The universal numbers. From Biology to Physics.

Progress in Biophysics and Molecular Biology xxx (2015) 1e14

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The universal numbers. From Biology to Physics Bruno Marchal IRIDIA, Universit e Libre de Bruxelles, Belgium

a r t i c l e i n f o

r es u m e

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I will explain how the mathematicians have discovered the universal numbers, or abstract computer, and I will explain some abstract biology, mainly self-reproduction and embryogenesis. Then I will explain how and why, and in which sense, some of those numbers can dream and why their dreams can glue together and must, when we assume computationalism in cognitive science, generate a phenomenological physics, as part of a larger phenomenological theology (in the sense of the greek theologians). The title should have been “From Biology to Physics, through the Phenomenological Theology of the Universal Numbers”, if that was not too long for a title. The theology will consist mainly, like in some (neo)platonist greek-indian-chinese tradition, in the truth about numbers' relative relations, with each others, and with themselves. The main difference between Aristotle and Plato is that Aristotle (especially in its common and modern christian interpretation) makes reality WYSIWYG (What you see is what you get: reality is what we observe, measure, i.e. the natural material physical science) where for Plato and the (rational) mystics, what we see might be only the shadow or the border of something else, which might be non physical (mathematical, arithmetical, theological, …). € del, we know that Truth, even just the Arithmetical Truth, is vastly bigger than what the Since Go machine can rationally justify. Yet, with Church's thesis, and the mechanizability of the diagonalizations involved, machines can apprehend this and can justify their limitations, and get some sense of what might be true beyond what they can prove or justify rationally. Indeed, the incompleteness phenomenon introduces a gap between what is provable by some machine € del saw already in 1931, the existence of that gap is and what is true about that machine, and, as Go accessible to the machine itself, once it is has enough provability abilities. Incompleteness separates truth and provable, and machines can justify this in some way. More importantly incompleteness entails the distinction between many intensional variants of € del's provability. For example, the absence of reflexion (beweisbar(£A·) / A with beweisbar being Go provability predicate) makes it impossible for the machine's provability to obey the axioms usually taken for a theory of knowledge. The most important consequence of this in the machine's possible phenomenology is that it provides sense, indeed arithmetical sense, to intensional variants of provability, like the logics of provability-andtruth, which at the propositional level can be mirrored by the logic of provable-and-true statements (beweisbar(£A·) ∧ A). It is incompleteness which makes this logic different from the logic of provability. Other variants, like provable-and-consistent, or provable-and-consistent-and-true, appears in the same way, and inherits the incompleteness splitting, unlike beweisbar(£A·) ∧ A. I will recall thought experience which motivates the use of those intensional variants to associate a knower and an observer in some canonical way to the machines or the numbers. We will in this way get an abstract and phenomenological theology of a machine M through the true logics of their true self-referential abilities (even if not provable, or knowable, by the machine itself), in those different intensional senses. Cognitive science and theoretical physics motivate the study of those logics with the arithmetical interpretation of the atomic sentences restricted to the “verifiable” (S1) sentences, which is the way to study the theology of the computationalist machine. This provides a logic of the observable, as expected by the Universal Dovetailer Argument, which will be recalled briefly, and which can lead to a comparison of the machine's logic of physics with the empirical logic of the physicists (like quantum logic). This leads also to a series of open problems. © 2015 Elsevier Ltd. All rights reserved.

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Please cite this article in press as: Marchal, B., The universal numbers. From Biology to Physics, Progress in Biophysics and Molecular Biology (2015), http://dx.doi.org/10.1016/j.pbiomolbio.2015.06.013

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B. Marchal / Progress in Biophysics and Molecular Biology xxx (2015) 1e14

1. The discovery of the universal numbers It all begun with Cantor set theory. Galilee and Gauss were already aware that the function which sends each non negative integers n on 2n was a bijection, that is a oneeone correspondence, although they did not use this terminology. They saw that infinite sets, like N, can be put in such a bijective correspondence with a subset of themselves, and concluded that we should better avoid to make such infinite set into actual infinite mathematical objects. This will remain so until Cantor, and it is not without courage that Cantor will reconsider this question and accept such infinite sets as legal citizen of the mathematical inquiry. Using a naive notion of set, Cantor will show the existence of bijection between the sets N (natural numbers, non negative integers), and Z (integers), and Q (rational numbers), and will discover that not all infinite set can be put in that bijective 1 1 correspondence. Indeed Cantor is famous for proving that R, the set of real numbers, or (equivalently) the set of infinite binary sequences, or the set of functions from N to {0,1} or from N to N, are not enumerable, where enumerable means that there is a bijection, or a surjection (if we allow repetition) from N to that set. This proof plays an important role in the mathematical discovery of computers, or universal numbers, as I will briefly illustrate. Cantor theorem: There is no bijection between N and the set of functions from N to N. Proof. Proof Let us suppose that there is a bijection from N to the set of functions from N to N. Let us denote by fi the function which is the image of i by that bijection. Then we can introduce the diagonal function g which sends n to fn(n) þ 1, that is, g(n) ¼ fn(n) þ 1. That is what I will call the first diagonalization act. g is obviously a function from N to N. Could g belong to the list f0,f1,f2,…? Well, if it could there would be some fk, such that g ¼ fk. In that case, by definition of the equality of function, we have that fk(x) ¼ g(x) for all x, and in particular fk(k) ¼ g(k). Applying fk on itself, or on some description of itself (which here will always be represented by some index number), is what I will call the second diagonalization act. Now, g(k) is equal to fk(k) þ 1, by definition of g. So we have, by Leibniz identity rule, that fk(k) ¼ fk(k) þ 1. As each fk is supposed to be a well-defined function, fk(k) is a number, and by subtracting it on both side of the last equation, we get 0 ¼ 1. Thus we can conclude that such a bijection cannot exist. Now, Cantor will generalize this procedure and will obtain many more similar results in set theory. Yet, difficulties will appear, like Galilee and Gauss did warn us. In fact defining a set by some properties leads to paradoxes, the most known being Russell's paradox. Let us write XY for the statement that X belongs to Y, then let us define the set E of all sets which does not belong to themselves: XE iff : XX, then we get EE iff : EE. In front of such paradoxes, there will be mainly three reactions by the mathematicians. One will be the impetus to formalize the notions, in a way which avoids the paradoxes. This will lead to a variety of set theories (ZermeloeFraenkel, Quine New Foundations, €del, etc). A second reaction will be more Von Neumann Bernays Go radical, and will throw away the use of the excluded middle principle (Brouwer's Intuitionism), and the last one, which can be related in many ways to the preceding one, will be an attempt to work on sets which are not too much large, and in particular to try, when possible, to restrict oneself to computable or constructive notions. This will lead many mathematicians to define what is a computable function, which is one step toward the discovery of the universal machine, or universal number. So what could be a computable function? The intuitive idea is that a function from N to N is computable when we can describe in a finite time, with a non ambiguous

language L, how to compute it, in a finite time, on each (finite) input. Such description are called algorithm or procedure. But is there a universal language capable to describe all computable functions from N to N? When Alonzo Church claimed that his “Lambda Calculus” provides such a universal language, his student Kleene was at first quite skeptical, and he tried to refute that claim by using Cantor's diagonal procedure (which enabled € del to show that there is no universal provability system). already Go Indeed, thought Kleene, if a universal language L (universal with respect of defining the notion of computable function) exists, then we know that such a set of computable functions has to be enumerable. The reason is that the finite descriptions of the procedures can be listed. Indeed, they are non ambiguously described, and thus the description have a simple checkable grammar, and so we can order them by lengthdand for those having the same length, we can sub-order them by alphabetical order, assuming some primitive order on the (finite) alphabet of the language. So if a universal language L exists, we would have an enumeration 40,41,42,… of all computable functions from N to N. All right, but then we can define (again) a function g such that g(n) ¼ 4n(n) þ 1. All 4n are computable, and “adding one” is without doubt computable. So g should be computable, and should admit a procedure and thus some description in the supposedly universal language L. This means that there is a k such that g ¼ 4k, and thus, in particular g(k) ¼ 4k(k), and again 4k(k) ¼ 4k(k) þ 1. All 4i are computable, so 4k(k) is a number that we can again subtract from both sides of the last equation, and get 0 ¼ 1. Now, it looks we are in trouble. We have certainly not prove that the set of computable functions is not enumerable, as the set of all strings in the alphabet, and the subset of the grammatically correct strings (describing procedures) are clearly enumerable by the argument given above. So it looks like Kleene has simply refuted Church's claim that his language, or any language, can describe all computable functions. But looking more closely, Kleene will understand that he has not done that. Kleene's proof just proves that there is no universal language L computing all and ONLY all computable functions from N to N. In particular, if L is built in such a way that all procedures compute functions from N to N, then indeed L is not universal. But, this shows that IF a universal language exists, then it must also computes other things. I guess Kleene already knew that the lambda-calculus expression(lx.xx)(lx.xx) gives a non terminating procedure, and those “other things” will be of that type. This saves the consistency of Church's claim: the apparent paradox in 4k(k) ¼ 4k(k) þ 1 does not obtained, because the computation of 4k(k) will just not terminate. In the computer's jargon, 4k(k) crashes the machine. Kleene said that overnight, after having been skeptical, he will become an ardent defender of Church's thesis. Indeed, he gave the most conceptual and profound argument in favor of the Church's thesis: the closure of the set of partial computable functions for Cantor's transcendental diagonalization procedure, where a partial computable function is now a function from a subset (perhaps equal to N, or empty) to N. We will say that a function from N to N is total if it is defined on all numbers, and we will use the term partial function if its domain is a subset of N. Partial functions generalize the notion of function, usually considered total on their domain. Cantor showed that the set of functions from N to N is not enumerable, Kleene did show that the subset of total computable functions, although enumerable, is not computably or recursively enumerable. If this seems a bit weird, keep in mind that although a subset cannot be bigger than the set in which it is a subset, it can be more complex, like the painting of the Mona Lisa is more complex than the paper area on which it is painted, or like the Mandelbrot set, a subset of the complex plane C, looks much more complex (apology for the pun) than C.



From that simple reasoning, we can easily derive, from Church's thesis, a general incompleteness theorem (Kleene, 1952, see also (Webb, 1980), (Kleene, 1987)): Kleene's theorem: If L is universal language with respect to computability, there is no effective complete theory capable of deciding if a number is the index of a total or partial function in the corresponding enumeration 4i. Proof. Indeed, if such a theory was possible, it could be used to enumerate all total functions, by testing the partialness, using that theory, on each index numbers 0, 1, 2, … By eliminating all codes/ index for the strictly partial (non total) computable functions, we would arrive at a computable enumeration of the total functions, and Kleene's diagonalization would be enacted again, leading to 0 ¼ 1. QED. The fact that a strong form of incompleteness is an easy consequence of Church's thesis is not well known. It has been exploited by Judson Webb to show that the incompleteness theo€ del somehow protects Church's thesis and the mechanist rem of Go philosophy. Indeed would incompleteness be false, there would be no universal language, and no universal machine (computer), as we will see below (Webb et al., 1983; Kleene, 1987, 1952). Emil Post, Alan Turing, Andrey Markov Jr. will independently submit similar, and indeed provably equivalent theses, with very different formalisms, and that provides the empirical argument in €del will miss it, and will be convinced of favor of Church's thesis. Go it only after reading Turing's paper (Davis, 1982). Church himself seems to have believed that it was a definition. The fact that Church's thesis entails a strong form of incompleteness shows that it is certainly not a definition, as it would make incompleteness a quasi-trivial (by the argument above) fact, which it is not. Of course, once we admit Church thesis, then we can indeed define the notion of (intuitively) computable function by the mathematical definition provided by some precise formal universal language proved equivalent to Church or Turing formalism. From now on, we will assume the existence of a universal language L, and 4i will denote its corresponding enumeration of partial (i.e. total or strictly partial) computable functions. The discovery. Turing will notice that among the 4i, one will compute a universal computable function:

du cx; y

4u ðx; yÞ ¼ 4x ðyÞ

Here (x,y) means either two inputs for a two variable computable function, or the coding into one number of those two inputs. It does not matter too much in what will follow. u is a finite thing, and is the index of a code, written in the formalism of L, or of Turing machine, or in any another formalism, of the (Turing) universal machine u. That machine/program/number u has the instructions making it able to understand the universal language. It can enumerate the 4i and apply them on the input given. x is called the program, and y is called the data, given to the computer u. It makes the intuitive definition of computable function given above more precise by stipulating who or what is supposed to understand the non ambiguous descriptions of the procedures. I use often the expression “universal number” in the place of “universal machine” to prevent the common belief that a universal machine needs to be infinite. The infinite tape of the Turing machine model might give that impression, but it is better to put the infinite tape in the environment, and to let the universal machines using the wall of the cavern in case they need more memory. The universal number is really the code of a universal program, capable of emulating all programs, as long as they can expand the memory space when they need to do so. I define a universal number, with respect to some universal enumeration, to be a number satisfying

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the relation described above: 4u(x,y) ¼ 4x(y). We can identify, on the syntactical level programs with machines, or with finitely describable entities or numbers, and I will say that u is a universal number, or universal using number instead of machine emphasize that we talk about a finitely describable entity. The notion of universality of a number is not intrinsic (like being prime) but is relational, and intensional (referring to form), yet mathematically well defined. Church's thesis is equivalent with Turing's thesis, and allows us to say “universal machine” instead or Turing universal machine. Many theorems characterize those notions and make them solid through some notions of Turing equivalence. Rogers found a neat characterization of a large class of such systems, that he called acceptable enumeration, and which are defined by sequences 4i of all partial computable functions verifying the existence of a universal number, as given by the equation just above, together with a parametrization theorem, known as the SMN theorem, for which I give the S21 presentation here:

ds ci cx; y

4i ðx; yÞ ¼ 44s ði;xÞ ðyÞ

Here s is the code or index, in the enumeration of the partial computable function, of a computable function which simply fix an argument in a code of a function. Its input is a code of a function, with two arguments, say lxly$x þ y, and a number, say 666, and it transforms the given code by fixing one variable with the number given to get the code/number for ly(666 þ y). It is really only a functional substitution, of a variable by an input, without further evaluation. It is easy, once we know a universal programming language, to write a program to that effect, and here the number s represents it in our unspecified universal language L. Similarly S31 would give an s capable of fixing two arguments in a function of three arguments, obtaining a function of one argument. In fact, the biology below applies to any enumeration verifying that second law, and the biology does not need the universal machine existence. This shows also that self-reproduction and the possibility of embryogenesis does not entail Turing universality per se. Many refinement of such results exists in the literature, and a good book is Royer's book (Royer, 1987), see also the treatise by Odifreddi (Odifreddi, 1989). 2. Biology The basic idea has appeared here with Russell paradox, It is the shape of many fake, and less fake paradoxes. A fake paradox is the barber paradox. In some village there is a shaver who shaves all and only all man who does not shave themselves. So Z shaves X iff X does not shave X. Then Z shaves Z iff Z does not shave Z. There is no paradox, here, as Z can be a woman, or if it is a man, it means that it does not exist, and there are no such village. With the notion of set, the paradox was less fake, as we do have the intuition that we can collect things by their properties, and this has led to many interesting developments, including the discovery of the universal machine u. The very shape of that paradox suggests an easy answer: indeed it looks like a song: if DX gives XX, what gives DD? Intuitively DD gives DD, which gives DD, which gives DD, … We might again have crashed the computer! Although the process above has its interest and importance, it remains that for a conceptually clear solution of Descartes selfreproduction problem, we need a stopping procedure: a machine which gives its own description/body and stop. This can be obtained by using a quoting procedure: if D£X· gives £X£X··, and if X still admits to be substituted in that quotation, then it will work: D£D· will give and stop on £D£D··.


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We can similarly build a machine capable of computing any transformation of itself. It is enough to build a D such that D£X· gives T(£X£X··), and such D£D· will give T(£D£D··).

A generalization of Kleene's recursion theorem, obtained by John Case (Case, 1971, 1974), solves that problem, for all sort of computable transformation t. Case theorem:

2.1. Self-reproduction/transformation

ct de cy; z

Let us do this in the setting of an acceptable universal enumeration. This is a version of the second recursion theorem of Kleene. It asserts that for all computable transformation 4t, it exists a number which computes that transformation, with index or code t, on itself. For example, we get self-reproduction by using the identity transformation id:4e() ¼ id(e) ¼ e. Kleene's second recursion theorem:

Proof. Let us use the SMN theorem, actually the S31 on 4t(x,y,z), so

ct de

4e ðyÞ ¼ 4t ðe; yÞ

44e ðyÞ ðzÞ ¼ 4t ðe; y; zÞ

4t ðx; y; zÞ ¼ 44s ðt;x;yÞ ðzÞ

By Church's thesis, there is a number g which computes 4s(t,x,y),

4g ðx; yÞ ¼ 4s ðt; x; yÞ so, by substituting this in the S31 relation, we get

Proof. To avoid the lambda notation, I will assume that x and y denote the variables. I will use freely Church's thesis, which implies that if some function f is computable, it exists a number r coding it, i.e. 4r ¼ f. Let us do the first diagonalization act, and consider the function 4t(4s(x,x),y). That function is computable, so by CT it exists a number r which computes it:

4r ðx; yÞ ¼ 4t ð4s ðx; xÞ; yÞ

4t ðx; y; zÞ ¼ 44g ðx;yÞ ðzÞ By the second recursion theorem on g, there is number e such that

4e ðyÞ ¼ 4g ðe; yÞ By using this relation, and substituting x by e in the relation above, we get

now, by the SMN law, indeed S21, we have

4t ðe; y; zÞ ¼ 44g ðe;yÞ ðzÞ ¼ 44e ðyÞ ðzÞ

4r ðx; yÞ ¼ 44s ðr;xÞ ðyÞ

So, to solve the problem, we need only to apply Case's theorem with 4t(x,y,z) defined by the following program

The second act of the diagonalization will consist in substituting x by the code r, and using Leibniz identity rule:

tðx; 1; ‘FOOÞ ¼ ð4x ð1Þ; 4x ð2ÞÞ

44s ðr;rÞ ðyÞ ¼ 4t ð4s ðr; rÞ; yÞ

tðx; 2; ‘FOOÞ ¼ ð4x ð1Þ; 4x ð2ÞÞ

So we get a solution:

tðx; 1; zÞ ¼ z þ 1

if z s ‘FOO

tðx; 2; zÞ ¼ z þ 2

if z s ‘FOO

e ¼ 4s ðr; rÞ

2.2. Abstract embryogenesis Embryogenesis can be seen as a particular case of a selfregeneration of a tissue, in which each cells can act as an egg, in the worst case of no adjacent cells being present, or partial embryo. This needs a more subtle nested recursion. What we want is a selfreferential collection of cells/programs/numbers 4e(y), or, more intensionally said, a self-referential meta-program 44e . For example, a simple two cells planaria (C1,C2) might be given by the cells 4e(1) and 4e(2), and e would be such that 4e(1), as a program, (e is a meta-program), would give (C1,C2) on some input ‘FOO, and do its normal cell job on other arguments. To illustre, I give some extensional “normal” roles to the cells: C1 ¼ lx$x þ 1, and C2 ¼ lx$x þ 2. So we want

44e ð1Þ ðyÞ ¼ ‘if z ¼ ‘FOO then output ð4e ð1Þ; 4e ð2ÞÞ else ouput z þ 1 and

44e ð2Þ ðyÞ ¼ ‘if z ¼ ‘FOO then outputð4e ð1Þ; 4e ð2ÞÞ else ouput z þ 2

Each cells has its own extensional role, in this case simple arithmetical tasks, and at the same time each individual cell incarnates the ability to regenerate the whole assembly of the two cells. See my long text (Marchal, 1994) to see illustration in LISP, with finite and infinite planaria (Marchal et al., 1992). Kleene's recursion theorem solves Descartes reproduction problem, and Case theorem refutes Driesch argument that embryogenesis cannot be a computable procedure. 3. The Universal Dovetailer Argument: physics is a branch of the theology of numbers The Universal Dovetailer Argument (UDA) is a simple argument for reminding us that we have not yet solved the mind-body problem. We will assume the computationalist thesis, in a very weak form, which makes possible to reformulate that problem in a precise way. It leads already to the general shape of the solution, in case it exists, which belongs to the platonician or platonist sort of solution. Indeed the UDA reduces the mind-body problem to a reduction of the physical science to a problem in machine's theology, which is both a branch of computer science and metaarithmetic, and in fact of arithmetic given that the relevant part of meta-arithmetic can be embedded in arithmetic in the usual €delian) way. I will just sketch the argument here and traditional (Go I refer the interested reader to my older publications for all details



(Marchal, 1988; Marchal et al., 1991; Marchal, 1994; Marchal, 1998; Marchal, 2004; Marchal, 2013). The computationalist hypothesis in cognitive science is the assumption that it exists a level of functional description of the brain, or body, with perhaps some part of the environment (what I call the generalized brain) such that my first person experience, or my current consciousness, would not see any difference by introspection in the case that generalized brain is replaced by a computing machine emulating the functioning at that level. To ease the reasoning, I assume also that the level of description is high and concerned only the biological brain, but it should be clear later, that this is not important to get the conclusion. That definition might aptly constitute the step 0 of the Argument. I sum it up by “Yes, Doctor”. A computationalist practitioner is someone agreeing to a doctor who proposed him an artificial brain made at the right level, and who is assumed to have enough expertise to do so. I have converged to a presentation of UDA in eight steps, which can also be put in the form of eight questions, where I will use freely the computer jargon, like cut, paste, etc. In step 7, I use the Universal Dovetailer (UD). Let 4zx ðyÞ denote the first zth computational steps of the computation of 4x(y). Then a Universal Dovetailer, which existence is assured by the existence of a universal number, and thus by Church's thesis, is given by the simple procedure described below, with x, y, and z representing natural numbers: FOR ALLx,y,z non negative integers: - compute the z first steps of the computation of 4x(y), that is compute 4zx ðyÞ. END. The UDA procedes then with the following eight questions: 1. Do you accept, assuming the computationalist hypothesis, the use of teleportation as a mean of locomotion. You are cut in one place, and pasted in another place. 2. Do you think you might be aware of a delay in the reconstitution (pasting) when not given any external clues? 3. What if you are pasted in two places at the same time? Do you agree that before this happens, knowing the protocol, you are unable to predict where you will feel to be after the double pasting? That is the computationalist First Person Indeterminacy (FPI). 4. Do you think that such an indeterminacy would be different if the pasting delay is bigger on one of the branch? 5. What if you are read only, and pasted in two places? Do you agree that this is equivalent with a triplication? 6. What if you are “pasted” in a virtual environment, instead of “real” one. You are “re-implemented” in a computer. Would that change any answer to the preceding questions, asked in this new virtual setting? 7. Assuming there is a (never ending) execution of a UD in the physical universe, do you see that your next first person experiences are determined by a statistics on all computations, made by that UD, going through your actual current states? 8. Do you think that assuming the existence of a small primitively real physical universe would help to avoid the consequences of the step 7, with “small” meaning that the physical universe cannot run any significant portion of UD* (the UD's computation)? At step seven, the reader is supposed to understand that if computationalism is true, and if we are in a physical universe running a UD, then the physical science are reduced to a statistics on infinitely many finite pieces of computations, realized in

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arithmetic, and seen from some first person points of view. This leads to a serious white rabbits problem, that is a possible inflation of predictions, with too many aberrant hallucinations. We will see in the next sections how the incompleteness phenomenon offers freely those phenomenological first person points of view, and how that might hunt the white rabbits away. Those finite pieces of computations can be proved to exist in arithmetic, indeed even without the induction axioms (see next section). The step 8 is more subtle, and is addressed to the person who reifies the physical existence and believe that a Turing machine has to be emulated in the physical universe (that he assumes to be primitively existing, part of the ontology) to be able to support consciousness. Up to the seventh step, such person can still save the epistemological consistency of that idea by invoking the existence of a small primitive physical universe, which would be incapable to run any statistically significant part of the UD. After step 8, we can grasp that this does not solve the white rabbit problem, or the measure problem, as the step 8, namely the Movie Graph Argument, implies that we are confronted with the non material dovetailing implicit in arithmetic or in any first order specification of any Turing complete theory. An argument can be used to show that those invoking an ontological small universe need some magic to do that in a way preserving any reasonable meaning to the computationalist hypothesis. That follows from the Movie Graph Argument (MGA) or Paradox (Marchal, 1988; Marchal, 2013), see also Maudlin (Maudlin, 1989) for a similar argument. For a more detailed treatment of the UDA, see (Marchal, 1994; Marchal, 1998). A detailed treatment of the UDA including the MGA, in the frame of Integral Biomathics (Simeonov et al., 2012) can be found in (Marchal, 2013). Somehow, a universal Turing machine, even physical, cannot distinguish between an arithmetical realization of a computation from a physical one, except, and that is the point, if given external clues. The quantum empirical clues happens to be serious hints that the physical emerges from an internally defined statistics on the numbers dreams or computations seen from inside. Too much White Rabbits? This generalizes the formulation of Everett of Quantum mechanics, which literally describes some universal quantum dovetailing. Quantum Mechanics seems to eliminate the white rabbits by a quantum randomization of the phases, as can be seen in Feynman formulation of quantum mechanics. Everett embedded the physicist into the physical reality, which is € ssler and CastiKarlqvist, 1987) for a deep very natural (see (Ro explanation of this fact), but here we see that the embedding has to be extended to some embedding of the mathematicians in some part of the arithmetical reality, the one which is enough rich to emulate all computations; as explained below. The UDA shows that to predict any first person assessment of any physical experience, that person needs to compare all computations going through his current state (before the experience is done), at a level lower than its substitution level, in arithmetic, to make the right statistics. From this many qualitative features of quantum mechanics are easy to explain, like the indeterminacy and the non-cloning. The problem is that arithmetic might contained to much aberrant dreams, so that computationalism might suffer from an inflation of weird predictions. The next section will shows that it might be hard to refute computationalism in this way, as the math suggests that we might been led to a statistics on computations similar to the quantum one. The quantum statistics might not be much different from the one we have to abstract and derive from UDA and the computationalist arithmetic, if we limit the ontology to the finite numbers or machine. In fact computationalism might explain the quantum nature of our most probable histories, at least, if like Everett and Feynman, we interpret the collapse of the wave function phenomenologically.


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€ bian number 4. From computability to provability: the Lo Here is an interface with the next section which comes from a non published older paper, and which describes the mathematics of the phenomenological theology and of the associated phenomenological physics of the “average universal number”, in the way prescribed by the UDA argument, and using the same selfreferential tools that those used in the abstract biology. The difference between self-reproduction and self-reference is only in the universal number that we are addressing.

4.1. Ontology: the basic assumptions or beliefs that we have to start from I give three examples of possible, and phenomenologically equivalent acceptable ontological theories. This means they lead to the same theology, and thus the same physics. Robinson Arithmetic: The following axioms are supposed to be added on the top of some presentation of classical predicate calculus (see (Kleene, 1952)) with identity. The inference rules is the modus ponens. The non logical symbols are þ, , s (for the successor of a number) and the constant 0.

0ssðxÞ sðxÞ ¼ sðyÞ/x ¼ y x ¼ 0∨dyðx ¼ sðyÞÞ xþ0¼x x þ sðyÞ ¼ sðx þ yÞ x0¼0 x sðyÞ ¼ ðx yÞ þ x SK-combinators: But, as long as the theory concerns only the ontology, we might chose any Turing complete theory, and some does not necessitate classical logic, although we still need some logic to define the reasoners, the subjects, the observers and the corresponding phenomenologies, from inside the ontology. An example of very simple ontology is provided by the theory of combinators (Barendregt, 1984; CurryFeys, 1958). A combinator is either K or S, or a combination (X Y) of combinators. So ((K$K)$((S$S)$K)) is a combinator for example. To ease the readability we don't write any left parenthesis, so the example given will be written KK(SSK). The theory is then given by the following axioms and rules:

AXIOMS :

Kxy ¼ x Sxyz ¼ xzðyzÞ

RULES :

ðx ¼ y∧x ¼ zÞ/y ¼ z x¼y xz ¼ yz x¼y zx ¼ zy

Such a theory is Turing universal. If we use K for representing a constant logical truth, and K(SKK) for a constant logical false, we get an elegant “if then else” structure by just appending three combinators ABC, with A supposingly given K or K(SKK) has truth and false value, as the reader can justify. Note that SKK is an identify combinator: SKKx ¼ x. See Smullyan's recreative book (Smullyan, 1985) for a gentle introduction to the combinators and their Turing universality as well as a recasting of incompleteness in that frame. Note that each acceptable enumeration makes N into a “concrete model” or what is called a combinatory algebra. Just define a new partial operation + on N such that x + y x 4x(y).

“x” means either both sides are undefined or they are both defined and equal. The algebra is said to be partial, as 4x(y) might be undefined (the price of universality). That 4x(y) defines a partial combinatory algebra comes from the fact that there are numbers k and s such that 4k(x,y) x x, and 4s ðx; y; zÞx44x ðzÞ 4y ðzÞ. The numbers k and s play the role of the combinator K and S. A universal Diophantine equations. Through the intense and long work made by Hillary Putnam, Martin Davis, Julia Robinson, Yuri Matiyasevich and James Jones, the known Turing universal relations x 2 Wy can be expressed by a unique diophantine polynomial equation with degree 4. James Jones provided a shorter system of diophantine relations, with a much higher degree, though, indeed the degree of one of the polynomial is 560, which is nice for not being to long to describe. The unknowns range on the non negative integers. There are 31 unknowns: A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,W,Z,U,Y,Al,Ga,Et,Th,La,Ta,Ph, and two parameters: Nu and X. See Matiyasevich book for more on this (Matiyasevich, 1993). Wi is defined as being the domain of 4i. The Wi can be proved to enumerated the recursively enumerable set and can be used as an alternative of the computable functions. It can be proved that X 2 WNu, or, put in another way, 4Nu(X) stops, if and only if

2 Nu ¼ ðZUYÞ2 þ U þ Y ELG2 þ Al ¼ ðB XYÞQ 2 60 Qu ¼ Bð5 Þ

La þ Qu4 ¼ 1 þ LaB5 Th þ 2Z ¼ B5 L ¼ U þ TTh E ¼ Y þ MTh N ¼ Q 16 4 R ¼ G þ EQ 3 þ LQ 5 þ 2ðE ZLaÞ 1 þ XB5 þ G þ LaB5 h þ þLaB5 Q 4 Q 4 N 2 N þ Q 3 BL þ L þ ThLaQ 3 i þ B5 2 Q 5 N 2 1 P ¼ 2W S2 R2 N 2 P 2 K 2 K 2 þ 1 ¼ Ta2 2 4 c KSN2 þ Et ¼ K 2 K ¼ R þ 1 þ HP H



A ¼ WN2 þ 1 RSN 2 C ¼ 2R þ 1 þ Ph D ¼ BW þ CA 2C þ 4AGa 5Ga D2 ¼ A2 1 C 2 þ 1 F 2 ¼ A2 1 I 2 C 4 þ 1 ðD þ OFÞ2 ¼

2 A þ F 2 D 2 A2 1 ð2R þ 1 þ JCÞ2 þ 1

This provides yet another purely equational theory, using the most elementary part of the integers theory: the diophantine polynomials. The positive solutions and their systematic search provide another example of universal dovetailing. Each such theories gives all 4i and their domain Wi. They provide particular, and non universal provability predicate (none are, with respect to provability) which are universal with respect to computatibility. Of course, we could have used any first order specification of a universal programming language for the ontology, but having simple and natural ontologies help to accept that the computable truth are independent of ourselves. It is natural to believe that the existence of primes is independent of the existence of humans, but that a Fortran or Lisp program behavior exists and does not depend on us seems less natural for many, as they conceive Lisp and Fortran as being purely human invention. €bian reasoners 4.2. The Lo We have the ontology: numbers, or combinators, obeying some (equational or not) laws. Such theories are very poor, even if Turing universal. For example it can easily be proven that Robinson Arithmetic cannot even prove that 0 þ x ¼ x. Nevertheless all the theories above can prove the existence of reasoners, having much more provability abilities. We will take Peano Arithmetic, and their recursively enumerable extensions, as typical reasoners. They believe in the RA axioms, but they also believe in the infinitely many induction axioms, with F denoting first-order arithmetical formula:

7

€bianity is defined by the provability of that completeness Lo €bian number, for each S1-sentence p. By formula by the Universal Lo abbreviating beweisbar by a modal box,1 we can sum up most important metamathematical theorems. They all use a formal version of the second recursion theorem of Kleene, already in € del's fundamental 1931 paper, known as the diagonal lemma. I Go refer the reader to the explanation above or to any good textbook in mathematical logic, notably Kleene's basic book (Kleene, 1952). For € del's second incompleteness theorem, is provable by example, Go € bian machine: the Lo

>T/:,>T with t being some always true arithmetical sentence, like 1 ¼1, and €b's gener> abbreviates :,:. Another important example is Lo €del's theorem: alization of Go

,ð,A/AÞ/,A €b has used to solve Henkin's problem (what can be said which Lo about an arithmetical formula asserting its own provability? Answer: it is true and provable). As explained in the next section, Solovay will prove that the propositional modal logic of provability bearing on the machine, and provable by the machine, is entirely axiomatized by the modal logic G (known also as KW4, Prl, GL, etc.). G is given by

AXIOMS :

RULES :

,ðA/BÞ/ð,A/,BÞ

K

,A/,,A

4

,ð,A/AÞ/,A

L

A ; A/B B

MP

A ,A

NEC

And Solovay will offer an unexpected gift, a modal logic G* which axiomatizes what is true about the provability by the machine, including what the machine cannot prove about that provability (like her consistency) >u. G* is given by:

AXIOMS :

Any Theorem of G ,A/A A ; A/B B

T

fFð0Þ∧cxðFðxÞ/Fðx þ 1Þg/cxFðxÞ

RULES :

Such reasoners, by extending RA, are Turing universal, but the induction axioms provide them with much more reasoning introspection powers: they can know, in some technical sense, that they are Turing universal. In fact it can be shown that a theory or a machine-prover is Turing universal iff it is S1-complete, which means that if a S1-sentence, that is, one with the shape dxP(x), with P decidable (recursive), is true, then the machine can prove it. €bian machines, or numbers, can reflect that fact: they can prove Lo p / beweisbar(£p·) for all such p S1-sentences. We can define, as € del did, the provability predicate of the Lo € bian number in the Go language of arithmetic, or of the combinators, or in term of some diophantine polynomial relations. The ontological theory is able to €bian machine can say, believe, etc. In fact, for simulate what the Lo most reasonable (effective, checkable mechanically, human verifiable (with Church's thesis)) definitions of proof, the proof predicate is itself S1, and Turing universality is given by the corresponding completeness relation, where p denotes S1-sentence:

See Smullyan's book (Smullyan, 1987) for a gentle introduction to the modal logic G. See Solovay's original paper (Solovay, 1976) for a proof that G and G* are decidable. Solovay found a nice representation theorem of G* in G, which in fact has a similar look to a version of toy computationalist philosophy (Marchal, 1994). Good textbooks on self-reference are the one by Boolos and ski, 1981; Smoryn ski, Smorynski (Boolos, 1979, 1993; Smoryn 1985).

p/beweisbarð£p·Þ

MP

4.3. Number's soul and number's matter hier, who suggests that Plotinus' I will follow a hint by Emile Bre explanation of the Soul makes it into basically a knower, which analytical philosophers, and intuition, makes it axiomatized by the modal logic known as S4:

1 see the next sections for more precision on the arithmetical interpretation of modal boxes, and more explanation on modal logics.


8


AXIOMS :

RULES :

,ðA/BÞ/ð,A/,BÞ

K

,A/,,A

4

,A/A A ; A/B B A ,A

T MP NEC

Here, “,A” has the intended interpretation “I know A”. The axiom named 4 is for a form of self-awareness: “if I know A then I know that I know A”, or, if A is knowable then it is knowable that A is knowable. The theory KT, without 4 is suited for more immediate form of knowledge or experience. € del already noticed that the arithmetical provability, which is Go mechanical, indeed S1-complete, does not obey the reflection principle T, which is the main thing for a notion of knowledge. That is sometimes abusively exploited to argue that we cannot be machine, but that is unfair, as the provability predicate only describes what a honest reasoner can say about itself, with itself given properly by us, or by the second recursion theorem. It is more the talk of a scientist about itself in the third person way. The question remains though: do the machine has a soul. It can be proved that if such a soul makes sense, it can't be described by anything in the third person way by the machine itself. The soul of the machine will not believe that she is a machine, if there is such a soul. So, if we can agree that the soul is the knower, we need a definition of knowledge. Socrates asked Theaetetus how to define what is knowledge. After confusing it with sensation, Theaetetus suggested to define it by the true justified opinion. This suggests defining the knowability of A by the provability of A conjuncted to €del and Tarski discovered, if such a predthe truth of A. But, as Go icate of truth was definable in arithmetic, the Epimenides paradox would became a frank contradiction. And, as I said, it was proved similarly that no predicate of knowledge is definable in the language of the machine (Kaplan and Montague, 1960). Yet, Theaetetus idea is still applicable, because, for each particular sentence p we can mirror “(p is provable) ∧ (p is true)” by “(p is provable) ∧ p”. See below for the precise translation, but the result is that this gives an S4 logic, describing a non nameable first person self, building its own mental and solipsist (intuitionist) space. To get the arithmetical completeness we need to add a formula, already discovered by Grzegorczyk. See below. See also (Marchal, 2012). The main point is that the incompleteness not only makes truth extending the justifiable, but it leads to giving sense to most modalities introduced by the rationalist open to mysticism, that is open to introspection and introspective experience. For the notion of matter, we need to add either truth, or consistency to transform the provability into a probability “one” on the possible extensions. We need also to restrict the accessible extensions on the computational one, UD-accessible, which here are obtained by restricting the propositional atomic formula arithmetical interpretation to the S1 sentences. The universal dovetailing can be seen as the proofs of all true S1 propositions dxyz 4x(y) ¼ z, with some sequences of such propositions mimicking the infinite failing of proving some false S1 propositions. A result by Goldblatt is used to extract some quantum logics from this approaches to physics, see more on this below. What follows is the older paper whose main goal was in formulating the open problems. It gives more precision. The conclusion section which follows is new.

5. Theology of the universal numbers. More details and the open problems I use the term “theology” in a sense closer to the original sense of Plato than the sense used in confessional religions, although there can be many relations. As Hirschberger (Hirschberger, 1987) recalls, the God of Plato is Truth, in the indefinite sense of the truth we are searching. Let M be an ideal machine having beliefs in some axioms which makes it complete for the provability of S1-sentences,2 and which is thus Turing-Universal, and thus complete with respect to computability. It is well known that such a machine's provability predicate is essentially incomplete3 with respect to arithmetical truth, which we will consider here as the set of all true arithmetical sentences, with “true” meaning “satisfied” by the standard model of arithmetic, i.e. the structure (N, 0, þ, *). Moreover, if the machine is able to prove her own S1-completeness, in the weak sense of being able to prove A / Beweisbar(£A·) for all S1-sentences A, then the machine has some ability to believe in its conditional incompletenessdthe machine can prove her own second incompleteness theorem and many of their consequences. This comes, as Judson Webb (Webb, 1980) already explained very well, €del’ from the mechanisability of the diagonalization involved in Go €del's proof, and this can be used to debunk many misuses of Go theorems against Mechanism, i.e. the thesis that machine can be conscious, or the stronger thesis that humans are Turing emulable at some level of functional description.4 On the contrary, once we take into account the mechanisability of the diagonalization, we get positive argument for both Church-Turing thesis and Mechanism. In fact, the gap between truth and provability can be used to associate to any machine a “theology” in the original greek (Platonist) sense of the word. Roughly speaking the theology of a machine is everything which is true about that machine, and the incompleteness imposes some transcendent aspect of such truth from the machine's point of view. 6. Modal preliminaries €del We suppose some knowledge about incompleteness like Go € b results (Go € del, 1931; Lo € b, 1955). See above for a summary and Lo of the main ideas and results. Modal Propositional Logic is Classical Propositional Logic (CPL) þ the unary symbol ,. All theories considered in this paper are close for the use of the modus ponens rule. Theories and proofs are defined in the usual way. The expression “>A” can be seen as an abbreviation of “:,:A”. Most modal logical theories, but not all, are closed for the necessitation rule: from A we can derive ,A. A Kripe model is a relational structure (a set of elements called “worlds”), together with an binary relation called relation of accessibility), with a valuation of the atomic sentences at each world. The world obeys classical logic. This means that CPL is verified in each world, so A∧B is true in a if both A and B are true in alpha, etc. The main non trivial clause is that a world a satisfies ,A if and only if all worlds accessible from alpha (that is: all worlds b such that aRb) satisfy A. In particular, a cul-de-sac world (accessing

2 Sentences having the shape dnP(n), or equivalent, with P being a decidable (S0) predicate. 3 “Essentially” means that not only the machine cannot prove all true sentences of arithmetic, but that none of its axiomatisable consistent extensions either will. The consistent extensions are incomplete or non-axiomatizable. 4 n, 2005), for a good description of the See the book by Torkel Franzen (Franze € del's in different domains. Despite the title of that book, there is main misuses of Go not much description of correct use of incompleteness described in the book. Torkel Franzen wrote also a very good book which illustrates other aspects of machine n, 2004). theology, notably its inexhaustibility (Franze



9

no worlds) satisfy trivially ,A for all formula A. A Kripke semantics can be given to any modal logics extending K, where K is the theory extending CPL with the axiom ,(A / B) / (,A / ,B), and close for the modus ponens and the necessitation rule. It is not very hard to verify that K wA iff A is satisfied in all Kripke models. Some key formula have weird but standard names: K is for ,(A / B) / (,A / ,B). So K is used both for the formula and the theory having that formula as axioms, and close for MP and Necessitation. That is why I use bold character for name of modal theories. T is the name of ,A / A, 4 is the name of the formula ,A / ,,A. B is the name of the formula A / ,>A. 5 is the name of >A / ,>A. C is the name of >A / :,>A. Triv is the name of A / ,A. L is the name of ,(,A / A) / ,A, D is the name of ,A / >A. Grz is the name of ,(,(A / ,A) / A) / A. Those formula participate to the formalization of the theories in which we are interested. We have the following theorems, and most proofs can be found in textbooks of modal logic or provability logic like (Chellas, 1980; Boolos, 1993). The accessibility relation is reflexive if for all a,aRa; transitive if for all a,b,g,aRb ∧ bRg / aRg, symmetric if aRb / bRa, euclidian if we have aRb ∧ aRg / bRg. A model is ideal if there is no cul-de-sac world. A model is realist if all transitory (non cul-de-sac) world access to a cul-de-sac world. We will say that a model is reflexive (resp. transitive, symmetric, etc.) if its accessibility relation is reflexive (resp. transitive, symmetric, etc.). Other abuses of language of that sort will be made to avoid unnecessary jargon.

€dels' provability predicate, where the modal box represents Go defined in the language of that machine, and A represents an arbitrary proposition, made into a sentence, in the language of €bian number as defined above. arithmetic or of any Lo This is made clear through the two important theorems by Solovay, which asserts the completeness and the soundness of the two modal logic G and G*, also known as GL and GLS. Precisely, we interpret modal logic in arithmetic, which is here supposed to be a subset of the language available to the machine. A realization r assigned to each atomic formula p,q,r,… a sentence of arithmetic. We define a translation T from modal logic to arithmetic by.

1. KD þ Triv ¼ KDTriv collapses into classical logic (the box and diamond becomes “verum” connector). 2.KwA iff A is satisfied by all Kripke models. 3.KTw A iff A is satisfied by all reflexive Kripke models. 4.K4w A iff A is satisfied by all transitive Kripke models. 5.K5w A iff A is satisfied by all euclidian Kripke models. 6.KDw A iff A is satisfied by all ideal Kripke models. 7.KCw A iff A is satisfied by all realist Kripke models. 8.KLw A iff A is satisfied by all irreflexive transitive finite Kripke models. 9.KTBw A iff A is satisfied by all reflexive and symmetric Kripke models. 10.KLw A iff A is satisfied by all irreflexive, transitive finite Kripke models. 11.KGrzw A iff A is satisfied by all reflexive, transitive and antisymmetric finite Kripke models.

Visser (Visser, 1985) proved a restricted form of Solovay theorem, where we restrict the arithmetical realization of the propositional atomic sentence on the arithmetical S1-sentences. Those sentences are such that if true, they are provable. S1-completeness is equivalent with Turing-completeness in Computability theory. “Rich” theories, like PA or ZF are able to prove their own S1completeness in the sense that for all S1 sentences p, they can prove that p / ,p, that is a particular form of Triv. V is G þ p / ,p. V* is defined accordingly. Visser proved:

1. 2. 3. 4. 5.

T(A) ¼ r(A) if A is atomic, T(A#B) T(A)# T(B) if # denotes a binary connector T(⊥) ¼ “0 ¼ 1” T(u) ¼ “0 ¼ 0” € del's provT(,A) ¼ Beweisbar(£T(A)·), where Beweisbar is Go ability predicate defined in the language of the machine, and £ € del number”) of T(A). T(A)· represents some coding (or “Go

N is the standard model of Arithmetic. Solovay (Solovay, 1976) proved two arithmetical completeness theorems: 1. GwA iff PA proves T(A) for all realizations r, 2. G*wA iff N ~ T(A) for all realizations r.

1. VwA iff PA proves T(A) for all S1-realizations r, 2. V*wA iff N¼ T(A) for all S1-realizations r. A semantics for V is provided by imposing on the Kripke models for G that if a satisfies p, then aRb implies b satisfies p (Visser, 1985). 8. The knower

7. The rational believer or reasoner Let M be some machine. I will say that the machine believes a proposition A, if it asserts a corresponding sentence, also noted A, soon or later. The machine is consistent if it never asserts A and :A for some A. I assume the machine to be platonist, which here will mean that the beliefs of the machine obey classical logic. I will € bian, by which I mean here that the suppose that the machine is Lo machine's provability ability extends the provability abilities of Peano Arithmetic. Such machine believes in elementary arithmetic, including the infinitely many induction axioms. It is that induction power which makes the machine able to prove its own incom€b. pleteness theorem, and notably to prove its generalization by Lo Indeed, induction makes such machine able to prove S / beweisbar(£S·) for all S1-sentences S. This makes also the formula 4, ,A / ,,A, into a theorem as Beweisbar(x) is itself S1, indeed S1-complete. The main modal formula is the formula L, € b: which is the modal rendering of the formal theorem of Lo

,ð,A/AÞ/,A

€del already knew in 1933 that his provability predicate does Go not obey to the axioms of the logic KT4, known as S4, which is, with KT, the most traditional formal account of knowledge or know€ del, 1933). Very typically, the reflection axiom ,A / A is ability (Go € del's predicate, as that would imply ,⊥ / ⊥, not satisfied by Go that is :,⊥, which is the machine's consistency, which is impossible to be proven by the consistent theory or the consistent ma€del's second incompleteness or Lo € b's theorem (Go €del, chine by Go € b, 1955). Now, we will limit ourself to correct machine, 1931; Lo so ,A / A is always, for all A, true, yet the machine only proves such reflection in the particular case where the machine can prove € b's theorem, and so the corresponding reflected proposition, by Lo the machine can't prove that the provability entails truth in general, or that provability leads to knowledge. But this is precisely what makes possible to apply the antic definition of knowledge by considering provability as a type of justified believability, and knowability as true-justified-provability, where truth is added per definition. When Socrates asked to Theaetetus to give a definition of knowledge, eventually Theaetetus did propose to equate


10


knowledge with justified true opinion (Platon, 1950; Burnyeat, 1990). To suggest only “justified opinion” succumbs to the fact that some justification can be invalid, as Socrates illustrated with some examples. defining knowledge by true justified opinion entails, by definition, the reflection property for all formula. Incompleteness allows one to try that definition with the provability predicate playing the role of the justifiability, so that we can define knowledge, or a form of machine's knowledge, in a similar way than Theaetetus. Knowledge is not just a belief which happens to be true, as the truth is build in the very definition of that knowledge. Precisely, we consider a translation of modal logic in arithmetic or in the language of the machine, as above, except that we transform the main close 5 by:

5bÞ

Tð,AÞ ¼ ðBeweisbarð£TðAÞ·Þ∧TðAÞ

Boolos, Goldblatt, Kuznetsov and Muravidskii (Boolos, 1980; Goldblatt, 1978; Kuznetsov and Muravitsky, 1977) have proven that S4, that is KT4, þ the axiom:

,ð,ðA / ,AÞ/AÞ/A

ðGrzÞ

formalizes soundly and completely the logic of provable-and-true, as we will name it hereafter. S4Grz proves A iff the correct machine proves T(A) for all realization r. The weird axiom Grz which needs to be added to get the completeness result comes from an older work of Grzegorczyk (Grzegorczyk, 1967), who showed that S4Grz, like S4, formalizes intuitionist logic. Indeed he proved that intuitionist logic proves A iff S4grz proves some transformation of A. For example the following transformation g translates intuitionist logic in modal logic S4Grz. g(p) is given by ,p, g(:A) is given by :,g(A), g(A ∨ B) by ,g(A) ∨ ,g(B), g(A ∧ B) by g(A) ∨ g(B) and g(A / B) by ,g(A) / ,g(B). There are other translations which €del's 1933 paper (Go €del, 1933). Such works. This one comes from Go a knowledge operator cannot be defined in the language of the machine, corroboring the mystical intuition that the knower has no name or third person description, and it corroborates also Brouwer's intuition that the subject is not axiomatizable. It has been shown that G and G* are equivalent for the proposition of SGrz. We can put it in this way: that is S4Grz* ¼ S4Grz. As Boolos (Boolos, 1993) suggests, this amazing fact might be a shadow of the intuitionist idea that provability and truth are (constructively) equivalent. In Marchal (1994) I explain that the difference of behavior of the logic G and S4Grz makes it possible to formalize the antic dream argument in metaphysics, and to retrieve the impossibility of distinguishing with certainty awakeness from dreaming, or to know if we belong in an arithmetical emulation or a physical or analytical emulation, which is of crucial importance for deriving the permitted logic of the observable in the computationalist context (Marchal, 1994; Marchal, 1998; Marchal, 2004; MarchalBarry Cooper et al., 2007). We see that not only incompleteness distinguishes provable from true extensionally, but incompleteness separates the logic of provable from the logic of “provable-and-true”. Indeed incompleteness and Solovay, and Visser, theorems distinguish basically all Theaetetus-like intensional variants of provability which definitions follow below. Incompleteness refutes Socrates refutation of Theaetetus' definition, and most critics of it, including modern writing like the one by Gerson (Gerson, 2009). Gerson claims that the Theaetetus' definition would make ”knowledge” into a propositional attitude, and that it would reduce it to mere belief, accidentally true, but this is not the case, like with Brouwer, the knower is so different from the believer that he is not even definable by the machine (see (MarchalBarry Cooper et al., 2007; Kaplan and Montague, 1960). This makes the machine's theology very close

to Plotinus' conception of reality. So, the definition by Theaetetus corroborates the point so dear to the ancient philosophers that knowledge is not entirely representational. It makes at the same time the Platonist argument from dreams coherent with the computationalist idea that a machine cannot a priori distinguish reality from possible dream, or possible video games (Marchal, 1994). 9. The observers If we are machine, we are duplicable, and in fact we, in the third person sense, have infinity of abstract incarnations in arithmetic, through the S1-sentences representing the computations in arithmetic. This introduces a relative indeterminacy, and from this we can predict that if a machine observes itself and its physical neighborhood below its substitution level, the machine has to be confronted to some fuzziness, which would be literally map of its accessible physical realities. See (Marchal, 2004; Marchal, 1994; Marchal, 1998) for much more explanation on this, and the reference therein. This is similar to quantum mechanics. The fuzziness would be corresponding to a quantum orbital, which can be seen as a map of where we can localize an electron in case we would decide to observe its position. But what could be the mathematics of that fuzinnes? Observation cannot be modeled by G, as it presupposes implicitly the existence of the accessible realities, avoiding the trivial truth of a boxed statement in a cul-de-sac-world. But could the knowable becomes the observable by simply restricting the arithmetical realization on the S1-sentences? In this case it follows easily from Visser theorem, and the preceding results, that S4Grz þ p / ,p, that is triv restricted to the atomic formula formalizes soundly and completely the logic of the knowable S1sentences. So what happens to S4Grz if we add the Triv axiom? At first sight it seems impossible to have both T (,A / A) and Triv (A / ,A), and the necessitation rule, without making the logic collapsing into classical logic, with both the box and the diamond becoming trivial verum connector. That is indeed easily shown to be the case, but it does not happen in our context. The reason why KTriv, K þ Triv, collapses is that the Triv formula is true for all sentences. In our case, we can admit it only on the atomic formula. The arithmetical reason is that the negation of a S1-sentences is not a S1-sentences. Then it is simple to find a Kripke counter-example to ,>A / A, indeed even with A being atomic. Take the model W ¼ {a,b}, R ¼ {(a,b)}, and no other accessibilities, and make :p statisfied by a. Then p is true in a, and ,>p is false, making ,>p / p false in a. The atomic p, obeys the B axioms in SGRz1. Now, we do have some empirical reasons to believe or assume that such a logic is given by the von NeumaneBirkhoff Quantum Logic, which admits itself, in a manner similar to intuitionist logic, a formalization by a modal logic, indeed by the system B, that is KTB, as shown by Goldblatt (Goldblatt, 1974). The following transformation t translates quantum logic in modal logic B: t(p) is given by ,>p, t(:A) is given by ,:t(A), and t(A ∧ B) is given by t(A) ∧ t(B) (Goldblatt, 1974; Dalla Chiara et al., 1986). In this way a form of classical computationalism leads to a testable platonist theory of matter, in the manner of Plotinus (see (MarchalBarry Cooper et al., 2007)). This leads also to an arithmetical interpretation of some quantum logic, by reversing Goldblatt transformation starting from S4Grz1, like Boolos and Goldblatt did for retrieving an arithmetical interpretation of intuitionist logic from the arithmetical interpretation of S4Grz. The S1-sentences admits a local proximity/ orthogonality structure, through a quantization of atomic formula p given by ,>p. Note that K4, and its extensions, proves ,>A 4 ,>,>A. It plays the role of an abstract projection in the Kripke modal setting, when restricted to the atomic formula.



In the frame of the computationalist assumption in the cognitive science (basically the assumption that our brain is Turing emulable) we need an intensional variant , of provability, weaker than the provable-and-true intensional variant, having still the formula D as axiom or theorem, and still restrict the arithmetical sentences to the S1 sentences. The basic idea is that machines are duplicable and appears infinitely many-times, with a high redundancy, in any constructive definition of the collection of *all* computations, and that all machines are indeterminate, from their first person point of view on such computations. We gave reasons, related to the mindbody problem, to claim that the logic of observability should be given by a logic of knowledge, or intensional weaken variants, when restricted to the S1-sentences. Moreover, if it is nice that SAGrz ¼ S4Grz*, which is extended to S4Grz1 ¼ S4Grz1*, it would be nice, to get the difference between qualia and quanta, to have logics of “observable” which do separate on the provable/true differences. Now, S4Grz admits reflexion, which implies D, as we want, but we could have taken a weakening of the provable-and-true translation, indeed we can take provableand-consistent translation. Then we can again reverse Goldblatt's translation, restricted to the s1-sentences, and compare to quantum logic. We obtain the arithmetical interpretation by replacing again the clause 5 above, by

5cÞ

11

This gives, with natural notations, two bi-reciprocable transformations zg and gz, given, with ⊥ the propositional constant falsity (:u), by:

,z A ¼ ,g A∧>g u;

>z A ¼ ,g A∨>g ⊥

and,

,g A ¼ ,z A∨>z ⊥;

>g A ¼ >z A∧,z u

This can be easily derived from the fact that G, and indeed K, proves ,gA ¼ ,zA ∨ >z⊥. The compositions of zg and gz can be proved as being identity relations. This leads to the following finite formalizations of Z, Z1, Z*, Z1*. For reason of simple readability, I abbreviate >⊥ ∨ ,A by ,gA, add the inference rule RE (rule of modal equivalence) of classical modal logic to get an arithmetically sound and complete formalization of Z:

AXIOMS :

,g ðA/BÞ/ ,g A/ ,g B ,g A/ ,g ,g A ,g ,g A/A / ,g A A ; A/B B A ,g A

RULES :

Tð,AÞ ¼ Beweisbarð£TðAÞ·Þ∧Consistentð£TðAÞ·Þ

Consistent(£T(A)·) is of course :Beweisbar( £:T(A)·). Amazingly we can then apply again the Theaetetus definition of such weak predicate and get a provable-and-consistent-and-true translation:

Kg 4g Lg MP NECg

A4B ,A 4 ,B

RE

and Z* is given by:

5dÞ

Tð,AÞ ¼ Beweisbarð£TðAÞ·Þ∧Consistentð£TðAÞ·Þ∧TðAÞ

We can replace Consistent(£T(A)·) by Consistent(£u·), with u being the propositional constant truth. This follows obviously from the Kripke semantics of G. Then we can limit again the arithmetical realization on the S1-sentences, for our goal of deriving the logic of observable propositions from self-reference, as motivated in the computationalist frame. In both case we still avoid the cul-de-sac problem, and get again a form of quantum logic, with the same quantization rule. What is nice at this stage, is that, contrarily to the non separation between provable and true for the knower, we inherit, thanks to the presence of explicit appeal to consistency, a G/ G* Solovay split. I call Z1 and X1 the corresponding modal logics. It is easy to show that Z1 and Z1* differs, and that X1 and X1* differ. This helps to separate the physical observable attribute that an observer can observe and justify to others and those that the observer can observe but not in a justifiable way. In this case we loss transitivity and we loss the necessitation rule at the “star level”, making us both closer and farer of the usual modal quantum logic B. X1* is particularly interesting as it is a refine use of the idea of Theaetetus, and it provides again a reason why a part of the observation process is non representational, making this very close to the account by Plotinus on the nature of matter in the platonist context. Solovay's G and G*, like Visser V and V*, are decidable. This makes all the Z, Z*, Z1, Z1* logic, and the X, X*, X1, and X1* logics decidable and well defined through their arithmetical interpretation. The problem of finding a finite axiomatization for the logics Z, Z*, Z1, and Z1* has been solved by Eric Vandenbusshe.5 Vandenbusshe discovered that we can reverse the transformation defining the modal operators of the Z logics, so that we can define the modal operators of G (and thus G*) from the modal operator of the Z logic:

5

http://iridia.ulb.ac.be/~marchal/Vandenbussche/AxiomatisationZ/scan01.html.

AXIOMS :

RULES :

Any Theorem of Z ,g A/A

Tg

A ; A/B B

MP

To get Z1 it is enough to add the axioms p / ,gp, for p atomic formula, and Z1*, has the theorems of Z1 together with the axioms ,gA / A$ Z1*, like G*, has lost the necessitation rule. Unfortunately this method for obtaining a finite formalization cannot been applied to the X, X*, X1, X1* logic. It is easy to see that those propositional modal logic are decidable, and thus recursively (computably) enumerable and thus axiomatizable, (by an argument due to Craig), but the question of a finite axiomatization remains open. All three candidates S4Grz1, Z1*, X1*, prove what is needed to have a quantization based on a reasonable abstract proximity relation (reflexive and symmetrical), or its dual, an abstract orthogonalization or perpendicularity (irreflexive and symmetrical), which in the starred context can be reinterpreted either with infinite sequences of Kripke models a la Solovay (Solovay, 1976), or with a ScotteMontague semantics (see (Marchal, 1994) for more details on this. Can we build a quantum NOR? The work by Rawling and Selesnick suggests that his could be possible (Rawling and Selesnick, 2000). S4Grz1 soundly axiomatizes by KGrz (which proves T and 4) þ p / ,p, that is Triv restricted on the atomic formula. This follows from Visser completeness results. We have also that S4Grz1 ¼ S4Grz1*. The question is do we have, for a reasonable quantum logic QL,

Q LwA

iff

S4Grz1wGoldbðAÞ;

and similarly for Z1* and X1*, with Goldblatt translation described above. If not, are there other transformations?


12


10. Open problems

11. Conclusion

I sum here the main open problems. The main open one is: are there finite axiomatizations for the logic X, X*, X1, X1*? Vandenbusshe's technic for solving this problem for Z, Z*, Z1, Z1* cannot be applied. Then the other problem consists in measuring how far those logics confirm or not a version of classical computationalismdwhich can be sum up by saying that it is a digital version of Descartes mechanism, (our bodies functions like machine at some level) together with the classical theory of knowledge (based on the modal logic S4. How does those arithmetical quantum logics compare with Birkhoff and von Neumann quantum logics? (Birkhoff and von Neumann, 1936). Do we find the quantum logic that von Neumann hoped for, which should be so constrained that all probabilities can be derived from the “Yes-No” measurement and their logic? They corresponds in quantum mechanics to the projection operators which provides the usual algebraical motivation for quantum logic. We need the equivalent of Gleason theorem for the measure induced by the modal quantum logics.6 Can we, in the manner of Rawling and Selesnick (Rawling and Selesnick, 2000) use the quantization ,>A to program some quantum universal gates? This would justify universal quantum computing ability and provides some confirmation of the computationalist hypothesis in the cognitive science. But, even if this works, we would still have only one qubit. We need a graded structure. A candidate for this is provided by the fact that the consistency condition, the (,A ∧ >u), can be strengthened into (,A ∧ >>u). Actually, it is easy to show that, with natural notation for nested modal boxes and diamonds, that each ,nA ∧ >mu brings a modal quantum logics similar to the Z logic when m n. Relation between quantum computing and the Jones polynomial made possible by the use of TemperleyeLieb algebra (Kauffman, 1991) provides some hope that this could be enough to get the appearance of space which would be able to percolate on the arithmetical truth in some sense. What about the derivation of the standard model of the theory of elementary particles. Most of it can be derived from “pure” quantum mechanics, together with some space-time geometry, and Noether fundamental theorem (see (Stenger, 2006) for a beautiful illustration of this). I hope we may find a possible TemperleyeLieb structure which would provide the means to extract space, and enough symmetries, so that the observer could break them, by their natural antisymmetrical views and lead to the equivalent of Noether theorem. But for this we need first to get a good measure and get it with the equivalent of Gleason theorem in our arithmetical quantum logics. But for this we need the S4Grz1, Z1*, and X1*, at least, to provide a quantization which is reasonable enough, and that seems to be the case. What about Z and X? The quantum logic appears only in Z* and X*. In fact this is good news, as it confirms the approach of Everett (Everett, 1957) in quantum mechanics, with a physical reality which is mainly a sharable first person plural reality. Details can be found in Marchal (1994). But the non-starred Z and X logics are interesting per se. The Z and X logics don't prove the formula 4. This rises the question: do we have the “projection property” ,>,>A 4 ,>A? It is an open question, even for only the atomic sentences. The nesting here prevents a simple computational approach on computer. See the appendix in my long text (in french) for theorem provers for most logics presented in this paper (Marchal, 1994).

In this section, I make an attempt to conclude, taking into account some points made by reviewers, which I thank for their attentive reading and remarks. A difficulty, to make a conclusion, is that we would like to be clear and crisp, when in reality things are subtle and nuanced. I proceed as if I was asked questions. Machine can dream. Is it a metaphor? No, it is not a metaphor. Computationalism, as I define it, means that there is a level of substitution such that I would not see, nor feel, any difference once my brain is replaced by a computer obtained by a digital functional substitution made at that level. In other word, to have computationalism false, we would need special actual infinities, and non computational primitive elements. Then, it is a theorem of mathematical logic that those computations are realized, emulated, implemented in the theoretical computer science sense, by special number relations, making a tiny part of the arithmetical reality full of life, dreams, emotions, including what the readers experience right now, and this in an infinitely distributed way, as those computations/number relations exist in a very redundant way in that tiny part of arithmetic. All what is said here should be taken literally, modulo some precision, which I often avoid to not add too much jargon. To say that a number dreams, for example means that, relatively to some universal number u, 4u(x,y), which computes 4x(y) emulates the working of a brain (humans, or aliens, …), at the right substitution level, for which we accept that some experience of consciousness can be related (with the computationalist working hypothesis). You seems to not aboard the question of emotion? I don't aboard any usual psychological stuff. I might have explained better that no psychological experiences are thrown away, as we would see a difference if one of them was not emulated by the artificial brain. But by definition of comp, no difference occur. Now, the very subject of emotion is beyond the scope of this paper, but the reader can guess that we have already the subject of the emotion, which is the first person, that we recover through the definition of the “knower” by Theaetetus. Indeed incompleteness provides variants of ”provability” obeying different logic, and we get a logic of knowledge exactly where we expect it, with the most famous intensional variant: the one given by ,p ∧ p, following Plato's Theaetetus. We get a sensible/sensitive subject, with qualia, with the intensional variant ,p ∧ >p ∧ p. We get in fact eight basic intensional variants. € delian incompleteness split between Three of them inherit the Go what is true about them, and what the machine can prove about them, which explains why we get 8 variants from those five distinctions. Emotion will then develops through some universal goal, or heuristic, which will separate the good and the bad. The universal goal or heuristic (like ”survive and multiply”) will separate the good (to eat) from the bad (being eaten) for example, accompanied by feeling of urgency, panic (do anything to avoid the current state), etc. The non justifiability of those emotion is then handled like the other non-justifiability: by the Solovay split between truth and proof inherited from the difference between G and G*. What is the computationalist mind-body problem? The main problem is provided by the Universal Dovetailer Argument (UDA): it shows that materialism and mechanism are incompatible. Materialism is taken in the weak sense of any doctrine assuming some matter exist in some primary way, or that some matter, or just some physical principle have to be assumed. The UDA makes the whole science of physics, and its object, the physical reality, emergent from arithmetic: indeed emergent of the biologypsychology-theology of the numbers. Physics becomes a statistics on computation seen from some internal point of view/hypostases. Again, those are not metaphor. Computationalism is theology, in

6 For a good introduction to Quantum Logic, including a relatively readable proof of Gleason theorem, see the book by Richard Hughes (Hughes, 1989).



the greek sense of the term: it concerned the truth, or falsity, of a form of possible reincarnation. At first that reincarnation is technical and based on the use of a physical (locally) digital artificial brain, and then we understand that it is necessarily prolonged in the arithmetical reality (actually a tiny part of it, which is assumed implicitly or explicitly in all scientific work). The miracle here is truly provided by incompleteness which makes possible to translated faithfully neoplatonist theologies into the self-referential discourse of ideally correct machine. Incompleteness separates the different intensional nuances, and separate some of them along the true or just provable parts. Let me give the lexicon (more is explained in the paper (MarchalBarry Cooper et al., 2007; Marchal, 2012). We have the 8 primary hypostases:

p ,p ,p∧p

Truth provability knowability

The One The Intellect The Universal Soul

And the “two matters”:

,p∧>t ,p∧>t∧p

The Intelligible Matter The Sensible Matter

Bet=Prediction Qualia

Three of them split along the G/G* differentiation, which gives eight hypostases from those five intensional distinctions. We have to keep in mind that the Arithmetical Truth is a highly non computable object, and that from the points of view of the machine, that object is not even nameable or definable by the machine. This does not prevent the machine to evoke it or use it in a variety of ways. Likewise, for the soul, and the sensible matter. By incompleteness, the adjunction of “∧ p” makes the intensional nuance obtained into something non definable by the machine. The machine cannot describe who she is, and we can show that the soul of the machine is able to correctly distinguish herself from any third person describable notion. Scientists who asserts that the soul does not exist are simply wrong, or they use a materialized notion of soul, like they use a materialized notion of matter, which indeed does not make sense with computationalism. The overall conclusion, is that computationalism leads to a complete reversal of the current Aristotelian7 paradigm, with a notion of primary matter and elements which would be at the origin of physics, and then of souls/observers/persons. What must be taken as primitive, once computationalism is assumed, is anything capable of emulating a Turing machine, like a tiny part of arithmetic, and all the rest is derived from the number or machine theology. I use the term primitive in the sense of being in need of being assumed at the start. 11.1. How would you summarize the consequence of the computationalist assumption in plain language? The Platonists and the mystics say that the truth is in our head, and nowhere else. Today the computationalists must say that the truth, including the physical truth, is in the “head” of all universal €del, machine, and that we can literally read it, in the manner of Go €b and the theoretical computer scientists and the mathematical Lo logicians. Then we can test different version of comp (classical, intuitionist) by comparing the physics ”in the head of the (mathematical) machine/number” and in the world around us. Technically, everything comes from the existence of the universal machine (a theorem in elementary arithmetic) and the

7 According to Gerson, Aristotle was still a platonist. The reification of Aristotle primary matter is mainly the work of its successors, notably the Christians, for making bodies into primitive realities.

13

parametrization theorem, from which we derive the self-reference abilities (second theorem of recursion, or its generalization, like the theorem of Case, or its formalization like the diagonal lemma of € del at the base of Solovay theorem). The proof is constructive and Go gives the way to derive physics (both the sharable quanta, and the non sharable qualia), and as we can compare the quanta of computationalism with the quanta inferred by observation, this provides a mean to test the computationalist thesis. € delian incompleteness and the Everett Thanks to both the Go quantum mechanics, a case can be given that computationalism fits remarkably with the facts. We get an explanation of the existence of undoubtable immediate but non justifiable (provable) truth, like consciousness, and an explanation of its physical appearance. It would be astonishing that the first interview with the machine gives the correct physics, but up to now, it seems to work. Our main meta-goal was to illustrate that some hypothesis (like computationalism) makes possible to reason modestly in field like philosophy of mind, theoretical cognitive science, biology, embryology, theology, metaphysics. This does not mean that computationalism is true, but it means that we can do science with it, and perhaps refute it, or improve it. Computationalism makes also possible to invalidate reductionism, as it shows precisely why the machine will invalidate the reductionist conception that we can have on machine. Mechanism could have been thought as a reductionism, and indeed, it leads to an ontological reductionism, but it justifies the impossibility to reduce the soul and the mind to anything purely third person describable, and indeed it illustrates that ideally correct machine have a rich theology, which can be of some use in comparative theology, independently of its truth or falsity. It has a very simple ontology, Pythagorean, and an unboundedly rich internal epistemology/theology. All the hypostases described the exact same part of the arithmetical reality, when fixating the machine, as G* will prove that ,p 4 ,p ∧ p, and so for each hypostasis, yet their logic is completely different from the machine's point of view. This can be related to some thought of Kant and Leibniz, but that would be another subject of research, and the similarity with neoplatonism is more striking, at this (propositional) level of generality. References Barendregt, H.P., 1984. The Lambda Calculus, second ed. Edition Originale: 1981, North-Holland, Amsterdam. Birkhoff, G., von Neumann, J., 1936. The logic of quantum mechanics. Ann. Math. 37 (4), 823e843. Boolos, G., 1979. The Unprovability of Consistency. Cambridge University Press, London. Boolos, G., 1980. Provability, truth, and modal logic. J. Philos. Log. 9, 1e7. Boolos, G., 1993. The Logic of Provability. Cambridge University Press, Cambridge. Burnyeat, M., 1990. In: Levett, M.J. (Ed.), The Theaetetus of Plato. Hackett Publishing Company, Indianapolis, Cambridge. Translation by. Case, J., 1971. A note on degrees of self-describing turing machines. JACM 18 (3), 329e338. Case, J., 1974. Periodicity in generations of automata. Math. Syst. Theory 8, 15e32. Chellas, Brian F., 1980. Modal Logic, an Introduction. Cambridge University Press, Cambridge. Curry, H.B., Feys, R., 1958. Combinatory Logic, vol. 1. North-Holland, Amsterdam. Dalla Chiara, M.L., 1986. Quantum logic. In: Gabbay, D., Guenthner, F. (Eds.), Handbook of Philosophical Logic, vol. III. D. Reidel Publishing Company, Dordrecht, pp. 427e469. €del didn't have Church's thesis. Inf. Control 54, 3e24. Davis, M., 1982. Why go Everett III, H., 1957. “Relative state” formulation of quantum mechanics. Rev. Mod. Phys. 9 (3), 454e462. n, T., 2004. Inexhaustibility, a Non-exhaustive Treatment. Number 16 in Franze Lectures Notes in Logic. Association for Symbolic Logic, Massachusetts. n, T., 2005. Go € del's Theorem, an Incomplete Guide to its Use and Abuse. A. K. Franze Peters, Natick, Massachusetts. Gerson, L.P., 2009. Ancien Epistemology. Cambridge University Press, UK, Cambridge. €del, K., 1931. Über formal unentscheidbare sa €tze der principia mathematica und Go verwandter systeme i. Monatsh. Math. Phys. 38, 173e198. Traduction ricaine dans Davis 1965, page 5þ. ame


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