148 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. |
| 20910 | Everything happens necessarily, and for a reason [Democritus] |
| Full Idea: Nothing happens at random, but everything for a reason and as the effect of necessity. | |
| From: Democritus (fragments/reports [c.431 BCE], B002), quoted by Pseudo-Plutarch - On the Doctrine of the Philosophers 1.25.4 | |
| A reaction: [In Aetius 'Stob'] This remark reminds us of the link between necessity and sufficient reason. Do all reasons arise for a reason? |
| 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) |
| 16146 | Two can't be a self-contained unit, because it would need to be one to do that [Democritus, by Aristotle] |
| Full Idea: Democritus claimed that one substance could not be composed from two nor two from one. …The same will clearly go for number, on the popular assumption that number is a combination of units. Unless two is one, it cannot contain a unit in actuality. | |
| From: report of Democritus (fragments/reports [c.431 BCE]) by Aristotle - Metaphysics 1039a15 | |
| A reaction: Chrysippus followed this up the first part with the memorable example of Dion and Theon. The problem with the second part is that 2, 3 and 4 are three numbers, so they can count as meta-units. |
| 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. |
| 20901 | True Being only occurs when it is completely full, with atoms and no void [Democritus, by Aristotle] |
| Full Idea: In response to defenders of the One, Democritus says that what is, in the proper sense, is being that is completely full, but that such a being is not one, but that they are unlimited in number and invisible because of the smallness of their masses. | |
| From: report of Democritus (fragments/reports [c.431 BCE], A007) by Aristotle - Coming-to-be and Passing-away (Gen/Corr) 325a28 | |
| A reaction: Democritus is in a tangle here. He says proper being has no void, having apparently conceded that motion needs void (which he admits is non-existent). So true being only occurs when everything grinds to a halt, which is not now. But Idea 20902. |
| 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. |
| 20902 | Being does not exist more than non-being [Democritus, by Aristotle] |
| Full Idea: They say that being does not exist more than non-being, because neither does the void exist more than the body. | |
| From: report of Democritus (fragments/reports [c.431 BCE], A006) by Aristotle - Metaphysics 0985b09 | |
| A reaction: The claim that Being and Non-Being are the same thing is pretty startling. It seems to be an expedient to get Void into the picture, even though it is taken to be wholly devoid of qualities. |
| 20904 | The non-existent exists as much as the existent, because it has causal powers [Democritus] |
| Full Idea: What exists does not exist at all more than what does not exist, and both are causes in a similar way for the things that come about. | |
| From: Democritus (fragments/reports [c.431 BCE], A008), quoted by Simplicius - On Aristotle's 'Physics' p.28.4-27 | |
| A reaction: [Simplicius actually attributes this to the shadowy Leucippus] You can see the point. If you drive into a pothole, that has considerable causal powers. |
| 20903 | The only distinctions are Configuration (shape), Disposition (order) and Turning (position) [Democritus, by Aristotle] |
| Full Idea: They say that what is differs only by Configuration ([rhusmos], which is the shape), by Disposition ([diathege], which is the order), and by Turning ([tropê], which is the position. | |
| From: report of Democritus (fragments/reports [c.431 BCE], A006) by Aristotle - Metaphysics 0985b16 | |
| A reaction: If you give the shape, structure and position of an object, there is no much more to say. Perhaps mention time. |
| 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. |
| 20893 | Nothing comes from non-existence, or passes into it [Democritus, by Diog. Laertius] |
| Full Idea: Nothing comes into being from what does not exist, nor is it destroyed into what does not exist. | |
| From: report of Democritus (fragments/reports [c.431 BCE], A001) by Diogenes Laertius - Lives of Eminent Philosophers 09.44 | |
| A reaction: [part of a concise summary of Democritus by DL] Probably an intuition about conservation laws, rather than a speculation about the Big Bang. |
| 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. |
| 20896 | It is not possible to know what sort each thing is [Democritus] |
| Full Idea: In reality [eteé] to recognise what sort each thing is, belongs to what is impracticable [aporos]. | |
| From: Democritus (fragments/reports [c.431 BCE], B008), quoted by Sextus Empiricus - Against the Logicians (two books) 7.137 | |
| A reaction: On the whole modern scientists (and the rest of us) shoehorn virtually everything into a specific category. It strikes me as wildly bad metaphysics to say that everything necessarily has its category. |
| 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. |
| 3357 | Democritus denies reality to large objects, because atomic entities can't combine to produce new ones [Benardete,JA on Democritus] |
| Full Idea: Democritus appears to rule out from his austere ontology all so-called emergent entities - even mountains and rivers - on the ground that two or more entities can never combine to produce a new one. | |
| From: comment on Democritus (fragments/reports [c.431 BCE]) by José A. Benardete - Metaphysics: the logical approach Ch.24 |
| 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. |
| 598 | Democritus said that substances could never be mixed, so atoms are the substances [Democritus, by Aristotle] |
| Full Idea: Democritus claimed that one substance could not be composed from two nor two from one; for him it is the atoms that are the substances. | |
| From: report of Democritus (fragments/reports [c.431 BCE]) by Aristotle - Metaphysics 1039a10 |
| 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? |
| 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)? |
| 3570 | Maybe knowledge is belief which 'tracks' the truth [Nozick, by Williams,M] |
| Full Idea: Nozick suggests that knowledge is just belief which 'tracks the truth' (hence leaving out justification). | |
| From: report of Robert Nozick (Philosophical Explanations [1981]) by Michael Williams - Problems of Knowledge Ch. 2 |
| 1532 | Sensible qualities can't be real if they appear different to different creatures [Democritus, by Theophrastus] |
| Full Idea: As proof of the fact that sensible qualities have no real existence he points to the fact that they do not appear the same to all creatures. | |
| From: report of Democritus (fragments/reports [c.431 BCE], A135) by Theophrastus - On the Senses 63 |
| 20894 | Man is separated from reality [Democritus] |
| Full Idea: It is necessary to recognise that man by virtue of this criterion is separated from reality. | |
| From: Democritus (fragments/reports [c.431 BCE], B006), quoted by Sextus Empiricus - Against the Logicians (two books) 7.137 | |
| A reaction: I don't know what 'this criterion' is, but it strikes me as quite a good slogan for fans (like myself) of the representative theory of perception. Critics say it is the big objection to the representative theory, but I say 'get over it'. |
| 20897 | Obscure knowledge belongs to the five senses, and genuine knowledge is the other type [Democritus] |
| Full Idea: There are two forms of knowledge [gnomé], the one genuine, the other obscure. And to the obscure one belongs all of these: sight, hearing, smell, taste, touch. The other is genuine, and is separated from this one. | |
| From: Democritus (fragments/reports [c.431 BCE], B011), quoted by Sextus Empiricus - Against the Logicians (two books) 7.139 | |
| A reaction: [Sextus goes on to make it clear that the 'genuine' one is knowledge acquired by thought]. I take Parmenides to be the first rationalist. It is interesting that Democritus, who devoted his life to finding causal explanations, seems to be a rationalist. |
| 517 | All evidence comes from senses, so they are indispensable to the mind [Democritus] |
| Full Idea: Mind must never reject the senses, because that is where it gets its evidence, and it would be the mind's downfall. | |
| From: Democritus (fragments/reports [c.431 BCE], B125), quoted by Galen - On Medical Experience 15.8 |
| 2748 | A true belief isn't knowledge if it would be believed even if false. It should 'track the truth' [Nozick, by Dancy,J] |
| Full Idea: Nozick says Gettier cases aren't knowledge because the proposition would be believed even if false. Proper justification must be more sensitive to the truth ("track the truth"). | |
| From: report of Robert Nozick (Philosophical Explanations [1981], 3.1) by Jonathan Dancy - Intro to Contemporary Epistemology 3.1 | |
| A reaction: This is a bad idea. I see a genuine tree in my garden and believe it is there, so I know it. That I might have believed it if I was in virtually reality, or observing a mirror, won't alter that. |
| 20895 | We actually know nothing, and opinions are mere flux [Democritus] |
| Full Idea: Certainly this argument too makes it clear that in reality [eteé] we know nothing about anything, but for each person opinion is a rhythmic afflux [epirhusmié]. | |
| From: Democritus (fragments/reports [c.431 BCE], B007), quoted by Sextus Empiricus - Against the Logicians (two books) 7.137 | |
| A reaction: This seems to pick 'all is flux' up from Heraclitus, and make Democritus (along with aspects of Socrates) the true source of ancient scepticism. |
| 1528 | We in fact know nothing, but we each restructure our reality with beliefs [Democritus] |
| Full Idea: In reality we know nothing about anything, but belief restructures things for each of us. | |
| From: Democritus (fragments/reports [c.431 BCE], B007), quoted by Sextus Empiricus - Against the Professors (six books) 7.136 |
| 492 | It is obviously impossible to understand the reality of each thing [Democritus] |
| Full Idea: It will be obvious that it is impossible to understand how in reality each thing is. | |
| From: Democritus (fragments/reports [c.431 BCE], B008), quoted by Sextus Empiricus - Against the Professors (six books) 7.137 |
| 515 | We know nothing in reality; for truth lies in an abyss [Democritus] |
| Full Idea: We know nothing in reality; for truth lies in an abyss. | |
| From: Democritus (fragments/reports [c.431 BCE], B117), quoted by Diogenes Laertius - Lives of Eminent Philosophers 09.72.10 |
| 577 | Democritus says there is either no truth, or it is concealed from us [Democritus, by Aristotle] |
| Full Idea: Democritus concludes that either there is no truth or it is concealed from us. | |
| From: report of Democritus (fragments/reports [c.431 BCE], A112) by Aristotle - Metaphysics 1009b12 |
| 20892 | Democritus was devoted to discovering causal explanations [Democritus, by Eusebius] |
| Full Idea: Democritus himself, as they say, stated that he would rather discover a single causal explanation [aitiologia] than become the King of the Persians. | |
| From: report of Democritus (fragments/reports [c.431 BCE], B118) by Eusebius - Preparation for the Gospel 14.27.4 | |
| A reaction: Democritus seems to be clearly the father of the physical sciences, because he focused single-mindedly on physical and causal explanations. David Lewis says all explanations are causal. |
| 5882 | Democritus says soul consists of smooth round bodies brought together in accidental collision [Democritus, by Cicero] |
| Full Idea: Since Democritus makes the soul consist of minute smooth round bodies brought together in some sort of accidental collision, let us pass him over. | |
| From: report of Democritus (fragments/reports [c.431 BCE]) by M. Tullius Cicero - Tusculan Disputations I.xi.23 | |
| A reaction: If we accept that Democritus thought the collision of atoms 'accidental', then it doesn't sound like a very good theory. What would Cicero say if we replaced 'accidental' with 'naturally selected'? |
| 6034 | Atomists say soul has a rational part in the chest, and a diffused non-rational part [Democritus, by Aetius] |
| Full Idea: Democritus and Epicurus say the soul has two parts, one which is rational and is situated in the chest area, and the other which is non-rational and is spread throughout the entire compound of the body. | |
| From: report of Democritus (fragments/reports [c.431 BCE]) by Aetius - fragments/reports 4.4.6 | |
| A reaction: The spread part corresponds to such things as feeling fear in the stomach, or excitement throughout the limbs. I can't think what grounds there would be for choosing the chest as the home of reason. I suppose you can hear reason thumping in there.. |
| 20912 | The soul is the same as the mind [Democritus, by Aristotle] |
| Full Idea: Democritus says the soul is the same thing as the mind. | |
| From: report of Democritus (fragments/reports [c.431 BCE], A101) by Aristotle - De Anima 405a10 | |
| A reaction: This is not in contrast to the Christian concept of the soul, but in contrast to the normal view of psuché, which is more like the life that permeates the whole body. Democritus was more in tune than Aristotle with modern thought on this. |
| 20916 | Animals have a share of reason [Democritus, by Porphyry] |
| Full Idea: Democritus recognised that animals have a share of reason. | |
| From: report of Democritus (fragments/reports [c.431 BCE]) by Porphyry - On Abstinence 3.6.7 | |
| A reaction: Since he considers thinking to be the interaction of atoms in the body, which animals evidently possess, this seems consistent. No one seems to observed animals closely before the 20th century, other than to exploit them. |
| 20914 | The directive centre is located in the whole head [Democritus, by Ps-Plutarch] |
| Full Idea: Democritus says [the directive centre is located] in the whole head. | |
| From: report of Democritus (fragments/reports [c.431 BCE], A105) by Pseudo-Plutarch - On the Doctrine of the Philosophers 4.5.1 | |
| A reaction: The whole head is not quite the brain, but he is getting very warm indeed, and long before anyone else got so close. |
| 6033 | Democritus said everything happens of necessity, by natural motion of atoms [Democritus, by Cicero] |
| Full Idea: Democritus, the founder of atomism, preferred to accept that all things happened by necessity than to tear from the atomic bodies their natural motions. | |
| From: report of Democritus (fragments/reports [c.431 BCE]) by M. Tullius Cicero - On Fate ('De fato') §22 | |
| A reaction: This is in opposition to Epicurus, who said that atoms can have a 'swerve', making free will possible. It is suggested that Epicurus was the first to really grasp the problem of free will. Democritus was just stating the (to him) obvious. |
| 5088 | Some say there is a determinate cause for every apparently spontaneous event [Democritus, by Aristotle] |
| Full Idea: Some people (Democritus?) say there is no such thing as a chance event; they claim that there is always a determinate cause for everything which is said to be a chance or a spontaneous event. | |
| From: report of Democritus (fragments/reports [c.431 BCE]) by Aristotle - Physics 195b37 | |
| A reaction: This is the mutual implication of physicalism and determinism, which strikes me as unavoidable. I say: don't panic about morality because determinism is true. Embrace determinism - it is harmless and true. Its opposite is a nonsense. |
| 21670 | Democritus said atoms only move by their natural motions, which are therefore necessary [Democritus, by Cicero] |
| Full Idea: The author of the atomic theory, Democritus, preferred to accept the view that all events are caused by necessity, rather than to deprive the atoms of their natural motions. | |
| From: report of Democritus (fragments/reports [c.431 BCE]) by M. Tullius Cicero - On Fate ('De fato') 10.23 | |
| A reaction: The 'deprivation' would have to be caused by mind, or by the later 'swerve' of Epicurus. |
| 20913 | Democritus says the soul is the body, and thinking is thus the mixture of the body [Democritus, by Theophrastus] |
| Full Idea: Democritus explains thinking by the mixture of the body, which is surely in accordance with his reasoning, since he makes the soul the body. | |
| From: report of Democritus (fragments/reports [c.431 BCE], A135) by Theophrastus - On the Senses 58 | |
| A reaction: I agree with Democritus. |
| 24041 | Democritus says spherical atoms are fire, and constitute the soul [psuche] [Democritus, by Aristotle] |
| Full Idea: Democritus says the soul is a sort of fire. For the shapes and atoms are unlimitied and those that are spherical he says are fire and soul - which are like the motes in the air when sunbeams come through the window. | |
| From: report of Democritus (fragments/reports [c.431 BCE], DK 67-68) by Aristotle - De Anima 403b31 | |
| A reaction: It's hard to see why the spherical atoms should be fire. Maybe because they move together quickly and easily. …At 404a5 Aristotle agrees with me! |
| 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. |
| 1540 | Pleasure and pain guide our choices of good and bad [Democritus] |
| Full Idea: The guides to what is good and bad for people are pleasure and pain. | |
| From: Democritus (fragments/reports [c.431 BCE], B188), quoted by John Stobaeus - Anthology 3.01.46 |
| 495 | Wisdom creates a healthy passion-free soul [Democritus] |
| Full Idea: Medicine heals diseases of the body, wisdom frees the soul from passions. | |
| From: Democritus (fragments/reports [c.431 BCE], B031), quoted by Clement - Pedagogue 1.6.2.1 | |
| A reaction: The interesting concept of a healthy mind seems to have got lost in modern moral philosophy. |
| 1537 | Happiness is identifying and separating the pleasures [Democritus, by Stobaeus] |
| Full Idea: Democritus thinks that happiness consists in the determination and separation of pleasures, and that this is what is both finest and most beneficial for people. | |
| From: report of Democritus (fragments/reports [c.431 BCE], A167) by John Stobaeus - Anthology 2.07.3 | |
| A reaction: A great deal of the strategy and ethics of living consists (if you are lucky) of discriminating among possible pleasures. Philosophers should produce criteria. |
| 20917 | Contentment comes from moderation and proportion in life [Democritus, by Stobaeus] |
| Full Idea: Contentment [euthumia] comes about for human beings from the moderation of enjoyment and proportion [summetria] in life. | |
| From: report of Democritus (fragments/reports [c.431 BCE], B191) by John Stobaeus - Anthology 3.1.210 | |
| A reaction: This is close to Aristotle's doctrine of the Mean. The majority of ethical ideas attributed to Democritus (presumably by the Epicureans) are thought to be spurious. This idea actually sounds rather stoic. |
| 13551 | Democritus says wealth is a burden to the virtuous mind [Democritus, by Seneca] |
| Full Idea: Democritus rejected wealth, regarding it as a burden to the virtuous mind. | |
| From: report of Democritus (fragments/reports [c.431 BCE]) by Seneca the Younger - On Providence §6 | |
| A reaction: The rival view is that wealth empowers a virtuous person to perform more fine deeds. Democritus seems to have a rather solitary view of virtue. |
| 1525 | Atomists say there are only three differences - in shape, arrangement and position [Democritus, by Aristotle] |
| Full Idea: Democritus and Leucippus say that there are only three differences - in shape, arrangement and position. | |
| From: report of Democritus (fragments/reports [c.431 BCE], A006) by Aristotle - Metaphysics 0985b15 |
| 493 | Experiences are merely convention; only atoms and the void are real [Democritus] |
| Full Idea: Sweet exists by convention, bitter by convention, colour by convention; atoms and void alone exist in reality. | |
| From: Democritus (fragments/reports [c.431 BCE], B009), quoted by Sextus Empiricus - Against the Logicians (two books) 7.135 |
| 17542 | 'Full' and 'Void' secularised Parmenides's Being and Not-being [Democritus, by Heisenberg] |
| Full Idea: In atomism, the antithesis of Being and Not-being of Parmenides is secularised into the antithesis of the 'Full' and the 'Void'. | |
| From: report of Democritus (fragments/reports [c.431 BCE]) by Werner Heisenberg - Physics and Philosophy 04 |
| 5947 | If only atoms are real and the rest is convention, we wouldn't bother to avoid pain [Democritus, by Diogenes of Oen.] |
| Full Idea: Democritus erred when he said that the atoms alone exist in truth among realities, but everything else is convention; for then, far from discovering the truth, we shall not even be able to live, since we shall avoid neither fire nor wounds. | |
| From: report of Democritus (fragments/reports [c.431 BCE]) by Diogenes (Oen) - fragments/reports F2 7 | |
| A reaction: The point is that we have to treat pain as a reality, not just as a convention. I suspect that Diogenes is making the same mistake made by modern attackers of 'eliminativism'. It is all about identity and reduction and levels of reality… |
| 13219 | When atoms touch, why don't they coalesce, like water drops? [Aristotle on Democritus] |
| Full Idea: Why, when they come into contact, do they [atoms] not coalesce into one, as drops of water run together when drop touches drop? | |
| From: comment on Democritus (fragments/reports [c.431 BCE]) by Aristotle - Coming-to-be and Passing-away (Gen/Corr) 326a33 | |
| A reaction: Presumably we must think of atoms as having basic ontological unity, rather than as being little globules of 'stuff'. They are more like monads than they are like mud. |
| 1533 | Because appearance is infinitely varied, atomists assume infinitely many shapes of atom [Democritus, by Aristotle] |
| Full Idea: They thought that truth lay in appearances, which they appreciated are contradictory and infinite, so they made the shapes of atoms infinite. Thus the infinite changes in compounds create the infinitely varies appearances. | |
| From: report of Democritus (fragments/reports [c.431 BCE], A009) by Aristotle - Coming-to-be and Passing-away (Gen/Corr) 315b9 |
| 20899 | Atoms cling together, until a stronger necessity disperses them [Democritus, by Aristotle] |
| Full Idea: Democritus thinks that the substances hold on to one another and remain together for a length of time until some stronger necessity arising from their surroundings shakes and disperses them. | |
| From: report of Democritus (fragments/reports [c.431 BCE], A037) by Aristotle - On Democritus (frag) | |
| A reaction: [quoted in Simplicius, Commentary on Aristotle's On the Heavens] He's not wrong. This seems to provide a mechanism for the Heracltean flux. Ancient critics wanted to know where the 'stronger necessity' came from. |
| 20898 | Atoms are irregular, hooked, concave, convex, and many other shapes [Democritus, by Aristotle] |
| Full Idea: Some substances are irregular, others hook-shaped, other concave, other convex, others provided with innumerable other differences. | |
| From: report of Democritus (fragments/reports [c.431 BCE], A037) by Aristotle - On Democritus (frag) | |
| A reaction: [quoted in Simplicius, Commentary on Aristotle's On the Heavens] 'Substance' here means a fundamental object, which for Democritus is an undividable atom. |
| 20908 | There could be an atom the size of the world [Democritus, by Ps-Plutarch] |
| Full Idea: Democritus say that it is possible that there exists an atom of the size of the world. | |
| From: report of Democritus (fragments/reports [c.431 BCE], A047) by Pseudo-Plutarch - On the Doctrine of the Philosophers 1.12.6 | |
| A reaction: The editor says this may have been a criticism of Democritus - presumably a reductio ad absurdum. But Democritus has no upper limit on the size of an atom. It challenges the imagination to treat such a huge thing as indivisible. |
| 1527 | There must be atoms, to avoid the absurdity of infinite division down to nothing [Democritus, by Aristotle] |
| Full Idea: If everything is infinitely divided, what survives the divisions? Alternatively, division would end at points with no magnitude, in which case bodies are composed of nothing. This is the argument claiming there are atoms of some magnitude. | |
| From: report of Democritus (fragments/reports [c.431 BCE], A048b) by Aristotle - Coming-to-be and Passing-away (Gen/Corr) 316a14- |
| 20909 | The basic atoms are without qualities - which only arise from encounters between atoms [Democritus, by Galen] |
| Full Idea: Democritus and the Epicureans posit that the first element is without quality, possessing by nature neither whiteness, blackness, sweetness or bitterness, warmth or cold. ...It is from the encounter of the atoms that all the sensible qualities come about. | |
| From: report of Democritus (fragments/reports [c.431 BCE], A049) by Galen - On Hippocrates and Plato 1.2 | |
| A reaction: Idea 493 comes in the middle of this summary by Galen. Hence atoms play the role that substrates play in object-based metaphysics. So atoms have the same problem. Is the shape of an atom a quality of an atom. Or are qualities what atoms DO? |
| 1536 | If a cone is horizontally sliced the surfaces can't be equal, so it goes up in steps [Democritus] |
| Full Idea: If a cone is cut parallel to the base are the two new surfaces equal or unequal? If they are unequal, the cone must have gone up in steps. If they are equal then the cone must have been a cylinder, which is absurd. | |
| From: Democritus (fragments/reports [c.431 BCE], B155), quoted by Plutarch - 72: Against Stoics on common Conceptions 1079e1 |
| 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. |
| 23314 | Greeks explained regularity by intellectual design, not by laws [Democritus, by Frede,M] |
| Full Idea: It is clear that Democritus had no idea of laws of nature …for in Greek thought regularity of behaviour is associated with design by an intellect. | |
| From: report of Democritus (fragments/reports [c.431 BCE]) by Michael Frede - A Free Will Intro | |
| A reaction: Ah. A simple realisation…! Seventeenth century laws of nature offered an explanation of natural order which didn't rely on God. Even though those scientists were all theists. |
| 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. |
| 24059 | Democritus is wrong: in a void we wouldn't see a distant ant in exact detail [Aristotle on Democritus] |
| Full Idea: Democritus did not speak correctly in supposing that if the intermediate space became a void, we would see an ant in exact detail if it were up in the heaven. …If the intermediate space became a void, rather nothing would be seen at all. | |
| From: comment on Democritus (fragments/reports [c.431 BCE]) by Aristotle - De Anima 419a15 | |
| A reaction: Depends what you mean by void, but Aristotle is nearer the truth. Is vision clearer in outer space than in our higher atmosphere? |
| 5101 | Movement is impossible in a void, because nothing can decide the direction of movement [Aristotle on Democritus] |
| Full Idea: Void makes it impossible for anything to move, since in a void there is nowhere for a thing to move to more or less than anywhere else, because the void by definition contains no differentiation. | |
| From: comment on Democritus (fragments/reports [c.431 BCE]) by Aristotle - Physics 214b32 | |
| A reaction: A lovely application of the Principle of Sufficient Reason. However this assumes that the cause of the movement is going to be in the void (telos?), rather than in the body which will move (modern causation?). |
| 20905 | Growth and movement would not exist if there were no void to receive them [Democritus] |
| Full Idea: They say that one argument for void is that otherwise local motion (that is, locomotion and growth) would not exist: for there would not seem to be motion if there were no void, for what is full is incapable of receiving anything. | |
| From: Democritus (fragments/reports [c.431 BCE], A019), quoted by Aristotle - Physics 213b03 | |
| A reaction: The modern concept of a 'field' seems to have removed the possibility of a genuine 'void'. |
| 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. |
| 20911 | There are unlimited worlds of varying sizes, some without life or water [Democritus, by Hippolytus] |
| Full Idea: Democritus says that there exist unlimited worlds and that they are different in magnitude. ...Some worlds are devoid of animals and plants and of all humidity. | |
| From: report of Democritus (fragments/reports [c.431 BCE], A010, A040) by Hippolytus - Refutation of All Heresies 1.12,13.2-4 | |
| A reaction: I'm not clear why Democritus came up with the idea of the Multicosmos. I don't suppose he meant the moon or planets, but another Cosmos. |
| 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) |
| 1535 | Democritus said people imagined gods as the source of what awed or frightened them [Democritus, by Sext.Empiricus] |
| Full Idea: Democritus thought that people imagined gods as responsible for the frightening and awesome things that happen in this world. | |
| From: report of Democritus (fragments/reports [c.431 BCE], A075) by Sextus Empiricus - Against the Professors (six books) 9.24 |
| 20915 | The soul is destroyed with the body [Democritus, by Ps-Plutarch] |
| Full Idea: Democritus says the soul is destructible, and is destroyed together with the whole body. | |
| From: report of Democritus (fragments/reports [c.431 BCE], A109) by Pseudo-Plutarch - On the Doctrine of the Philosophers 4.7.4 | |
| A reaction: This is the only belief possible for Democritus, since everything, including life and soul, is just the confluence of atoms, and they are regularly dispersed. This is the epitome of materialist philosophy. |