On Friday, September 28, 2018 at 12:03:35 AM UTC-5, Bruce wrote: > > From: Brent Meeker <[email protected] <javascript:>> > > > On 9/27/2018 9:29 PM, Bruce Kellett wrote: > > From: Bruno Marchal <[email protected] <javascript:>> > > > But now, let us move forward. Stop saying “realism or platonism”, in pour > metaphysical context this lead to misunderstanding. > > Assuming classical arithmetic = arithmetical realism. > > > It is becoming clear that we have very different understandings of what is > meant by arithmetical (or mathematical) realism. I gathered a few > statements about realism from your recent post -- included here: > > "Realism = classical. Realism means that I use the axiom (A v ~A)." > > "All scientific theories use arithmetical realism, but you are still using > it in a philosophical/metaphysical sense, when it means simply that we > accept the excluded middle principle in arithmetic." > > "Of course, given what I mean by arithmetical realism (which I thought I > already told you), this would mean that you reject the use of (A v ~A) in > arithmetic" > > > My understanding of 'realism' comes from the idea of scientific realism. > This can have a number of nuanced interpretations, but the basic idea of > scientific realism is a cluster of views about the nature of scientific > theories and theorizing. A common core might be the following: > (1) The aim of scientific inquiry is to produce theories that provide > description of the world that are literally true. > (2) Theories in the 'mature sciences' are usually approximately true, and > the entities postulated by those theories usually exist. > > One of the most popular arguments for scientific realism is the so-called > 'miracle argument'. Following Putnam, scientific realism is capable of > explaining why a predictively successful theory is predictively successful, > whereas the success of a theory would be miraculous if scientific realism > were not true. > > Stathis Psillos adds a metaphysical component: the world has a definite > mind-independent structure; and a semantic component: scientific theories > are truth-conditioned descriptions of their intended domain, so the > theoretical terms in theories have factual reference -- the unobservable > entities they posit populate the world -- form the 'furniture' of reality. > > The Oxford Dictionary of Philosophy, in the section on the philosophy of > mathematics, gives the following definitions: > > "There are two distinct types of realism in the philosophy of mathematics. > Realism-in-ontology is the view that the subject matter of mathematics is > the realm of objects that exist independent of the mind, conventions, and > language of the mathematician. Most advocates of this view hold that > mathematical objects -- numbers, functions, points, sets, etc. -- are > abstract, eternal, and do not enter into causal relationships with material > objects. Because of this, realism-in-ontology is sometimes called platonism. > "Realism-in-truth-value is the view that unambiguous assertions of > mathematics are non-vacuously true or false, independent of the mind, > language, and conventions of the mathematician. (This would seem to be > close to the view that you, Bruno, espouse.) > > "There is a natural connection between the two varieties of realism. > Consider the following statement: > > 'There is a prime number greater than 1,000,000.' > > "The realist-in-truth-value holds that this is an objective truth. But > what does it mean? Prima facie, '1,000,000' is a singular terms, and 'prime > number' is a common noun. If the surface grammar of this sentence reflects > its logical form, and if 'there is' means 'there exists', then the sentence > entails that both the number 1,000,000 and a greater prime number exist. > For the realist-in-truth-value, this existence is objective, and so we are > led to realism-in-ontology. In sum, if one is a realist-in-truth-value, > then realism-in-ontology is the result of taking mathematical assertions at > face value." > > Other references that I have looked up, such as entries in the Stanford > Encyclopedia of Philosophy on "Realism" and "Platonism in the Philosophy of > Mathematics", say similar things. Though, of course, there are probably > more nuances in the understanding of mathematical realism than there are > philosophers of mathematics. > > > Given the above references, I think it should be clear why I say "realism > or platonism", and refer to "an independently existing mathematical realm". > In Western philosophy at least, that is what realism in mathematics means > -- although things might be different in Gallic philosophy. > > It seems that your idea of arithmetical (mathematical) realism is entirely > from classical logic and is, therefore, essentially a > 'realism-in-truth-value' understanding. It is interesting, in that case, > that you make no reference to mathematical objects. You claim that the > truth of propositions such as '2+2=4' is independent of the mind, language, > and conventions of arithmetic, as in the definition of > 'realist-in-truth-value' above. But you do not seem to go the additional > step of saying that mathematical objects, numbers and so on, are objects > that actually exist (which would be a form of platonism). If you want to > reject platonism, and the idea that mathematical concepts are objects that > actually exist -- that there is a mathematical realm of objects that exist > independently of any physical existence -- then I suppose you are entitled > to any view that you wish to hold. But you cannot claim that any such view > is uniquely necessary. > > If you reject platonism, it is hard to see how you can make sense of > claims such as "All calculations exist in arithmetic", or that physics > arises from the statistics of computations in the universal dovetailer. > Since I reject all forms of arithmetical realism, particularly platonism, I > do not think that your arguments for 'comp' have any merit. > > However, the philosophy of mathematics is not an area in which I have had > any particular interest, so apart from rejecting mathematical realism and > platonism, I do not have any strong views about which of the many > alternatives on offer might be an acceptable philosophical attitude to > arithmetic. > > Bruce > > > I don't see any problem in saying that "2+2=4" while denying that numbers > exist. ISTM analogous to saying "A unicorn has a single horn." without > implying that unicorns exist. The truth that is maintained by mathematical > proof is just a marker which is analytically preserved by the rules of > inference. The fact that there are true but unprovable sentences in > arithmetic, in just a logical inference in meta-mathematics. It's not > something outside mathematics that is true in the sense that ice is cold. > > > That is essentially a fictionalist philosophy of mathematics. Or it could > relate to a logicist account in which theorems of mathematics are viewed as > tautologies -- true by virtue of the meaning of the terms involved. I am > not unsympathetic to either of these views. > > Bruce > > Although one can have fictional truths, constructive truths, material truths ... perhaps a whole type-system of truths, at the end of the day, what Rorty said is about it could be all one can say:
*Rorty on truth* (1 min 41 sec) https://www.youtube.com/watch?v=CzynRPP9XkY - pt -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
