42 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) |
| 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. |
| 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? |
| 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) |
| 9572 | Realists about sets say there exists a null set in the real world, with no members [Chihara] |
| Full Idea: In the Gödelian realistic view of set theory the statement that there is a null set as the assertion of the existence in the real world of a set that has no members. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 11.6) | |
| A reaction: It seems to me obvious that such a claim is nonsense on stilts. 'In the beginning there was the null set'? |
| 9550 | We only know relational facts about the empty set, but nothing intrinsic [Chihara] |
| Full Idea: Everything we know about the empty set is relational; we know that nothing is the membership relation to it. But what do we know about its 'intrinsic properties'? | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 01.5) | |
| A reaction: Set theory seems to depend on the concept of the empty set. Modern theorists seem over-influenced by the Quine-Putnam view, that if science needs it, we must commit ourselves to its existence. |
| 9562 | In simple type theory there is a hierarchy of null sets [Chihara] |
| Full Idea: In simple type theory, there is a null set of type 1, a null set of type 2, a null set of type 3..... (Quine has expressed his distaste for this). | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 07.4) | |
| A reaction: It is bad enough trying to individuate the unique null set, without whole gangs of them drifting indistinguishably through the logical fog. All rational beings should share Quine's distaste, even if Quine is wrong. |
| 9573 | The null set is a structural position which has no other position in membership relation [Chihara] |
| Full Idea: In the structuralist view of sets, in structures of a certain sort the null set is taken to be a position (or point) that will be such that no other position (or point) will be in the membership relation to it. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 11.6) | |
| A reaction: It would be hard to conceive of something having a place in a structure if nothing had a relation to it, so is the null set related to singeton sets but not there members. It will be hard to avoid Platonism here. Set theory needs the null set. |
| 9551 | What is special about Bill Clinton's unit set, in comparison with all the others? [Chihara] |
| Full Idea: What is it about the intrinsic properties of just that one unit set in virtue of which Bill Clinton is related to just it and not to any other unit sets in the set-theoretical universe? | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 01.5) | |
| A reaction: If we all kept pet woodlice, we had better not hold a wood louse rally, or we might go home with the wrong one. My singleton seems seems remarkably like yours. Could we, perhaps, swap, just for a change? |
| 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. |
| 9549 | The set theorist cannot tell us what 'membership' is [Chihara] |
| Full Idea: The set theorist cannot tell us anything about the true relationship of membership. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 01.5) | |
| A reaction: If three unrelated objects suddenly became members of a set, it is hard to see how the world would have changed, except in the minds of those thinking about it. |
| 9571 | ZFU refers to the physical world, when it talks of 'urelements' [Chihara] |
| Full Idea: ZFU set theory talks about physical objects (the urelements), and hence is in some way about the physical world. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 11.5) | |
| A reaction: This sounds a bit surprising, given that the whole theory would appear to be quite unaffected if God announced that idealism is true and there are no physical objects. |
| 9563 | A pack of wolves doesn't cease when one member dies [Chihara] |
| Full Idea: A pack of wolves is not thought to go out of existence just because some member of the pack is killed. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 07.5) | |
| A reaction: The point is that the formal extensional notion of a set doesn't correspond to our common sense notion of a group or class. Even a highly scientific theory about wolves needs a loose notion of a wolf pack. |
| 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) |
| 9561 | The mathematics of relations is entirely covered by ordered pairs [Chihara] |
| Full Idea: Everything one needs to do with relations in mathematics can be done by taking a relation to be a set of ordered pairs. (Ordered triples etc. can be defined as order pairs, so that <x,y,z> is <x,<y,z>>). | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 07.2) | |
| A reaction: How do we distinguish 'I own my cat' from 'I love my cat'? Or 'I quite like my cat' from 'I adore my cat'? Nevertheless, this is an interesting starting point for a discussion of relations. |
| 9552 | Sentences are consistent if they can all be true; for Frege it is that no contradiction can be deduced [Chihara] |
| Full Idea: In first-order logic a set of sentences is 'consistent' iff there is an interpretation (or structure) in which the set of sentences is true. ..For Frege, though, a set of sentences is consistent if it is not possible to deduce a contradiction from it. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 02.1) | |
| A reaction: The first approach seems positive, the second negative. Frege seems to have a higher standard, which is appealing, but the first one seems intuitively right. There is a possible world where this could work. |
| 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 naďve. What to do? |
| 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. |
| 9553 | Analytic geometry gave space a mathematical structure, which could then have axioms [Chihara] |
| Full Idea: With the invention of analytic geometry (by Fermat and then Descartes) physical space could be represented as having a mathematical structure, which could eventually lead to its axiomatization (by Hilbert). | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 02.3) | |
| A reaction: The idea that space might have axioms seems to be pythagoreanism run riot. I wonder if there is some flaw at the heart of Einstein's General Theory because of this? |
| 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? |
| 10192 | We can replace existence of sets with possibility of constructing token sentences [Chihara, by MacBride] |
| Full Idea: Chihara's 'constructability theory' is nominalist - mathematics is reducible to a simple theory of types. Instead of talk of sets {x:x is F}, we talk of open sentences Fx defining them. Existence claims become constructability of sentence tokens. | |
| From: report of Charles Chihara (A Structural Account of Mathematics [2004]) by Fraser MacBride - Review of Chihara's 'Structural Acc of Maths' p.81 | |
| A reaction: This seems to be approaching the problem in a Fregean way, by giving an account of the semantics. Chihara is trying to evade the Quinean idea that assertion is ontological commitment. But has Chihara retreated too far? How does he assert existence? |
| 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. |
| 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) |
| 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'. |
| 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. |
| 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. |
| 9559 | If a successful theory confirms mathematics, presumably a failed theory disconfirms it? [Chihara] |
| Full Idea: If mathematics shares whatever confirmation accrues to the theories using it, would it not be reasonable to suppose that mathematics shares whatever disconfirmation accrues to the theories using it? | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 05.8) | |
| A reaction: Presumably Quine would bite the bullet here, although maths is much closer to the centre of his web of belief, and so far less likely to require adjustment. In practice, though, mathematics is not challenged whenever an experiment fails. |
| 9566 | No scientific explanation would collapse if mathematical objects were shown not to exist [Chihara] |
| Full Idea: Evidently, no scientific explanations of specific phenomena would collapse as a result of any hypothetical discovery that no mathematical objects exist. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 09.1) | |
| A reaction: It is inconceivable that anyone would challenge this claim. A good model seems to be drama; a play needs commitment from actors and audience, even when we know it is fiction. The point is that mathematics doesn't collapse either. |
| 7091 | The argument from analogy is not a strong inference, since the other being might be an actor or a robot [Grayling] |
| Full Idea: The argument from analogy is a weak one, because it does not logically guarantee the inference I draw to the other's inner states, for he might be dissimulating or acting, or may even be a cleverly contrived robot which feels nothing. | |
| From: A.C. Grayling (Wittgenstein [1988], Ch.3) | |
| A reaction: This gives the impression that for an argument to be strong it must logically guarantee its inference. It strikes me that analogy is a good reason for believing in other minds, but that is because I am looking for the best explanation, not logical proof. |
| 9568 | I prefer the open sentences of a Constructibility Theory, to Platonist ideas of 'equivalence classes' [Chihara] |
| Full Idea: What I refer to as an 'equivalence class' (of line segments of a particular length) is an open sentence in my Constructibility Theory. I just use this terminology of the Platonist for didactic purposes. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 09.10) | |
| A reaction: This is because 'equivalence classes' is committed to the existence of classes, which is Quinean Platonism. I am with Chihara in wanting a story that avoids such things. Kit Fine is investigating similar notions of rules of construction. |
| 9547 | Mathematical entities are causally inert, so the causal theory of reference won't work for them [Chihara] |
| Full Idea: Causal theories of reference seem doomed to failure for the case of reference to mathematical entities, since such entities are evidently causally inert. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 01.3) | |
| A reaction: Presumably you could baptise a fictional entity such as 'Polonius', and initiate a social causal chain, with a tradition of reference. You could baptise a baby in absentia. |
| 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. |
| 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) |
| 9574 | 'Gunk' is an individual possessing no parts that are atoms [Chihara] |
| Full Idea: An 'atomless gunk' is defined to be an individual possessing no parts that are atoms. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], App A) | |
| A reaction: [Lewis coined it] If you ask what are a-toms made of and what are ideas made of, the only answer we can offer is that the a-toms are made of gunk, and the ideas aren't made of anything, which is still bad news for the existence of ideas. |
| 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) |