42 ideas
| 7920 | Descriptive metaphysics aims at actual structure, revisionary metaphysics at a better structure [Strawson,P] |
| Full Idea: Descriptive metaphysics (e.g. Aristotle and Kant) is content to describe the actual structure of our thought about the world; revisionary metaphysics (e.g. Descartes, Leibniz, Berkeley) is concerned to produce a better structure. | |
| From: Peter F. Strawson (Individuals:Essay in Descript Metaphysics [1959], Intro) | |
| A reaction: This distinction by Strawson was incredibly helpful in reinstating metaphysics as a feasible activity. I don't want to abandon the revisionary version. We can hammer the current metaphysics into a more efficient shape, or even create new concepts. |
| 7922 | Descriptive metaphysics concerns unchanging core concepts and categories [Strawson,P] |
| Full Idea: Descriptive metaphysics is primarily concerned with categories and concepts which, in their fundamental character, change not at all. They are the commonplaces of the least refined thinking, and the indispensable core for the most sophisticated humans. | |
| From: Peter F. Strawson (Individuals:Essay in Descript Metaphysics [1959], Intro) | |
| A reaction: This seems to be the basic premise for a modern metaphysician such as E.J.Lowe, though such thinkers are not averse to suggesting clarifications of our conceptual scheme. The aim must be good foundations for a successful edifice of knowledge. |
| 7921 | Close examination of actual word usage is the only sure way in philosophy [Strawson,P] |
| Full Idea: Up to a point, the reliance upon a close examination of the actual use of words is the best, and indeed the only sure, way in philosophy. | |
| From: Peter F. Strawson (Individuals:Essay in Descript Metaphysics [1959], Intro) | |
| A reaction: Probably the last bold assertion of ordinary language philosophy, though Strawson goes on the defend his 'deeper' version of the activity, which he says is 'descriptive metaphysics', rather than mere 'analysis'. Mere verbal analysis now looks hopeless. |
| 15924 | Predicative definitions are acceptable in mathematics if they distinguish objects, rather than creating them? [Zermelo, by Lavine] |
| Full Idea: On Zermelo's view, predicative definitions are not only indispensable to mathematics, but they are unobjectionable since they do not create the objects they define, but merely distinguish them from other objects. | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Shaughan Lavine - Understanding the Infinite V.1 | |
| A reaction: This seems to have an underlying platonism, that there are hitherto undefined 'objects' lying around awaiting the honour of being defined. Hm. |
| 10842 | The fact which is stated by a true sentence is not something in the world [Strawson,P] |
| Full Idea: The fact which is stated by a true sentence is not something in the world. | |
| From: Peter F. Strawson (Truth [1950], §2) | |
| A reaction: Everything is in the world. This may just be a quibble over how we should use the word 'fact'. At some point the substance of what is stated in a sentence must eventually be out there, or we would never act on what we say. |
| 10843 | Facts aren't exactly true statements, but they are what those statements say [Strawson,P] |
| Full Idea: Facts are what statements (when true) state; they are not what statements are about. ..But it would be wrong to identify 'fact' and 'true statement' for these expressions have different roles in our language. | |
| From: Peter F. Strawson (Truth [1950], §2) | |
| A reaction: Personally I like to reserve the word 'facts' for what is out there, independent of any human thought or speech. As a realist, I believe that the facts are quite independent of our attempts to understand the facts. True statements attempt to state facts. |
| 10844 | The statement that it is raining perfectly fits the fact that it is raining [Strawson,P] |
| Full Idea: What could fit more perfectly the fact that it is raining than the statement that it is raining? | |
| From: Peter F. Strawson (Truth [1950], §2) |
| 10841 | The word 'true' always refers to a possible statement [Strawson,P] |
| Full Idea: It is of prime importance to distinguish the fact that the use of 'true' always glances backwards or forwards to the actual or envisaged making of a statement by someone. | |
| From: Peter F. Strawson (Truth [1950], §1) | |
| A reaction: 'The truth of this matter will never be known'. Strawson is largely right, but it is crazy for any philosopher to use the word 'always' if they can possibly avoid it. |
| 17608 | We take set theory as given, and retain everything valuable, while avoiding contradictions [Zermelo] |
| Full Idea: Starting from set theory as it is historically given ...we must, on the one hand, restrict these principles sufficiently to exclude as contradiction and, on the other, take them sufficiently wide to retain all that is valuable in this theory. | |
| From: Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908], Intro) | |
| A reaction: Maddy calls this the one-step-back-from-disaster rule of thumb. Zermelo explicitly mentions the 'Russell antinomy' that blocked Frege's approach to sets. |
| 17607 | Set theory investigates number, order and function, showing logical foundations for mathematics [Zermelo] |
| Full Idea: Set theory is that branch whose task is to investigate mathematically the fundamental notions 'number', 'order', and 'function', taking them in their pristine, simple form, and to develop thereby the logical foundations of all of arithmetic and analysis. | |
| From: Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908], Intro) | |
| A reaction: At this point Zermelo seems to be a logicist. Right from the start set theory was meant to be foundational to mathematics, and not just a study of the logic of collections. |
| 10870 | ZFC: Existence, Extension, Specification, Pairing, Unions, Powers, Infinity, Choice [Zermelo, by Clegg] |
| Full Idea: Zermelo-Fraenkel axioms: Existence (at least one set); Extension (same elements, same set); Specification (a condition creates a new set); Pairing (two sets make a set); Unions; Powers (all subsets make a set); Infinity (set of successors); Choice | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.15 |
| 13012 | Zermelo published his axioms in 1908, to secure a controversial proof [Zermelo, by Maddy] |
| Full Idea: Zermelo proposed his listed of assumptions (including the controversial Axiom of Choice) in 1908, in order to secure his controversial proof of Cantor's claim that ' we can always bring any well-defined set into the form of a well-ordered set'. | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1 | |
| A reaction: This is interesting because it sometimes looks as if axiom systems are just a way of tidying things up. Presumably it is essential to get people to accept the axioms in their own right, the 'old-fashioned' approach that they be self-evident. |
| 17609 | Set theory can be reduced to a few definitions and seven independent axioms [Zermelo] |
| Full Idea: I intend to show how the entire theory created by Cantor and Dedekind can be reduced to a few definitions and seven principles, or axioms, which appear to be mutually independent. | |
| From: Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908], Intro) | |
| A reaction: The number of axioms crept up to nine or ten in subsequent years. The point of axioms is maximum reduction and independence from one another. He says nothing about self-evidence (though Boolos claimed a degree of that). |
| 9565 | Zermelo made 'set' and 'member' undefined axioms [Zermelo, by Chihara] |
| Full Idea: The terms 'set' and 'is a member of' are primitives of Zermelo's 1908 axiomatization of set theory. They are not given model-theoretic analyses or definitions. | |
| From: report of Ernst Zermelo (works [1920]) by Charles Chihara - A Structural Account of Mathematics 7.5 | |
| A reaction: This looks like good practice if you want to work with sets, but not so hot if you are interested in metaphysics. |
| 3339 | For Zermelo's set theory the empty set is zero and the successor of each number is its unit set [Zermelo, by Blackburn] |
| Full Idea: For Zermelo's set theory the empty set is zero and the successor of each number is its unit set. | |
| From: report of Ernst Zermelo (works [1920]) by Simon Blackburn - Oxford Dictionary of Philosophy p.280 |
| 17832 | Zermelo showed that the ZF axioms in 1930 were non-categorical [Zermelo, by Hallett,M] |
| Full Idea: Zermelo's paper sets out to show that the standard set-theoretic axioms (what he calls the 'constitutive axioms', thus the ZF axioms minus the axiom of infinity) have an unending sequence of different models, thus that they are non-categorical. | |
| From: report of Ernst Zermelo (On boundary numbers and domains of sets [1930]) by Michael Hallett - Introduction to Zermelo's 1930 paper p.1209 | |
| A reaction: Hallett says later that Zermelo is working with second-order set theory. The addition of an Axiom of Infinity seems to have aimed at addressing the problem, and the complexities of that were pursued by Gödel. |
| 13017 | Zermelo introduced Pairing in 1930, and it seems fairly obvious [Zermelo, by Maddy] |
| Full Idea: Zermelo's Pairing Axiom superseded (in 1930) his original 1908 Axiom of Elementary Sets. Like Union, its only justification seems to rest on 'limitations of size' and on the 'iterative conception'. | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1.3 | |
| A reaction: Maddy says of this and Union, that they seem fairly obvious, but that their justification is of prime importance, if we are to understand what the axioms should be. |
| 13028 | Replacement was added when some advanced theorems seemed to need it [Zermelo, by Maddy] |
| Full Idea: Zermelo included Replacement in 1930, after it was noticed that the sequence of power sets was needed, and Replacement gave the ordinal form of the well-ordering theorem, and justification for transfinite recursion. | |
| From: report of Ernst Zermelo (On boundary numbers and domains of sets [1930]) by Penelope Maddy - Believing the Axioms I §1.8 | |
| A reaction: Maddy says that this axiom suits the 'limitation of size' theorists very well, but is not so good for the 'iterative conception'. |
| 13015 | Zermelo used Foundation to block paradox, but then decided that only Separation was needed [Zermelo, by Maddy] |
| Full Idea: Zermelo used a weak form of the Axiom of Foundation to block Russell's paradox in 1906, but in 1908 felt that the form of his Separation Axiom was enough by itself, and left the earlier axiom off his published list. | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1.2 | |
| A reaction: Foundation turns out to be fairly controversial. Barwise actually proposes Anti-Foundation as an axiom. Foundation seems to be the rock upon which the iterative view of sets is built. Foundation blocks infinite descending chains of sets, and circularity. |
| 13020 | The Axiom of Separation requires set generation up to one step back from contradiction [Zermelo, by Maddy] |
| Full Idea: The most characteristic Zermelo axiom is Separation, guided by a new rule of thumb: 'one step back from disaster' - principles of set generation should be as strong as possible short of contradiction. | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1.4 | |
| A reaction: Why is there an underlying assumption that we must have as many sets as possible? We are then tempted to abolish axioms like Foundation, so that we can have even more sets! |
| 13486 | Not every predicate has an extension, but Separation picks the members that satisfy a predicate [Zermelo, by Hart,WD] |
| Full Idea: Zermelo assumes that not every predicate has an extension but rather that given a set we may separate out from it those of its members satisfying the predicate. This is called 'separation' (Aussonderung). | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by William D. Hart - The Evolution of Logic 3 |
| 8358 | There are no rules for the exact logic of ordinary language, because that doesn't exist [Strawson,P] |
| Full Idea: Neither Aristotelian nor Russellian rules give the exact logic of any expression of ordinary language; for ordinary language has no exact logic. | |
| From: Peter F. Strawson (On Referring [1950], §5) | |
| A reaction: This seems to imply that it is impossible to find precise logical forms, because of the pragmatic element in language, but I don't see why. Even more extreme modern pragmatics (where meaning is shifted) doesn't rule out precise underlying propositions. |
| 6413 | 'The present King of France is bald' presupposes existence, rather than stating it [Strawson,P, by Grayling] |
| Full Idea: Strawson argues that in saying 'the present King of France is bald' one is not stating that a present King of France exists, but presupposing or assuming that it does. | |
| From: report of Peter F. Strawson (On Referring [1950]) by A.C. Grayling - Russell Ch.2 | |
| A reaction: We have the notion of a leading question, such as 'when did you stop beating your wife?' But is a presupposition not simply an implied claim, as Russell said it was? |
| 8354 | Russell asks when 'The King of France is wise' would be a true assertion [Strawson,P] |
| Full Idea: The way in which Russell arrived at his analysis was by asking himself what would be the circumstances in which we would say that anyone who uttered the sentence 'The King of France is wise' had made a true assertion. | |
| From: Peter F. Strawson (On Referring [1950], §1) | |
| A reaction: This seems to connect Russell's theory of definite descriptions with the truth conditions theory of meaning which is associated (initially) with Frege. Truth will require some reference to what actually exists. |
| 17626 | The antinomy of endless advance and of completion is resolved in well-ordered transfinite numbers [Zermelo] |
| Full Idea: Two opposite tendencies of thought, the idea of creative advance and of collection and completion (underlying the Kantian 'antinomies') find their symbolic representation and their symbolic reconciliation in the transfinite numbers based on well-ordering. | |
| From: Ernst Zermelo (On boundary numbers and domains of sets [1930], §5) | |
| A reaction: [a bit compressed] It is this sort of idea, from one of the greatest set-theorists, that leads philosophers to think that the philosophy of mathematics may offer solutions to metaphysical problems. As an outsider, I am sceptical. |
| 13487 | In ZF, the Burali-Forti Paradox proves that there is no set of all ordinals [Zermelo, by Hart,WD] |
| Full Idea: In Zermelo's set theory, the Burali-Forti Paradox becomes a proof that there is no set of all ordinals (so 'is an ordinal' has no extension). | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by William D. Hart - The Evolution of Logic 3 |
| 15897 | Zermelo realised that Choice would facilitate the sort of 'counting' Cantor needed [Zermelo, by Lavine] |
| Full Idea: Zermelo realised that the Axiom of Choice (based on arbitrary functions) could be used to 'count', in the Cantorian sense, those collections that had given Cantor so much trouble, which restored a certain unity to set theory. | |
| From: report of Ernst Zermelo (Proof that every set can be well-ordered [1904]) by Shaughan Lavine - Understanding the Infinite I |
| 18178 | For Zermelo the successor of n is {n} (rather than n U {n}) [Zermelo, by Maddy] |
| Full Idea: For Zermelo the successor of n is {n} (rather than Von Neumann's successor, which is n U {n}). | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Naturalism in Mathematics I.2 n8 | |
| A reaction: I could ask some naive questions about the comparison of these two, but I am too shy about revealing my ignorance. |
| 13027 | Zermelo believed, and Von Neumann seemed to confirm, that numbers are sets [Zermelo, by Maddy] |
| Full Idea: Zermelo was a reductionist, and believed that theorems purportedly about numbers (cardinal or ordinal) are really about sets, and since Von Neumann's definitions of ordinals and cardinals as sets, this has become common doctrine. | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1.8 | |
| A reaction: Frege has a more sophisticated take on this approach. It may just be an updating of the Greek idea that arithmetic is about treating many things as a unit. A set bestows an identity on a group, and that is all that is needed. |
| 9627 | Different versions of set theory result in different underlying structures for numbers [Zermelo, by Brown,JR] |
| Full Idea: In Zermelo's set-theoretic definition of number, 2 is a member of 3, but not a member of 4; in Von Neumann's definition every number is a member of every larger number. This means they have two different structures. | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by James Robert Brown - Philosophy of Mathematics Ch. 4 | |
| A reaction: This refers back to the dilemma highlighted by Benacerraf, which was supposed to be the motivation for structuralism. My intuition says that the best answer is that they are both wrong. In a pattern, the nodes aren't 'members' of one another. |
| 16980 | We need a logical use of 'object' as predicate-worthy, and an 'ontological' use [Strawson,P] |
| Full Idea: There is a good case for a conservative reform of the word 'object'. Objects in the 'logical' sense would be all predicate-worthy identifiabilia whatever. Objects in the 'ontological' sense would form one ontological category among many others. | |
| From: Peter F. Strawson (Entity and Identity [1978], I n4) | |
| A reaction: This ambiguity has caused me no end of confusion (and irritation!). I wish philosophers wouldn't hijack perfectly good English words and give them weird meanings. Nice to have a distinguished fellow like Strawson make this suggestion. |
| 16979 | It makes no sense to ask of some individual thing what it is that makes it that individual [Strawson,P] |
| Full Idea: For no object is there a unique character or relation by which it must be identified if it is to be identified at all. This is why it makes no sense to ask, impersonally and in general, of some individual object what makes it the individual object it is. | |
| From: Peter F. Strawson (Entity and Identity [1978], I) | |
| A reaction: He links this remark with the claim that there is no individual essence, but he seems to view an individual essence as indispensable to recognition or individuation of the object, which I don't see. Recognise it first, work out its essence later. |
| 9282 | I can only apply consciousness predicates to myself if I can apply them to others [Strawson,P] |
| Full Idea: One can ascribed states of consciousness to oneself only if one can ascribe them to others. One can ascribe them to others only if one can identify other subjects of experience, and they cannot be identified only as subjects of experience. | |
| From: Peter F. Strawson (Individuals:Essay in Descript Metaphysics [1959], 3.4) | |
| A reaction: A neat linguistic twist on the analogy argument, but rather dubious, if it is actually meant to prove that other minds exist. It is based on his view of predicates - see Idea 9281. If the rest of humanity are zombies, why would I not apply them? |
| 9263 | A person is an entity to which we can ascribe predicates of consciousness and corporeality [Strawson,P] |
| Full Idea: What I mean by the concept of a person is the concept of a type of entity such that both predicates ascribing states of consciousness and predicates ascribing corporeal characteristics are equally applicable to a single individual of that single type. | |
| From: Peter F. Strawson (Individuals:Essay in Descript Metaphysics [1959], 3.4) | |
| A reaction: As Frankfurt points out, merely requiring the entity to be 'conscious' is a grossly inadequate definition of what we mean by a person, which is typically a being that is self-aware and capable of rational decisions between alternatives. |
| 17613 | We should judge principles by the science, not science by some fixed principles [Zermelo] |
| Full Idea: Principles must be judged from the point of view of science, and not science from the point of view of principles fixed once and for all. Geometry existed before Euclid's 'Elements', just as arithmetic and set theory did before Peano's 'Formulaire'. | |
| From: Ernst Zermelo (New Proof of Possibility of Well-Ordering [1908], §2a) | |
| A reaction: This shows why the axiomatisation of set theory is an ongoing and much-debated activity. |
| 8356 | The meaning of an expression or sentence is general directions for its use, to refer or to assert [Strawson,P] |
| Full Idea: To give the meaning of an expression is to give general directions for its use to refer to or mention particular objects or persons; in like manner, sentences are for use to make true or false assertions. | |
| From: Peter F. Strawson (On Referring [1950], §2) | |
| A reaction: The influence of Wittgenstein? I don't like it. The general idea that you can say what something is by giving directions for its use is what I think of as the Functional Fallacy: confusing the role of x with its inherent nature. Shirt as goalpost. |
| 10430 | Reference is mainly a social phenomenon [Strawson,P, by Sainsbury] |
| Full Idea: Strawson's early work gave a new direction to the study of reference by stressing that it is a social phenomenon. | |
| From: report of Peter F. Strawson (On Referring [1950]) by Mark Sainsbury - The Essence of Reference 18.2 | |
| A reaction: The question is whether speakers refer, or sentences, or expressions, or propositions. The modern consensus seems to be that some parts of language are inherently referring, but speakers combine such tools with context. Sounds right. |
| 10448 | If an expression can refer to anything, it may still instrinsically refer, but relative to a context [Bach on Strawson,P] |
| Full Idea: Strawson claimed that virtually any expression that can be used to refer to one thing in one context can be used to refer to something else in another context. Maybe expressions still refer, but only relative to a context. | |
| From: comment on Peter F. Strawson (On Referring [1950]) by Kent Bach - What Does It Take to Refer? 22.2 | |
| A reaction: If there is complete freedom, then Bach's criticism doesn't sound plausible. If something is semantically referential, that should impose pretty tight restrictions on speakers. Why distinguish names as intrinsically referential, and descriptions as not? |
| 8355 | Expressions don't refer; people use expressions to refer [Strawson,P] |
| Full Idea: 'Mentioning', or 'referring', is not something an expression does; it is something that someone can use an expression to do. | |
| From: Peter F. Strawson (On Referring [1950], §2) | |
| A reaction: That can't be whole story, because I might make a mistake when referring, so that I used the expression to refer to x, but unfortunately the words themselves referred to y. The power of language exceeds the intentions of speakers. |
| 8357 | If an utterance fails to refer then it is a pseudo-use, though a speaker may think they assert something [Strawson,P] |
| Full Idea: If an utterance is not talking about anything, then the speaker's use is not a genuine one, but a spurious or pseudo-use; he is not making either a true or a false assertion, though he may think he is. | |
| From: Peter F. Strawson (On Referring [1950], §2) | |
| A reaction: This is Strawson's verdict on 'The present King of France is bald'. His view puts speculative statements in no man's land. What do we make of 'Elvis lives' or 'phlogiston explains fire'? |
| 9281 | The idea of a predicate matches a range of things to which it can be applied [Strawson,P] |
| Full Idea: The idea of a predicate is correlative with a range of distinguishable individuals of which the predicate can be significantly, though not necessarily truly, affirmed. | |
| From: Peter F. Strawson (Individuals:Essay in Descript Metaphysics [1959], 3.4 n1) | |
| A reaction: Said to be one of Strawson's most important ideas. The idea is that you understand a predicate if you understand its range, not just a one-off application. So you must understand the implied universal, whatever that is. |
| 1513 | The Egyptians were the first to say the soul is immortal and reincarnated [Herodotus] |
| Full Idea: The Egyptians were the first to claim that the soul of a human being is immortal, and that each time the body dies the soul enters another creature just as it is being born. | |
| From: Herodotus (The Histories [c.435 BCE], 2.123.2) |