162 ideas
| 19693 | There is practical wisdom (for action), and theoretical wisdom (for deep understanding) [Aristotle, by Whitcomb] |
| Full Idea: Aristotle takes wisdom to come in two forms, the practical and the theoretical, the former of which is good judgement about how to act, and the latter of which is deep knowledge or understanding. | |
| From: report of Aristotle (works [c.330 BCE]) by Dennis Whitcomb - Wisdom Intro | |
| A reaction: The interesting question is then whether the two are connected. One might be thoroughly 'sensible' about action, without counting as 'wise', which seems to require a broader view of what is being done. Whitcomb endorses Aristotle on this idea. |
| 13466 | We are all post-Kantians, because he set the current agenda for philosophy [Hart,WD] |
| Full Idea: We are all post-Kantians, ...because Kant set an agenda for philosophy that we are still working through. | |
| From: William D. Hart (The Evolution of Logic [2010], 2) | |
| A reaction: Hart says that the main agenda is set by Kant's desire to defend the principle of sufficient reason against Hume's attack on causation. I would take it more generally to be the assessment of metaphysics, and of a priori knowledge. |
| 13477 | The problems are the monuments of philosophy [Hart,WD] |
| Full Idea: The real monuments of philosophy are its problems. | |
| From: William D. Hart (The Evolution of Logic [2010], 2) | |
| A reaction: Presumably he means '....rather than its solutions'. No other subject would be very happy with that sort of claim. Compare Idea 8243. A complaint against analytic philosophy is that it has achieved no consensus at all. |
| 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. |
| 13515 | To study abstract problems, some knowledge of set theory is essential [Hart,WD] |
| Full Idea: By now, no education in abstract pursuits is adequate without some familiarity with sets. | |
| From: William D. Hart (The Evolution of Logic [2010], 10) | |
| A reaction: A heart-sinking observation for those who aspire to study metaphysics and modality. The question is, what will count as 'some' familiarity? Are only professional logicians now allowed to be proper philosophers? |
| 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. |
| 1575 | For Aristotle logos is essentially the ability to talk rationally about questions of value [Roochnik on Aristotle] |
| Full Idea: For Aristotle logos is the ability to speak rationally about, with the hope of attaining knowledge, questions of value. | |
| From: comment on Aristotle (works [c.330 BCE]) by David Roochnik - The Tragedy of Reason p.26 |
| 1589 | Aristotle is the supreme optimist about the ability of logos to explain nature [Roochnik on Aristotle] |
| Full Idea: Aristotle is the great theoretician who articulates a vision of a world in which natural and stable structures can be rationally discovered. His is the most optimistic and richest view of the possibilities of logos | |
| From: comment on Aristotle (works [c.330 BCE]) by David Roochnik - The Tragedy of Reason p.95 |
| 8200 | Aristotelian definitions aim to give the essential properties of the thing defined [Aristotle, by Quine] |
| Full Idea: A real definition, according to the Aristotelian tradition, gives the essence of the kind of thing defined. Man is defined as a rational animal, and thus rationality and animality are of the essence of each of us. | |
| From: report of Aristotle (works [c.330 BCE]) by Willard Quine - Vagaries of Definition p.51 | |
| A reaction: Compare Idea 4385. Personally I prefer the Aristotelian approach, but we may have to say 'We cannot identify the essence of x, and so x cannot be defined'. Compare 'his mood was hard to define' with 'his mood was hostile'. |
| 4385 | Aristotelian definition involves first stating the genus, then the differentia of the thing [Aristotle, by Urmson] |
| Full Idea: For Aristotle, to give a definition one must first state the genus and then the differentia of the kind of thing to be defined. | |
| From: report of Aristotle (works [c.330 BCE]) by J.O. Urmson - Aristotle's Doctrine of the Mean p.157 | |
| A reaction: Presumably a modern definition would just be a list of properties, but Aristotle seeks the substance. How does he define a genus? - by placing it in a further genus? |
| 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'. |
| 13469 | Tarski showed how we could have a correspondence theory of truth, without using 'facts' [Hart,WD] |
| Full Idea: It is an ancient and honourable view that truth is correspondence to fact; Tarski showed us how to do without facts here. | |
| From: William D. Hart (The Evolution of Logic [2010], 2) | |
| A reaction: This is a very interesting spin on Tarski, who certainly seems to endorse the correspondence theory, even while apparently inventing a new 'semantic' theory of truth. It is controversial how far Tarski's theory really is a 'correspondence' theory. |
| 13504 | Truth for sentences is satisfaction of formulae; for sentences, either all sequences satisfy it (true) or none do [Hart,WD] |
| Full Idea: We explain truth for sentences in terms of satisfaction of formulae. The crux here is that for a sentence, either all sequences satisfy it or none do (with no middle ground). For formulae, some sequences may satisfy it and others not. | |
| From: William D. Hart (The Evolution of Logic [2010], 4) | |
| A reaction: This is the hardest part of Tarski's theory of truth to grasp. |
| 13503 | A first-order language has an infinity of T-sentences, which cannot add up to a definition of truth [Hart,WD] |
| Full Idea: In any first-order language, there are infinitely many T-sentences. Since definitions should be finite, the agglomeration of all the T-sentences is not a definition of truth. | |
| From: William D. Hart (The Evolution of Logic [2010], 4) | |
| A reaction: This may be a warning shot aimed at Davidson's extensive use of Tarski's formal account in his own views on meaning in natural language. |
| 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. |
| 13500 | Conditional Proof: infer a conditional, if the consequent can be deduced from the antecedent [Hart,WD] |
| Full Idea: A 'conditional proof' licenses inferences to a conditional from a deduction of its consequent from its antecedent. | |
| From: William D. Hart (The Evolution of Logic [2010], 4) | |
| A reaction: That is, a proof can be enshrined in an arrow. |
| 13502 | ∃y... is read as 'There exists an individual, call it y, such that...', and not 'There exists a y such that...' [Hart,WD] |
| Full Idea: When a quantifier is attached to a variable, as in '∃(y)....', then it should be read as 'There exists an individual, call it y, such that....'. One should not read it as 'There exists a y such that...', which would attach predicate to quantifier. | |
| From: William D. Hart (The Evolution of Logic [2010], 4) | |
| A reaction: The point is to make clear that in classical logic the predicates attach to the objects, and not to some formal component like a quantifier. |
| 13456 | Set theory articulates the concept of order (through relations) [Hart,WD] |
| Full Idea: It is set theory, and more specifically the theory of relations, that articulates order. | |
| From: William D. Hart (The Evolution of Logic [2010]) | |
| A reaction: It would seem that we mainly need set theory in order to talk accurately about order, and about infinity. The two come together in the study of the ordinal numbers. |
| 13497 | Nowadays ZFC and NBG are the set theories; types are dead, and NF is only useful for the whole universe [Hart,WD] |
| Full Idea: The theory of types is a thing of the past. There is now nothing to choose between ZFC and NBG (Neumann-Bernays-Gödel). NF (Quine's) is a more specialized taste, but is a place to look if you want the universe. | |
| From: William D. Hart (The Evolution of Logic [2010], 3) |
| 13443 | ∈ relates across layers, while ⊆ relates within layers [Hart,WD] |
| Full Idea: ∈ relates across layers (Plato is a member of his unit set and the set of people), while ⊆ relates within layers (the singleton of Plato is a subset of the set of people). This distinction only became clear in the 19th century. | |
| From: William D. Hart (The Evolution of Logic [2010], 1) | |
| A reaction: Getting these two clear may be the most important distinction needed to understand how set theory works. |
| 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. |
| 13442 | Without the empty set we could not form a∩b without checking that a and b meet [Hart,WD] |
| Full Idea: Without the empty set, disjoint sets would have no intersection, and we could not form a∩b without checking that a and b meet. This is an example of the utility of the empty set. | |
| From: William D. Hart (The Evolution of Logic [2010], 1) | |
| A reaction: A novice might plausibly ask why there should be an intersection for every pair of sets, if they have nothing in common except for containing this little puff of nothingness. But then what do novices know? |
| 13493 | In the modern view, foundation is the heart of the way to do set theory [Hart,WD] |
| Full Idea: In the second half of the twentieth century there emerged the opinion that foundation is the heart of the way to do set theory. | |
| From: William D. Hart (The Evolution of Logic [2010], 3) | |
| A reaction: It is foundation which is the central axiom of the iterative conception of sets, where each level of sets is built on previous levels, and they are all 'well-founded'. |
| 13495 | Foundation Axiom: an nonempty set has a member disjoint from it [Hart,WD] |
| Full Idea: The usual statement of Foundation is that any nonempty set has a member disjoint from it. This phrasing is ordinal-free and closer to the primitives of ZFC. | |
| From: William D. Hart (The Evolution of Logic [2010], 3) |
| 13461 | We can choose from finite and evident sets, but not from infinite opaque ones [Hart,WD] |
| Full Idea: When a set is finite, we can prove it has a choice function (∀x x∈A → f(x)∈A), but we need an axiom when A is infinite and the members opaque. From infinite shoes we can pick a left one, but from socks we need the axiom of choice. | |
| From: William D. Hart (The Evolution of Logic [2010], 1) | |
| A reaction: The socks example in from Russell 1919:126. |
| 13462 | With the Axiom of Choice every set can be well-ordered [Hart,WD] |
| Full Idea: It follows from the Axiom of Choice that every set can be well-ordered. | |
| From: William D. Hart (The Evolution of Logic [2010], 1) | |
| A reaction: For 'well-ordered' see Idea 13460. Every set can be ordered with a least member. |
| 13516 | If we accept that V=L, it seems to settle all the open questions of set theory [Hart,WD] |
| Full Idea: It has been said (by Burt Dreben) that the only reason set theorists do not generally buy the view that V = L is that it would put them out of business by settling their open questions. | |
| From: William D. Hart (The Evolution of Logic [2010], 10) | |
| A reaction: Hart says V=L breaks with the interative conception of sets at level ω+1, which is countable is the constructible view, but has continuum many in the cumulative (iterative) hierarch. The constructible V=L view is anti-platonist. |
| 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 |
| 13441 | Naïve set theory has trouble with comprehension, the claim that every predicate has an extension [Hart,WD] |
| Full Idea: 'Comprehension' is the assumption that every predicate has an extension. Naïve set theory is the theory whose axioms are extensionality and comprehension, and comprehension is thought to be its naivety. | |
| From: William D. Hart (The Evolution of Logic [2010], 1) | |
| A reaction: This doesn't, of course, mean that there couldn't be a more modest version of comprehension. The notorious difficulty come with the discovery of self-referring predicates which can't possibly have extensions. |
| 13494 | The iterative conception may not be necessary, and may have fixed points or infinitely descending chains [Hart,WD] |
| Full Idea: That the iterative sets suffice for most of ZFC does not show they are necessary, nor is it evident that the set of operations has no fixed points (as 0 is a fixed point for square-of), and no infinitely descending chains (like negative integers). | |
| From: William D. Hart (The Evolution of Logic [2010], 3) | |
| A reaction: People don't seem to worry that they aren't 'necessary', and further measures are possible to block infinitely descending chains. |
| 13457 | A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets [Hart,WD] |
| Full Idea: We say that a binary relation R 'partially orders' a field A just in case R is irreflexive (so that nothing bears R to itself) and transitive. When the set is {a,b}, its subsets {a} and {b} are incomparable in a partial ordering. | |
| From: William D. Hart (The Evolution of Logic [2010], 1) |
| 13458 | A partial ordering becomes 'total' if any two members of its field are comparable [Hart,WD] |
| Full Idea: A partial ordering is a 'total ordering' just in case any two members of its field are comparable, that is, either a is R to b, or b is R to a, or a is b. | |
| From: William D. Hart (The Evolution of Logic [2010], 1) | |
| A reaction: See Idea 13457 for 'partial ordering'. The three conditions are known as the 'trichotomy' condition. |
| 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) |
| 13460 | 'Well-ordering' must have a least member, so it does the natural numbers but not the integers [Hart,WD] |
| Full Idea: A total order 'well-orders' its field just in case any nonempty subset B of its field has an R-least member, that is, there is a b in B such that for any a in B different from b, b bears R to a. So less-than well-orders natural numbers, but not integers. | |
| From: William D. Hart (The Evolution of Logic [2010], 1) | |
| A reaction: The natural numbers have a starting point, but the integers are infinite in both directions. In plain English, an order is 'well-ordered' if there is a starting point. |
| 13490 | Von Neumann defines α<β as α∈β [Hart,WD] |
| Full Idea: One of the glories of Von Neumann's theory of numbers is to define α < β to mean that α ∈ β. | |
| From: William D. Hart (The Evolution of Logic [2010], 3) |
| 13481 | Maybe sets should be rethought in terms of the even more basic categories [Hart,WD] |
| Full Idea: Some have claimed that sets should be rethought in terms of still more basic things, categories. | |
| From: William D. Hart (The Evolution of Logic [2010], 2) | |
| A reaction: [He cites F.William Lawvere 1966] It appears to the the context of foundations for mathematics that he has in mind. |
| 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). |
| 13282 | Aristotle relativises the notion of wholeness to different measures [Aristotle, by Koslicki] |
| Full Idea: Aristotle proposes to relativise unity and plurality, so that a single object can be both one (indivisible) and many (divisible) simultaneously, without contradiction, relative to different measures. Wholeness has degrees, with the strength of the unity. | |
| From: report of Aristotle (works [c.330 BCE]) by Kathrin Koslicki - The Structure of Objects 7.2.12 | |
| A reaction: [see Koslicki's account of Aristotle for details] As always, the Aristotelian approach looks by far the most promising. Simplistic mechanical accounts of how parts make wholes aren't going to work. We must include the conventional and conceptual bit. |
| 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. |
| 4730 | For Aristotle, the subject-predicate structure of Greek reflected a substance-accident structure of reality [Aristotle, by O'Grady] |
| Full Idea: Aristotle apparently believed that the subject-predicate structure of Greek reflected the substance-accident nature of reality. | |
| From: report of Aristotle (works [c.330 BCE]) by Paul O'Grady - Relativism Ch.4 | |
| A reaction: We need not assume that Aristotle is wrong. It is a chicken-and-egg. There is something obvious about subject-predicate language, if one assumes that unified objects are part of nature, and not just conventional. |
| 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. |
| 13506 | The universal quantifier can't really mean 'all', because there is no universal set [Hart,WD] |
| Full Idea: All the main set theories deny that there is a set of which everything is a member. No interpretation has a domain with everything in it. So the universal quantifier never gets to mean everything all at once; 'all' does not mean all. | |
| From: William D. Hart (The Evolution of Logic [2010], 4) | |
| A reaction: Could you have an 'uncompleted' universal set, in the spirit of uncompleted infinities? In ordinary English we can talk about 'absolutely everything' - we just can't define a set of everything. Must we 'define' our domain? |
| 13512 | Modern model theory begins with the proof of Los's Conjecture in 1962 [Hart,WD] |
| Full Idea: The beginning of modern model theory was when Morley proved Los's Conjecture in 1962 - that a complete theory in a countable language categorical in one uncountable cardinal is categorical in all. | |
| From: William D. Hart (The Evolution of Logic [2010], 9) |
| 13505 | Model theory studies how set theory can model sets of sentences [Hart,WD] |
| Full Idea: Modern model theory investigates which set theoretic structures are models for which collections of sentences. | |
| From: William D. Hart (The Evolution of Logic [2010], 4) | |
| A reaction: So first you must choose your set theory (see Idea 13497). Then you presumably look at how to formalise sentences, and then look at the really tricky ones, many of which will involve various degrees of infinity. |
| 13511 | Model theory is mostly confined to first-order theories [Hart,WD] |
| Full Idea: There is no developed methematics of models for second-order theories, so for the most part, model theory is about models for first-order theories. | |
| From: William D. Hart (The Evolution of Logic [2010], 9) |
| 13513 | Models are ways the world might be from a first-order point of view [Hart,WD] |
| Full Idea: Models are ways the world might be from a first-order point of view. | |
| From: William D. Hart (The Evolution of Logic [2010], 9) |
| 13496 | First-order logic is 'compact': consequences of a set are consequences of a finite subset [Hart,WD] |
| Full Idea: First-order logic is 'compact', which means that any logical consequence of a set (finite or infinite) of first-order sentences is a logical consequence of a finite subset of those sentences. | |
| From: William D. Hart (The Evolution of Logic [2010], 3) |
| 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. |
| 13484 | Berry's Paradox: we succeed in referring to a number, with a term which says we can't do that [Hart,WD] |
| Full Idea: Berry's Paradox: by the least number principle 'the least number denoted by no description of fewer than 79 letters' exists, but we just referred to it using a description of 77 letters. | |
| From: William D. Hart (The Evolution of Logic [2010], 3) | |
| A reaction: I struggle with this. If I refer to 'an object to which no human being could possibly refer', have I just referred to something? Graham Priest likes this sort of idea. |
| 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 |
| 13482 | The Burali-Forti paradox is a crisis for Cantor's ordinals [Hart,WD] |
| Full Idea: The Burali-Forti Paradox was a crisis for Cantor's theory of ordinal numbers. | |
| From: William D. Hart (The Evolution of Logic [2010], 3) |
| 13507 | The machinery used to solve the Liar can be rejigged to produce a new Liar [Hart,WD] |
| Full Idea: In effect, the machinery introduced to solve the liar can always be rejigged to yield another version the liar. | |
| From: William D. Hart (The Evolution of Logic [2010], 4) | |
| A reaction: [He cites Hans Herzberger 1980-81] The machinery is Tarski's device of only talking about sentences of a language by using a 'metalanguage'. |
| 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. |
| 13491 | The axiom of infinity with separation gives a least limit ordinal ω [Hart,WD] |
| Full Idea: The axiom of infinity with separation yields a least limit ordinal, which is called ω. | |
| From: William D. Hart (The Evolution of Logic [2010], 3) |
| 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. |
| 13463 | There are at least as many infinite cardinals as transfinite ordinals (because they will map) [Hart,WD] |
| Full Idea: Since we can map the transfinite ordinals one-one into the infinite cardinals, there are at least as many infinite cardinals as transfinite ordinals. | |
| From: William D. Hart (The Evolution of Logic [2010], 1) |
| 13492 | Von Neumann's ordinals generalise into the transfinite better, because Zermelo's ω is a singleton [Hart,WD] |
| Full Idea: It is easier to generalize von Neumann's finite ordinals into the transfinite. All Zermelo's nonzero finite ordinals are singletons, but if ω were a singleton it is hard to see how if could fail to be the successor of its member and so not a limit. | |
| From: William D. Hart (The Evolution of Logic [2010], 3) |
| 13459 | The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers [Hart,WD] |
| Full Idea: We can show (using the axiom of choice) that the less-than relation, <, well-orders the ordinals, ...and that it partially orders the ordinals, ...and that it totally orders the ordinals. | |
| From: William D. Hart (The Evolution of Logic [2010], 1) |
| 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) |
| 13446 | 19th century arithmetization of analysis isolated the real numbers from geometry [Hart,WD] |
| Full Idea: The real numbers were not isolated from geometry until the arithmetization of analysis during the nineteenth century. | |
| From: William D. Hart (The Evolution of Logic [2010], 1) |
| 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) |
| 13509 | We can establish truths about infinite numbers by means of induction [Hart,WD] |
| Full Idea: Mathematical induction is a way to establish truths about the infinity of natural numbers by a finite proof. | |
| From: William D. Hart (The Evolution of Logic [2010], 5) | |
| A reaction: If there are truths about infinities, it is very tempting to infer that the infinities must therefore 'exist'. A nice, and large, question in philosophy is whether there can be truths without corresponding implications of existence. |
| 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. |
| 13474 | Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several [Hart,WD] |
| Full Idea: There is a familiar comparison between Euclid (unique parallel) and 'spherical' geometry (no parallel) and 'saddle' geometry (several parallels). | |
| From: William D. Hart (The Evolution of Logic [2010], 2) |
| 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. |
| 13471 | Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD] |
| Full Idea: The thesis that no existence proposition is analytic is one of the few constants in philosophical consciences, but there are many existence claims in mathematics, such as the infinity of primes, five regular solids, and certain undecidable propositions. | |
| From: William D. Hart (The Evolution of Logic [2010], 2) |
| 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. |
| 13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD] |
| Full Idea: Jespersen calls a noun a mass word when it has no plural, does not take numerical adjectives, and does not take 'fewer'. | |
| From: William D. Hart (The Evolution of Logic [2010], 3) | |
| A reaction: Jespersen was a great linguistics expert. |
| 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. |
| 13276 | The unmoved mover and the soul show Aristotelian form as the ultimate mereological atom [Aristotle, by Koslicki] |
| Full Idea: Aristotle's discussion of the unmoved mover and of the soul confirms the suspicion that form, when it is not thought of as the object represented in a definition, plays the role of the ultimate mereological atom within his system. | |
| From: report of Aristotle (works [c.330 BCE]) by Kathrin Koslicki - The Structure of Objects 6.6 | |
| A reaction: Aristotle is concerned with which things are 'divisible', and he cites these two examples as indivisible, but they may be too unusual to offer an actual theory of how Aristotle builds up wholes from atoms. He denies atoms in matter. |
| 13277 | The 'form' is the recipe for building wholes of a particular kind [Aristotle, by Koslicki] |
| Full Idea: Thus in Aristotle we may think of an object's formal components as a sort of recipe for how to build wholes of that particular kind. | |
| From: report of Aristotle (works [c.330 BCE]) by Kathrin Koslicki - The Structure of Objects 7.2.5 | |
| A reaction: In the elusive business of pinning down what Aristotle means by the crucial idea of 'form', this analogy strikes me as being quite illuminating. It would fit DNA in living things, and the design of an artifact. |
| 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)? |
| 5991 | For Aristotle, knowledge is of causes, and is theoretical, practical or productive [Aristotle, by Code] |
| Full Idea: Aristotle thinks that in general we have knowledge or understanding when we grasp causes, and he distinguishes three fundamental types of knowledge - theoretical, practical and productive. | |
| From: report of Aristotle (works [c.330 BCE]) by Alan D. Code - Aristotle | |
| A reaction: Productive knowledge we tend to label as 'knowing how'. The centrality of causes for knowledge would get Aristotle nowadays labelled as a 'naturalist'. It is hard to disagree with his three types, though they may overlap. |
| 11239 | The notion of a priori truth is absent in Aristotle [Aristotle, by Politis] |
| Full Idea: The notion of a priori truth is conspicuously absent in Aristotle. | |
| From: report of Aristotle (works [c.330 BCE]) by Vassilis Politis - Aristotle and the Metaphysics 1.5 | |
| A reaction: Cf. Idea 11240. |
| 13480 | Fregean self-evidence is an intrinsic property of basic truths, rules and definitions [Hart,WD] |
| Full Idea: The conception of Frege is that self-evidence is an intrinsic property of the basic truths, rules, and thoughts expressed by definitions. | |
| From: William D. Hart (The Evolution of Logic [2010], p.350) | |
| A reaction: The problem is always that what appears to be self-evident may turn out to be wrong. Presumably the effort of arriving at a definition ought to clarify and support the self-evident ingredient. |
| 13476 | The failure of key assumptions in geometry, mereology and set theory throw doubt on the a priori [Hart,WD] |
| Full Idea: In the case of the parallels postulate, Euclid's fifth axiom (the whole is greater than the part), and comprehension, saying was believing for a while, but what was said was false. This should make a shrewd philosopher sceptical about a priori knowledge. | |
| From: William D. Hart (The Evolution of Logic [2010], 2) | |
| A reaction: Euclid's fifth is challenged by infinite numbers, and comprehension is challenged by Russell's paradox. I can't see a defender of the a priori being greatly worried about these cases. No one ever said we would be right - in doing arithmetic, for example. |
| 23312 | Aristotle is a rationalist, but reason is slowly acquired through perception and experience [Aristotle, by Frede,M] |
| Full Idea: Aristotle is a rationalist …but reason for him is a disposition which we only acquire over time. Its acquisition is made possible primarily by perception and experience. | |
| From: report of Aristotle (works [c.330 BCE]) by Michael Frede - Aristotle's Rationalism p.173 | |
| A reaction: I would describe this process as the gradual acquisition of the skill of objectivity, which needs the right knowledge and concepts to evaluate new experiences. |
| 16111 | Aristotle wants to fit common intuitions, and therefore uses language as a guide [Aristotle, by Gill,ML] |
| Full Idea: Since Aristotle generally prefers a metaphysical theory that accords with common intuitions, he frequently relies on facts about language to guide his metaphysical claims. | |
| From: report of Aristotle (works [c.330 BCE]) by Mary Louise Gill - Aristotle on Substance Ch.5 | |
| A reaction: I approve of his procedure. I take intuition to be largely rational justifications too complex for us to enunciate fully, and language embodies folk intuitions in its concepts (especially if the concepts occur in many languages). |
| 16971 | Plato says sciences are unified around Forms; Aristotle says they're unified around substance [Aristotle, by Moravcsik] |
| Full Idea: Plato's unity of science principle states that all - legitimate - sciences are ultimately about the Forms. Aristotle's principle states that all sciences must be, ultimately, about substances, or aspects of substances. | |
| From: report of Aristotle (works [c.330 BCE], 1) by Julius Moravcsik - Aristotle on Adequate Explanations 1 |
| 11243 | Aristotelian explanations are facts, while modern explanations depend on human conceptions [Aristotle, by Politis] |
| Full Idea: For Aristotle things which explain (the explanantia) are facts, which should not be associated with the modern view that says explanations are dependent on how we conceive and describe the world (where causes are independent of us). | |
| From: report of Aristotle (works [c.330 BCE]) by Vassilis Politis - Aristotle and the Metaphysics 2.1 | |
| A reaction: There must be some room in modern thought for the Aristotelian view, if some sort of robust scientific realism is being maintained against the highly linguistic view of philosophy found in the twentieth century. |
| 3320 | Aristotle's standard analysis of species and genus involves specifying things in terms of something more general [Aristotle, by Benardete,JA] |
| Full Idea: The standard Aristotelian doctrine of species and genus in the theory of anything whatever involves specifying what the thing is in terms of something more general. | |
| From: report of Aristotle (works [c.330 BCE]) by José A. Benardete - Metaphysics: the logical approach Ch.10 |
| 12000 | Aristotle regularly says that essential properties explain other significant properties [Aristotle, by Kung] |
| Full Idea: The view that essential properties are those in virtue of which other significant properties of the subjects under investigation can be explained is encountered repeatedly in Aristotle's work. | |
| From: report of Aristotle (works [c.330 BCE]) by Joan Kung - Aristotle on Essence and Explanation IV | |
| A reaction: What does 'significant' mean here? I take it that the significant properties are the ones which explain the role, function and powers of the object. |
| 23300 | Aristotle and the Stoics denied rationality to animals, while Platonists affirmed it [Aristotle, by Sorabji] |
| Full Idea: Aristotle, and also the Stoics, denied rationality to animals. …The Platonists, the Pythagoreans, and some more independent Aristotelians, did grant reason and intellect to animals. | |
| From: report of Aristotle (works [c.330 BCE]) by Richard Sorabji - Rationality 'Denial' | |
| A reaction: This is not the same as affirming or denying their consciousness. The debate depends on how rationality is conceived. |
| 13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
| Full Idea: A Fregean concept is a function that assigns to each object a truth value. So instead of the colour green, the concept GREEN assigns truth to each green thing, but falsity to anything else. | |
| From: William D. Hart (The Evolution of Logic [2010], 2) | |
| A reaction: This would seem to immediately hit the renate/cordate problem, if there was a world in which all and only the green things happened to be square. How could Frege then distinguish the green from the square? Compare Idea 8245. |
| 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. |
| 11240 | The notion of analytic truth is absent in Aristotle [Aristotle, by Politis] |
| Full Idea: The notion of analytic truth is conspicuously absent in Aristotle. | |
| From: report of Aristotle (works [c.330 BCE]) by Vassilis Politis - Aristotle and the Metaphysics 1.5 | |
| A reaction: Cf. Idea 11239. |
| 6559 | Aristotle never actually says that man is a rational animal [Aristotle, by Fogelin] |
| Full Idea: To the best of my knowledge (and somewhat to my surprise), Aristotle never actually says that man is a rational animal; however, he all but says it. | |
| From: report of Aristotle (works [c.330 BCE]) by Robert Fogelin - Walking the Tightrope of Reason Ch.1 | |
| A reaction: When I read this I thought that this database would prove Fogelin wrong, but it actually supports him, as I can't find it in Aristotle either. Descartes refers to it in Med.Two. In Idea 5133 Aristotle does say that man is a 'social being'. But 22586! |
| 11150 | It is the mark of an educated mind to be able to entertain an idea without accepting it [Aristotle] |
| Full Idea: It is the mark of an educated mind to be able to entertain an idea without accepting it. | |
| From: Aristotle (works [c.330 BCE]) | |
| A reaction: The epigraph on a David Chalmers website. A wonderful remark, and it should be on the wall of every beginners' philosophy class. However, while it is in the spirit of Aristotle, it appears to be a misattribution with no ancient provenance. |
| 3037 | Aristotle said the educated were superior to the uneducated as the living are to the dead [Aristotle, by Diog. Laertius] |
| Full Idea: Aristotle was asked how much educated men were superior to those uneducated; "As much," he said, "as the living are to the dead." | |
| From: report of Aristotle (works [c.330 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 05.1.11 |
| 8660 | There are potential infinities (never running out), but actual infinity is incoherent [Aristotle, by Friend] |
| Full Idea: Aristotle developed his own distinction between potential infinity (never running out) and actual infinity (there being a collection of an actual infinite number of things, such as places, times, objects). He decided that actual infinity was incoherent. | |
| From: report of Aristotle (works [c.330 BCE]) by Michèle Friend - Introducing the Philosophy of Mathematics 1.3 | |
| A reaction: Friend argues, plausibly, that this won't do, since potential infinity doesn't make much sense if there is not an actual infinity of things to supply the demand. It seems to just illustrate how boggling and uncongenial infinity was to Aristotle. |
| 12058 | Aristotle's matter can become any other kind of matter [Aristotle, by Wiggins] |
| Full Idea: Aristotle's conception of matter permits any kind of matter to become any other kind of matter. | |
| From: report of Aristotle (works [c.330 BCE]) by David Wiggins - Substance 4.11.2 | |
| A reaction: This is obviously crucial background information when we read Aristotle on matter. Our 92+ elements, and fixed fundamental particles, gives a quite different picture. Aristotle would discuss form and matter quite differently now. |
| 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. |
| 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. |
| 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) |
| 22729 | The concepts of gods arose from observing the soul, and the cosmos [Aristotle, by Sext.Empiricus] |
| Full Idea: Aristotle said that the conception of gods arose among mankind from two originating causes, namely from events which concern the soul and from celestial phenomena. | |
| From: report of Aristotle (works [c.330 BCE], Frag 10) by Sextus Empiricus - Against the Physicists (two books) I.20 | |
| A reaction: The cosmos suggests order, and possible creation. What do events of the soul suggest? It doesn't seem to be its non-physical nature, because Aristotle is more of a functionalist. Puzzling. (It says later that gods are like the soul). |