> On 28 Sep 2018, at 06:29, Bruce Kellett <[email protected]> wrote: > > From: Bruno Marchal <[email protected] <mailto:[email protected]>> >> >> 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.
That is doing philosophy (of science) before doing the science. You need to understand that I am a scientist. What I do does not ask for any philosophy, just understanding. In science, there is no agreement or disagreement, but understanding or no understanding. That is a point that some people do not grasp, and it is normal, as we used “philosophical terms (to ease the understanding, especially for those who does not do the math). I will still comments, but the idea is that such philosophy should be done only after a good understanding of the “scientific theory” (the metaphysics or theology of numbers). > 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 is what we cannot do? The idea that there is a world, or not, is part of the inquiry when we do metaphysics with the scientific method. The very notion of world cannot be used. I can be OK, by enlarging probably the sense for “world” (the logician large definition is an element of a set). > that are literally true. Same problem for the notion of “true”. Here, with Mechanism, we can use Tarski theory of truth. Eventually, the only notion of truth which plays a role will be the notion of arithmetical truth, and even this one will eventually be limited to sigma_1-truth (which id definable by Peano arithmetic). > (2) Theories in the 'mature sciences' are usually approximately true, and the > entities postulated by those theories usually exist. Same problem here. The notion of “existence” will be refine a lot through the Mechanist hypothesis. The ontological existence is given by the “use” of the existential quantifier in the base theory (combinators or Robinson Q theory, for example, but a diophantine polynomial equation is as good). All other existence will appear to be phenomenological, and mathematically are the existential quantifiers in the modal logis extracted from incompleteness. But that is for the second part, when we translate the UDA (Universal Dovetailer Argument) in arithmetic. > > 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. Unfortunately that is not true, and use a brain-mind identity thesis which cannot work with Mechanism. We can “attach” a mind or a person to a computer, but a computer cannot attach its mind to a computer, only to an infinity of computer (in arithmetic). > > Stathis Psillos adds a metaphysical component: the world has a definite > mind-independent structure; If by world we mean the arithmetical reality, or the sigma_1 arithmetical reality, then I am OK with this, but the physical reality will appear to be more “mind-dependent”. Mind refers here to the mind of all universal machine/number, not to the human mind. The physical reality can be explained to be largely independent of the human mind, but not independent of the mind of *all* universal numbers. > 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. Assuming already some metaphysics. But that is exactly what we need to avoid when doing metaphysics with the scientific attitude. As a scientist, I want first a formula which makes the right prediction. That is what I gave. Then we can philosophise from there. > Because of this, realism-in-ontology is sometimes called platonism. Yes, and that is why I avoid the term. I use Platonism only for the theology, and use realism for mathematics, but it will not be different from, technically, having the (A v ~A) principle in the postulate. > "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.) Yes. To be sure, existence is defined by the truth value of the existential proposition in the base theory. So only 0, s(0), s(s(0)), … are considered to exist. Or only K and S and their combination. The theology, including physics, does not depend on the choice of the base theory, if that base theory is 1) Turing complete, and 2) without induction axioms, nor infinity axiom. > > "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.” Yes. That is what I was saying. But in metaphysics, such interpretation of existence can be done only on the base theory. Physics is not a base theory, here, it has to be phenomenological (by the UDA argument). All other existence, even “god” are enforced by the gap between truth and belief/provability. > > 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. No problem, but the vocabulary “platonism” is dangerous in metaphysics, as it has a different sense than in philosophy of mathematics. But exactly like we cannot demolish Einstein’s theory by criticising the fact that he has never defined what he meant by "2” in E = mc^2, we cannot attack “realism in arithmetic” before getting the theory right. Standford dictionary is rather good (compared to many Wikis) but still unaware of the recent work in the mathematical theology of the machine, which remains unknown for unknown reason (say) Tegmarl and Bostrom have not been able to refer to it). > > It seems that your idea of arithmetical (mathematical) realism is entirely > from classical logic and is, therefore, essentially a > 'realism-in-truth-value' understanding. That is right, but as you say, once the base theory is made precise, existence is truth value of existential proposition. > 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. Like Einstein and everybody ins science. Nobody attacked Einstein by telling him that he assumes the existence of the number two in his theory. > 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). I use “exist” in the same sense as “it exist a number x such that x + 7 = 8”. The intuition is only that I do not put any doubt on what we learn in primary school about natural numbers. I take the table of addition and multiplication at face value, and I derive the existence of some number n verifying a predicate P from the truth or the proof of a proposition with shape “ExP(x)”. It is realist in the sense that the proof can use the excluded middle principle, and thus be non constructive. > 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. Arithmetical realism is just the acceptance of classical (with “A v ~A”) + 1) 0 ≠ s(x) 2) x ≠ y -> s(x) ≠ s(y) 3) x ≠ 0 -> Ey(x = s(y)) 4) x+0 = x 5) x+s(y) = s(x+y) 6) x*0=0 7) x*s(y)=(x*y)+x The “philosophy” of the machine is extracted entirely from that, and that is why doing philosophy at the start would be equivalent of ciricizing Einstein because he is unclear on what he meant by “2”. It would be what I called a “1004 fallacy”. Asking non relevant precision at the wrong place. > > If you reject platonism, it is hard to see how you can make sense of claims > such as "All calculations exist in arithmetic”, It has the same sense than “prime numbers exists”. A computation is just a special number. It exists in the sense that we can say that each of 0, s(0) s(s(0)) … exists. It is a number participating in a Turing universal relation. Unlike being prime, It is just a relative notion, but its existence is only due to the fact that some “ExP(x)” is true, in the usual primary school sense. > or that physics arises from the statistics of computations in the universal > dovetailer. It is a logical consequence of the axiom above, together with Mechanism. > Since I reject all forms of arithmetical realism, particularly platonism, I > do not think that your arguments for 'comp' have any merit. Then you reject any meaning in “E = mc^2”. I think you just do philosophy at the wrong place. You need to study the mathematical theory, and how it is testable, before using philosophy to put doubt on what every one accept, like when verifying their taxes. > > However, the philosophy of mathematics is not an area in which I have had any > particular interest, Me neither. I work on the mind body problem. > 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. I don’t do philosophy in that sense, although I am not criticising it, except if misused to avoid a scientific theory, like the creationist do with evolution or like Bergson did with Einstein’s relativity, or Göethe did with Newton’s theory of color. Bruno > > Bruce > > > -- > 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] > <mailto:[email protected]>. > To post to this group, send email to [email protected] > <mailto:[email protected]>. > Visit this group at https://groups.google.com/group/everything-list > <https://groups.google.com/group/everything-list>. > For more options, visit https://groups.google.com/d/optout > <https://groups.google.com/d/optout>. -- 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.
