Dear Mr. Punin,

Thank you very much for taking the time to read my essay carefully and writing thoughtful and enriching comments here.

I need more time to think about your considerations, and as soon as I have something to add, I will comment here. Your essay sounds intriguing and I shall read it opportunely. No need to apologize for your English, which I find very good. I am not a native speaker as well. As for Cavailles' work, I read it in a Portuguese edition, as I indicated in the references. I can read French, but somewhat poorly. I was fortunate to have found that volume on Cavailles' works (I was completely unaware of this philosopher/mathematician just about a year ago). Yes, I am also under time pressure too, and it has been difficult for me not only to find time to write the present essay, but also, as it happened with previous contests, to read the other essays with dedicated attention. Hopefully, this will be a better year for reading.

Best regards,

Christine

Dear Christine

Thank you for your kind reply. I understand that you do not have much time. Of course, you will read my paper when the opportunity arises. Can I just ask you to report briefly on my own page your reactions to the various essays which may concern me, so I could participate in the discussion?

For my part, I will recommend your essay to Mr. Al Schneider claiming that mathematics would be "simple descriptions, 'something like English.'"

Best regards

Peter

    Great, Christine, I do like your vision here, already for its general approach to mathematics as something at least seriously comparable to natural self-referential systems (including life and consciousness). Myself, I would insist even that it should eventually "mimic" them well enough, by reproducing all their observed specific features in their unreduced quality (implying also respective technological advances). I am not sure that you adhere to that ultimately strong statement, as it is evident, for example, that the standard mathematics is not really at that level. By the way, I propose an extended mathematics attempt with that "total self-reference" property at this competition (Extended Mathematics). In any case, I like the logic of your approach and its universality.

      Dear Andrei,

      Thanks a lot for you comment and for taking the time to read my essay. I have just posted over at your entry page.

      Best wishes and good luck in the competition!

      Christine

      Dear Peter,

      Oh, if I find the time, yes, of course, I will report over at your entry page. Your essay is on top of my list. I have browsed it already, but I need more time to read it carefully, as it has many details.

      Yes, the view of "mathematics being just a language" is shared by (perhaps) most people. I do not claim this to be entirely incorrect, but just a way to see it, which nevertheless is incomplete (as stated in my essay).

      Best,

      Christine

      Christine - A magnificent essay, striking to the heart of the issue - self-reference.

      Could you explain the nature of the origin (or creation) of an ASR? I see the statement that "an autonomous self-referential system must contain an assertion "on top" that is not fundamentally self-referential, in order to close the ladder of all self-referential expressions, thus admitting an external reference to which it can be defined as autonomous". Does this suggest that ASRs are not self-starting, but flow, in some sense, from something "on top"? To use an analogy, they are "jump-started" rather than "boot-strapped."

      I think we are saying very much the same thing but in quite different language. I would love to have your thoughts on my essay.

      With great respect - George Gantz

        Dear Christine Dantas

        An interesting essay on a topic with which I am very familiar. Your conjecture about self-referential systems in physics is indeed correct. There-exists a whole class of self-referential systems for which I have proven any physics theory modelling the system components directly will be subject to Gödel's incompleteness theorem (see my paper). In my 2012 FQXi essay

        I discussed the consequences of every particle in physics being a self-referential dynamic state of particle interactions (what quantum theory says of particles) - any non-quantum theory attempt to model the components of the dynamic particle interaction state would be subject to Gödel's incompleteness. Since Gödel's result depends on natural-number based maths it can be bypassed by switching to a real-number basis in a physics theory. If you do this for the particle case, what you find is mathematically the same as quantum theory. I discuss further in my 2015 FQXi essay how quantum theory is inevitably what you get from having to bypass Gödel's incompleteness to get a usable physics theory for particles because of their self-referential character.

        In my 2013 FQXi essay I discussed how other self-referential systems in system would also be subject to Gödel's incompleteness. I also gave an overview of the physical conditions required for self-reference and how they divide physics into Object Physics and Agent Physics. The trick that can be used for quantum theory won't work in other areas of science and in Agent Physics I look for alternative approaches to self-reference and independently arrived at the same overall approach discussed by Andrei Kirilyuk. I am impressed by his work and it looks a very promising approach to address self-reference in science - beyond that underlying quantum theory.

        Regards

        Michael Goodband

          Dear Michael Goodband,

          Thanks for your comment, and I find it very interesting to see that various approaches and ideas do have some common, maybe intuitive or qualitative intersections with respect to self-reference notions. Yes, I should be reading your essay(s) with great interest as time allows. Good luck!

          Best wishes,

          Christine

          Dear George,

          Thanks for your comments, they are appreciated. It is difficult to give meaning and instant clarification to deep and so little understood concepts using few words in a comment section. But in an intuitive sense, and forgive the lack of formal/rigorous response, I would rather not address the "origin" of a self-referential system such as Nature. But I do think that irreducibility, as explained in my essay, points towards the need of a generating "seed" as the most elementary self-referential element. This is a quite fundamental and necessary property, but I cannot tell why. All the implications that I have seen are stated in my essay.

          Yes, I have several essays to read, as time allows. Every one who comments here will have their essay in my list, apart from others that I find interesting. My problem is to find the time. Thanks.

          Best,

          Christine

          Dear Dr. Dantas,

          Could you please explain to me why you thought that my comment about the real Universe was inappropriate?

          You are I hope aware that suppression of the truth is unethical.

          Eagerly awaiting your answer,

          Joe Fisher

            Dear Joe Fisher,

            I did not report it as inappropriate. It is true however that I have stubbed it as I considered that your comments did not add any value to the discussion of my essay, as it only promoted yours. To my understanding, to stub only means not to show it entirely in the thread.

            Best,

            Christine

            Dear Christine,

            I just recommend your essay to Aleksandar Miković. Since it would be interesting that you reformulate your own essay in terms of modified block universe, I also recommend you the paper of Mr Mikovic. http://fqxi.org/community/forum/topic/2443

            There are complementarities.

            Best regards

            Peter

              Dear Peter,

              Thank you for the kind attention. I shall read it opportunely.

              Best wishes,

              Christine

              Dear Dr. Dantas,

              Thank you for your explanation. How may I stub a few of the aggravating comments from my essay's site? Why does the moderator of the sites not know when an inappropriate item has been removed and when it has been stubbed?

              Joe Fisher

              Hello. My impression from your essay is that you need to express conjectures about the foundations of mathematics just because you did not understand clearly enough what these foundations actually are. Since years I work to explain the foundations of mathematics as clearly as possible. This includes philosophical aspects, however these aspects are expressed so clearly that they can almost be qualified as mathematical expressions. I say "almost" because of course absolutely mathematical formulations would lack their self-justifications as they would be subject to the incompleteness theorem, which I avoid by using concepts not purely mathematical, however I still manage to make them clear.

              I'm not sure what you mean by "It is clearly very hard to develop an independent methodology to avoid the ironic situation of using mathematical principles themselves in order to explain mathematics. This is not, evidently, the purpose here, so we limit to qualitative statements, on a more "meta", abstract level."

              The second sentence claims to continue the first, yet seems to actually contradict it, since your following statements being "qualitative", i.e. vague, are not mathematical. In this sense, you do (claim to) avoid using mathematical principles to explain mathematics, which the first sentence qualifies as "very hard".

              On your first conjecture of "irreducibility". Sorry I don't agree, as I consider the mathematical reality as a pervasive one, i.e. it cannot (or can hardly) be absent from anything, including non-mathematical realities : we can (often) obtain mathematical structures by taking non-mathematical things and then discerning mathematical structures there as approximations ; in my exposition of the foundations of maths I start by explanations in natural language to progressively build mathematics. The claim I hold is the opposite one : non-mathematical realities, namely consciousness, are irreducible to mathematics.

              On your second conjecture : "Mathematics cannot, as a whole, be constructed from a "master impredicative". First, you did not rigorously define what you mean by "impredicative" or "self-referential system", as, first, what do you mean by "system" ? Do you mean a formula, a theory, or possibly something else ? I don't see the need to state vague conjectures on how mathematics can be constructed, since I find it much better to specify how mathematics is actually constructed, then you can just look at this construction to figure out how you wish to qualify it. Namely, I found mathematics to be constructed from an interplay between set theory and model theory. As I wrote in my introduction : Set theory describes the universe of «all mathematical objects» ; Model theory is the general theory of theories ; each one is the natural framework to formalize the other. The incompleteness here is in the fact that an arbitrary choice is needed for the axiomatization of set theory, and any specific choice is unable to formally prove its own consistency. Yet I found a way to introduce all concepts in a seemingly logical order.

              We have similar ideas on the nature of time. Indeed I see the incompleteness of mathematics as an expression if a time in mathematics, where only the past actually "exists" or is known ; this time is similar but independent of our time.

              You wrote "an autonomous self-referential system is irreducible to anything else that is not itself self-referential". In which sense is this not directly refuted by what I call the Self-quotation theorem (which is an intermediate step to the proof of the incompleteness theorem) ? I mean, my formulation there seems to assume the ability for a theory to describe itself, but this does not really require self-reference since it only results from the general ability to describe formulas, among which the copies of its own formulas come as particular cases. Actually the hard work of Godel was to prove the possibility for arithmetic to describe formulas, including the formulas of arithmetic itself, while the axioms of arithmetic do not formally contain any self-reference.

              I wrote a page of general review of many essays of this contest, with a list of best essays, and for the defense of mathematical Platonism.

              Dear Sylvain Poirier,

              Thank you for you detailed comments. If you are an expert on the foundations of mathematics, having studied it for years, then I congratulate you for the hard work. I looked at your site, but not in detail. I shall do that opportunely.

              My response to your questions and comments follows.

              -------------------

              1) You wrote that I "need to express conjectures about the foundations of mathematics just because you did not understand clearly enough what these foundations actually are."

              I do not claim to be an expert, but I have been studying and thinking about the matter for a while. My ideas are here exposed to the open criticism of others and I am glad that you read them and criticized them. I do not claim that these ideas are completely or formally developed, and I clearly stated that in my essay. But I do think they bring a level of originality and relevance, otherwise I would have not submitted them.

              You do not seem to have realized that the conjectures that I wrote are not claimed to be *the* conjectures of mathematics, in the sense that they would be a kind of substitution of current foundations. My proposed conjectures offer a different way to see mathematics.

              2) You wrote: "I'm not sure what you mean by "It is clearly very hard to develop an independent methodology (...)"

              If you explain what mathematics is by using mathematics, then you are being circular. This is what I meant.

              3) You wrote: "On your first conjecture of "irreducibility". Sorry I don't agree, as I consider the mathematical reality as a pervasive one, i.e. it cannot (or can hardly) be absent from anything, including non-mathematical realities (...)"

              Your criticism shows that you read my essay only superficially. First, see my paragraph associated with footnote 3. Second, your criticism indicates that you did not understand my conjecture. It does not refer to non-mathematical things, whatever your definition of them. My conjecture refers to mathematics. If you associate "consciousness" with non-mathematics, that is your conjecture. Nothing about consciousness or whatever is stated in my conjecture. I only state and explain why mathematics is irreductible.

              4) You wrote: "First, you did not rigorously define what you mean by "impredicative" or "self-referential system", as, first, what do you mean by "system" ?"

              I do not rigorously define "impredicative" and "self-referential systems", but I do define them briefly and link to references in footnote 6 for further details. Again, you seem to have not read my essay carefully. About what "system" is, I did not find it necessary to go down to that level. One cannot write a short essay if having to engage into infinitely regressive semantical inquiries. All common terms are fixed to the dictionary meaning, unless otherwise specified. So "system" has the definition meaning according to the context of where it is used in the text. Note that there is no occurrence of the word "system" in the second conjecture, but "self-referencing mathematical formulations". Hence you can infer the nature of "self-referential system" from that.

              Following the above comment, I can only find references to your theories, so I will not comment them now.

              5) You wrote: "You wrote "an autonomous self-referential system is irreducible to anything else that is not itself self-referential". In which sense is this not directly refuted by what I call the Self-quotation theorem (...)".

              I do not see a contradiction with the "Self-quotation theorem" that you describe; actually, both share a correspondence, although I am not certain which of them is more general (see that my footnote 13 could possibly absorb the theorem you quote). In any case, the point of my note on page 5 was exactly to emphasize the qualification "autonomous". See also my footnote 7 for assumptions made.

              ----------------

              Thank you for sharing your link with your considerations. I am not sure I would place myself in the classification that you defined, because I do not claim that mathematics is a kind of "ultimate reality", as in Tegmark's view. I only address the correspondences between mathematics and nature (as perceived by physics, the requirement of the present contest). I admit the possibility that mathematics or the universe as we perceive it might be not the ultimate stratification, if there is one at all.

              I hope my response clarify your questions.

              Best,

              Christine

                Dear Christine,

                I beg to differ on the claim that my remark would reflect a mere superficial view of your essay. Maybe your essay just carries your own paradoxes which were not so clearly stated, or maybe I was just not so clear in my comment.

                If you also consider mathematics to be pervasive in all reality, by reading the remarkable usefulness of mathematics in physics in the sense of qualifying all reality as "mathematically expressible", well why not, in which case it seems that you are just making your "conjecture" that "Mathematics is irreducible to anything else that is not itself mathematically expressible" trivially true by the mere fact that everything that exists would be mathematically expressible, so that of course, everything which exists, such as mathematics, will be irreducible to any non-existing thing. Which is then not such an interesting conjecture anymore. In this case it would have been interesting to specify which non-trivial sense you intended to give to your conjecture, which would require to discuss the possibility of non-mathematical systems and what they may look like.

                Do you mean, then, that the circularity problem you point out with defining mathematics in mathematical ways, is a problem that probably cannot be resolved ? The paradox (misunderstanding) then, was your non-mathematical way of discussing the foundations of mathematics without seeing this as any try to explain mathematics in any non-mathematical manner.

                I did look at the wikipedia article on "impredicativity" however I consider that just because there is a wikipedia article about a word, does not mean that this word makes any clear sense. In particular after reading wikipedia on impredicativity I keep the feeling that it does not make so much sense.

                "About what "system" is, I did not find it necessary to go down to that level". This is precisely what I meant when I complained about the lack of sense of "self-referential systems".

                You wrote "One cannot write a short essay if having to engage into infinitely regressive semantical inquiries". I recognize the difficulty of making a clear and short essay, however I do not consider it a justification for writing nonsense; I cared myself to be clear in my own essay in spite of the very hard constraint of making so sort the large number of ideas I decided to express. The problem is whether your short unclear presentation really means some clear sense that exists behind and that you are able to explain (or that you did explain in another article), or if it looks unclear just because it really is unclear and that is really all what you have to say about it.

                There is a clear sense for "system" in mathematics, as I explained in my site, however the adjective "self-referential" is not applicable to it. To make this meaningful you need to specify what kind of system you are talking about.

                "Note that there is no occurrence of the word "system" in the second conjecture, but "self-referencing mathematical formulations". Hence you can infer the nature of "self-referential system" from that. "

                If you mean here that by "system" you always meant "formula", it should have been made clearer from the start. I know what is a self-quoting formula. This is clear as I presented in my site. What is unclear is whether you specifically meant this or if you could mean something else as well, but what ?

                You wrote : "I do not see a contradiction with the "Self-quotation theorem" that you describe".

                This theorem precisely shows how to construct self-quoting formulas out of any theory just able to generally handle formulas without any axiom of self-reference ; in particular, as Godel showed, the theory of first-order arithmetic suffices to make this, despite the absence of self-reference in the axioms of arithmetic. So, self-quoting formulas can be written inside (thus are reducible to) theories just able to express formulas, such as arithmetic, which are not themselves self-referential. For this to not refute your claim that "an autonomous self-referential system is irreducible to anything else that is not itself self-referential", is it because these self-quoting formulas are not autonomous systems (as they describe natural numbers, and belong to the theory of arithmetic) ? But in this case I do not see what you are talking about with your sentence "the most elementary self-referential expression must be the primordial one (the generating "seed"), otherwise the system cannot be autonomous, in the sense of self-generating". I'm not even sure if you really mean something or if you may just be telling nonsense. Can you give any effective description of what do "autonomous self-referential systems" and this "generating seed" look like ? And if you cannot give any example of what you are discussing, then why not dismiss all your essay as nonsense ?

                Your footnote 7 just confirms my above remark that the very notion of impredicativity is there in the dictionary as a mere fossil of some old struggles with foundations, from a time when the irrelevance of this notion was not clear yet.

                "I am not sure I would place myself in the classification that you defined" No problem, I just removed you from my classification then :-/

                Best,

                Sylvain

                Dear Sylvain Poirier,

                1) You wrote: "(...)it seems that you are just making your "conjecture" that "Mathematics is irreducible to anything else that is not itself mathematically expressible" trivially true by the mere fact that everything that exists would be mathematically expressible, so that of course, everything which exists, such as mathematics, will be irreducible to any non-existing thing."

                You seem to be making a confusion between object and property. I wrote on page 3:

                "When a physical law is expressed mathematically, no evidence for the "truth" regarding the law actually surfaces, but only a guarantee for the logical foundation of the outcomes resulting from the implemented assumptions. In fact, any application of mathematics (seen as a form of language) to physical problems just brings forth a relative valorization of their logical evidence, but not the evidence, per se, of their truth."

                The observational fact that everything can be mathematically expressed does not imply that everything is fundamentally mathematical. Such observations only make evident a phenomenological recognition made by our brains that sensible objects are logically intelligible, hence mathematically expressible. But to claim that there is an ultimate reality that is mathematical because everything that we perceive can be mathematically expressed is not implied, as far as I see. My conjecture does not make any reference to that.

                2) "Do you mean, then, that the circularity problem you point out with defining mathematics in mathematical ways, is a problem that probably cannot be resolved ?"

                I only claimed it to be very hard.

                3) "I recognize the difficulty of making a clear and short essay, however I do not consider it a justification for writing nonsense; I cared myself to be clear in my own essay(...)

                Congratulations to you for having succeeded. You have the right to claim that my essay is nonsense, and I am trying my best to respond to your criticisms. However, I am not unexperienced nor malicious. I would never waste my time or that of potential readers by submitting "nonsense".

                I wrote: "The proposed conjectures are philosophical, a fact that could be unattractive to some readers. However, these ideas could eventually be expressed in a more concrete or formal way, so they should be regarded as preliminary for the purposes of the present essay."

                I have clearly stated that my ideas are preliminary, under development, and are not presented in a formal way. You have the right to criticize my essay, to dislike it and call it nonsense. However, I find your tone a diminishing one, and unfortunately I do not see how we can continue in a constructive way. Thank you for removing my name from your site.

                Regards,

                Christine

                Hi Christine,

                It is a pleasure to meet you again in FQXi Essay Contest. You wrote an intriguing Essay also this year. Here are my comments:

                1) Your statement that "A rational mind, able of consciousness, is a manifestation of the self-referentiality of the Universe" seems an interesting definition of the anthropic principle.

                2) From the point b) in page 3 it seems that you consider mathematics as intermediate between trick and truth. This seems confirmed by your statement that you "believe that mathematics is more than a "transformation machine"".

                3) If your conjecture of insaturation is partially based on Godel theorems, it is something more than a conjecture.

                4) Your principle that "Mathematics represents the ultimate tactics of self-referential systems to mimic themselves" is interesting, but a question emerges: why does a self-referential system need to mimic itself?

                5) I have been always fascinated by the Golden Ratio. You could be interested on an intriguing connection between Golden Ratio and hydrogen atom which was found by Petrusevski.

                In any case, the reading of your nice Essay gave me a lot of fun. It deserves a very high score that I am going to give you.

                Based on your scientific interests, you could be interested on my Essay, which finds a new proof for the general theory of relativity trough a pure geometrical analysis.

                I wish you best luck in the Contest.

                Cheers, Ch.

                Hi Christian,

                It's very nice to meet you again, my near-namesake fellow! I look forward to read your essay, it was already in my list. Thank you for reading my essay with the right "spirit" and for placing your comments here.

                I think you saw the point of my essay (equally valid to previous contests): to bring forth some unusual ideas (at least, I think they are unusual). I do not see the point of participating in the FQXi contests if one cannot bring unusual, different ideas, even if they are not fully developed. If they are fully developed, there is no place for them in a FQXi contest, at least the way I see it. So it is true that I sacrifice rigor but I think it is worthy to a point. Thanks a lot to have understood that. I also think you share the same ideal in your essays, so I am looking forward to reading yours opportunely.

                As for your question #4, it is implicit, deep down in what I wrote, that this must be a result from an autonomy property, and deeper down, it is about freedom. I do not make this clear, as I cannot develop such ideas any further, not in the present moment. This relates to free-will.

                Good luck!

                Christine