Bruno Marchal wrote:

> Le 29-août-06, à 20:45, 1Z a écrit : > > > > > The version of AR that is supported by comp > > only makes a commitment about mind-independent *truth*. The idea > > that the mind-independent truth of mathematical propositions > > entails the mind-independent *existence* of mathematical objects is > > a very contentious and substantive claim. > > > You have not yet answered my question: what difference are you making > between "there exist a prime number in platonia" and "the truth of the > proposition asserting the *existence* of a prime number is independent > of me, you, and all contingencies" ? "P is true" is not different to "P". That is not the difference I making. I'm making a difference between what "exists" means in mathematical sentences and what it means in empiricial sentences (and what it means in fictional contexts...) The logical case for mathematical Platonism is based on the idea that mathematical statements are true, and make existence claims. That they are true is not disputed by the anti-Platonist, who must therefore claim that mathematical existence claims are somehow weaker than other existence claims -- perhaps merely metaphorical. That the the word "exists" means different things in different contexts is easily established. For one thing, this is already conceded by Platonists! Platonists think Platonic existence is eternal, immaterial non-spatial, and so on, unlike the Earthly existence of material bodies. For another, we are already used to contextualising the meaning of "exists". We agree with both: "helicopters exist"; and "helicopters don't exist in Middle Earth". (People who base their entire anti-Platonic philosophy are called fictionalists. However, mathematics is not a fiction because it is not a free creation. Mathematicians are constrained by consistency and non-contradiction in a way that authors are not. Dr Watson's fictional existence is intact despite the fact that he is sometimes called John and sometimes James in Conan Doyle's stories). The epistemic case for mathematical Platonism is be argued on the basis of the objective nature of mathematical truth. Superficially, it seems persuasive that objectivity requires objects. However, the basic case for the objectivity of mathematics is the tendency of mathematicians to agree about the answers to mathematical problems; this can be explained by noting that mathematical logic is based on axioms and rules of inference, and different mathematicians following the same rules will tend to get the same answers , like different computers running the same problem. (There is also disagreement about some axioms, such as the Axiom of Choice, and different mathematicians with different attitudes about the AoC will tend to get different answers -- a phenomenon which is easily explained by the formalist view I am taking here). The semantic case for mathematical Platonism is based on the idea that the terms in a mathematical sentence must mean something, and therefore must refer to objects. It can be argued on general linguistic grounds that not all meaning is reference to some kind of object outside the head. Some meaning is sense, some is reference. That establishes the possibility that mathematical terms do not have references. What establishes it is as likely and not merely possible is the obeservation that nothing like empirical investigation is needed to establish the truth of mathematical statements. Mathematical truth is arrived at by a purely conceptual process, which is what would be expected if mathematical meaning were restricted to the Sense, the "in the head" component of meaning. A possible counter argument by the Platonist is that the downgrading of mathematical existence to a mere metaphor is arbitrary. The anti-Platonist must show that a consistent standard is being applied. This it is possible to do; the standard is to take the meaning of existence in the context of a particular proposition to relate to the means of justification of the proposition. Since ordinary statements are confirmed empirically, "exists" means "can be perceived" in that context. Since sufficient grounds for asserting the existence of mathematical objects are that it is does not contradict anything else in mathematics, mathematical existence just amounts to concpetual non-contradictoriness. (Incidentally, this approach answers a question about mathematical and empirical truth. The anti-Platonists want sthe two kinds of truth to be different, but also needs them to be related so as to avoid the charge that one class of statement is not true at all. This can be achieved because empirical statements rest on non-contradiction in order to achive correspondence. If an empricial observation fails co correspond to a statemet, there is a contradiction between them. Thus non-contradiciton is a necessary but insufficient justification for truth in empircal statements, but a sufficient one for mathematical statements). > > Where is it shown the UD exists ? > > > If you agree that the number 0, 1, 2, 3, 4, ... exist (or again, if you > prefer, that the truth of the propositions: > > Ex(x = 0), > Ex(x = s(0)), > Ex(x = s(s(0))), > ... > > is independent of me), then it can proved that the UD exists. It can be > proved also that Peano Arithmetic (PA) can both define the UD and prove > that it exists. But again this is just "mathematical existence". You need some reason to assert that mathematical existence is not a mere metaphor implying no real existence, as anti-Platonist mathematicians claim. I do not think that is given by computationalism. > >> Tell me also this, if you don't mind: are you able to doubt about the > >> existence of "primary matter"? I know it is your main fundamental > >> postulate. Could you imagine that you could be wrong? > > > > It is possible that I am wrong. It is possible that I am right. > > But you are -- or were -- telling me matter is impossible. > > > Only when I use Occam. Occam does not support conclusions of impossibility. It could be a brute fact that the universe is more complicated than strcitly necessary. > Without Occam I say only that the notion of > primary matter is necessarily useless i.e. without explanatory purposes > (even concerning just the belief in the physical proposition only) . > This is a non trivial consequence of the comp hyp. (cf UDA). As is the way with these things, we anti-Platonists appeal to Occam as well (although not qua impossibilia). All the facts about mathematical truth and methodology can be established without appeal to the actual existence of mathematical objects. In fact, the lack of such objects actually explains the objectivity and necessity of maths. Mathematical statements are necessarily true because there are no possible circumstances that make them false; there are no possible circumstances that would make them false because they do not refer to anything external. This is much simpler than the Platonist alternative that mathematical statements : 1) have referents which are 2) unchanging and eternal, unlike anything anyone has actuall seen and thereby 3) explain the necessity (invariance) of mathematical statements without 4) performing any other role -- they are not involved in mathematical proof. > > But the negative integers exist (or "exist"), so it has > > an existing predecessor. > > > Yes. But the axiom Q1 "Ax ~(0 = s(x)" is not made wrong just because > you define the negative integer in Robinson Arithmetic. The "x" are > still for "natural number". The integer are new objects defined from > the natural number. All right? To take another example, you can define > in RA all partial recursive functions, but obviously they does not obey > to the Q axioms, they are just constructs, definable in RA. So the specialness of Time depends on the specialness of nautral numbers, depends on the specialness of Robinson Arithemtic ? > Bruno > > > http://iridia.ulb.ac.be/~marchal/ --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~----------~----~----~----~------~----~------~--~---