53 ideas
| 12105 | Our knowledge starts in theology, passes through metaphysics, and ends in positivism [Comte] |
| Full Idea: Our principal conceptions, each branch of our knowledge, passes in succession through three different theoretical states: the theological or fictitious state, the metaphysical or abstract state, and the scientific or positive state. | |
| From: Auguste Comte (Intro to Positive Philosophy [1830], Ch.1) | |
| A reaction: See Idea 5077 for the abstraction step. The idea that there is a 'law' here, as Comte thinks, is daft, but something of what he describes is undeniable. I suspect, though, that science rests on abstractions, so the last part is wrong. |
| 12104 | All ideas must be understood historically [Comte] |
| Full Idea: No idea can be properly understood apart from its history. | |
| From: Auguste Comte (Intro to Positive Philosophy [1830], Ch.1) | |
| A reaction: This is somewhat dubious. Comte is preparing the ground for asserting positivism by rejecting out-of-date theology and metaphysics. The history is revealing, but can be misleading, when a meaning shifts. Try 'object' in logic. |
| 12112 | Metaphysics is just the oversubtle qualification of abstract names for phenomena [Comte] |
| Full Idea: The development of positivism was caused by the concept of metaphysical agents gradually becoming so empty through oversubtle qualification that all right-minded persons considered them to be only the abstract names of the phenomena in question. | |
| From: Auguste Comte (Intro to Positive Philosophy [1830], Ch.1) | |
| A reaction: I have quite a lot of sympathy with this thesis, but not couched in this negative way. I take abstraction to be essential to scientific thought, and wisdom to occur amongst the higher reaches of the abstractions. |
| 12106 | Positivism gives up absolute truth, and seeks phenomenal laws, by reason and observation [Comte] |
| Full Idea: In the positive state, the human mind, recognizing the impossibility of obtaining absolute truth, gives up the search for hidden and final causes. It endeavours to discover, by well-combined reasoning and observation, the actual laws of phenomena. | |
| From: Auguste Comte (Intro to Positive Philosophy [1830], Ch.1) | |
| A reaction: [compressed] Positivism attempted to turn the Humean regularity view of laws into a semi-religion. It is striking how pessimistic Comte was (as was Hume) about the chances of science revealing deep explanations. He would be astoundeds. |
| 7491 | The phases of human thought are theological, then metaphysical, then positivist [Comte, by Watson] |
| Full Idea: The first phase of humanity was theological, attributing phenomena to a deity, the second metaphysical stage attributed them to abstract forms, the third positive stage abandons ultimate causes and just searches for regularities. | |
| From: report of Auguste Comte (Course of Positive Philosophy [1846]) by Peter Watson - Ideas Ch.32 | |
| A reaction: This is obviously a highly empirical programme, which reasserts Hume's view of the laws of nature. Effectively, positivism just is the rejection of metaphysics. |
| 12111 | Positivism is the final state of human intelligence [Comte] |
| Full Idea: The positive philosophy represents the true final state of human intelligence. | |
| From: Auguste Comte (Intro to Positive Philosophy [1830], Ch.1) | |
| A reaction: This is the sort of remark which made Comte notorious, and it looks a bit extravagant now, but the debate about his view is still ongoing. I am certainly sympathetic to his general drift. |
| 12114 | Science can drown in detail, so we need broad scientists (to keep out the metaphysicians) [Comte] |
| Full Idea: Getting lost in a mass of detail is the weak side of positivism, where partisans of theology and metaphysics may attack with some hope of success. ...We must train scientists who will consider all the different branches of positive science. | |
| From: Auguste Comte (Intro to Positive Philosophy [1830], Ch.1) | |
| A reaction: This would be Comte's answer now to those who claim there is still a role for metaphysics within the scientific world view. I would say that metaphysics not only takes an overview, but also deals with higher generalisations than Comte's general scientist. |
| 12116 | Only positivist philosophy can terminate modern social crises [Comte] |
| Full Idea: We may look upon the positive philosophy as constituting the only solid basis for the social reorganisation that must terminate the crisis in which the most civilized nations have found themselves for so long. | |
| From: Auguste Comte (Intro to Positive Philosophy [1830], Ch.1) | |
| A reaction: He is proposing not only to use positivist methods to solve social problems (he coined the word 'sociology'), but is also proposing that positivism itself should act as the unifying belief-system for future society. Science will be our religion. |
| 9724 | Until the 1960s the only semantics was truth-tables [Enderton] |
| Full Idea: Until the 1960s standard truth-table semantics were the only ones that there were. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.10.1) | |
| A reaction: The 1960s presumably marked the advent of possible worlds. |
| 9703 | 'dom R' indicates the 'domain' of objects having a relation [Enderton] |
| Full Idea: 'dom R' indicates the 'domain' of a relation, that is, the set of all objects that are members of ordered pairs and that have that relation. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9705 | 'fld R' indicates the 'field' of all objects in the relation [Enderton] |
| Full Idea: 'fld R' indicates the 'field' of a relation, that is, the set of all objects that are members of ordered pairs on either side of the relation. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9704 | 'ran R' indicates the 'range' of objects being related to [Enderton] |
| Full Idea: 'ran R' indicates the 'range' of a relation, that is, the set of all objects that are members of ordered pairs and that are related to by the first objects. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9710 | We write F:A→B to indicate that A maps into B (the output of F on A is in B) [Enderton] |
| Full Idea: We write F : A → B to indicate that A maps into B, that is, the domain of relating things is set A, and the things related to are all in B. If we add that F = B, then A maps 'onto' B. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9707 | 'F(x)' is the unique value which F assumes for a value of x [Enderton] |
|
Full Idea:
F(x) is a 'function', which indicates the unique value which y takes in |
|
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 13206 | A 'linear or total ordering' must be transitive and satisfy trichotomy [Enderton] |
| Full Idea: A 'linear ordering' (or 'total ordering') on A is a binary relation R meeting two conditions: R is transitive (of xRy and yRz, the xRz), and R satisfies trichotomy (either xRy or x=y or yRx). | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 3:62) |
| 13201 | ∈ says the whole set is in the other; ⊆ says the members of the subset are in the other [Enderton] |
| Full Idea: To know if A ∈ B, we look at the set A as a single object, and check if it is among B's members. But if we want to know whether A ⊆ B then we must open up set A and check whether its various members are among the members of B. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 1:04) | |
| A reaction: This idea is one of the key ideas to grasp if you are going to get the hang of set theory. John ∈ USA ∈ UN, but John is not a member of the UN, because he isn't a country. See Idea 12337 for a special case. |
| 13204 | The 'ordered pair' <x,y> is defined to be {{x}, {x,y}} [Enderton] |
| Full Idea: The 'ordered pair' <x,y> is defined to be {{x}, {x,y}}; hence it can be proved that <u,v> = <x,y> iff u = x and v = y (given by Kuratowski in 1921). ...The definition is somewhat arbitrary, and others could be used. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 3:36) | |
| A reaction: This looks to me like one of those regular cases where the formal definitions capture all the logical behaviour of the concept that are required for inference, while failing to fully capture the concept for ordinary conversation. |
| 9699 | The 'powerset' of a set is all the subsets of a given set [Enderton] |
| Full Idea: The 'powerset' of a set is all the subsets of a given set. Thus: PA = {x : x ⊆ A}. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9700 | Two sets are 'disjoint' iff their intersection is empty [Enderton] |
| Full Idea: Two sets are 'disjoint' iff their intersection is empty (i.e. they have no members in common). | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9702 | A 'domain' of a relation is the set of members of ordered pairs in the relation [Enderton] |
| Full Idea: The 'domain' of a relation is the set of all objects that are members of ordered pairs that are members of the relation. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9701 | A 'relation' is a set of ordered pairs [Enderton] |
| Full Idea: A 'relation' is a set of ordered pairs. The ordering relation on the numbers 0-3 is captured by - in fact it is - the set of ordered pairs {<0,1>,<0,2>,<0,3>,<1,2>,<1,3>,<2,3>}. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) | |
| A reaction: This can't quite be a definition of order among numbers, since it relies on the notion of a 'ordered' pair. |
| 9706 | A 'function' is a relation in which each object is related to just one other object [Enderton] |
| Full Idea: A 'function' is a relation which is single-valued. That is, for each object, there is only one object in the function set to which that object is related. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9708 | A function 'maps A into B' if the relating things are set A, and the things related to are all in B [Enderton] |
| Full Idea: A function 'maps A into B' if the domain of relating things is set A, and the things related to are all in B. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9709 | A function 'maps A onto B' if the relating things are set A, and the things related to are set B [Enderton] |
| Full Idea: A function 'maps A onto B' if the domain of relating things is set A, and the things related to are set B. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9711 | A relation is 'reflexive' on a set if every member bears the relation to itself [Enderton] |
| Full Idea: A relation is 'reflexive' on a set if every member of the set bears the relation to itself. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9712 | A relation is 'symmetric' on a set if every ordered pair has the relation in both directions [Enderton] |
| Full Idea: A relation is 'symmetric' on a set if every ordered pair in the set has the relation in both directions. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9717 | A set is 'dominated' by another if a one-to-one function maps the first set into a subset of the second [Enderton] |
| Full Idea: A set is 'dominated' by another if a one-to-one function maps the first set into a subset of the second. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9713 | A relation is 'transitive' if it can be carried over from two ordered pairs to a third [Enderton] |
| Full Idea: A relation is 'transitive' on a set if the relation can be carried over from two ordered pairs to a third. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9714 | A relation satisfies 'trichotomy' if all pairs are either relations, or contain identical objects [Enderton] |
| Full Idea: A relation satisfies 'trichotomy' on a set if every ordered pair is related (in either direction), or the objects are identical. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 13200 | Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ [Enderton] |
| Full Idea: Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ. A man with an empty container is better off than a man with nothing. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 1.03) |
| 13199 | The empty set may look pointless, but many sets can be constructed from it [Enderton] |
| Full Idea: It might be thought at first that the empty set would be a rather useless or even frivolous set to mention, but from the empty set by various set-theoretic operations a surprising array of sets will be constructed. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 1:02) | |
| A reaction: This nicely sums up the ontological commitments of mathematics - that we will accept absolutely anything, as long as we can have some fun with it. Sets are an abstraction from reality, and the empty set is the very idea of that abstraction. |
| 13203 | The singleton is defined using the pairing axiom (as {x,x}) [Enderton] |
| Full Idea: Given any x we have the singleton {x}, which is defined by the pairing axiom to be {x,x}. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 2:19) | |
| A reaction: An interesting contrivance which is obviously aimed at keeping the axioms to a minimum. If you can do it intuitively with a new axiom, or unintuitively with an existing axiom - prefer the latter! |
| 9715 | An 'equivalence relation' is a reflexive, symmetric and transitive binary relation [Enderton] |
| Full Idea: An 'equivalence relation' is a binary relation which is reflexive, and symmetric, and transitive. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9716 | We 'partition' a set into distinct subsets, according to each relation on its objects [Enderton] |
| Full Idea: Equivalence classes will 'partition' a set. That is, it will divide it into distinct subsets, according to each relation on the set. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 13202 | Fraenkel added Replacement, to give a theory of ordinal numbers [Enderton] |
| Full Idea: It was observed by several people that for a satisfactory theory of ordinal numbers, Zermelo's axioms required strengthening. The Axiom of Replacement was proposed by Fraenkel and others, giving rise to the Zermelo-Fraenkel (ZF) axioms. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 1:15) |
| 13205 | We can only define functions if Choice tells us which items are involved [Enderton] |
| Full Idea: For functions, we know that for any y there exists an appropriate x, but we can't yet form a function H, as we have no way of defining one particular choice of x. Hence we need the axiom of choice. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 3:48) |
| 9722 | Inference not from content, but from the fact that it was said, is 'conversational implicature' [Enderton] |
| Full Idea: The process is dubbed 'conversational implicature' when the inference is not from the content of what has been said, but from the fact that it has been said. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.7.3) |
| 9718 | Validity is either semantic (what preserves truth), or proof-theoretic (following procedures) [Enderton] |
| Full Idea: The point of logic is to give an account of the notion of validity,..in two standard ways: the semantic way says that a valid inference preserves truth (symbol |=), and the proof-theoretic way is defined in terms of purely formal procedures (symbol |-). | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.1.3..) | |
| A reaction: This division can be mirrored in mathematics, where it is either to do with counting or theorising about things in the physical world, or following sets of rules from axioms. Language can discuss reality, or play word-games. |
| 9721 | A logical truth or tautology is a logical consequence of the empty set [Enderton] |
| Full Idea: A is a logical truth (tautology) (|= A) iff it is a semantic consequence of the empty set of premises (φ |= A), that is, every interpretation makes A true. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.3.4) | |
| A reaction: So the final column of every line of the truth table will be T. |
| 9994 | A truth assignment to the components of a wff 'satisfy' it if the wff is then True [Enderton] |
| Full Idea: A truth assignment 'satisfies' a formula, or set of formulae, if it evaluates as True when all of its components have been assigned truth values. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.2) | |
| A reaction: [very roughly what Enderton says!] The concept becomes most significant when a large set of wff's is pronounced 'satisfied' after a truth assignment leads to them all being true. |
| 9719 | A proof theory is 'sound' if its valid inferences entail semantic validity [Enderton] |
| Full Idea: If every proof-theoretically valid inference is semantically valid (so that |- entails |=), the proof theory is said to be 'sound'. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.1.7) |
| 9720 | A proof theory is 'complete' if semantically valid inferences entail proof-theoretic validity [Enderton] |
| Full Idea: If every semantically valid inference is proof-theoretically valid (so that |= entails |-), the proof-theory is said to be 'complete'. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.1.7) |
| 9995 | Proof in finite subsets is sufficient for proof in an infinite set [Enderton] |
| Full Idea: If a wff is tautologically implied by a set of wff's, it is implied by a finite subset of them; and if every finite subset is satisfiable, then so is the whole set of wff's. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 2.5) | |
| A reaction: [Enderton's account is more symbolic] He adds that this also applies to models. It is a 'theorem' because it can be proved. It is a major theorem in logic, because it brings the infinite under control, and who doesn't want that? |
| 9996 | Expressions are 'decidable' if inclusion in them (or not) can be proved [Enderton] |
| Full Idea: A set of expressions is 'decidable' iff there exists an effective procedure (qv) that, given some expression, will decide whether or not the expression is included in the set (i.e. doesn't contradict it). | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.7) | |
| A reaction: This is obviously a highly desirable feature for a really reliable system of expressions to possess. All finite sets are decidable, but some infinite sets are not. |
| 9997 | For a reasonable language, the set of valid wff's can always be enumerated [Enderton] |
| Full Idea: The Enumerability Theorem says that for a reasonable language, the set of valid wff's can be effectively enumerated. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 2.5) | |
| A reaction: There are criteria for what makes a 'reasonable' language (probably specified to ensure enumerability!). Predicates and functions must be decidable, and the language must be finite. |
| 9723 | Sentences with 'if' are only conditionals if they can read as A-implies-B [Enderton] |
| Full Idea: Not all sentences using 'if' are conditionals. Consider 'if you want a banana, there is one in the kitchen'. The rough test is that a conditional can be rewritten as 'that A implies that B'. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.6.4) |
| 12108 | All real knowledge rests on observed facts [Comte] |
| Full Idea: All competent thinkers agree with Bacon that there can be no real knowledge except that which rests upon observed facts. | |
| From: Auguste Comte (Intro to Positive Philosophy [1830], Ch.1) | |
| A reaction: Are there any unobservable facts? If so, can we know them? The only plausible route is to add 'best explanation' to the positivist armoury. With positivism, empiricism became - for a while - a quasi-religion. |
| 12109 | We must observe in order to form theories, but connected observations need prior theories [Comte] |
| Full Idea: There is a difficulty: the human mind had to observe in order to form real theories; and yet it had to form theories of some sort before it could apply itself to a connected series of observations. | |
| From: Auguste Comte (Intro to Positive Philosophy [1830], Ch.1) | |
| A reaction: Comte's view is that we get started by forming a silly theory (religion), and then refine the theory once the observations get going. Note that Comte has sort of anticipated the Quine-Duhem thesis. |
| 12107 | Positivism explains facts by connecting particular phenomena with general facts [Comte] |
| Full Idea: In positivism the explanation of facts consists only in the connection established between different particular phenomena and some general facts, the number of which the progress of science tends more and more to diminish. | |
| From: Auguste Comte (Intro to Positive Philosophy [1830], Ch.1) | |
| A reaction: This seems to be the ancestor of Hempel's more precisely formulated 'covering law' account, which became very fashionably, and now seems fairly discredited. It is just a fancy version of Humeanism about laws. |
| 20653 | Six reduction levels: groups, lives, cells, molecules, atoms, particles [Putnam/Oppenheim, by Watson] |
| Full Idea: There are six 'reductive levels' in science: social groups, (multicellular) living things, cells, molecules, atoms, and elementary particles. | |
| From: report of H.Putnam/P.Oppenheim (Unity of Science as a Working Hypothesis [1958]) by Peter Watson - Convergence 10 'Intro' | |
| A reaction: I have the impression that fields are seen as more fundamental that elementary particles. What is the status of the 'laws' that are supposed to govern these things? What is the status of space and time within this picture? |
| 12115 | Introspection is pure illusion; we can obviously observe everything except ourselves [Comte] |
| Full Idea: The pretended direct contemplation of the mind by itself is a pure illusion. ...It is clear that, by an inevitable necessity, the human mind can observe all phenomena directly, except its own. | |
| From: Auguste Comte (Intro to Positive Philosophy [1830], Ch.1) | |
| A reaction: I recently heard of a university psychology department which was seeking skilled introspectors to help with their researches. I take introspection to be very difficult, but partially possible. Read Proust. |
| 12113 | The search for first or final causes is futile [Comte] |
| Full Idea: We regard the search after what are called causes, whether first or final, as absolutely inaccessible and unmeaning. | |
| From: Auguste Comte (Intro to Positive Philosophy [1830], Ch.1) | |
| A reaction: This remark lies behind Russell's rejection of the notion of cause in scientific thinking. Personally it seems to me indispensable, even if we accept that the pursuit of 'final' causes is fairly hopeless. We don't know where the quest will lead. |
| 12110 | We can never know origins, purposes or inner natures [Comte] |
| Full Idea: The inner nature of objects, or the origin and purpose of all phenomena, are the most insoluble questions. | |
| From: Auguste Comte (Intro to Positive Philosophy [1830], Ch.1) | |
| A reaction: I take it that this Humean pessimism about science ever penetrating below the surface is precisely what is challenged by modern science, and that 'scientific essentialism' is catching up with what has happened. 'Inner' is knowable, bottom level isn't. |