrasing Cantor's original account of ω. |
| 14138 | You can't get a new transfinite cardinal from an old one just by adding finite numbers to it [Russell] |
| Full Idea: It must not be supposed that we can obtain a new transfinite cardinal by merely adding one to it, or even by adding any finite number, or aleph-0. On the contrary, such puny weapons cannot disturb the transfinite cardinals. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §288) | |
| A reaction: If you add one, the original cardinal would be a subset of the new one, and infinite numbers have their subsets equal to the whole, so you have gone nowhere. You begin to wonder whether transfinite cardinals are numbers at all. |
| 14140 | For every transfinite cardinal there is an infinite collection of transfinite ordinals [Russell] |
| Full Idea: For every transfinite cardinal there is an infinite collection of transfinite ordinals, although the cardinal number of all ordinals is the same as or less than that of all cardinals. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §290) | |
| A reaction: Sort that one out, and you are beginning to get to grips with the world of the transfinite! Sounds like there are more ordinals than cardinals, and more cardinals than ordinals. |
| 14124 | Axiom of Archimedes: a finite multiple of a lesser magnitude can always exceed a greater [Russell] |
| Full Idea: The Axiom of Archimedes asserts that, given any two magnitudes of a kind, some finite multiple of the lesser exceeds the greater. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §168 n*) |
| 7530 | Russell tried to replace Peano's Postulates with the simple idea of 'class' [Russell, by Monk] |
| Full Idea: What Russell tried to show [at this time] was that Peano's Postulates (based on 'zero', 'number' and 'successor') could in turn be dispensed with, and the whole edifice built upon nothing more than the notion of 'class'. | |
| From: report of Bertrand Russell (The Principles of Mathematics [1903]) by Ray Monk - Bertrand Russell: Spirit of Solitude Ch.4 | |
| A reaction: (See Idea 5897 for Peano) Presumably you can't afford to lose the notion of 'successor' in the account. If you build any theory on the idea of classes, you are still required to explain why a particular is a member of that class, and not another. |
| 18246 | Dedekind failed to distinguish the numbers from other progressions [Shapiro on Russell] |
| Full Idea: Dedekind's demonstrations nowhere - not even where he comes to cardinals - involve any property distinguishing numbers from other progressions. | |
| From: comment on Bertrand Russell (The Principles of Mathematics [1903], p.249) by Stewart Shapiro - Philosophy of Mathematics 5.4 | |
| A reaction: Shapiro notes that his sounds like Frege's Julius Caesar problem, of ensuring that your definition really does capture a number. Russell is objecting to mathematical structuralism. |
| 14147 | Denying mathematical induction gave us the transfinite [Russell] |
| Full Idea: The transfinite was obtained by denying mathematical induction. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §310) | |
| A reaction: This refers to the work of Dedekind and Cantor. This raises the question (about which thinkers have ceased to care, it seems), of whether it is rational to deny mathematical induction. |
| 14125 | Finite numbers, unlike infinite numbers, obey mathematical induction [Russell] |
| Full Idea: Finite numbers obey the law of mathematical induction: infinite numbers do not. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §183) |
| 14116 | Numbers were once defined on the basis of 1, but neglected infinities and + [Russell] |
| Full Idea: It used to be common to define numbers by means of 1, with 2 being 1+1 and so on. But this method was only applicable to finite numbers, made a tiresome different between 1 and the other numbers, and left + unexplained. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §109) | |
| A reaction: Am I alone in hankering after the old approach? The idea of a 'unit' is what connected numbers to the patterns of the world. Russell's approach invites unneeded platonism. + is just 'and', and infinities are fictional extrapolations. Sounds fine to me. |
| 14117 | Numbers are properties of classes [Russell] |
| Full Idea: Numbers are to be regarded as properties of classes. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §109) | |
| A reaction: If properties are then defined extensionally as classes, you end up with numbers as classes of classes. |
| 9977 | Ordinals can't be defined just by progression; they have intrinsic qualities [Russell] |
| Full Idea: It is impossible that the ordinals should be, as Dedekind suggests, nothing but the terms of such relations as constitute a progression. If they are anything at all, they must be intrinsically something. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §242) | |
| A reaction: This is the obvious platonist response to the incipient doctrine of structuralism. We have a chicken-and-egg problem. Bricks need intrinsic properties to make a structure. A structure isomorphic to numbers is not thereby the numbers. |
| 14162 | Mathematics doesn't care whether its entities exist [Russell] |
| Full Idea: Mathematics is throughout indifferent to the question whether its entities exist. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §434) | |
| A reaction: There is an 'if-thenist' attitude in this book, since he is trying to reduce mathematics to logic. Total indifference leaves the problem of why mathematics is applicable to the real world. |
| 14103 | Pure mathematics is the class of propositions of the form 'p implies q' [Russell] |
| Full Idea: Pure mathematics is the class of all propositions of the form 'p implies q', where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §001) | |
| A reaction: Linnebo calls Russell's view here 'deductive structuralism'. Russell gives (§5) as an example that Euclid is just whatever is deduced from his axioms. |
| 21555 | For 'x is a u' to be meaningful, u must be one range of individuals (or 'type') higher than x [Russell] |
| Full Idea: In his 1903 theory of types he distinguished between individuals, ranges of individuals, ranges of ranges of individuals, and so on. Each level was a type, and it was stipulated that for 'x is a u' to be meaningful, u must be one type higher than x. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], App) | |
| A reaction: Russell was dissatisfied because this theory could not deal with Cantor's Paradox. Is this the first time in modern philosophy that someone has offered a criterion for whether a proposition is 'meaningful'? |
| 18003 | In 'x is a u', x and u must be of different types, so 'x is an x' is generally meaningless [Russell, by Magidor] |
| Full Idea: Russell argues that in a statement of the form 'x is a u' (and correspondingly, 'x is a not-u'), 'x must be of different types', and hence that ''x is an x' must in general be meaningless'. | |
| From: report of Bertrand Russell (The Principles of Mathematics [1903], App B:524) by Ofra Magidor - Category Mistakes 1.2 | |
| A reaction: " 'Word' is a word " comes to mind, but this would be the sort of ascent to a metalanguage (to distinguish the types) which Tarski exploited. It is the simple point that a classification can't be the same as a member of the classification. |
| 11010 | Being is what belongs to every possible object of thought [Russell] |
| Full Idea: Being is that which belongs to every conceivable, to every possible object of thought. | |
| From: Bertrand Russell (The Principles of Mathematics [1903]), quoted by Stephen Read - Thinking About Logic Ch.5 | |
| A reaction: I take Russell's (or anyone's) attempt to distinguish two different senses of the word 'being' or 'exist' to be an umitigated metaphysical disaster. |
| 14161 | Many things have being (as topics of propositions), but may not have actual existence [Russell] |
| Full Idea: Numbers, the Homeric gods, relations, chimeras and four-dimensional space all have being, for if they were not entities of a kind, we could not make propositions about them. Existence, on the contrary, is the prerogative of some only amongst the beings. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §427) | |
| A reaction: This is the analytic philosophy account of being (a long way from Heidegger). Contemporary philosophy seems to be full of confusions on this, with many writers claiming existence for things which should only be awarded 'being' status. |
| 14173 | What exists has causal relations, but non-existent things may also have them [Russell] |
| Full Idea: It would seem that whatever exists at any part of time has causal relations. This is not a distinguishing characteristic of what exists, since we have seen that two non-existent terms may be cause and effect. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §449) | |
| A reaction: Presumably he means that the non-existence of something (such as a safety rail) might the cause of an event. This is a problem for Alexander's Principle, in Idea 3534. I think we could redescribe his problem cases, to save Alexander. |
| 6780 | Anti-realism is more plausible about laws than about entities and theories [Bird] |
| Full Idea: There is anti-realism with regard to unobservable entities and the theories that purport to mention them, but the more plausible version attaches to theories concerning what laws of nature are. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.4) | |
| A reaction: This sounds right. I certainly find anti-realism about the entities of science utterly implausible. I also doubt whether there is any such thing as a law, above and beyond the behaviour of matter. Theories float between the two. |
| 14163 | Four classes of terms: instants, points, terms at instants only, and terms at instants and points [Russell] |
| Full Idea: Among terms which appear to exist, there are, we may say, four great classes: 1) instants, 2) points, 3) terms which occupy instants but not points, 4) terms which occupy both points and instants. Analysis cannot explain 'occupy'. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §437) | |
| A reaction: This is a massively reductive scientific approach to categorising existence. Note that it homes in on 'terms', which seems a rather linguistic approach, although Russell is cautious about such things. |
| 21341 | Philosophers of logic and maths insisted that a vocabulary of relations was essential [Russell, by Heil] |
| Full Idea: Relations were regarded with suspicion, until philosophers working in logic and mathematics advanced reasons to doubt that we could provide anything like an adequate description of the world without developing a relational vocabulary. | |
| From: report of Bertrand Russell (The Principles of Mathematics [1903], Ch.26) by John Heil - Relations | |
| A reaction: [Heil cites Russell as the only reference] A little warning light, that philosophers describing the world managed to do without real relations, and it was only for the abstraction of logic and maths that they became essential. |
| 10586 | 'Reflexiveness' holds between a term and itself, and cannot be inferred from symmetry and transitiveness [Russell] |
| Full Idea: The property of a relation which insures that it holds between a term and itself is called by Peano 'reflexiveness', and he has shown, contrary to what was previously believed, that this property cannot be inferred from symmetry and transitiveness. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §209) | |
| A reaction: So we might say 'this is a sentence' has a reflexive relation, and 'this is a wasp' does not. While there are plenty of examples of mental properties with this property, I'm not sure that it makes much sense of a physical object. Indexicality... |
| 10585 | Symmetrical and transitive relations are formally like equality [Russell] |
| Full Idea: Relations which are both symmetrical and transitive are formally of the nature of equality. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §209) | |
| A reaction: This is the key to the whole equivalence approach to abstraction and Frege's definition of numbers. Establish equality conditions is the nearest you can get to saying what such things are. Personally I think we can say more, by revisiting older views. |
| 7781 | I call an object of thought a 'term'. This is a wide concept implying unity and existence. [Russell] |
| Full Idea: Whatever may be an object of thought, or occur in a true or false proposition, or be counted as one, I call a term. This is the widest word in the philosophical vocabulary, which I use synonymously with unit, individual, entity (being one, and existing). | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §047) | |
| A reaction: The claim of existence begs many questions, such as whether the non-existence of the Loch Ness Monster is an 'object' of thought. |
| 14166 | Unities are only in propositions or concepts, and nothing that exists has unity [Russell] |
| Full Idea: It is sufficient to observe that all unities are propositions or propositional concepts, and that consequently nothing that exists is a unity. If, therefore, it is maintained that things are unities, we must reply that no things exist. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §439) | |
| A reaction: The point, I presume, is that you end up as a nihilist about identities (like van Inwagen and Merricks) by mistakenly thinking (as Aristotle and Leibniz did) that everything that exists needs to have something called 'unity'. |
| 14164 | The only unities are simples, or wholes composed of parts [Russell] |
| Full Idea: The only kind of unity to which I can attach any precise sense - apart from the unity of the absolutely simple - is that of a whole composed of parts. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §439) | |
| A reaction: This comes from a keen student of Leibniz, who was obsessed with unity. Russell leaves unaddressed the question of what turns some parts into a whole. |
| 14112 | A set has some sort of unity, but not enough to be a 'whole' [Russell] |
| Full Idea: In a class as many, the component terms, though they have some kind of unity, have less than is required for a whole. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §070) | |
| A reaction: This is interesting because (among many other things), sets are used to stand for numbers, but numbers are usually reqarded as wholes. |
| 14170 | Change is obscured by substance, a thing's nature, subject-predicate form, and by essences [Russell] |
| Full Idea: The notion of change is obscured by the doctrine of substance, by a thing's nature versus its external relations, and by subject-predicate form, so that things can be different and the same. Hence the useless distinction between essential and accidental. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §443) | |
| A reaction: He goes on to object to vague unconscious usage of 'essence' by modern thinkers, but allows (teasingly) that medieval thinkers may have been precise about it. It is a fact, in common life, that things can change and be the same. Explain it! |
| 14107 | Terms are identical if they belong to all the same classes [Russell] |
| Full Idea: Two terms are identical when the second belongs to every class to which the first belongs. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §026) |
| 11849 | It at least makes sense to say two objects have all their properties in common [Wittgenstein on Russell] |
| Full Idea: Russell's definition of '=' is inadequate, because according to it we cannot say that two objects have all their properties in common. (Even if this proposition is never correct, it still has a sense). | |
| From: comment on Bertrand Russell (The Principles of Mathematics [1903]) by Ludwig Wittgenstein - Tractatus Logico-Philosophicus 5.5302 | |
| A reaction: This is what now seems to be a standard denial of the bizarre Leibniz claim that there never could be two things with identical properties, even, it seems, in principle. What would Leibniz made of two electrons? |
| 6796 | Subjective probability measures personal beliefs; objective probability measures the chance of an event happening [Bird] |
| Full Idea: Subjective probability measures a person's strength of belief in the truth of a proposition; objective probability concerns the chance a certain sort of event has of happening, independently of whether anyone thinks it is likely to occur or not. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.6) | |
| A reaction: The challenge to the second one is that God would know for certain whether a meteor will hit the Earth next week. The impact looks like 'bad luck' to us, but necessary to one who really knows. |
| 6797 | Objective probability of tails measures the bias of the coin, not our beliefs about it [Bird] |
| Full Idea: In tossing a coin, the objective probability of tails is a measure of the bias of the coin; the bias and the probability are objective features of the coin, like its mass and shape; these properties have nothing to do with our beliefs about the coin. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.6) | |
| A reaction: Despite my reservation that God would not seem to be very interested in the probabilities of coin-tossing, since he knows each outcome with certaintly, this is fairly convincing. God might say that the coin has a 'three-to-two bias'. |
| 22303 | It makes no sense to say that a true proposition could have been false [Russell] |
| Full Idea: There seems to be no true proposition of which it makes sense to say that it might have been false. One might as well say that redness might have been a taste and not a colour. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §430), quoted by Michael Potter - The Rise of Analytic Philosophy 1879-1930 29 'Analy' | |
| A reaction: Few thinkers agree with this rejection of counterfactuals. It seems to rely on Moore's idea that true propositions are facts. It also sounds deterministic. Does 'he is standing' mean he couldn't have been sitting (at t)? |
| 6800 | Many philosophers rate justification as a more important concept than knowledge [Bird] |
| Full Idea: Many philosophers take the notion of justification to be more important or more basic than the concept of knowledge. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.7) | |
| A reaction: Intriguing. Given the obvious social and conventional element in 'knowledge' ("do we agree that the candidate really knows the answer?"), justification may well be closer to where the real action is. 'Logos', after all, is at the heart of philosophy. |
| 6786 | As science investigates more phenomena, the theories it needs decreases [Bird] |
| Full Idea: A remarkable fact about modern science is that as the number of phenomena which science has investigated has grown, the number of theories needed to explain them has decreased. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.4) | |
| A reaction: This rebuts the idea that theories are probably false because we are unlikely to have thought of the right one (Idea 6784). More data suggests more theories, yet we end up with fewer theories. Why is simplification of theories possible? |
| 6792 | If theories need observation, and observations need theories, how do we start? [Bird] |
| Full Idea: If we cannot know the truth of theories without observation, and we cannot know the truth of observations without theories, where do we start? | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.5) | |
| A reaction: See Idea 6793. You make a few observations, under the illusion that they are objective, then formulate a promising theory, then go back and deconstruct the observations, then tighten up the theory, and so on. |
| 6757 | Explanation predicts after the event; prediction explains before the event [Bird] |
| Full Idea: Explanation is prediction after the event and prediction is explanation before the event. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.2) | |
| A reaction: A nice slogan, fitting Hempel's 'covering law' view of explanation. It doesn't seem quite right, because explanations and predictions are couched in very different language. Prediction implies an explanation; explanation implies a prediction. |
| 6777 | Realists say their theories involve truth and the existence of their phenomena [Bird] |
| Full Idea: A realist says of their theories that they can be evaluated according to truth, they aim at truth, their success favours their truth, their unobserved entities probably exist, and they would explain the observable phenomena. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.4) | |
| A reaction: This seems to me to be the only sensible attitude towards scientific theories, even if they do become confusing down at the level of quantum theory. Theories aim to be true explanations. |
| 6804 | There is no agreement on scientific method - because there is no such thing [Bird] |
| Full Idea: I find little concurrence as to what scientific method might actually be - the reason being, I conclude, that there is no such thing. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.8) | |
| A reaction: I take the essence of science to be two things: first, becoming very fussy about empirical evidence; second, setting up controlled conditions to get at the evidence that seems to be needed. I agree that there seems to be no distinctive way of thinking. |
| 6805 | Relativity ousted Newtonian mechanics despite a loss of simplicity [Bird] |
| Full Idea: The theories of relativity ousted Newtonian mechanics despite a loss of simplicity. | |
| From: Alexander Bird (Philosophy of Science [1998]) | |
| A reaction: This nicely demonstrates that simplicity is not essential, even if it is desirable. The point applies to the use of Ockham's Razor (Idea 6806), and to Hume's objection to miracles (Idea 2227), where strange unnatural events may be the truth. |
| 6778 | Instrumentalists regard theories as tools for prediction, with truth being irrelevant [Bird] |
| Full Idea: Instrumentalism is so called because it regards theories not as attempts to describe or explain the world, but as instruments for making predictions; for the instrumentalist, asking about the truth of a theory is a conceptual mistake. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.4) | |
| A reaction: It cannot be denied that theories are used to make predictions, and there is nothing wrong with being solely interested in predictions. I cannot make head or tail of the idea that truth is irrelevant. Why is a given theory so successful? |
| 6775 | Induction is inference to the best explanation, where the explanation is a law [Bird] |
| Full Idea: Induction can be seen as inference to the best explanation, where the explanation is a law. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.3) | |
| A reaction: I like this. I increasingly think of explanation as central to rational thought, as the key route for empiricists to go beyond their immediate and verifiable experience. Laws can be probabilistic. |
| 6791 | If Hume is right about induction, there is no scientific knowledge [Bird] |
| Full Idea: If Hume is right about induction then there is no scientific knowledge. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.5) | |
| A reaction: The first step is to recognise that induction is not deductively valid, but that does not make it irrational. If something happens five times, get ready for the sixth. If we discover the necessary features of nature, we can predict the future. |
| 6790 | Anything justifying inferences from observed to unobserved must itself do that [Bird] |
| Full Idea: Whatever could do the job of justifying an inference from the observed to the unobserved must itself be an inference from the observed to the unobserved. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.5) | |
| A reaction: We must first accept that the unobserved might not be like the observed, no matter how much regularity we have, so it can't possibly be a logical 'inference'. Essences generate regularities, but non-essences may not. |
| 6738 | Any conclusion can be drawn from an induction, if we use grue-like predicates [Bird] |
| Full Idea: It looks as if any claim about the future can be made to be a conclusion of an inductive argument from any premises about the past, as long as we use a strange enough grue-like predicate. | |
| From: Alexander Bird (Philosophy of Science [1998], Intro) | |
| A reaction: So don't use strange grue-like predicates. If all our predicates randomly changed their reference each day, we would be unable to talk to one another at all. Emeralds don't change their colour-properties, so why change the predicates that refer to them? |
| 6739 | Several months of observing beech trees supports the deciduous and evergreen hypotheses [Bird] |
| Full Idea: If someone were to observe beech trees every day over one summer they would have evidence that seems to support both the hypothesis that beech trees are deciduous and the hypothesis that they are evergreens. | |
| From: Alexander Bird (Philosophy of Science [1998], Intro) | |
| A reaction: Bird offers this to anyone who (like me) is tempted to dismiss the 'grue' problem as ridiculous. Obviously he is right; 'deciduous' works like 'grue'. But we invented the predicate 'deciduous' to match an observed property. |
| 6799 | We normally learn natural kinds from laws, but Goodman shows laws require prior natural kinds [Bird] |
| Full Idea: We know what natural kinds there are by seeing which properties appear in the laws of nature. But one lesson of Goodman's problem is that we cannot identify the laws of nature without some prior identification of natural kinds. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.7) | |
| A reaction: For Goodman's problem, see Idea 4783. The essentialist view is that the natural kinds come first, and the so-called 'laws' are just regularities in events that arise from the interaction of stable natural kinds. (Keep predicates and properties separate). |
| 6798 | Bayesianism claims to find rationality and truth in induction, and show how science works [Bird] |
| Full Idea: Keen supporters of Bayesianism say it can show how induction is rational and can lead to truth, and it can reveal the underlying structure of actual scientific reasoning. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.6) | |
| A reaction: See Idea 2798 for Bayes' Theorem. I find it intuitively implausible that our feeling for probabilities could be reduced to precise numbers, given the subjective nature of the numbers we put into the equation. |
| 6752 | The objective component of explanations is the things that must exist for the explanation [Bird] |
| Full Idea: There is an 'objective', non-epistemic component to explanations, consisting of the things that must exist for A to be able to explain B, and the relations those things have to one another. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.2) | |
| A reaction: There seems to be some question-begging here, in that you have to decide what explanation you are after before you can decide which existences are of interest. There are objective facts, though, about what causally links to what. |
| 6754 | We talk both of 'people' explaining things, and of 'facts' explaining things [Bird] |
| Full Idea: We talk both of 'people' explaining things, and of 'facts' explaining things. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.2) | |
| A reaction: An important point, and it is the job of philosophers to pull the two apart. How we talk does not necessarily show how it is. The concept of explanation is irrelevant in a universe containing no minds, or one containing only God. People seek the facts. |
| 6750 | Explanations are causal, nomic, psychological, psychoanalytic, Darwinian or functional [Bird] |
| Full Idea: Explanations can be classified as causal, nomic, psychological, psychoanalytic, Darwinian and functional. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.2) | |
| A reaction: These could be subdivided, perhaps according to different types of cause. Personally, being a reductionist (like David Lewis, see Idea 3989), I suspect that all of these explanations could be reduced to causation. Essences explain causes. |
| 6761 | Contrastive explanations say why one thing happened but not another [Bird] |
| Full Idea: A 'contrastive explanation' explains why one thing happened but not another. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.2) | |
| A reaction: If I explain why the ship sank, is this contrastive, or just causal, or both? Am I explaining why it sank rather than turned into a giraffe? An interesting concept, but I can't see myself making use of it. |
| 6758 | 'Covering law' explanations only work if no other explanations are to be found [Bird] |
| Full Idea: The fact that something fits the 'covering law' model of explanation is no guarantee that it is an explanation, for that depends on what other explanations are there to be found. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.2) | |
| A reaction: He gives Achinstein's example of a poisoned man who is run over by a bus. It has to be a basic requirement of explanations that they are the 'best', and not just something that fits a formula. |
| 6759 | Livers always accompany hearts, but they don't explain hearts [Bird] |
| Full Idea: All animals with a liver also have a heart; so we can deduce from this plus the existence of Fido's liver that he also has a heart, but his liver does not explain why he has a heart. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.2) | |
| A reaction: This is a counterexample to Hempel's deductive-nomological view of explanation. It seems a fairly decisive refutation of any attempt to give a simple rule for explaining things. Different types of explanation compete, and there is a subjective element. |
| 6756 | Probabilistic-statistical explanations don't entail the explanandum, but makes it more likely [Bird] |
| Full Idea: The probabilistic-statistical view of explanation (also called inductive-statistical explantion) is similar to deductive-nomological explanation, but instead of entailing the explanandum a probabilistic-statistical explantion makes it very likely. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.2) | |
| A reaction: If people have umbrellas up, does that explain rain? Does the presence of a psychopath in the audience explain why I don't go to a rock concert? Still, it has a point. |
| 6760 | An operation might reduce the probability of death, yet explain a death [Bird] |
| Full Idea: An operation for cancer might lead to a patient's death, and so it explains the patient's death while at the same time reducing the probability of death. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.2) | |
| A reaction: This attacks Hempel's 'covering law' approach. Increasing probability of something clearly does not necessarily explain it, though it often will. Feeding you contaminated food will increase the probability of your death, and may cause it. |
| 6785 | Inference to the Best Explanation is done with facts, so it has to be realist [Bird] |
| Full Idea: Explanation of a fact is some other fact or set of facts. And so Inference to the Best Explanation is inference to facts; someone who employs it cannot but take a realist attitude to a theory which is preferred on these grounds. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.4) | |
| A reaction: So my personal commitment to abduction is entailed by my realism, and my realism is entailed by my belief in the possibility of abduction. We can't explain the properties of a table just by referring to our experiences of tables. |
| 6788 | Maybe bad explanations are the true ones, in this messy world [Bird] |
| Full Idea: It is objected to 'best explanation' that this may well not be the best of all possible worlds - so why think that the best explanation is true? Maybe bad (complicated, unsystematic and weak) explanations are true. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.4) | |
| A reaction: The only rebuttal of this objection to best explanation seems to be a priori. It would just seem an odd situation if very simple explanations fitted the facts and yet were false, like the points on a graph being a straight line by pure coincidence. |
| 6787 | Which explanation is 'best' is bound to be subjective, and no guide to truth [Bird] |
| Full Idea: It is objected to 'best explanation' that beauty is in the eye of the beholder - the goodness of possible explanations is subjective, and so the choice of best explanation is also subjective, and hence not a suitable guide to truth. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.4) | |
| A reaction: Explanation is indeed dependent both on the knowledge of the person involved, and on their interests. That doesn't, though, mean that you can choose any old explanation. Causal networks are features of the world. |
| 6751 | Maybe explanation is so subjective that it cannot be a part of science [Bird] |
| Full Idea: Some philosophers have thought that explanation is hopelessly subjective, so subjective even that it is should have no part in proper science. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.2) | |
| A reaction: God requires no explanations, and children require many. If fundamental explanations are causal, then laying bare the causal chains is the explanation, whether you want it or not. God knows all the explanations. See Idea 6752. |
| 10583 | Abstraction principles identify a common property, which is some third term with the right relation [Russell] |
| Full Idea: The relations in an abstraction principle are always constituted by possession of a common property (which is imprecise as it relies on 'predicate'), ..so we say a common property of two terms is any third term to which both have the same relation. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §157) | |
| A reaction: This brings out clearly the linguistic approach of the modern account of abstraction, where the older abstractionism was torn between the ontology and the epistemology (that is, the parts of objects, or the appearances of them in the mind). |
| 10582 | The principle of Abstraction says a symmetrical, transitive relation analyses into an identity [Russell] |
| Full Idea: The principle of Abstraction says that whenever a relation with instances is symmetrical and transitive, then the relation is not primitive, but is analyzable into sameness of relation to some other term. ..This is provable and states a common assumption. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §157) | |
| A reaction: At last I have found someone who explains the whole thing clearly! Bertrand Russell was wonderful. See other ideas on the subject from this text, for a proper understanding of abstraction by equivalence. |
| 10584 | A certain type of property occurs if and only if there is an equivalence relation [Russell] |
| Full Idea: The possession of a common property of a certain type always leads to a symmetrical transitive relation. The principle of Abstraction asserts the converse, that such relations only spring from common properties of the above type. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §157) | |
| A reaction: The type of property is where only one term is applicable to it, such as the magnitude of a quantity, or the time of an event. So symmetrical and transitive relations occur if and only if there is a property of that type. |
| 14110 | Proposition contain entities indicated by words, rather than the words themselves [Russell] |
| Full Idea: A proposition, unless it happens to be linguistic, does not itself contain words: it contains the entities indicated by words. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §051) | |
| A reaction: Russell says in his Preface that he took over this view of propositions from G.E. Moore. They are now known as 'Russellian' propositions, which are mainly distinguished by not being mental event, but by being complexes out in the world. |
| 19164 | If propositions are facts, then false and true propositions are indistinguishable [Davidson on Russell] |
| Full Idea: Russell often treated propositions as facts, but discovered that correspondence then became useless for explaining truth, since every meaningful expression, true or false, expresses a proposition. | |
| From: comment on Bertrand Russell (The Principles of Mathematics [1903]) by Donald Davidson - Truth and Predication 6 | |
| A reaction: So 'pigs fly' would have to mean pigs actually flying (which they don't). They might correspond to possible situations, but only if pigs might fly. What do you make of 'circles are square'? Russell had many a sleepless night over that. |
| 14111 | A proposition is a unity, and analysis destroys it [Russell] |
| Full Idea: A proposition is essentially a unity, and when analysis has destroyed the unity, no enumeration of constituents will restore the proposition. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §054) | |
| A reaction: The question of the 'unity of the proposition' led to a prolonged debate. |
| 19157 | Russell said the proposition must explain its own unity - or else objective truth is impossible [Russell, by Davidson] |
| Full Idea: Moore and Russell reacted strongly against the idea that the unity of the proposition depended on human acts of judgement. ...Russell decided that unless the unity is explained in terms of the proposition itself, there can be no objective truth. | |
| From: report of Bertrand Russell (The Principles of Mathematics [1903], p.42) by Donald Davidson - Truth and Predication 5 | |
| A reaction: Put like this, the Russellian view strikes me as false. Effectively he is saying that a unified proposition is the same as a fact. I take a proposition to be a brain event, best labelled by Frege as a 'thought'. Thoughts may not even have parts. |
| 6776 | Natural kinds are those that we use in induction [Bird] |
| Full Idea: Natural kinds are the kinds one should make use of in inductive inference (if that is explanation which leads to laws). | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.3) | |
| A reaction: The problem with this is that it is epistemological rather than ontological. In induction we use superficial resemblences that are immediately obvious, whereas the nature of kinds can be buried deep in the chemistry or physics. |
| 6767 | Rubies and sapphires are both corundum, with traces of metals varying their colours [Bird] |
| Full Idea: Both rubies (valuable) and sapphires (less valuable) are corundum (Al2O3), differing only in their colours, for which traces of iron, titanium and chomium are responsible. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.3) | |
| A reaction: A nice example which illustrates how natural kinds determined by nominal essence could be drastically different from those suggested by real essence. It certainly suggests that corundum might be a natural kind, but ruby isn't. |
| 6768 | Tin is not one natural kind, but appears to be 21, depending on isotope [Bird] |
| Full Idea: If real essences are decided by microstructure, then what we call the element tin is not a natural kind, but a mixture of 21 different kinds, one for each isotope. There also exist two different allotropes of tin - white tin and grey tin. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.3) | |
| A reaction: This example vividly brings out the difficulties of the Kripke-Putnam view. If natural kinds 'overlap', then there would be a very extensive overlap among the 21 isotopes of tin. |
| 6770 | Membership of a purely random collection cannot be used as an explanation [Bird] |
| Full Idea: One might randomly collect diverse things and give the collection a name, but one would not expect it to explain anything to say that a certain object belonged to this collection. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.3) | |
| A reaction: This is in support of Bird's view that natural kinds are formulated because of their explanatory role. There is, though, an undeniable subjective aspect to explanation, in that explanations arise from the ignorance and interests of persons. |
| 6771 | Natural kinds may overlap, or be sub-kinds of one another [Bird] |
| Full Idea: It seems clear that in some cases one natural kind may be a subkind of another, while in other cases natural kinds may overlap without one being the subkind of another. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.3) | |
| A reaction: Given the enormous difficulty of pinpointing natural kinds (e.g. Idea 6768), it is hard to know whether the comment is correct or not. Ellis says natural kinds come 'in hierarchies', which would make subkinds normal, but overlapping unlikely. |
| 6773 | If F is a universal appearing in a natural law, then Fs form a natural kind [Bird] |
| Full Idea: The proposal is that if F is a universal appearing in some natural law, then Fs form a natural kind. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.3) | |
| A reaction: Such proposals always invite the question 'What is it about F that enables it to be a universal in a natural law?' Nothing can be ultimately defined simply by its role. The character (essence, even) of the thing makes the role possible. |
| 6769 | In the Kripke-Putnam view only nuclear physicists can know natural kinds [Bird] |
| Full Idea: In the Kripke-Putnam view, it is very difficult for anyone except nuclear physicists to pick out natural kinds, since everything else is made out of compounds of different isotopes. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.3) | |
| A reaction: The concept of a rigid 'natural kind' does not have to be sacred. Tin might be considered a natural kind, despite having 21 isotopes. What matters is protons, not the neutrons. |
| 6774 | Darwinism suggests that we should have a native ability to detect natural kinds [Bird] |
| Full Idea: Creatures that are able to recognise natural kinds and laws have a selective advantage, so Darwinism suggests that we should have some native ability to detect natural kinds. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.3) | |
| A reaction: This seems right, but it makes 'natural kind' a rather instrumental concept, relative to our interests. True natural kinds cut across our interests, as when we discover by anatomy that whales are not fish, or that rubies and sapphires are both corundum. |
| 6764 | Nominal essence of a natural kind is the features that make it fit its name [Bird] |
| Full Idea: The nominal essence of a natural kind K consists of those features a thing must have to deserve the name 'a K' by virtue of the meaning of that name. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.3) | |
| A reaction: Some people think 'nominal essence' is the only essence there is, which would make it relative to human languages. The rival view is that there are 'real essences'. I favour the latter view. |
| 6766 | Jadeite and nephrite are superficially identical, but have different composition [Bird] |
| Full Idea: There might be more than one natural kind that shares the same superficial features, …jade, for example, has two forms, jadeite and nephrite, which are similar in superficial properties, but have different chemical composition and structure. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.3) | |
| A reaction: It might be questioned whether jadeite and nephrite really are natural kinds, either together or separately. |
| 6808 | Reference to scientific terms is by explanatory role, not by descriptions [Bird] |
| Full Idea: I propose that reference to scientific terms, such as natural kinds and theoretical terms, is not determined by a sense or description attached to the term, but by its explanatory role. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.8) | |
| A reaction: He gives the example of an electron, which had the same role in electrical theory, despite changes in understanding its nature. One might talk of its 'natural' (causal) role, rather than its 'explanatory' role (which implies a human viewpoint). |
| 14175 | We can drop 'cause', and just make inferences between facts [Russell] |
| Full Idea: On the whole it is not worthwhile preserving the word 'cause': it is enough to say, what is far less misleading, that any two configurations allow us to infer any other. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §460) | |
| A reaction: Russell spelled this out fully in a 1912 paper. This sounds like David Hume, but he prefers to talk of 'habit' rather than 'inference', which might contain a sneaky necessity. |
| 14172 | Moments and points seem to imply other moments and points, but don't cause them [Russell] |
| Full Idea: Some people would hold that two moments of time, or two points of space, imply each other's existence; yet the relation between these cannot be said to be causal. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §449) | |
| A reaction: Famously, Russell utterly rejected causation a few years after this. The example seems clearer if you say that two points or moments can imply at least one point or instant between them, without causing them. |
| 6753 | Laws are more fundamental in science than causes, and laws will explain causes [Bird] |
| Full Idea: I think laws are fundamental and where there is a cause there is always a set of laws that encompasses the cause; identifying a cause will never be the final word in an scientific investigation, but will be open to supplementation by the underlying law. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.2) | |
| A reaction: I think this is wrong. I would say (from the essentialist angle) that essences have causes, and the laws are the regularities that are caused by the essences. If laws are the lowest level of explanation, why these laws and not others? God? |
| 6762 | Newton's laws cannot be confirmed individually, but only in combinations [Bird] |
| Full Idea: None of Newton's laws individually records anything that can be observed; it is only from combinations of Newton's laws that we can derive the measurable motions of bodies. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.2) | |
| A reaction: This certainly scuppers any traditional positivist approach to how we confirm laws of nature. It invites the possibility that a different combination might fit the same observations. Experiments attempt to isolate laws. |
| 6763 | Parapsychology is mere speculation, because it offers no mechanisms for its working [Bird] |
| Full Idea: Wegener's theory of continental drift was only accepted when the theory of plate tectonics was developed, providing a mechanism. While some correlations exist for parapsychology, lack of plausible mechanisms leaves it as speculation. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.2) | |
| A reaction: But parapsychology is not even on a par with Wegener's speculation, because his was consistent with known physical laws, whereas parapsychology flatly contradicts them. The so-called correlations are also not properly established. |
| 6772 | Existence requires laws, as inertia or gravity are needed for mass or matter [Bird] |
| Full Idea: I suspect that what we mean by 'mass' and 'matter' depends on our identifying the existence of laws of inertia and gravity; hence the idea of a world without laws is incoherent, for there to be anything at all there must be some laws and some kinds. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.3) | |
| A reaction: I find this counterintuitive. Reasonably stable existence requires something reasonably like laws. We only understand the physical world because we interact with it. But neither of those is remotely as strong as Bird's claim. |
| 6740 | 'All uranium lumps are small' is a law, but 'all gold lumps are small' is not [Bird] |
| Full Idea: 'Uranium lumps have mass of less than 1000 kg' is a law, but 'gold lumps have mass of less than 1000kg' is not a law. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.1) | |
| A reaction: A nice example. Essentialists talk about the nature of the substances; regularity theorists prefer to talk of nested or connected regularities (e.g. about explosions). In induction, how do you decide what your duty requires you to observe? |
| 6741 | There can be remarkable uniformities in nature that are purely coincidental [Bird] |
| Full Idea: Bode's non-law (of 1772, about the gaps between the planets) shows that there can be remarkable uniformities in nature that are purely coincidental. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.1) | |
| A reaction: If Bode's law really were confirmed, even for asteroids and newly discovered planets, it might suggest that an explanation really is required, and there is some underlying cause. How likely is the coincidence? Perhaps we have no way of telling. |
| 6742 | A law might have no instances, if it was about things that only exist momentarily [Bird] |
| Full Idea: A law might have no instances at all; for example, about the chemical and electrical behaviour of the transuranic elements, which only exist briefly in laboratories. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.1) | |
| A reaction: Nice example. We need to distinguish, though, (as Bird reminds us) between laws and theories. We have no theories in this area, but there are counterfactual truths about what the transuranic elements would do in certain circumstances. |
| 6743 | If laws are just instances, the law should either have gaps, or join the instances arbitrarily [Bird] |
| Full Idea: For the simple regularity theorist, the function ought to be a gappy one, leaving out values not actually instantiated; …one function would fit the actual points on the graph as well as any other. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.1) | |
| A reaction: The 'simple' theorist says there is nothing more to a law than its instances. Clearly Bird is right; if the points line up, we join them with a straight line, making counterfactual assumptions about points which were not actually observed. |
| 6744 | Where is the regularity in a law predicting nuclear decay? [Bird] |
| Full Idea: If a law of nuclear physics says that nuclei of a certain kind have a probability p of decaying within time t, what is the regularity here? | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.1) | |
| A reaction: Hume gives an answer, in terms of regularities observed among previous instances. Nevertheless the figure p given in the law does not itself have any instances, so the law is predicting something that may never have actually happened before. |
| 6747 | Laws cannot explain instances if they are regularities, as something can't explain itself [Bird] |
| Full Idea: It can be objected that laws cannot do the job of explaining their instances if they are merely regularities, ...because something cannot explain itself. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.1) | |
| A reaction: A nice point. The objection assumes that a law should explain things, rather than just describing them. I take the model to be smoking-and-cancer; the statistics describe what is happening, but only lung biochemistry will explain it. |
| 6746 | There may be many laws, each with only a few instances [Bird] |
| Full Idea: It might be that there is a large number of laws each of which has only a small number of instances. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.1) | |
| A reaction: This is a problem for the Ramsey-Lewis view (Idea 6745) that the laws of nature are a simple, powerful and coherent system. We must be cautious about bringing a priori principles like Ockham's Razor (Idea 3667) to bear on the laws of nature. |
| 6748 | Similar appearance of siblings is a regularity, but shared parents is what links them [Bird] |
| Full Idea: There may be a regularity of siblings looking similar, but the tie that binds them is not their similarity, but rather their being born of the same parents. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.1) | |
| A reaction: A nice objection to the regularity view. Regularities, as so often in philosophy (e.g. Idea 1364), may be the evidence or test for a law, rather than the law itself, which requires causal mechanisms, ultimately based (I think) in essences. |
| 6749 | We can only infer a true regularity if something binds the instances together [Bird] |
| Full Idea: We cannot infer a regularity from its instances unless there is something stronger than the regularity itself binding the instances together. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.1) | |
| A reaction: Spells out the implication of the example in Idea 6748. The reply to this criticism would be that no account can possibly be given of the 'something stronger' than further regularities, at a lower level (e.g. in the physics). |
| 6803 | If we only infer laws from regularities among observations, we can't infer unobservable entities. [Bird] |
| Full Idea: If the naïve inductivist says we should see well-established regularities among our observations, and take that to be the law or causal connection…this will not help us to infer the existence of unobservable entities. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.8) | |
| A reaction: The obvious solution to this difficulty is an appeal to 'best explanation'. Bird is obviously right that we couldn't survive in the world, let alone do science, if we only acted on what we had actually observed (e.g. many bodies, but not the poison). |
| 6801 | Accidental regularities are not laws, and an apparent regularity may not be actual [Bird] |
| Full Idea: Many actual regularities are not laws (accidental regularities), and many perceived regularities are not actual ones (a summer's worth of observing green leaves). | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.8) | |
| A reaction: These problems are not sufficient to refute the regularity view of laws. Accidental regularities can only be short-lived, and perceived regularities support laws without clinching them. There is an awful lot of regularity behind laws concerning gravity. |
| 6745 | A regularity is only a law if it is part of a complete system which is simple and strong [Bird] |
| Full Idea: The systematic (Ramsey-Lewis) regularity theory says that a regularity is a law of nature if and only if it appears as a theorem or axiom in that true deductive system which achieves a best combination of simplicity and strength. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.1) | |
| A reaction: Personally I don't accept the regularity view of laws, but this looks like the best account anyone has come up with. Individual bunches of regularities can't add up to or demonstrate a law, but coherence with all regularities might do it. |
| 6802 | With strange enough predicates, anything could be made out to be a regularity [Bird] |
| Full Idea: We learned from Goodman's problem that with strange enough predicates anything could be made out to be a regularity. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.8) | |
| A reaction: For Goodman's problem, see Idea 4783. The point, as I see it, is that while predicates can be applied arbitrarily (because they are just linguistic), properties cannot, because they are features of the world. Emeralds are green. |
| 6789 | If flame colour is characteristic of a metal, that is an empirical claim needing justification [Bird] |
| Full Idea: I might say that flame colours are a characteristic feature of metals, but this is an empirical proposition which is in part about the unobserved, and stands in need of justification. | |
| From: Alexander Bird (Philosophy of Science [1998], Ch.5) | |
| A reaction: This draws attention to the fact that essentialism is not just a metaphysical theory, but is also part of the scientific enterprise. Among things to research about metals is the reason why they have a characteristic flame. |
| 14174 | The laws of motion and gravitation are just parts of the definition of a kind of matter [Russell] |
| Full Idea: For us, as pure mathematicians, the laws of motion and the law of gravitation are not properly laws at all, but parts of the definition of a certain kind of matter. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §459) | |
| A reaction: The 'certain kind of matter' is that which has 'mass'. Since these are paradigm cases of supposed laws, this is the beginning of the end for real laws of nature, and good riddance say I. See Mumford on this. |
| 14168 | Occupying a place and change are prior to motion, so motion is just occupying places at continuous times [Russell] |
| Full Idea: The concept of motion is logically subsequent to that of occupying as place at a time, and also to that of change. Motion is the occupation, by one entity, of a continuous series of places at a continuous series of times. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §442) | |
| A reaction: This is Russell's famous theory of motion, which came to be called the 'At-At' theory (at some place at some time). It seems to mathematically pin down motion all right, but seems a bit short on the poetry of the thing. |
| 14171 | Force is supposed to cause acceleration, but acceleration is a mathematical fiction [Russell] |
| Full Idea: A force is the supposed cause of acceleration, ...but an acceleration is a mere mathematical fiction, a number, not a physical fact. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §448) | |
| A reaction: This rests on his at-at theory of motion, in Idea 14168. I'm not sure that if I fell off a cliff I could be reassured on the way down that my acceleration was just a mathematical fiction. |
| 6807 | In Newton mass is conserved, but in Einstein it can convert into energy [Bird] |
| Full Idea: According to Newton mass is conserved, while in Einstein's theory mass is not conserved but can be converted into and from energy. | |
| From: Alexander Bird (Philosophy of Science [1998]) | |
| A reaction: Perhaps this is the most fundamental difference between the theories. It certainly suggests that 'mass' was a conventional concept rather than a natural one. Maybe the relative notion of 'weight' is more natural than 'mass'. |
| 14160 | Space is the extension of 'point', and aggregates of points seem necessary for geometry [Russell] |
| Full Idea: I won't discuss whether points are unities or simple terms, but whether space is an aggregate of them. ..There is no geometry without points, nothing against them, and logical reasons in their favour. Space is the extension of the concept 'point'. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §423) |
| 14156 | Mathematicians don't distinguish between instants of time and points on a line [Russell] |
| Full Idea: To the mathematician as such there is no relevant distinction between the instants of time and the points on a line. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §387) | |
| A reaction: This is the germ of the modern view of space time, which is dictated by the mathematics, rather than by our intuitions or insights into what is actually going on. |
| 14169 | The 'universe' can mean what exists now, what always has or will exist [Russell] |
| Full Idea: The universe is a somewhat ambiguous term: it may mean all the things that exist at a single moment, or all things that ever have existed or will exist, or the common quality of whatever exists. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §442) |
| 16713 | Philosophers are the forefathers of heretics [Tertullian] |
| Full Idea: Philosophers are the forefathers of heretics. | |
| From: Tertullian (works [c.200]), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 20.2 |
| 6610 | I believe because it is absurd [Tertullian] |
| Full Idea: I believe because it is absurd ('Credo quia absurdum est'). | |
| From: Tertullian (works [c.200]), quoted by Robert Fogelin - Walking the Tightrope of Reason n4.2 | |
| A reaction: This seems to be a rather desperate remark, in response to what must have been rather good hostile arguments. No one would abandon the support of reason if it was easy to acquire. You can't deny its engaging romantic defiance, though. |