Dear Thomas,

Thank you for your comments on my essay. I have read your essay with great interest and will comment soon on your forum. I agree that the voting scheme is a bit bumpy... if I keep track correctly, I think I've gotten three 10 votes that were followed within a few hours with 1's or 2's... Maybe there are some voters who give a rating that will "adjust" the cumulative rating of any one essay to what they think it should be, instead of giving a fair grade to an essay irrespective of the rating it already has... Objectively, I think that very few of the essays that were submitted in this contest only deserve a 1 or a 2!

I am glad that the rules have been changed so that 10 out of 40 essays will make it to the final round at the discretion of the judges. It will make it less likely that some interesting essays will be purposefully eliminated from the finals by some unfair downgraders...

Marc

Dear Joe,

Thank you for saying that my essay was exceptionally well written... But of course, you are aware that we can see what you write to other participants, and that everyone knows by now that you basically say the same thing to everybody! :) :) :)

Your views are by now quite familiar to me (because you submit essentially the same essay to every contest, no matter what the subject is), but they are also a complete mystery to me, because I do not understand the meaning of the (abstract) words that you use to describe the (abstract) concepts that you (abstractly) mention in your essays (like "speed", "surface" and "subsurface") and that you seem to be using in a particular way that is unique, once, in all of known theorizing on the nature of the Universe... Nevertheless, I will soon go to your forum to ask you about clarifications about the basic concepts of your THEOREM OF INERT LIGHT. I hope I will be luckier than others who have enquired about your abstraction-filled view of the universe!

Playfully,

Marc

Dear Thomas,

One more thing... In your comment above, your wrote "Where our arguments diverge, is on the limit of mathematics -- I agree with Tegmark's limit, that I define in a purely objective and local physical context." You are quite right about this divergence, but I think it is a consequence of the different ways we chose to interpret this year's FQXi's contest question. I have noticed that essays in this year's contest seem to divide in two camps: those who tackled the issue of the relationship between known (or potentially) known mathematics and the observable (or potentially observable) universe, and those (such as myself) who tackled the issue of the relationship between "All-of-Math" and "All-of-Existence", defined in a necessarily philosophical way (because these universal concepts are not observable).

Marc

Dear Marc,

Thank you for the answers and the interesting additional information and scenarios you propose about immortality in maxiverse!

I replied to the comments you posted on my forum.

Good luck at the contest!

Cheers!

Cristi

Marc,

Thanks for shedding light on the topic of this essay contest in such an interesting and unique way.

I will like to take you up on a couple of things.

1. I like the way you clearly defined the term, emergent, viz. "Most of the properties that we associate with matter at our scale (like texture and color) are emergent properties that do not exist at the level of electrons or quarks". Many use this term without properly clarifying what they mean by it. In this regard, what is your view whether space is a mathematical structure or physical structure, i.e. whether space is emergent?

2. I find intriguing this question you posed, "No matter what the ultimate cause of existence is, we know that it has been able to create an actual world at least once, since we observe such a world. What could prevent this cause from acting again to create another world, and another, and another? And even if a given cause eventually "runs out of steam", being an ultimate cause, it exists by itself: if it instantiated itself once, what could prevent it from instantiating itself once more, creating other worlds?"

I have agonized over this a lot and invite you to do same.

a) What is a world and does it have a size? That is, is a world an extended thing?

b) Can a world perish or is it eternally existing?

c) As your statement suggests, if your answer in 2) is Yes, what could prevent this from occurring again and again? Indeed, would the creation and perishing of worlds be the most fundamental event?

You can equate your Clones to my extended point, if your clone and worlds have a size of some very tiny limit. In my essay, I discuss the creation and perishing of extended geometric points. You may find it interesting.

All the best,

Akinbo

    That is a very interesting and insightful comment, Marc. It makes me realize that my love of limits probably reflects my love of analysis. :-)

    Tom

    Dear Akinbo,

    You ask interesting and deep questions! Is space purely mathematical, or is there such a thing as a physical space (independent from the processes that take place within it) that emerges from the math? What is a world and does it have a size?

    I think there are always many different ways to look at the same thing (that's why I believe that we live simultaneously in an infinite number of different larger contexts), and I think that space is one of those things that can be seen either as physically real (so that space can bend or stretch according to relativity and Big Bang cosmology), or as "merely" a convenient mathematical construct to make sense of the phenomena that we observe. The concept of world also depends on your point of view: from one perspective, your world is the sum of all your sense impressions, from another, your world is all that lies within the cosmic horizon of the observable universe, from another, it is even larger, being the totality of what could potentially causally connect with you.

    You ask if a world can perish. Once again, it depends on your point of view. I believe that capital-E Existence ("All that exists") exists in an "atemporal and eternal" way: I don't think it makes sense to say that it can be different at different times, because it would mean that there is a "time-counter" outside of capital-E Existence to make sense of this change, and this is impossible because capital-E Existence is all there is! On the other hand, when you look at a subset of reality, at a "local" world, it is quite possible to define a time-counter outside this world, and relative to this time-counter, this world can be born, evolve and perish.

    The fact that worlds are born and perish is certainly one of many properties that worlds can have, but I don't see it as fundamental. For instance, I have no problems with eternal physical worlds, and mathematical structures are, by themselves, "atemporal and eternal". Of course, it is possible to define a mathematical structure that is related to another structure that acts as a time-counter, and relative to that time-counter, the first structure can evolve, even appear and disappear, so I think it is possible to define a structure made of geometric points (or extended geometric objects) that, in some sense, can be "born" and "perish". (That's why I have no problem in believing that a physical universe that is born, evolves and ultimately perish can be thought as nothing more than a mathematical structure.)

    I've read you essay and I know that "perishable geometry" is a crucial part of the theory you propose, and I will soon post my comments about your essay on your forum.

    Marc

    Your inclination only when questioned further it asks about super conscious universal truth which classifies various sub-junctions in question?

    - Best regards,

    Miss. Sujatha Jagannathan

      Dear Sujatha,

      Thank you for taking the time to comment on my essay. I will take a look at yours. Good luck in the contest!

      Marc

      Dear Marc,

      I was not sure whether or not to view with suspicion any comments dated April 1st.

      Nevertheless, thank you first of all for reading my essay, and for treating it kindly. I took a brief look at your own essay so I could better understand your comment (I will read your essay in great detail and respond on your page with my comment this week).

      Consequently, the following lines are very preliminary and should only be taken as an indication of "first thoughts."

      It appears that there is a difference between what you understand by "mathematics" and what I think it is. But we may at least gain a mutual understanding of each other's position if we compare our respective general understanding of what exists and how we know things (I didn't want to use "them" big words here).

      Of course, the issue you describe in the 2nd half of your third paragraph will be understood more clearly once we have done what I have just said.

      For the sake of continuity, I also posted this comment on my essay page.

      En

      Hi Marc,

      "Digital Physics" has been submitted to the Fantasia Film Festival in Montréal, which takes place in July and August, so if you or the Collège de Maisonneuve have any connections there, please let them know of your interest in the film. Even if you don't know anyone connected to the festival, an email on behalf of your physics department might go a long way in terms of increasing its chances of being programmed:) Even though it is a fun movie, some of the ideas presented in it are challenging enough that it may be passed over for more traditional genre films. Sometimes movies that don't have a clearly defined genre can struggle. ("Pi" and "Computer Chess" are two movies that might fall in its genre.) The movie does have a little French spoken in it, so hopefully the festival programmers see that as as an additional reason to program it. I hope you have a chance to see it this summer at the Fantasia Film Festival, but if you don't, you may have to wait until it gets released online in late 2015 or early 2016.

      Now on to the mathematics/physics:)...

      Thanks for the explanation. I agree substructures can always be extended, with an unlimited number of possibilities, but I guess I am still curious as to what you think the substructure that represents you/self might be in this sense. When reading your paper, I first considered something like once you try to sever the observer from its environment, the notion of self might unravel, but maybe that was based on a misunderstanding of what substructure you really had in mind for a person/observer/entity. Do you imagine a substructure for "self" being something similar to the arrangement of atoms that make up your body or more akin to everything in your past light cone/observable universe?

      I agree personal identity is probably one on the hardest problems to actually address, but it's fun to give it shot:)

      Talk to you later,

      Jon

      Dear Marc,

      It is interesting to see that while you push the logical consequences of the MUH to the extreme, you still accept them, while I see them as a proof by absurdity against this hypothesis. Also I found it interesting that you have some common ideas with my view (see my essay): seeing consciousness as playing a fundamental role in giving mathematical universes a physical existence, and as you say "when we succeed at something in our universe (...), we do not change the Maxiverse in any way, we merely "visit" preexisting mathematical structures that have always been part of the Maxiverse." However we differ by the fact you consider consciousness as a particular mathematical structure, while I see it as non-mathematical.

      Here are some remarks on the details:

      You mentioned Occam's razor, and that the MUH is basically unfalsifiable (though you mentioned changing your mind about the latter in your comments). I consider both concepts of Occam's razor and falsifiability as different aspects of the more precise and general condition of quality for a scientific theory, as I mentioned in my essay: to give a probability law on observables which minimizes their entropy, i.e. presents the observed data in their most compressed form, where the size of the compression algorithm (or equivalently, the algorithm of probability law) is itself counted as added to the size of compressed data. Here, Occam's razor is about insisting on the need of simplicity of the algorithm.

      The MUH is relatively unfalsifiable in the sense that it somehow gives a non-zero probability to every possibility, thus preventing the risk for an observation to contradict the theory : an observation which a theory excludes would mean the occurrence of a finite piece of data with infinite entropy as interpreted by the theory (theoretical probability 0 implies entropy = - ln 0 = infinity if it occurs). However, that a theory is prevented against infinite entropy, does not mean that it reaches a minimal entropy in comparison with other theories, which leaves the possibility to be relatively refuted by finding another theory doing a better job at explaining (compressing) the data, with a more precise and verified probability law. In any case, I consider that trying to avoid the issue of specifying probabilities by admitting the existence of all possibilities and qualifying the whole probability issue as a kind of mystery waiting for future elucidation, is not a very good theoretical job.

      On this topic, you wrote "If you want to explain why one or only a few universes exist, you must specify the precise laws they obey and their initial conditions (at least). You must also specify and justify the rules which select these universes to be real while relegating all other possibilities to the dustbin of existence. Specifying the initial conditions alone might necessitate a mind boggling amount of information. On the other hand, to describe completely the Level IV multiverse, one short sentence is enough".

      Later, you wrote: "Another argument against the Maxiverse hypothesis (in fact, against any theory which incorporates seriously the notion of a multiverse) is the belief that it critically undermines the future of theoretical physics.".

      It seems to be the same character of the MUH that you first present as a quality (of satisfying Occam's razor) and then as a defect, isn't it ? I see the second view as quite amusing, as if, by principle, the Universe ought to have been well-designed for the purpose of giving jobs to physicists :) That reminds me the attitude of some climate-skeptics, which look as if the physical properties of the atmosphere had to be well-designed by God to ensure that free market structures will remain the best solution to all problems (the famous Invisible Hand) and thus for letting libertarians always remain the good guys in their defense of liberties.

      In fact, a more detailed analysis of this question would let both alternatives roughly equivalent: given a piece of data that looks random as we could not find an explanation (compression), both hypothesis "It is really random" and "It only looks random but has a hidden pattern yet to be discovered" are as bad as each other at the job of actually compressing that piece of data. Chaitin's theorem ensures that the second hypothesis remains irrefutable even if it is false. However, the first hypothesis is falsifiable (by the act of finding an explanation), so that persisting failures to find any pattern (time passes without any discovery of explanation) progressively leans to the first hypothesis, while discoveries of patterns (traces of design, even if not well understood yet, as expressed in the essay of A&L.Burov) leans to the second hypothesis (if we find some patterns then there should be some metaphysical reasons for them).

      You wrote : "I do not think it's possible to imagine an abstract structure which could not, in some way, be described by mathematics" I think there is, that is called feelings or qualia, discussed by the famous hard problem of consciousness. For example, what is the sensation of the red color ? It is not expressible as a mathematical structure. The physical object of red light can be described by mathematics; the sensation of the red color can't.

      "Moravec explains that we observe that our universe stays lawful and predictable, even if there are many scenarios where it doesn't, because in these scenarios, our consciousness immediately ceases to exist". This reasoning is not applicable without probabilistic assumptions that beg for justifications. Namely, to imagine that just because a regular law was needed to reach some result, it will therefore continue to apply, means that just because some regularity happened, it will be more likely to happen again. But where does that law itself come from ? If I play heads and tails and happen to get 10 heads successively, will it make it more likely that I still get heads next times ? If I won at Lotto first, would it make it more likely that I will win again another time ? Is the Born rule of quantum probabilities, made more likely to be obeyed by future observations, by the fact it seemed to be obeyed in past observations ? And what other conclusion should we draw instead if it didn't seem so ?

      "I expect to wake up in my bed and lead a more or less ordinary day, which must indicate that somehow (despite the measure problem), my F-clones which correspond to these ordinary scenarios greatly outnumber the other ones."

      "Outnumber" : is it a matter of number ? It is well-known among specialists of the Many-worlds interpretation, such as David Wallace, that the Born rule cannot be justified as a measure of the ratio between numbers of distinct worlds with the different given states of a subsystem, because, first, there is no such a thing as a number of possibilities; second, even if there was, anyway such quantities do not fit.

      Last autumn I wrote a description of the Many-worlds interpretation, with what I see as its necessary assumptions and consequences pushed to their extreme, which have a lot in common with your own description of the MUH. I do not subscribe to this interpretation but I think it is important to examine it, as part of the understanding of the one I support (mind makes collapse), because I include this many-world interpretation as part of the picture : it describes what happens in the absence conscious observation. And what remains of the physical universe when removing conscious observers, is precisely a purely mathematical world, which is ontologically equivalent to the MUH or Maxiverse (with the "small" difference that in the many-worlds interpretation of QM, the physical laws with the values of physical constants are fixed).

        Dear Sir, I would have taken the time to read your entire essay, but after a page or so, I can see that it is simply preposterous and so a waste of time to continue reading. Sorry but the thesis is beyond absurd. It postulates the idea that mathematics is the universe and then postulates multiple universes because multiple mathematical structures are possible. While there is not a single concept in this essay that imposes any kind of common sense rules. So it is a waste of my time to continue reading. I think that this is an example of bad thinking and I hope that you are not teaching such nonsense to any empty headed students who might be beguiled into actually establishing a academic career based on promoting such an absurd approach to human thinking.

          Dear Mr. Ricker,

          I find your comment interesting, and I agree with you that the Mathematical Universe Hypothesis does not meshes easily with "common sense rules". In his 2007 article "The Mathematical Universe", Max Tegmark (who holds many of the views that I present in my essay) had this to say about the Mathematical Universe Hypothesis:

          "[T]he MUH (...) forms a logical extreme in a broad spectrum of philosophical interpretations of physics. It is arguably extreme in the sense of being maximally offensive to human vanity."

          By the distressed tone of your comment, I can only conclude that he is right.

          Sorry of having wasted your time... :(

          Marc

          Dear Sylvain,

          Thank you for your detailed analysis of my essay. It is very welcome!

          I have read your essay, the more complete version on your website ("Specifications for a Mind Makes Collapse interpretation of quantum physics", that you suggested to Tommaso Bolognesi) and your very interesting description of the Many-worlds interpretation that you suggested in your post above.

          You certainly have many interesting and provocative ideas, and some of them do correspond to the way I (currently) see things. Like you, I think it makes a lot of sense to say that a mathematical structure becomes "physical" (whatever that means) when it is consciously observed. I really like the way you put in in your analysis of Laurence Hitterdale's essay:

          "But what do you think it might mean for a universe to "physically exist" when considered independently of the presence of consciousness? How would such an "independent existence" differ from purely mathematical existence? In my view, that is what it is: the physical aspects of the universe, insofar as we examine what "exists independently of minds", turn out to be mathematical because the mathematical nature is exactly what remains of the physical universe when the role of consciousness is removed."

          Where our views differ, is that your system is essentially dualistic, with consciousness existing in a separate realm, being NON-mathematical and NOT being subject to physical laws. I agree that our world does seem dualistic, with consciousness residing in a separate plane, but I believe that an Ultimate theory of Existence should reconcile all aspects of existence in a single category (so ultimately it would be monistic instead of dualistic). In keeping with the theme of this year's FQXi essay, I argued in my essay that everything that exists (including consciousness) can be understood in terms of mathematical structures, but I think one could also make an interesting case that everything can be understood in terms of mental structures. So in my view, Existence is monistic, but this monism can be interpreted, from one point of view, as "all is math", and from another point of view, as "all is mind".

          In your comment above, you make very interesting remarks about what gives "quality" to a theory: compressing the observed data, or, equivalently, minimizing the "entropy" of the explanation. You go on to say that the position I argue for in my essay, by avoiding "the issue of specifying probabilities by admitting the existence of all possibilities and qualifying the whole probability issue as a kind of mystery waiting for future elucidation", does not constitute a "very good theoretical job". I am fully aware of this, but what can I do? Suppose it is true that capital-E Existence ("All that exists") does indeed follow from the most simple rule imaginable, "Everything Exists", but that we are not advanced (or intelligent) enough to solve the riddle of the measure problem. Should we just refrain to make that hypothesis?

          As an historical analogy, suppose a philosopher in the time of Kepler was skeptical of Kepler's attempt to explain the fact there were 6 planets by linking the geometry of the solar system with the 5 regular solids, and said instead that the simplest possibility was that every possible solar system exists, but that the hypothesis was, of course, unverifiable with 17th century technology. If you had lived at that time, you would probably have said that the philosopher's idea did not make a "very good theoretical job". Yet, we now know that his idea was closer to reality than Kepler's hopeless attempt to explain "from first principles" the contingent details of our solar system!

          You claim that qualia (sensations and feelings) are an example of something that is not mathematical, and in your essay you also claim that the flow of time (the distinction that consciousness makes between past and future) cannot possibly be described by mathematics. But if mathematics is the general study of structures, that would mean that qualia and the flow of time are not structures. What are they then? I will address this issue when I review your essay on your forum (hopefully, within the next few days).

          About the observed regularity of our universe... you criticize Moravec's explanation (that our universe stays lawful and predictable, even if there are many scenarios where it doesn't, because in these scenarios, our consciousness immediately ceases to exist) with the analogy of the lottery ("If I won at Lotto first, why would it make me more likely to win again another time?"). But suppose that the lottery's prize is Existence itself. Then, you never become aware of the scenarios where you lose, and you always keep winning, against all odds. This is, of course, the main reasoning behind the "Maxiverse Immortality Hypothesis" that I describe in my essay.

          As I said, I read your webpage on the Many-worlds interpretation, and I agree with you that there is no such thing as a number of possibilities, and that there are better ways to describe the thorny issue of the "amount of existence" of the different possibilities within a Many-worlds or Maxiverse model. I will try to be more precise next time when I discuss these issues! :) By the way, I really like the way you describe the Many-worlds worldview, especially when you deal with the issue of how the past evolutionary history of the universe, relative to a given individual, is largely undetermined (a view that is similar to how I view things, and that I have seldom seen so explicitely and clearly stated).

          More to come on your own forum!

          Marc

          Dear Marc,

          Indeed I might also describe my view as a monism, more precisely a mental one, since, as I wrote, consciousness can understand mathematics, but mathematics cannot describe consciousness. However, mathematics is then a remarkably stable part of this mental realm. And we (I mean, some people) have straightforward abilities to describe mathematical systems, that cannot be applied with such a success to non-mathematical ones.

          "I am fully aware of this, but what can I do? Suppose it is true that capital-E Existence ("All that exists") does indeed follow from the most simple rule imaginable, "Everything Exists","

          What can be done: notice that science is no more in its infancy, and that we do already have some effective materials to consider. Namely : is existence described by a probability law or not ? We did find one important probability law, that is the Born rule. So we know that a probability law exists, and we can already describe it. Then, in case we feel lost in our speculations on the nature of existence, we should not forget that there is this thing we know, that needs to be taken into account, so that our ideas remain compatible with it. Now what I criticize about Moravec's explanation, is its incoherent way of presenting arguments that are actually based on probabilistic assumptions (outside which no such arguments can make sense), while trying to deny having any clue on probabilities. My suggestion is to work on clarifying the structure of the probabilistic assumptions that these arguments are implicitly based on. Then, are such probabilistic assumptions compatible with the Born rule ? If you admit a non-probabilistic distribution of existence (i.e. not described by a measure), then in such a framework, how can any probabilistic law such as the Born rule make sense at all ?

          About the flow of time, I wrote that we have an available analogy with the time of the foundations of mathematics, which I described in my site. So time has a sort of structure for which we have a mathematical analogy (better than the pure geometric view of linear order), yet it is only an analogy which does not suffice to describe the effective contents of consciousness and its time flow.

          "qualia and the flow of time are not structures. What are they then?"

          What sort of answer do you expect ? If a mathematical description was given, it would make these things mathematical. However it does not mean that it is a complete mystery, since, anyway they are things we somehow know by personal experience (unlike the concept of "nature" put forwards by materialists).

          Dear Marc,

          If I were to deal with every (from my point of view) objectionable idea, it would simply take too long. Instead, I selected three sections from your essay that I think might be representative of our differences. After each quote, I have added my response to it.

          After these three sections, I continue with a general discussion of the issue, and explain (necessarily without much detail) my own views of these things.

          "Imagine there's only math -- physics is nothing more than mathematics, we are self-aware mathematical substructures, and our physical universe is nothing more than a mathematical structure "seen from the inside.""

          But this starting assumption is just that - an assumption. I don't deny that if you start with this assumption, then your conclusions (and likely a few more) can be said to follow.

          "According to the MUH, physical reality is a web of relationships between entities that are themselves purely abstract: it's "all structure, no stuff," a view that Jim Holt [4] calls cosmic structuralism."

          Only "stuff" can have real-world structure. Abstract structures (represented by drawings, computer simulations, or mathematics) contain no "stuff," and therefore cannot give rise to the (material) universe.

          "But if you accept that a living being can be thought of as nothing more than a complex arrangement of atoms obeying the laws of physics, is it really that hard to accept that a physical universe can be thought of as nothing more than a complex mathematical structure?"

          I think the implied equivalency is extremely weak.

          We are free to define words like "existence" any way we like, but it is most useful to stick with accepted dictionary (usage) definitions. Ambiguity in philosophy or physics is not desirable, so I define "existence" as follows. Only those things that we can detect with our senses (even in principle) or those that we can detect with an enhancement of our senses (with instruments etc., again, even in principle) should be said to exist. If we included abstractions in "existence," it would be mixing very different things. (Of course, without abstractions, we would be leading a very primitive life.) Mathematics, per se, does not exist under this definition (what you see on paper or a computer screen is just ink or "pixels," the rest is happening inside your head).

          Let me try a real world analogy. Consider an ordinary pocket calculator. It has a certain organization of atoms arranged so that when you press certain keys, you get a "desired" display as a consequence. We "say" that it has an internal logic, but what we actually did (in building it) is to utilize how the universe behaves on its own (in this case how silicon and electricity, etc. behave) and then used those behaviors to get the functioning we wanted. We are "exploiting" already built-in behaviors of the universe, but we think that we imputed logic into the calculator (and in an abstract way this is how we speak) while in fact we are only riding on the universe's coattails (to borrow an idiom from another discipline). The calculator is not doing any "mathematics" (although we call it that, and it is useful) - it is simply streaming electricity along different paths, depending on the keys you press.

          The MUH is essentially saying that the implicit rules guiding the various electron streams (we abstract those rules and call them "mathematics") is the "cause" of the calculator itself (or better yet, is the calculator itself). The rules (which don't have a physical existence) cannot create the calculator. And the calculator is not doing any more "math" than a bicycle chain interacting with the gear it is engaged in. The bicycle chain only needs to have the right dimensions to fit the gear, and the rest is done "via that very fact" (no calculations are performed as you ride the bicycle).

          My view is diametrically opposed to the MUH. The universe does no calculations of any kind. It just behaves that way without any mathematics. To put it more colloquially: "the universe does not even know what mathematics is." Mathematics just helps physics to describe (and economically codify) how the universe behaves. Of course, math has applications everywhere, and is not "subservient" to any other discipline.

          Marc, I will give you a high rating on your essay. I like it. I don't feel any need to convince anyone about my views, and I don't think that a belief in what you said in your essay is in any way detrimental to science. I am sure you can think of many counterarguments to what I said, and after all, you could be right.

          En

            Dear Marc,

            thanks a lot for these very interesting and in a sense maximally extreme thoughts. I read through many comments and your replies as well, but haven't found an answer yet to the following questions:

            - How come the F-clone? Assuming I am a mathematical structure, what does it mean that there is a clone of me? I would say a mathematical structure is a set of equations, inequalities, or of axioms, and possibly the full set of theorems that follow from them. So what is a clone of that supposed to be? The same set of equations, axioms, theories and so on once more? I would think, one equation A=B is enough.

            Or do you think of them defining universes in each of which a F-clone of me would possibly live? If so, then some more information is required that defines me in each universe, I could incorporate that information in the definition of "me", and then ask again: how come the F-clone?

            - Is the number of F-clones countably infinite, or which infinity of the many different ones?

            In fact, in writing my own essay, I initially intended to find out the proportion of the set of physical theories in the set of all mathematical theories, the latter corresponding probably to your Maxiverse. But it appears to me that this set is ill-defined, as set theory is a mathematical structure, and if applied to itself runs into logical problems. Do you avoid these problems by adding the qualifier "imaginable"? But who imagines? Is it some mathematical structure that imagines another (in which case the definition is cyclic and difficult to accept), or is it me or you? Then the whole thing becomes very anthropocentric and is difficult to accept as well.

            More generally, I believe that mathematics as we know it is more of a human creation than we might want to believe - already judging from the fact that it uses Boolean logic in which logical variables have a well defined value 0 or 1, and at least the math established from finite axiomatic systems can be obtained from a classical computer program - as opposed to a quantum one. Will a new "quantum math" arise now that we are quite sure that not all observables have well defined value 0 or 1 at all times and that quantum computers will soon be on sale at Macy's? :-)

            Of course you might argue that if so, this just extends the Maxiverse to so far unthought levels. But it makes me suspect that math has a human component to it - which would make the idea of the Maxiverse be too anthropocentric to be acceptable.

            I would be happy to read your thoughts on this. In my own essay I finally settled down to the much more humble and down-to-Earth question what the "size" (cardinality, really) of the set of possible physical theories is, resting with the structure of those that we know. It is infinite as well. But which infinity? If you want to find out you will have to read it :-)

            Daniel

              Dear Daniel,

              Thank you for your comments on my essay.

              Your comments made me realize that I did use the expression "all imaginable [something]" quite a few times in my essay, which begs the question "who does the imagining", and can give the impression that my system is fundamentally subjective, or, as you say, "anthropomorphic". Perhaps I should have used "possible" (or "logically possible") instead of "imaginable" --- that's indeed what I meant to convey.

              You ask a very interesting question about what I mean when I talk about mathematical structures that are different "clones" of myself. You define a mathematical structure as "a set of equations, inequalities, or of axioms, and possibly the full set of theorems that follow from them". What you describe I would rather call a "mathematical system", as I use the term "mathematical stucture" in a looser way. For instance, I would say that the digits of pi are a mathematical stucture, even if they are not in themselves a set of axioms that generates a system of theorems, so, for me, a "mathematical structure" is a generic term that can be used to describe any abstract structure. In my view, what makes up "me" at a given time (my sense impressions at that time, my thoughts at that time and my memories) is a mathematical "sub"-structure that can be embedded in different larger mathematical structures, or universes. In the space of all possible mathematical structures, there are related sub-structures that correspond to slightly "later" versions of me (having memories that correspond to my current direct sense impressions), and the set of all these sub-structures constitutes my F-clones. (This view of "timeless" structures that taken together give rise to a subjective flow of consciousness in time is described in Julian Barbour's book "The End of Time", and also in the sub-section "Living in the moment", on pages 284 to 289 of Max Tegmark's book "Our Mathematical Universe".)

              I have really enjoyed your essay about the ambitious undertaking of defining the cardinality of all possible physical theories. I will leave comments soon on your forum.

              Marc

              Dear "En Passant",

              Thank you for your detailed comments. They are much appreciated!

              Indeed, we all start with basic assumptions. Mine is that there is a "monist" way to understand the world, a fundamental level of reality that can account for all that exists and is self-existing and self-explanatory --- which I identify with "All of Mathematics", an infinite structure that globally does not contain any information (like the Library of Babel of Borges' short story). On the other hand, you start with the assumption that only those things that we can detect with our senses or with an enhancement of our senses should be said to exist. Then, as I said in my original message on your essay's page, your conclusions are very well argued and follow naturally. I really like the way you put it in the your post above, with the example of the pocket calculator. When I put on my "pragmatic physicist" hat (to borrow the expression from Sophia Magnusdottir's essay), I completely agree with you!

              I hope your essay finds the audience it deserves and does well in the contest. All the best!

              Marc

              P.S. I will post this reply on your essay's page also.