Re: Combinator joker (a note in passing on formal combinator theory)
Well, you must assume the principle of identity at the beginning, otherwise all your other assumptions will be their own negations. -- 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 everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
Re: Mathematical Universe Hypothesis
Bruno, as I see it, my ontology, whose relational aspect is defined by the relation of similarity (and its special kinds - instantiation and composition), includes your ontology, because pure set theory includes arithmetic. You may be right that arithmetic is sufficient to define physics but reality may also contain more than arithmetic. On the other hand, if I understand Godel's second incompleteness theorem correctly, as far as the relational/mathematical aspect of reality is concerned, we will never be able to prove that there exists more than arithmetic (because we will never be able to prove that it is consistent). And if we are not able to interact with infinite objects, we will never be able to observe them either. But I don't see a reason to exclude infinite objects from existence. Some say that an infinite collection can never be "completed", as if mathematical objects are created by some kind of *process *that must reach completion*. *They are not created by a process; they exist timelessly; there is nothing to complete. Only inconsistency would prevent their existence. You said you don't really believe in sets. But a set is just a combination of objects, where the combination is another object, isn't it? Everything you see around you is structurally a set. About category theory vs. set theory, this is how I understand it: more general (more abstract) mathematical objects are instantiated in more specific mathematical objects (e.g. "geometric object" is instantiated in "triangle") and ultimately in concrete mathematical objects (e.g. in concrete triangles), which are not instantiated in anything else. (Those objects that can be instantiated in other objects are also called properties.) All concrete objects are concrete collections, that is, collections of concrete objects, so all mathematical objects are ultimately instantiated in concrete collections. This fact is used in set theory, where every mathematical object is represented as a collection (set), and that's how set theory can be a foundation of mathematics. The collections referred to in set theory are not concrete collections though but abstract collections (generalized collections), because differences between concrete collections of the same kind are not relevant for mathematical purposes. So for example, set theory does not refer to concrete empty sets but to one abstract empty set (which is instantiated in all concrete empty sets). (Although I have also heard of the extension of set theory to so-called "multiset" theory, which admits copies (instances) of the same object as distinct members of a set.) The approach of category theory is not to represent mathematical objects as collections but to study similarities (morphisms) directly between mathematical objects themselves. Collections, there, are treated just as one of many kinds of mathematical objects. About qualia, some time ago I imagined that maybe Godel sentences could explain qualia, as Godel sentences depend on an axiomatic system and yet cannot be proved from that system, similarly like qualia seem to depend on a neural system and yet cannot be proved from it. But then I grew skeptical of this idea because it seemed to me that numbers will always be just numbers, even if they are infinitely big, and an infinitely big number may be beyond our grasp in a sense but it will not somehow turn into red color, for example. Gradually I started to lean to the idea that numbers and mathematics in general are about the relation of similarity; that mathematics basically says that something is similar to something else but never says what that "something" is. So now it seems more plausible to me that qualia are those "somethings" that stand in similarity relations. Russellian monism is a similar explanation of qualia. -- 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 everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
Combinator joker (a note in passing on formal combinator theory)
Thomas Pavel wanted that something is identical to itself. In most formal combinator theory x = x is given as an axiom, yet I did not take it. Likewise, the rule: if x = y then y = x is also given, usually, but you can prove it too! That is very simple, but admittedly subtle, probably more difficult than all the exercises given so far, so don’t worry and feel free to look at the solution. Note that the meta-logic is the usual informal classical logic. Here is the formal theory of combinators: Three rules and two reduction axioms: 1) If x = y and x = z, then y = z 2) If x = y then xz = yz 3) If x = y then zx = zy 4) Kxy = x 5) Sxyz = xz(yz) Exercise(*): a) prove that x = x, Hint use 1) and 4). 2) prove that the rule which follows is correct: If x = y then y = x. Bruno (*) Solution (coming from a book by Rosser) below: ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 1) x = x Proof: Kxy = y (by 4) Thus we have Kxy = x and Kxy = x, so by “1)” we have that x = x. 2) if x = y then y = x Proof Let us suppose x = y. But by the exercise just above, x = x, so now we have x = y and x = x, so by “1)” again, we have y = x. -- 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 everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
Re: Mathematical Universe Hypothesis
Hi Thomas, Sorry for the delay, busy days, see the mail only now. > On 22 Oct 2018, at 19:15, Tomas Pales wrote: > > > > On Monday, October 22, 2018 at 1:41:23 PM UTC+2, Bruno Marchal wrote: > > The computable universe hypothesis cannot make sense. To define “computable” > you need to assume arithmetic. But arithmetic executes all computations, and > the measure problem will have to involve infinities. > So the correct passage from “mathematical universe” to computationalism > consists in > > 1) distinguish well the ontology and the phenomenology. Restrict the ontology > to the finite and computable finite objects, > > In his paper, Tegmark included in CUH only mathematical structures defined by > halting computations. So I guess this would include only finite computable > objects? > > > 2) allowing the infinite in the phenomenology, where indeed the physical > universe will appear. > > The infinities ruins physics only if they are put in the ontology. This is > explained in details in basically all my papers (on this subject). I can give > the reference (again) if you are interested. Tegmark missed the mind-body > problem. > > > How do you include the infinite in the phenomenology when you only have > finite objects in ontology? I imagine that we cannot really experience or > perceive the infinite, but we may infer it inductively from finite objects. > And this inductive inference may perhaps give us a kind of feeling or sense > of "the infinite", but it would be a feeling from the inferential process > rather than from the infinite itself. If we are machine at some substitution level, we are duplicable at that level, and this leads to a first person duplication, which itself leads to a reduction of physics to a probability/credibility calculus on all computations going trough our actual relative state. There are infinitely many computations doing that, in arithmetic, so that the first person indeterminacy domain is infinite. Machine are “locally finite” as seen through their histories. > > Anyway, I should tell you that I am no mathematician, so I am afraid I can't > digest the technicalities in your papers. I am not even a physicist or a > professional philosopher, I just dabble in philosophy. My meta-goal consists in showing that with suitable hypothesis, we can proceed with the scientific attitude, in theology/metaphysics/philosophy. Unfortunately, this is not yet well seen in may places. There is no problem with musing and even with metaphysical poetry. Some poet can even be more rdigoruous than philosopher, like with Borges, Valery and Galouye. What cannot be done is to use a personal conviction against a logical argument. That cannot be done with all hypothesis, but with the ChurcTuring thesis, the notion of computation becomes a precise mathematical notion, and we can, by thought experiment and mathematical proof proceed and get results, some of which can lead to testable consequences. > > In my ontological musings, I try to get to the bottom of what is necessary > and avoid arbitrary assumptions. That is the best attitude. > First, what is existence? All definitions of existence should follow the > principle of logical consistency, or in other words, the principle of > identity: an object (that which exists) should be identical to itself. x = x I agree. > It should be what it is and not be what it is not. This also means that the > object should be defined consistently in relation to everything else, > otherwise its identity would be violated. I know there are people who believe > in the existence of inconsistently defined objects (dialetheists), but that > seems like craziness to me, sorry. I rather follow you on this. Paraconsistent logic are interesting for the natural language, and the psychology of lies, but it would be insane to use it in metaphysics, unless a very good argument is provided. > Moreover, unless you arbitrarily block logical explosion, such an > inconsistency would render all ontology meaningless, erasing even the > difference between existence and non-existence. OK. > > So, logical consistency is a necessary criterion of existence. Hmm… I agree that theory of the fundamental things that we will assume has better to be consistent, but that is a meta-assumption. Logical consistency is an attribute of some being, it assumes already some ontology. With the mechanist assumption, we will assume only the numbers, or the combinators, or any basic Turing complete/universal system. Then some machine will be consistent, but some machine can be inconsistent also. Arithmetic if full of inconsistent machines, which eventually get trivial. > Is any other criterion necessary? I don't think so. Adding any other > criterion seems like an arbitrary restriction on what exists. If an object is > identical to itself, then it is something rather than nothing and so it is > there in some
Re: The hard problem of matter
I already published this elsewhere. On Tue, Oct 23, 2018 at 9:22 PM Brent Meeker wrote: > > > On 10/23/2018 9:23 AM, Bruno Marchal wrote: > >> On 23 Oct 2018, at 04:30, Brent Meeker wrote: > >> > >> > >> > >> On 10/22/2018 6:54 AM, Bruno Marchal wrote: > >>> The mathematical reality has noting to do with languages, except that > languages are needed if machine/people want to share the results of their > exploration. > >> So how do you prove theorems without a language? > > Of course, proving a theorem requires a theory, and a language. I was > saying (see the quote) that the *arithmetical reality* does not require a > language. > > > > The arithmetical reality does not require a language more than dinosaurs > needed the word “dinosaur” to exist. The prime character of 17 does not > need a mathematician to assert it, or to think about. > > > > To prove a theorem requires a theory, which requires a language. We can > only ope that our theory is in relation with truth, but the truth of 17 is > prime, assuming it true, does not need a proof to be true. A proof is > neither necessary, nor sufficient. The arithmetical reality is independent > of the big-bang. It is more plausible than an event like the big-bang > requires some part of the arithmetical reality. > > But you are basing our shared reality in what is provable, which is > therefore dependent on having language. Right? > > Brent > > -- > 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 everything-list+unsubscr...@googlegroups.com. > To post to this group, send email to everything-list@googlegroups.com. > Visit this group at https://groups.google.com/group/everything-list. > For more options, visit 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 everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
Re: The hard problem of matter
Consciousness is IDENTITY like a fingerprint or voice print. Every consciousness is unique but each needs something to be conscious OF like a human body or perhaps a larger (virtual?) program (hologram?) It can be stored on a database. That's as far as I've got. On Tue, Oct 23, 2018 at 9:22 PM Brent Meeker wrote: > > > On 10/23/2018 9:23 AM, Bruno Marchal wrote: > >> On 23 Oct 2018, at 04:30, Brent Meeker wrote: > >> > >> > >> > >> On 10/22/2018 6:54 AM, Bruno Marchal wrote: > >>> The mathematical reality has noting to do with languages, except that > languages are needed if machine/people want to share the results of their > exploration. > >> So how do you prove theorems without a language? > > Of course, proving a theorem requires a theory, and a language. I was > saying (see the quote) that the *arithmetical reality* does not require a > language. > > > > The arithmetical reality does not require a language more than dinosaurs > needed the word “dinosaur” to exist. The prime character of 17 does not > need a mathematician to assert it, or to think about. > > > > To prove a theorem requires a theory, which requires a language. We can > only ope that our theory is in relation with truth, but the truth of 17 is > prime, assuming it true, does not need a proof to be true. A proof is > neither necessary, nor sufficient. The arithmetical reality is independent > of the big-bang. It is more plausible than an event like the big-bang > requires some part of the arithmetical reality. > > But you are basing our shared reality in what is provable, which is > therefore dependent on having language. Right? > > Brent > > -- > 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 everything-list+unsubscr...@googlegroups.com. > To post to this group, send email to everything-list@googlegroups.com. > Visit this group at https://groups.google.com/group/everything-list. > For more options, visit 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 everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
