141 ideas
| 12330 | In ontology, logic dominated language, until logic was mathematized [Badiou] |
| Full Idea: From Aristotle to Hegel, logic was the philosophical category of ontology's dominion over language. The mathematization of logic has authorized language to become that which seizes philosophy for itself. | |
| From: Alain Badiou (Briefings on Existence [1998], 8) |
| 9808 | Philosophy aims to reveal the grandeur of mathematics [Badiou] |
| Full Idea: Philosophy's role consists in informing mathematics of its own speculative grandeur. | |
| From: Alain Badiou (Mathematics and Philosophy: grand and little [2004], p.11) | |
| A reaction: Revealing the grandeur of something sounds more like a rhetorical than a rational exercise. How would you reveal the grandeur of a sunset to someone? |
| 9408 | Science studies phenomena, but only metaphysics tells us what exists [Mumford] |
| Full Idea: Science deals with the phenomena, ..but it is metaphysics, and only metaphysics, that tells us what ultimately exists. | |
| From: Stephen Mumford (Laws in Nature [2004], 01.2) |
| 12318 | The female body, when taken in its entirety, is the Phallus itself [Badiou] |
| Full Idea: The female body, when taken in its entirety, is the Phallus itself. | |
| From: Alain Badiou (Briefings on Existence [1998]) | |
| A reaction: Too good to pass over, too crazy to file sensibly, too creepy to have been filed under humour, my candidate for the weirdest remark I have ever read in a serious philosopher, but no doubt if you read Lacan etc for long enough it looks deeply wise. |
| 12325 | Philosophy has been relieved of physics, cosmology, politics, and now must give up ontology [Badiou] |
| Full Idea: Philosophy has been released from, even relieved of, physics, cosmology, and politics, as well as many other things. It is important for it to be released from ontology per se. | |
| From: Alain Badiou (Briefings on Existence [1998], 3) | |
| A reaction: A startling proposal, for anyone who thought that ontology was First Philosophy. Badiou wants to hand ontology over to mathematicians, but I am unclear what remains for the philosophers to do. |
| 9429 | Many forms of reasoning, such as extrapolation and analogy, are useful but deductively invalid [Mumford] |
| Full Idea: There are many forms of reasoning - extrapolation, interpolation, and other arguments from analogy - that are useful but deductively invalid. | |
| From: Stephen Mumford (Laws in Nature [2004], 04.4) | |
| A reaction: [He cites Molnar for this] |
| 12324 | Consensus is the enemy of thought [Badiou] |
| Full Idea: Consensus is the enemy of thought. | |
| From: Alain Badiou (Briefings on Existence [1998], 2) | |
| A reaction: A nice slogan for bringing Enlightenment optimists to a halt. I am struck. Do I allow my own thinking to always be diverted towards something which might result in a consensus? Do I actually (horror!) prefer consensus to truth? |
| 8721 | An 'impredicative' definition seems circular, because it uses the term being defined [Friend] |
| Full Idea: An 'impredicative' definition is one that uses the terms being defined in order to give the definition; in some way the definition is then circular. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], Glossary) | |
| A reaction: There has been a big controversy in the philosophy of mathematics over these. Shapiro gives the definition of 'village idiot' (which probably mentions 'village') as an example. |
| 8680 | Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects [Friend] |
| Full Idea: In classical logic definitions are thought of as revealing our attempts to refer to objects, ...but for intuitionist or constructivist logics, if our definitions do not uniquely characterize an object, we are not entitled to discuss the object. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.4) | |
| A reaction: In defining a chess piece we are obviously creating. In defining a 'tree' we are trying to respond to fact, but the borderlines are vague. Philosophical life would be easier if we were allowed a mixture of creation and fact - so let's have that. |
| 3678 | Reductio ad absurdum proves an idea by showing that its denial produces contradiction [Friend] |
| Full Idea: Reductio ad absurdum arguments are ones that start by denying what one wants to prove. We then prove a contradiction from this 'denied' idea and more reasonable ideas in one's theory, showing that we were wrong in denying what we wanted to prove. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |
| A reaction: This is a mathematical definition, which rests on logical contradiction, but in ordinary life (and philosophy) it would be enough to show that denial led to absurdity, rather than actual contradiction. |
| 8705 | Anti-realists see truth as our servant, and epistemically contrained [Friend] |
| Full Idea: For the anti-realist, truth belongs to us, it is our servant, and as such, it must be 'epistemically constrained'. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.1) | |
| A reaction: Put as clearly as this, it strikes me as being utterly and spectacularly wrong, a complete failure to grasp the elementary meaning of a concept etc. etc. If we aren't the servants of truth then we jolly we ought to be. Truth is above us. |
| 8713 | In classical/realist logic the connectives are defined by truth-tables [Friend] |
| Full Idea: In the classical or realist view of logic the meaning of abstract symbols for logical connectives is given by the truth-tables for the symbol. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007]) | |
| A reaction: Presumably this is realist because it connects them to 'truth', but only if that involves a fairly 'realist' view of truth. You could, of course, translate 'true' and 'false' in the table to empty (formalist) symbols such a 0 and 1. Logic is electronics. |
| 8708 | Double negation elimination is not valid in intuitionist logic [Friend] |
| Full Idea: In intuitionist logic, if we do not know that we do not know A, it does not follow that we know A, so the inference (and, in general, double negation elimination) is not intuitionistically valid. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.2) | |
| A reaction: That inference had better not be valid in any logic! I am unaware of not knowing the birthday of someone I have never heard of. Propositional attitudes such as 'know' are notoriously difficult to explain in formal logic. |
| 8694 | Free logic was developed for fictional or non-existent objects [Friend] |
| Full Idea: Free logic is especially designed to help regiment our reasoning about fictional objects, or nonexistent objects of some sort. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 3.7) | |
| A reaction: This makes it sound marginal, but I wonder whether existential commitment shouldn't be eliminated from all logic. Why do fictional objects need a different logic? What logic should we use for Robin Hood, if we aren't sure whether or not he is real? |
| 8665 | A 'proper subset' of A contains only members of A, but not all of them [Friend] |
| Full Idea: A 'subset' of A is a set containing only members of A, and a 'proper subset' is one that does not contain all the members of A. Note that the empty set is a subset of every set, but it is not a member of every set. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: Is it the same empty set in each case? 'No pens' is a subset of 'pens', but is it a subset of 'paper'? Idea 8219 should be borne in mind when discussing such things, though I am not saying I agree with it. |
| 8672 | A 'powerset' is all the subsets of a set [Friend] |
| Full Idea: The 'powerset' of a set is a set made up of all the subsets of a set. For example, the powerset of {3,7,9} is {null, {3}, {7}, {9}, {3,7}, {3,9}, {7,9}, {3,7,9}}. Taking the powerset of an infinite set gets us from one infinite cardinality to the next. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: Note that the null (empty) set occurs once, but not in the combinations. I begin to have queasy sympathies with the constructivist view of mathematics at this point, since no one has the time, space or energy to 'take' an infinite powerset. |
| 12337 | There is 'transivity' iff membership ∈ also means inclusion ⊆ [Badiou] |
| Full Idea: 'Transitivity' signifies that all of the elements of the set are also parts of the set. If you have α∈Β, you also have α⊆Β. This correlation of membership and inclusion gives a stability which is the sets' natural being. | |
| From: Alain Badiou (Briefings on Existence [1998], 11) |
| 8677 | Set theory makes a minimum ontological claim, that the empty set exists [Friend] |
| Full Idea: As a realist choice of what is basic in mathematics, set theory is rather clever, because it only makes a very simple ontological claim: that, independent of us, there exists the empty set. The whole hierarchy of finite and infinite sets then follows. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |
| A reaction: Even so, for non-logicians the existence of the empty set is rather counterintuitive. "There was nobody on the road, so I overtook him". See Ideas 7035 and 8322. You might work back to the empty set, but how do you start from it? |
| 8666 | Infinite sets correspond one-to-one with a subset [Friend] |
| Full Idea: Two sets are the same size if they can be placed in one-to-one correspondence. But even numbers have one-to-one correspondence with the natural numbers. So a set is infinite if it has one-one correspondence with a proper subset. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: Dedekind's definition. We can match 1 with 2, 2 with 4, 3 with 6, 4 with 8, etc. Logicians seem happy to give as a definition anything which fixes the target uniquely, even if it doesn't give the essence. See Frege on 0 and 1, Ideas 8653/4. |
| 8682 | Major set theories differ in their axioms, and also over the additional axioms of choice and infinity [Friend] |
| Full Idea: Zermelo-Fraenkel and Gödel-Bernays set theory differ over the notions of ordinal construction and over the notion of class, among other things. Then there are optional axioms which can be attached, such as the axiom of choice and the axiom of infinity. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.6) | |
| A reaction: This summarises the reasons why we cannot just talk about 'set theory' as if it was a single concept. The philosophical interest I would take to be found in disentangling the ontological commitments of each version. |
| 12321 | The axiom of choice must accept an indeterminate, indefinable, unconstructible set [Badiou] |
| Full Idea: The axiom of choice actually amounts to admitting an absolutely indeterminate infinite set whose existence is asserted albeit remaining linguistically indefinable. On the other hand, as a process, it is unconstructible. | |
| From: Alain Badiou (Briefings on Existence [1998], 2) | |
| A reaction: If only constructible sets are admitted (see 'V = L') then there is a contradiction. |
| 12342 | Topos theory explains the plurality of possible logics [Badiou] |
| Full Idea: Topos theory explains the plurality of possible logics. | |
| From: Alain Badiou (Briefings on Existence [1998], 14) | |
| A reaction: This will because logic will have a distinct theory within each 'topos'. |
| 12341 | Logic is a mathematical account of a universe of relations [Badiou] |
| Full Idea: Logic should first and foremost be a mathematical thought of what a universe of relations is. | |
| From: Alain Badiou (Briefings on Existence [1998], 14) |
| 8709 | The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend] |
| Full Idea: The law of excluded middle is purely syntactic: it says for any well-formed formula A, either A or not-A. It is not a semantic law; it does not say that either A is true or A is false. The semantic version (true or false) is the law of bivalence. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.2) | |
| A reaction: No wonder these two are confusing, sufficiently so for a lot of professional philosophers to blur the distinction. Presumably the 'or' is exclusive. So A-and-not-A is a contradiction; but how do you explain a contradiction without mentioning truth? |
| 8711 | Intuitionists read the universal quantifier as "we have a procedure for checking every..." [Friend] |
| Full Idea: In the intuitionist version of quantification, the universal quantifier (normally read as "all") is understood as "we have a procedure for checking every" or "we have checked every". | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.5) | |
| A reaction: It seems better to describe this as 'verificationist' (or, as Dummett prefers, 'justificationist'). Intuition suggests an ability to 'see' beyond the evidence. It strikes me as bizarre to say that you can't discuss things you can't check. |
| 8675 | Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets' [Friend] |
| Full Idea: The realist meets the Burali-Forti paradox by saying that all the ordinals are a 'class', not a set. A proper class is what we discuss when we say "all" the so-and-sos when they cannot be reached by normal set-construction. Grammar is their only limit. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |
| A reaction: This strategy would be useful for Class Nominalism, which tries to define properties in terms of classes, but gets tangled in paradoxes. But why bother with strict sets if easy-going classes will do just as well? Descartes's Dream: everything is rational. |
| 8674 | The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal [Friend] |
| Full Idea: The Burali-Forti paradox says that if ordinals are defined by 'gathering' all their predecessors with the empty set, then is the set of all ordinals an ordinal? It is created the same way, so it should be a further member of this 'complete' set! | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |
| A reaction: This is an example (along with Russell's more famous paradox) of the problems that began to appear in set theory in the early twentieth century. See Idea 8675 for a modern solution. |
| 9812 | In mathematics, if a problem can be formulated, it will eventually be solved [Badiou] |
| Full Idea: Only in mathematics can one unequivocally maintain that if thought can formulate a problem, it can and will solve it, regardless of how long it takes. | |
| From: Alain Badiou (Mathematics and Philosophy: grand and little [2004], p.17) | |
| A reaction: I hope this includes proving the Continuum Hypothesis, and Goldbach's Conjecture. It doesn't seem quite true, but it shows why philosophers of a rationalist persuasion are drawn to mathematics. |
| 12335 | Numbers are for measuring and for calculating (and the two must be consistent) [Badiou] |
| Full Idea: Number is an instance of measuring (distinguishing the more from the less, and calibrating data), ..and a figure for calculating (one counts with numbers), ..and it ought to be a figure of consistency (the compatibility of order and calculation). | |
| From: Alain Badiou (Briefings on Existence [1998], 11) |
| 12334 | There is no single unified definition of number [Badiou] |
| Full Idea: Apparently - and this is quite unlike old Greek times - there is no single unified definition of number. | |
| From: Alain Badiou (Briefings on Existence [1998], 11) |
| 12333 | Each type of number has its own characteristic procedure of introduction [Badiou] |
| Full Idea: There is a heterogeneity of introductory procedures of different classical number types: axiomatic for natural numbers, structural for ordinals, algebraic for negative and rational numbers, topological for reals, mainly geometric for complex numbers. | |
| From: Alain Badiou (Briefings on Existence [1998], 11) |
| 12322 | Must we accept numbers as existing when they no longer consist of units? [Badiou] |
| Full Idea: Do we have to confer existence on numbers whose principle is to no longer consist of units? | |
| From: Alain Badiou (Briefings on Existence [1998], 2) | |
| A reaction: This very nicely expresses what seems to me perhaps the most important question in the philosophy of mathematics. I am reluctant to accept such 'unitless' numbers, but I then feel hopelessly old-fashioned and naïve. What to do? |
| 8667 | The 'integers' are the positive and negative natural numbers, plus zero [Friend] |
| Full Idea: The set of 'integers' is all of the negative natural numbers, and zero, together with the positive natural numbers. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: Zero always looks like a misfit at this party. Credit and debit explain positive and negative nicely, but what is the difference between having no money, and money being irrelevant? I can be 'broke', but can the North Pole be broke? |
| 8668 | The 'rational' numbers are those representable as fractions [Friend] |
| Full Idea: The 'rational' numbers are all those that can be represented in the form m/n (i.e. as fractions), where m and n are natural numbers different from zero. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: Pythagoreans needed numbers to stop there, in order to represent the whole of reality numerically. See irrational numbers for the ensuing disaster. How can a universe with a finite number of particles contain numbers that are not 'rational'? |
| 8670 | A number is 'irrational' if it cannot be represented as a fraction [Friend] |
| Full Idea: A number is 'irrational' just in case it cannot be represented as a fraction. An irrational number has an infinite non-repeating decimal expansion. Famous examples are pi and e. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: There must be an infinite number of irrational numbers. You could, for example, take the expansion of pi, and change just one digit to produce a new irrational number, and pi has an infinity of digits to tinker with. |
| 8661 | The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend] |
| Full Idea: The natural numbers are quite primitive, and are what we first learn about. The order of objects (the 'ordinals') is one level of abstraction up from the natural numbers: we impose an order on objects. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.4) | |
| A reaction: Note the talk of 'levels of abstraction'. So is there a first level of abstraction? Dedekind disagrees with Friend (Idea 7524). I would say that natural numbers are abstracted from something, but I'm not sure what. See Structuralism in maths. |
| 8664 | Cardinal numbers answer 'how many?', with the order being irrelevant [Friend] |
| Full Idea: The 'cardinal' numbers answer the question 'How many?'; the order of presentation of the objects being counted as immaterial. Def: the cardinality of a set is the number of members of the set. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: If one asks whether cardinals or ordinals are logically prior (see Ideas 7524 and 8661), I am inclined to answer 'neither'. Presenting them as answers to the questions 'how many?' and 'which comes first?' is illuminating. |
| 8671 | The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps [Friend] |
| Full Idea: The set of 'real' numbers, which consists of the rational numbers and the irrational numbers together, represents "the continuum", since it is like a smooth line which has no gaps (unlike the rational numbers, which have the irrationals missing). | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: The Continuum is the perfect abstract object, because a series of abstractions has arrived at a vast limit in its nature. It still has dizzying infinities contained within it, and at either end of the line. It makes you feel humble. |
| 9813 | Mathematics shows that thinking is not confined to the finite [Badiou] |
| Full Idea: Mathematics teaches us that there is no reason whatsoever to confne thinking within the ambit of finitude. | |
| From: Alain Badiou (Mathematics and Philosophy: grand and little [2004], p.19) | |
| A reaction: This would perhaps make Cantor the greatest thinker who ever lived. It is an exhilarating idea, but we should ward the reader against romping of into unrestrained philosophical thought about infinities. You may be jumping without your Cantorian parachute. |
| 12327 | The undecidability of the Continuum Hypothesis may have ruined or fragmented set theory [Badiou] |
| Full Idea: As we have known since Paul Cohen's theorem, the Continuum Hypothesis is intrinsically undecidable. Many believe Cohen's discovery has driven the set-theoretic project into ruin, or 'pluralized' what was once presented as a unified construct. | |
| From: Alain Badiou (Briefings on Existence [1998], 6) | |
| A reaction: Badiou thinks the theorem completes set theory, by (roughly) finalising its map. |
| 8663 | Raising omega to successive powers of omega reveal an infinity of infinities [Friend] |
| Full Idea: After the multiples of omega, we can successively raise omega to powers of omega, and after that is done an infinite number of times we arrive at a new limit ordinal, which is called 'epsilon'. We have an infinite number of infinite ordinals. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.4) | |
| A reaction: When most people are dumbstruck by the idea of a single infinity, Cantor unleashes an infinity of infinities, which must be the highest into the stratosphere of abstract thought that any human being has ever gone. |
| 8662 | The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend] |
| Full Idea: The first 'limit ordinal' is called 'omega', which is ordinal because it is greater than other numbers, but it has no immediate predecessor. But it has successors, and after all of those we come to twice-omega, which is the next limit ordinal. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.4) | |
| A reaction: This is the gateway to Cantor's paradise of infinities, which Hilbert loved and defended. Who could resist the pleasure of being totally boggled (like Aristotle) by a concept such as infinity, only to have someone draw a map of it? See 8663 for sequel. |
| 8669 | Between any two rational numbers there is an infinite number of rational numbers [Friend] |
| Full Idea: Since between any two rational numbers there is an infinite number of rational numbers, we could consider that we have infinity in three dimensions: positive numbers, negative numbers, and the 'depth' of infinite numbers between any rational numbers. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: This is before we even reach Cantor's staggering infinities (Ideas 8662 and 8663), which presumably reside at the outer reaches of all three of these dimensions of infinity. The 'deep' infinities come from fractions with huge denominators. |
| 8676 | Is mathematics based on sets, types, categories, models or topology? [Friend] |
| Full Idea: Successful competing founding disciplines in mathematics include: the various set theories, type theory, category theory, model theory and topology. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |
| A reaction: Or none of the above? Set theories are very popular. Type theory is, apparently, discredited. Shapiro has a version of structuralism based on model theory (which sound promising). Topology is the one that intrigues me... |
| 8678 | Most mathematical theories can be translated into the language of set theory [Friend] |
| Full Idea: Most of mathematics can be faithfully redescribed by classical (realist) set theory. More precisely, we can translate other mathematical theories - such as group theory, analysis, calculus, arithmetic, geometry and so on - into the language of set theory. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |
| A reaction: This is why most mathematicians seem to regard set theory as foundational. We could also translate football matches into the language of atomic physics. |
| 8701 | The number 8 in isolation from the other numbers is of no interest [Friend] |
| Full Idea: There is no interest for the mathematician in studying the number 8 in isolation from the other numbers. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.4) | |
| A reaction: This is a crucial and simple point (arising during a discussion of Shapiro's structuralism). Most things are interesting in themselves, as well as for their relationships, but mathematical 'objects' just are relationships. |
| 8702 | In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend] |
| Full Idea: Structuralists give a historical account of why the 'same' number occupies different structures. Numbers are equivalent rather than identical. 8 is the immediate predecessor of 9 in the whole numbers, but in the rationals 9 has no predecessor. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.4) | |
| A reaction: I don't become a different person if I move from a detached house to a terraced house. This suggests that 8 can't be entirely defined by its relations, and yet it is hard to see what its intrinsic nature could be, apart from the units which compose it. |
| 8699 | Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? [Friend] |
| Full Idea: Structuralists disagree over whether objects in structures are 'ante rem' (before reality, existing independently of whether the objects exist) or 'in re' (in reality, grounded in the real world, usually in our theories of physics). | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.4) | |
| A reaction: Shapiro holds the first view, Hellman and Resnik the second. The first view sounds too platonist and ontologically extravagant; the second sounds too contingent and limited. The correct account is somewhere in abstractions from the real. |
| 12329 | If mathematics is a logic of the possible, then questions of existence are not intrinsic to it [Badiou] |
| Full Idea: If mathematics is a logic of the possible, then questions of existence are not intrinsic to it (as they are for the Platonist). | |
| From: Alain Badiou (Briefings on Existence [1998], 7) | |
| A reaction: See also Idea 12328. I file this to connect it with Hellman's modal (and nominalist) version of structuralism. Could it be that mathematics and modal logic are identical? |
| 8696 | Structuralist says maths concerns concepts about base objects, not base objects themselves [Friend] |
| Full Idea: According to the structuralist, mathematicians study the concepts (objects of study) such as variable, greater, real, add, similar, infinite set, which are one level of abstraction up from prima facie base objects such as numbers, shapes and lines. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.1) | |
| A reaction: This still seems to imply an ontology in which numbers, shapes and lines exist. I would have thought you could eliminate the 'base objects', and just say that the concepts are one level of abstraction up from the physical world. |
| 8695 | Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend] |
| Full Idea: Structuralism says we study whole structures: objects together with their predicates, relations that bear between them, and functions that take us from one domain of objects to a range of other objects. The objects can even be eliminated. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.1) | |
| A reaction: The unity of object and predicate is a Quinean idea. The idea that objects are inessential is the dramatic move. To me the proposal has very strong intuitive appeal. 'Eight' is meaningless out of context. Ordinality precedes cardinality? Ideas 7524/8661. |
| 8700 | 'In re' structuralism says that the process of abstraction is pattern-spotting [Friend] |
| Full Idea: In the 'in re' version of mathematical structuralism, pattern-spotting is the process of abstraction. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.4) | |
| A reaction: This might work for non-mathematical abstraction as well, if we are allowed to spot patterns within sensual experience, and patterns within abstractions. Properties are causal patterns in the world? No - properties cause patterns. |
| 12328 | Platonists like axioms and decisions, Aristotelians like definitions, possibilities and logic [Badiou] |
| Full Idea: A Platonist's interest focuses on axioms in which the decision of thought is played out, where an Aristotelian or Leibnizian interest focuses on definitions laying out the representation of possibilities (...and the essence of mathematics is logic). | |
| From: Alain Badiou (Briefings on Existence [1998], 7) | |
| A reaction: See Idea 12323 for the significance of the Platonist approach. So logicism is an Aristotelian project? Frege is not a true platonist? I like the notion of 'the representation of possibilities', so will vote for the Aristotelians, against Badiou. |
| 8681 | The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? [Friend] |
| Full Idea: The main philosophical problem with the position of platonism or realism is the epistemic problem: of explaining what perception or intuition consists in; how it is possible that we should accurately detect whatever it is we are realists about. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.5) | |
| A reaction: The best bet, I suppose, is that the mind directly perceives concepts just as eyes perceive the physical (see Idea 8679), but it strikes me as implausible. If we have to come up with a special mental faculty for an area of knowledge, we are in trouble. |
| 8712 | Mathematics should be treated as true whenever it is indispensable to our best physical theory [Friend] |
| Full Idea: Central to naturalism about mathematics are 'indispensability arguments', to the effect that some part of mathematics is indispensable to our best physical theory, and therefore we ought to take that part of mathematics to be true. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 6.1) | |
| A reaction: Quine and Putnam hold this view; Field challenges it. It has the odd consequence that the dispensable parts (if they can be identified!) do not need to be treated as true (even though they might follow logically from the dispensable parts!). Wrong! |
| 12331 | Logic is definitional, but real mathematics is axiomatic [Badiou] |
| Full Idea: Logic is definitional, whereas real mathematics is axiomatic. | |
| From: Alain Badiou (Briefings on Existence [1998], 10) |
| 8716 | Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend] |
| Full Idea: There are not enough constraints in the Formalist view of mathematics, so there is no way to select a direction for trying to develop mathematics. There is no part of mathematics that is more important than another. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 6.6) | |
| A reaction: One might reply that an area of maths could be 'important' if lots of other areas depended on it, and big developments would ripple big changes through the interior of the subject. Formalism does, though, seem to reduce maths to a game. |
| 8706 | Constructivism rejects too much mathematics [Friend] |
| Full Idea: Too much of mathematics is rejected by the constructivist. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.1) | |
| A reaction: This was Hilbert's view. This seems to be generally true of verificationism. My favourite example is that legitimate speculations can be labelled as meaningless. |
| 8707 | Intuitionists typically retain bivalence but reject the law of excluded middle [Friend] |
| Full Idea: An intuitionist typically retains bivalence, but rejects the law of excluded middle. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.2) | |
| A reaction: The idea would be to say that only T and F are available as truth-values, but failing to be T does not ensure being F, but merely not-T. 'Unproven' is not-T, but may not be F. |
| 9427 | For Humeans the world is a world primarily of events [Mumford] |
| Full Idea: For Humeans the world is a world primarily of events. | |
| From: Stephen Mumford (Laws in Nature [2004], 03.6) |
| 12340 | There is no Being as a whole, because there is no set of all sets [Badiou] |
| Full Idea: The fundamental theorem that 'there does not exist a set of all sets' designates the inexistence of Being as a whole. ...A crucial consequence of this property is that any ontological investigation is irremediably local. | |
| From: Alain Badiou (Briefings on Existence [1998], 14) | |
| A reaction: The second thought pushes Badiou into Topos Theory, where the real numbers (for example) have a separate theory in each 'topos'. |
| 9809 | Mathematics inscribes being as such [Badiou] |
| Full Idea: Mathematics inscribes being as such. | |
| From: Alain Badiou (Mathematics and Philosophy: grand and little [2004], p.12) | |
| A reaction: I don't pretend to understand that, but there is something about the purity and certainty of mathematics that makes us feel we are grappling with the core of existence. Perhaps. The same might be said of stubbing your toe on a bedpost. |
| 12323 | Existence is Being itself, but only as our thought decides it [Badiou] |
| Full Idea: Existence is precisely Being itself in as much as thought decides it. And that decision orients thought essentially. ...It is when you decide upon what exists that you bind your thought to Being. | |
| From: Alain Badiou (Briefings on Existence [1998], 2) | |
| A reaction: [2nd half p.57] Helpful for us non-Heideggerians to see what is going on. Does this mean that Being is Kant's noumenon? |
| 12332 | The modern view of Being comes when we reject numbers as merely successions of One [Badiou] |
| Full Idea: The saturation and collapse of the Euclidean idea of the being of number as One's procession signs the entry of the thought of Being into modern times. | |
| From: Alain Badiou (Briefings on Existence [1998], 11) | |
| A reaction: That is, by allowing that not all numbers are built of units, numbers expand widely enough to embrace everything we think of as Being. The landmark event is the acceptance of the infinite as a number. |
| 12326 | The primitive name of Being is the empty set; in a sense, only the empty set 'is' [Badiou] |
| Full Idea: In Set Theory, the primitive name of Being is the void, the empty set. The whole hierarchy takes root in it. In a certain sense, it alone 'is'. | |
| From: Alain Badiou (Briefings on Existence [1998], 6) | |
| A reaction: This is the key to Badiou's view that ontology is mathematics. David Lewis pursued interesting enquiries in this area. |
| 9811 | It is of the essence of being to appear [Badiou] |
| Full Idea: It is of the essence of being to appear. | |
| From: Alain Badiou (Mathematics and Philosophy: grand and little [2004], p.16) | |
| A reaction: Nice slogan. In my humble opinion 'continental' philosophy is well worth reading because, despite the fluffy rhetoric and the shameless egotism and the desire to shock the bourgeoisie, they occasionally make wonderfully thought-provoking remarks. |
| 12320 | Ontology is (and always has been) Cantorian mathematics [Badiou] |
| Full Idea: Enlightened by the Cantorian grounding of mathematics, we can assert ontology to be nothing other than mathematics itself. This has been the case ever since its Greek origin. | |
| From: Alain Badiou (Briefings on Existence [1998], 1) | |
| A reaction: There seems to be quite a strong feeling among mathematicians that new 'realms of being' are emerging from their researches. Only a Platonist, of course, is likely to find this idea sympathetic. |
| 14334 | Modest realism says there is a reality; the presumptuous view says we can accurately describe it [Mumford] |
| Full Idea: The claim of modest realism is that there is a subject-independent reality; the presumptuous claim is that we are capable of describing that reality accurately. | |
| From: Stephen Mumford (Dispositions [1998], 09.1) | |
| A reaction: And the super-presumptuous claim is that there only exists one ultimate accurate description of reality. I am happy to call myself a Modest Realist on this one. |
| 14306 | Anti-realists deny truth-values to all statements, and say evidence and ontology are inseparable [Mumford] |
| Full Idea: The anti-realist declines to permit that all statements have truth-values. ...The essence of the anti-realist position is that evidence and ontology cannot be separated. | |
| From: Stephen Mumford (Dispositions [1998], 03.6) | |
| A reaction: [second half on p.51] The idea that evidence and ontology are 'inseparable' strikes me as an absurd idea. The proposal that you should not speculate about ontology without some sort of evidence is, of course, not unreasonable. |
| 14333 | Dispositions and categorical properties are two modes of presentation of the same thing [Mumford] |
| Full Idea: The dispositional and the categorical are correctly understood just as two modes of presentation of the same instantiated properties. | |
| From: Stephen Mumford (Dispositions [1998], 08.6) | |
| A reaction: This is Mumford's own conclusion, after discussing the views of Armstrong. How about 'a disposition is the modal profile' of a categorical property? |
| 14336 | Categorical predicates are those unconnected to functions [Mumford] |
| Full Idea: A predicate which is conceptually connected to no function ... is a categorical predicate. | |
| From: Stephen Mumford (Dispositions [1998], 09.7) | |
| A reaction: This is an expansion of Mumford's own theory of dispositions, as functional. Does a cork in a wine bottle have a function, but without doing anything? It seems to achieve its function purely through its structure. |
| 14315 | Categorical properties and dispositions appear to explain one another [Mumford] |
| Full Idea: Though categorical properties provide explanations for dispositions, categorical properties are also explained by dispositions; hence neither category uniquely explains the other. | |
| From: Stephen Mumford (Dispositions [1998], 05.3) | |
| A reaction: The conclusion doesn't seem to follow. It depends which one is found at the bottom level. It can go up from a basic disposition, to a categorical property, to another disposition - or the other way around. |
| 14332 | There are four reasons for seeing categorical properties as the most fundamental [Mumford] |
| Full Idea: Four reasons for reducing everything to the categorical are: categorical predicates have wider scope; dispositions are variably realised by the categorical; categorical is 1st order, dispositions 2nd; categorical properties are explanatorily basic. | |
| From: Stephen Mumford (Dispositions [1998], 08.5) | |
| A reaction: I particularly reject the fourth reason, as I take categorical properties as still in need of explanation. The categorical view is contingent (and Humean), but I take the categorical properties to be necessitated by the underlying powers. |
| 14302 | A lead molecule is not leaden, and macroscopic properties need not be microscopically present [Mumford] |
| Full Idea: Though lead is said to be composed of molecules of lead, these molecules are not leaden in the everyday sense of the word. This suggests that a property need not be present at the microscopic level in order to be present at the macroscopic level. | |
| From: Stephen Mumford (Dispositions [1998], 02.3) | |
| A reaction: [He quotes Joske] This strikes me as a key principle to grasp about properties. One H2O molecule is not water, any more than a brick is a house! Nearly all properties (or all?) are 'emergent' (in the sensible, non-mystical use of that word). |
| 9446 | Properties are just natural clusters of powers [Mumford] |
| Full Idea: The view of properties I find most attractive is one in which they are natural clusters of, and exhausted by, powers (plus other connections to other properties). | |
| From: Stephen Mumford (Laws in Nature [2004], 10.6) |
| 14294 | Dispositions are attacked as mere regularities of events, or place-holders for unknown properties [Mumford] |
| Full Idea: Dispositions are attacked as either just saying how something will behave (logical fictions about regularities of events), or as primitive pre-scientific terms like 'phlogiston', place-holders used when we are ignorant of real properties. | |
| From: Stephen Mumford (Dispositions [1998], 01.1) | |
| A reaction: [compressed] The first view he calls the Ryle-Wittgenstein view, which seems to track back to Hume. |
| 14310 | Dispositions are classifications of properties by functional role [Mumford] |
| Full Idea: A dispositional property is the classification of a property according to its functional role....[p.85] What is essential to a disposition - its identity condition - is its functional role. | |
| From: Stephen Mumford (Dispositions [1998], 04.5) | |
| A reaction: This is Mumford's view of dispositions. I am wary of any proposal to define something according to its role, because it must have an intrinsic nature which equips it to have that role. |
| 14317 | I say the categorical base causes the disposition manifestation [Mumford] |
| Full Idea: The view I promote is one where the categorical base is a cause of the disposition manifestation. | |
| From: Stephen Mumford (Dispositions [1998], 05.5) | |
| A reaction: It seems to me (I think) that the most basic thing has to be a power, whose nature is intrinsically beyond our grasp, and that categorical properties are the result of these powers. Powers are dispositional in character. |
| 14316 | If dispositions have several categorical realisations, that makes the two separate [Mumford] |
| Full Idea: We might claim that dispositions are variably realized by a number of categorical bases; therefore they must be distinct from those bases. | |
| From: Stephen Mumford (Dispositions [1998], 05.4) | |
| A reaction: Cars can be realised by a variety of models, therefore models are not cars? This might work if dispositions are only characterised functionally, as Mumford proposes, but I'm not convinced. |
| 14313 | All properties must be causal powers (since they wouldn't exist otherwise) [Mumford] |
| Full Idea: It seems that every property must be a causal power, since every property must be causally potent (as a necessary condition of its very existence). | |
| From: Stephen Mumford (Dispositions [1998], 04.7) | |
| A reaction: Mumford cautiously endorses this idea, which seems to rest on the thesis that 'to exist is to have causal powers'. I think I am even keener on it than Mumford is. Powers and properties need to be disentangled, however. |
| 14318 | Intrinsic properties are just causal powers, and identifying a property as causal is then analytic [Mumford] |
| Full Idea: Understanding intrinsic properties as being causal powers is likely to be most profitable, and, if true, renders the causal criterion of property existence true analytically. | |
| From: Stephen Mumford (Dispositions [1998], 06.2) | |
| A reaction: [He cites E.Fales on this] I'm inclined to think that in the ultimate ontology the notion of a 'property' drops out. There are true causal powers, and then conventional human ways of grouping such powers together and naming them. |
| 14293 | Dispositions are ascribed to at least objects, substances and persons [Mumford] |
| Full Idea: Dispositions are ascribed to at least three distinguishable classes of things: objects, substances, and persons. | |
| From: Stephen Mumford (Dispositions [1998], 01.1) | |
| A reaction: Are dispositions not also ascribed to properties? Magnetism has a disposition to attract iron filings? |
| 14326 | Unlike categorical bases, dispositions necessarily occupy a particular causal role [Mumford] |
| Full Idea: The idea of a disposition occupying a different causal role involves a conceptual confusion, ...but there is no conceptual or logical absurdity in a categorical base occupying a different causal role. | |
| From: Stephen Mumford (Dispositions [1998], 07.3) | |
| A reaction: This is the core of Mumford's theory of dispositions. I'm beginning to think that dispositions are merely ways we have of describing and labelling functional mechanisms, and so 'dispositions' drop out of the final story. |
| 14298 | Dispositions can be contrasted either with occurrences, or with categorical properties [Mumford] |
| Full Idea: For some the notion of a disposition is contrasted with the notion of an occurrence; for others, it is contrasted with that of a categorical property. | |
| From: Stephen Mumford (Dispositions [1998], 01.6) | |
| A reaction: I vote for dispositions over the other two, but I take the categorical properties to be the main rival. |
| 14314 | If dispositions are powers, background conditions makes it hard to say what they do [Mumford] |
| Full Idea: The realist says that disposition ascriptions are ascriptions of real powers. This leaves unanswered the question, 'power to do what?' The problem of background conditions means that the realist cannot say what it is that a power is a power to do. | |
| From: Stephen Mumford (Dispositions [1998], 04.9) | |
| A reaction: It is hard to say what a disposition will do, under any other account of dispositions. I would take a power to be defined by a 'modal profile', rather than an actual account of what it will lead to. |
| 14325 | Maybe dispositions can replace powers in metaphysics, as what induces property change [Mumford] |
| Full Idea: Dispositions can regain the metaphysical role traditionally ascribed to real powers: the that-in-virtue-of-which-something-will-G, if F. | |
| From: Stephen Mumford (Dispositions [1998], 06.9) | |
| A reaction: The attraction is that dispositions can be specified a little more clearly (especially in Mumford's functional version) whereas there may be no more to say about a power once it has been located and named. |
| 14312 | Orthodoxy says dispositions entail conditionals (rather than being equivalent to them) [Mumford] |
| Full Idea: The orthodox realist view has it that what makes an ascription a disposition ascription is not that it is equivalent to a conditional proposition but that it entails one. | |
| From: Stephen Mumford (Dispositions [1998], 04.7) | |
| A reaction: Mumford says that Martin has shown that dispositions need not entail conditionals (when a 'fink' is operating, something which intervenes between disposition and outcome). |
| 14291 | Dispositions are not just possibilities - they are features of actual things [Mumford] |
| Full Idea: Dispositions should correctly be understood as more than mere possibilities. To say something has a disposition is to say something about how it is actually. | |
| From: Stephen Mumford (Dispositions [1998], Pref) | |
| A reaction: To me this is a basic axiom of metaphysics. The word 'power' serves well for the actual embodiment of a disposition. A power gives rise to one or more dispositions. Or one or more powers give rise to a disposition? |
| 14299 | There could be dispositions that are never manifested [Mumford] |
| Full Idea: It seems plausible that a disposition could be possessed though no manifestation events occur. | |
| From: Stephen Mumford (Dispositions [1998], 01.6) | |
| A reaction: It is more than 'plausible' - it is screamingly obvious to everybody, apart from a few philosophers. "Some mute inglorious Milton here may rest" (Gray's Elegy). |
| 14323 | If every event has a cause, it is easy to invent a power to explain each case [Mumford] |
| Full Idea: Given any event, and the assumption that every event has a cause, then some power can always be invented as the cause of that event. | |
| From: Stephen Mumford (Dispositions [1998], 06.6) | |
| A reaction: This is a useful warning, and probably explains why 'powers' fell out of fashion in scientifice theorising. They seem to make a return, though, as an appropriate term for the bottom level of each of our explanations. |
| 14328 | Traditional powers initiate change, but are mysterious between those changes [Mumford] |
| Full Idea: In the old-fashioned sense, 'powers' are real potentialities that initiate changes but seem to have a mysterious existence in between those changes. | |
| From: Stephen Mumford (Dispositions [1998], 07.10) | |
| A reaction: What is a person when they are asleep? What is a dishwasher when it isn't running? What is gunpowder when it doesn't explode? We all understand latent powers. To see them as a 'mystery' is to want to know too much. |
| 14331 | Categorical eliminativists say there are no dispositions, just categorical states or mechanisms [Mumford] |
| Full Idea: The categorical eliminativist claims that there are no dispositional properties. All properties must be conceived of as categorical states or mechanisms, in the spirit of Boyle's explanation of powers. | |
| From: Stephen Mumford (Dispositions [1998], 08.3A) | |
| A reaction: What is the difference between a structure and a mechanism? How do we distinguish an active from an inactive mechanism? Without powers or dispositions, nature is dead junk. |
| 9435 | A 'porridge' nominalist thinks we just divide reality in any way that suits us [Mumford] |
| Full Idea: A 'porridge' nominalist denies natural kinds, and thinks there are no objective divisions in reality, so concepts or words can be used by a community to divide the world up in any way that suits their purposes. | |
| From: Stephen Mumford (Laws in Nature [2004], 07.3) |
| 9447 | If properties are clusters of powers, this can explain why properties resemble in degrees [Mumford] |
| Full Idea: If a cluster of ten powers exhausts property F, and property G differs in respect of just one power, this might explain why properties can resemble other properties and in different degrees. | |
| From: Stephen Mumford (Laws in Nature [2004], 10.6) | |
| A reaction: I love this. The most intractable problem about properties and universals is that of abstract reference - pink resembles red more than pink resembles green. If colours are clusters of powers, red and pink share nine out of ten of them. |
| 8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
| Full Idea: What the mathematician labels an 'object' in her discipline, is called 'a place in a structure' by the structuralist. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.5) | |
| A reaction: This is a strategy for dispersing the idea of an object in the world of thought, parallel to attempts to eliminate them from physical ontology (e.g. Idea 614). |
| 18617 | Substances, unlike aggregates, can survive a change of parts [Mumford] |
| Full Idea: Substances can survive a change in their parts in a way that a mere aggregate of parts. | |
| From: Stephen Mumford (Metaphysics: a very short introduction [2012], 3) | |
| A reaction: A simple but very important idea. If we then distinguish between 'substances' and 'aggregates' we get a much clearer grip on things. Is the Ship of Theseus a substance or an aggregate? There is no factual answer to that. What do you want to explain? |
| 14295 | Many artefacts have dispositional essences, which make them what they are [Mumford] |
| Full Idea: Thermostats, thermometers, axes, spoons, and batteries have dispositional essences, which make them what they are. | |
| From: Stephen Mumford (Dispositions [1998], 01.2 iv) | |
| A reaction: I would have thought that we could extend this proposal well beyond artefacts, but it certainly seems particularly clear in artefacts, where a human intention seems to be inescapably involved. |
| 12248 | How can we show that a universally possessed property is an essential property? [Mumford] |
| Full Idea: Essentialists fail to show how we ascend from being a property universally possessed, by all kind members, to the status of being an essential property. | |
| From: Stephen Mumford (Laws in Nature [2004], 07.5) | |
| A reaction: This is precisely where my proposal comes in - the essential properties, as opposed to the accidentaly universals, are those which explain the nature and behaviour of each kind of thing (and each individual thing). |
| 18618 | Maybe possibilities are recombinations of the existing elements of reality [Mumford] |
| Full Idea: It has been suggested that we could think of possibilities as recombinations of all the existing elements of reality. | |
| From: Stephen Mumford (Metaphysics: a very short introduction [2012], 8) | |
| A reaction: [Armstrong 1989 is the source] The obvious problem would be that the existence of an entirely different reality would be impossible, if this was all possibility could be. It seems to cramp the style of the possible too much. Are properties elements? |
| 18619 | Combinatorial possibility has to allow all elements to be combinable, which seems unlikely [Mumford] |
| Full Idea: The combinatorial account only works if you allow that the elements are recombinable. ...But could Lincoln really have been green? It seems possible that you could jump to the moon, unless we impose some restrictions. | |
| From: Stephen Mumford (Metaphysics: a very short introduction [2012], 8) | |
| A reaction: Mumford suggests different combination rules for logical and natural possibility. The general objection is that combinatorial possibility is too permissive - which it clearly is. |
| 18620 | Combinatorial possibility relies on what actually exists (even over time), but there could be more [Mumford] |
| Full Idea: Can combinatorial possibility deliver enough possibilities? It uses the existing elements, but there might have been one more particular or one more property. Even extended over time, the elements seem finite, yet there could have been more. | |
| From: Stephen Mumford (Metaphysics: a very short introduction [2012], 8) | |
| A reaction: [compressed] One objection is that the theory allows too much, and now the objection is that it allows too little. Both objections are correct, so that's the end of that. But I admire the attempt to base modality on actuality. |
| 14309 | Truth-functional conditionals can't distinguish whether they are causal or accidental [Mumford] |
| Full Idea: If a conditional remains truth-functional it is incapable of expressing the fact that the connection between antecedent and consequent in the conditional is a causal one rather than merely accidental | |
| From: Stephen Mumford (Dispositions [1998], 03.8) | |
| A reaction: This is the first step towards an account of conditionals which will work in real life rather than merely in classical logic. |
| 14311 | Dispositions are not equivalent to stronger-than-material conditionals [Mumford] |
| Full Idea: The conclusion that disposition ascriptions are not equivalent to stronger-than-material conditionals is largely to be accepted. | |
| From: Stephen Mumford (Dispositions [1998], 04.7) | |
| A reaction: [he attributes the view to C.B.Martin 1994] It is hard to see how to describe a disposition in anything other than conditional terms. Mumford's 'functional role' probably has to be described conditionally. It is how the conditional cashes out. |
| 14319 | Nomothetic explanations cite laws, and structural explanations cite mechanisms [Mumford] |
| Full Idea: A nomothetic explanation appeals to laws where the explanandum is shown to be an instance of a general law. ...The alternative is a structural explanation, which postulates a mechanism, opening up a hidden world. | |
| From: Stephen Mumford (Dispositions [1998], 06.4) | |
| A reaction: [He cites E.McMullin 1978] I am very much in favour of structural explanations, and opposed to nomothetic ones. That is, nomothetic accounts are only the first step towards an explanation - perhaps a mere identification of the explanandum. |
| 14342 | General laws depend upon the capacities of particulars, not the other way around [Mumford] |
| Full Idea: Laws, qua true generalities, if they exist at all, are ontologically parasitic upon the capacities of particulars, rather than the other way round. | |
| From: Stephen Mumford (Dispositions [1998], 10.6) | |
| A reaction: Quite so. And hence trying to explain a particular behaviour by saying that it falls under a law is absurdly circular and vacuous. |
| 14322 | If fragile just means 'breaks when dropped', it won't explain a breakage [Mumford] |
| Full Idea: If fragile means nothing more than 'breaks when dropped', then it is no explanation of why something breaks when dropped. | |
| From: Stephen Mumford (Dispositions [1998], 06.5) | |
| A reaction: His point is that you have to unpack the notion of fragile, which presumably cites underlying mechanisms. This is the 'virtus dormitiva' problem - but that explanation of opium's dormitive powers is not entirely stupid. |
| 14337 | Maybe dispositions can replace the 'laws of nature' as the basis of explanation [Mumford] |
| Full Idea: I will consider the case for an ontology of real dispositions replacing the so-called laws of nature as the basic building blocks of explanation. | |
| From: Stephen Mumford (Dispositions [1998], 10.1) | |
| A reaction: This precisely summarises the view I am exploring, with a particular focus on real essences. I certainly think the 'laws of nature' must go. See Mumford's second book on this. |
| 14343 | To avoid a regress in explanations, ungrounded dispositions will always have to be posited [Mumford] |
| Full Idea: The nature of explanation is such that ungrounded dispositions will always have to be posited in order to avoid a regress of explanation. | |
| From: Stephen Mumford (Dispositions [1998], 10.6) | |
| A reaction: This seems to be right, but leaves it open to mock the proposals as 'virtus dormitiva' - empty place-holders that ground explanations but do no explanatory work. What else can be done, though? |
| 14320 | Subatomic particles may terminate explanation, if they lack structure [Mumford] |
| Full Idea: The behaviour of subatomic particles cannot be further analysed into structures and this may tempt us to regard these as instances of 'brute' ungrounded dispositions which end any possible regress of explanation. | |
| From: Stephen Mumford (Dispositions [1998], 06.4) | |
| A reaction: This seems right, if it is 'structural' explanations we are after (as I think we are) which look for mechanisms. An electron seems to be just three dispositions and no structure, so there is nothing more to say. Ladyman scorns this account. |
| 14324 | Ontology is unrelated to explanation, which concerns modes of presentation and states of knowledge [Mumford] |
| Full Idea: Nothing about ontology is at stake in questions of explanation, for explanatory success is contingent upon the modes of presentation of explanans and explananda, and relative states of knowledge and ignorance. | |
| From: Stephen Mumford (Dispositions [1998], 06.8) | |
| A reaction: There are real facts about the immediate and unusual causes which immediately precede an event, and these might be candidates for a real explanation. There are also real mechanisms and powers which dictate a things behaviour. |
| 8685 | Studying biology presumes the laws of chemistry, and it could never contradict them [Friend] |
| Full Idea: In the hierarchy of reduction, when we investigate questions in biology, we have to assume the laws of chemistry but not of economics. We could never find a law of biology that contradicted something in physics or in chemistry. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 3.1) | |
| A reaction: This spells out the idea that there is a direction of dependence between aspects of the world, though we should be cautious of talking about 'levels' (see Idea 7003). We cannot choose the direction in which reduction must go. |
| 8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend] |
| Full Idea: The extensional presentation of a concept is just a list of the objects falling under the concept. In contrast, an intensional presentation of a concept gives a characterization of the concept, which allows us to pick out which objects fall under it. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 3.4) | |
| A reaction: Logicians seem to favour the extensional view, because (in the standard view) sets are defined simply by their members, so concepts can be explained using sets. I take this to be a mistake. The intensional view seems obviously prior. |
| 12338 | We must either assert or deny any single predicate of any single subject [Badiou] |
| Full Idea: There can be nothing intermediate to an assertion and a denial. We must either assert or deny any single predicate of any single subject. | |
| From: Alain Badiou (Briefings on Existence [1998], 1011b24) | |
| A reaction: The first sentence seems to be bivalence, and the second sentence excluded middle. |
| 9814 | All great poetry is engaged in rivalry with mathematics [Badiou] |
| Full Idea: Like every great poet, Mallarmé was engaged in a tacit rivalry with mathematics. | |
| From: Alain Badiou (Mathematics and Philosophy: grand and little [2004], p.20) | |
| A reaction: I love these French pronouncements! Would Mallarmé have agreed? If poetry and mathematics are the poles, where is philosophy to be found? |
| 12316 | For Enlightenment philosophers, God was no longer involved in politics [Badiou] |
| Full Idea: For the philosophers of the Enlightenment politics is strictly the affair of humankind, an immanent practice from which recourse to the All Mighty's providential organization had to be discarded. | |
| From: Alain Badiou (Briefings on Existence [1998], Prol) |
| 14344 | Natural kinds, such as electrons, all behave the same way because we divide them by dispositions [Mumford] |
| Full Idea: Regularities exist because we classify kinds on the basis of their dispositions, not on pre-established divisions of kinds. The dispositions are the basis for the division into kinds, which is why all electrons behave in the same way. | |
| From: Stephen Mumford (Dispositions [1998], 10.7) | |
| A reaction: This strikes me as being so obvious that it is hardly worth saying, and yet an enormous number of philosophers seem to have been led up the garden path by the notion of a 'kind', probably under the influence of Kripke, Putnam and Wiggins. |
| 19068 | Causation interests us because we want to explain change [Mumford] |
| Full Idea: Like Aristotle, the reason we are really interested in causation is because we want to be able to explain change. | |
| From: Stephen Mumford (Contemporary Efficient Causation: Aristotelian themes [2014], 8) | |
| A reaction: This pinpoints a very important and simple idea. It raises the question (among others) of whether we have just invented this thing called 'causation', because no explanation of change was visible. Hume certainly couldn't see any explanation. |
| 9430 | Singular causes, and identities, might be necessary without falling under a law [Mumford] |
| Full Idea: One might have a singularist view of causation in which a cause necessitates its effect, but they need not be subsumed under a law, ..and there are identities which are metaphysically necessary without being laws of nature. | |
| From: Stephen Mumford (Laws in Nature [2004], 04.5) |
| 9445 | We can give up the counterfactual account if we take causal language at face value [Mumford] |
| Full Idea: If we take causal language at face value and give up reducing causal concepts to non-causal, non-modal concepts, we can give up the counterfactual dependence account. | |
| From: Stephen Mumford (Laws in Nature [2004], 10.5) |
| 9443 | It is only properties which are the source of necessity in the world [Mumford] |
| Full Idea: If laws do not give the world necessity, what does? I argue the positive case for it being properties, and properties alone, that do the job (so we might call them 'modal properties'). | |
| From: Stephen Mumford (Laws in Nature [2004], 10.1) |
| 14338 | In the 'laws' view events are basic, and properties are categorical, only existing when manifested [Mumford] |
| Full Idea: In the 'laws' world view, events are the basic ontological unit and properties are parasitic upon them. Properties exist only in virtue of their instantiation in events. Properties are categorical, because they are only manifested in the present. | |
| From: Stephen Mumford (Dispositions [1998], 10.2) | |
| A reaction: Mumford rejects this view, and I am with him all the way. The first requirement is that properties be active, and not inert. See Leibniz on this. |
| 9444 | There are four candidates for the logical form of law statements [Mumford] |
| Full Idea: The contenders for the logical form of a law statement are 1) a universally quantified conditional, 2) a second-order relation between first-order universals, 3) a functional equivalence, and 4) a dispositional characteristic of a natural kind. | |
| From: Stephen Mumford (Laws in Nature [2004], 10.3) |
| 14339 | Without laws, how can a dispositionalist explain general behaviour within kinds? [Mumford] |
| Full Idea: The problem is how, without general laws, can the dispositionalist explain why generalities in behaviour are true of kinds. | |
| From: Stephen Mumford (Dispositions [1998], 10.3) | |
| A reaction: And the answer is to make kinds depend on individuals, and not vice versa, and then point to the necessary patterns that arise from conjunctions of individual dispositions, given their identity in many individuals. |
| 14341 | Dretske and Armstrong base laws on regularities between individual properties, not between events [Mumford] |
| Full Idea: The improved Dretske/Armstrong regularity view of laws dispenses with the empiricist articulation of them in terms of events, and construes them as singular statements of fact that describe relations between properties. | |
| From: Stephen Mumford (Dispositions [1998], 10.4) | |
| A reaction: They then seem to go a bit mystical, by insisting that the properties are 'universals' (even if they have to be instantiated). Universals explain nothing. |
| 9431 | Pure regularities are rare, usually only found in idealized conditions [Mumford] |
| Full Idea: Pure regularities are not nearly as common as might have been thought, and are usually only to be found in simplified or idealized conditions. | |
| From: Stephen Mumford (Laws in Nature [2004], 05.3) | |
| A reaction: [He cites Nancy Cartwright 1999 for this view] |
| 9416 | Regularities are more likely with few instances, and guaranteed with no instances! [Mumford] |
| Full Idea: It seems that the fewer the instances, the more likely it is that there be a regularity, ..and if there are no cases at all, and no S is P, that is a regularity. | |
| From: Stephen Mumford (Laws in Nature [2004], 03.3) | |
| A reaction: [He attributes the second point to Molnar] |
| 9415 | Would it count as a regularity if the only five As were also B? [Mumford] |
| Full Idea: While it might be true that for all x, if Ax then Bx, would we really want to count it as a genuine regularity in nature if only five things were A (and all five were also B)? | |
| From: Stephen Mumford (Laws in Nature [2004], 03.3) |
| 9441 | Regularity laws don't explain, because they have no governing role [Mumford] |
| Full Idea: A regularity-law does not explain its instances, because such laws play no role in determining or governing their instances. | |
| From: Stephen Mumford (Laws in Nature [2004], 09.7) | |
| A reaction: Good. It has always seemed to me entirely vacuous to explain an event simply by saying that it falls under some law. |
| 14340 | It is a regularity that whenever a person sneezes, someone (somewhere) promptly coughs [Mumford] |
| Full Idea: It is no doubt a true regularity that every time I sneeze, someone, somewhere in the world, immediately coughs. | |
| From: Stephen Mumford (Dispositions [1998], 10.4) | |
| A reaction: Not a huge problem for the regularity theory of laws, but the first challenge that it must meet. |
| 9422 | If the best system describes a nomological system, the laws are in nature, not in the description [Mumford] |
| Full Idea: If the world really does have its own nomological structure, that a systematization merely describes, why are the laws not to be equated with the nomological structure itself, rather than with the system that describes it? | |
| From: Stephen Mumford (Laws in Nature [2004], 03.4) |
| 9421 | The best systems theory says regularities derive from laws, rather than constituting them [Mumford] |
| Full Idea: The best systems theory (of Mill-Ramsey-Lewis) says that laws are not seen as regularities but, rather, as those things from which regularities - or rather, the whole world history including the regularities and everything else - can be derived. | |
| From: Stephen Mumford (Laws in Nature [2004], 03.4) | |
| A reaction: Put this way, the theory invites questions about ontology. Regularities are just patterns in physical reality, but axioms are propositions. So are they just features of human thought, or do these axioms actuallyr reside in reality. Too weak or too strong. |
| 9432 | Laws of nature are necessary relations between universal properties, rather than about particulars [Mumford] |
| Full Idea: The core of the Dretske-Tooley-Armstrong view of the late 70s is that we have a law of nature when we have a relation of natural necessitation between universals. ..The innovation was that laws are about properties, and only indirectly about particulars. | |
| From: Stephen Mumford (Laws in Nature [2004], 06.2) | |
| A reaction: It sounds as if we should then be able to know the laws of nature a priori, since that was Russell's 1912 definition of a priori knowledge. |
| 9433 | If laws can be uninstantiated, this favours the view of them as connecting universals [Mumford] |
| Full Idea: If there are laws that are instantiated in no particulars, then this would seem to favour the theory that laws connect universals rather than particulars. | |
| From: Stephen Mumford (Laws in Nature [2004], 06.4) | |
| A reaction: There is a dispute here between the Platonic view of uninstantiated universals (Tooley) and the Aristotelian instantiated view (Armstrong). Mumford and I prefer the dispositional account. |
| 14345 | The necessity of an electron being an electron is conceptual, and won't ground necessary laws [Mumford] |
| Full Idea: The logical necessity of physical laws is not required by dispositional essentialism. An electron would not be an electron if its behaviour were different from the behaviour it has in the actual world, but this necessity is purely conceptual. | |
| From: Stephen Mumford (Dispositions [1998], 10.8) | |
| A reaction: [He is particularly aiming this at Ellis and Lierse 1994] This may be missing the point. Given those electron dispositions, the electrons necessitate law-like happenings. Whether a variable entity is called an 'electron' is trivial. |
| 9434 | Laws of nature are just the possession of essential properties by natural kinds [Mumford] |
| Full Idea: If dispositional essentialism is granted, then there is a law of nature wherever there is an essential property of a natural kind; laws are just the havings of essential properties by natural kinds. | |
| From: Stephen Mumford (Laws in Nature [2004], 07.2) | |
| A reaction: [He is expounding Ellis's view] |
| 14307 | Some dispositions are so far unknown, until we learn how to manifest them [Mumford] |
| Full Idea: It seems reasonable to assume that there are some dispositions of some things of which we are not aware because we have not yet discovered the way to get these dispositions to manifest. | |
| From: Stephen Mumford (Dispositions [1998], 03.7) | |
| A reaction: This strikes me as a pretty good description of what scientists are currently doing when, for example, they build a new particle accelerator. |
| 9437 | To distinguish accidental from essential properties, we must include possible members of kinds [Mumford] |
| Full Idea: Where properties are possessed by all kind members, we must distinguish the accidental from essential ones by considering all actual and possible kind members. | |
| From: Stephen Mumford (Laws in Nature [2004], 07.5) | |
| A reaction: This is why we must treat possibilities as features of the actual world. |
| 9439 | The Central Dilemma is how to explain an internal or external view of laws which govern [Mumford] |
| Full Idea: The Central Dilemma about laws of nature is that, if they have some governing role, then they must be internal or external to the things governed, and it is hard to give a plausible account of either view. | |
| From: Stephen Mumford (Laws in Nature [2004], 09.2) | |
| A reaction: This dilemma is the basis of Mumford's total rejection of 'laws of nature'. I think I agree. |
| 9412 | You only need laws if you (erroneously) think the world is otherwise inert [Mumford] |
| Full Idea: Laws are a solution to a problem that was misconceived. Only if you think that the world would be otherwise inactive or inanimate, do you have the need to add laws to your ontology. | |
| From: Stephen Mumford (Laws in Nature [2004], 01.5) | |
| A reaction: This is a bold and extreme view - and I agree with it. I consider laws to be quite a useful concept when discussing nature, but they are not part of the ontology, and they don't do any work. They are metaphysically hopeless. |
| 9411 | There are no laws of nature in Aristotle; they became standard with Descartes and Newton [Mumford] |
| Full Idea: Laws do not appear in Aristotle's metaphysics, and it wasn't until Descartes and Newton that laws entered the intellectual mainstream. | |
| From: Stephen Mumford (Laws in Nature [2004], 01.5) | |
| A reaction: Cf. Idea 5470. |
| 12317 | The God of religion results from an encounter, not from a proof [Badiou] |
| Full Idea: The God of metaphysics makes sense of existing according to a proof, while the God of religion makes sense of living according to an encounter | |
| From: Alain Badiou (Briefings on Existence [1998], Prol) |