Essay Abstract

Mathematics is usually regarded as a kind of language. The essential behavior of physical phenomena can be expressed by mathematical laws, providing descriptions and predictions. In the present essay I argue that, although mathematics can be seen, in a first approach, as a language, it goes beyond this concept. I conjecture that mathematics presents two extreme features, denoted here by irreducibility and insaturation, representing delimiters for self-referentiality. These features are then related to physical laws by realizing that nature is a self-referential system obeying bounds similar to those respected by mathematics. Self-referential systems can only be autonomous entities by a kind of metabolism that provides and sustains such an autonomy. A rational mind, able of consciousness, is a manifestation of the self-referentiality of the Universe. Hence mathematics is here proposed to go beyond language by actually representing the most fundamental existence condition for self-referentiality. This idea is synthesized in the form of a principle, namely, that mathematics is the ultimate tactics of self-referential systems to mimic themselves. That is, well beyond an effective language to express the physical world, mathematics uncovers a deep manifestation of the autonomous nature of the Universe, wherein the human brain is but an instance.

Author Bio

Christine C. Dantas has an undergraduate degree in Data Processing Technology (PUC-RJ/Brazil, 1991), BS in Astronomy (UFRJ/Brazil, 1993), MSc in Astrophysics (INPE/Brazil, 1996) and PhD in Astrophysics (INPE/Brazil, 2001). She is interested in all areas of science and philosophy. Scientific papers can be downloaded at http://arxiv.org/a/dantas_c_1 .

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Your work moreover relates the recommended common views in comparison to the work of mine.

Great Sir!

Best Regards,

Miss. Sujatha Jagannathan

Christine,

I've read your essay with great interest. I'm intrigued by your comparison of mathematics and the physical universe as "autonomous self-referential systems" -- on the one hand not derivable from anything beyond themselves, in a sense all-inclusive, and yet always open to further possibilities. And also by your suggestion that this same special logical structure is "mimicked" in our quest to understand the world we participate in.

There were some points I wasn't able to follow, though they may be important... e.g. that this kind of system "must contain an assertion 'on top' that is not self-referential, to close the ladder of all self-referential expressions, thus admitting an external reference to which it can be defined as autonomous." Could you clarify what such an assertion might be, and what "external" viewpoint you have in mind?

I think the theme of my essay may be closely related to yours - I tried to describe the unique kind of "semantic closure" that pertains to the mathematical language of physics. My emphasis was different, though. Rather than discussing the nature of mathematics per se, I focused on the highly specific and diverse forms of math we find in the physical universe. Also, instead of exploring the logical self-referentiality of the system as a whole, I emphasized the inter-referentiality of the many different parameters and distinct mathematical structures that make up this language. If you get a chance to look at it, I'd be very interested in your thoughts on my argument in relation to yours.

Thanks for a challenging and very thoughtful piece of work.

    Conrad,

    Thanks for reading my essay and your comments.

    Concerning your question, in the context of my essay, autonomous self-referential systems are intrinsically bounded, but I found it necessary to give them an extrinsic meaning as well. This is just to set a reference to which those systems are "seen" as autonomous, apart from being also self-referential. Perhaps "external" is not the best word, but extrinsic. We tend to think of those properties spatially, but there is no spatial meaning at all in the present case, there is no "inside" or "outside", but what can be referred to within the system, by the system, and what cannot be proved consistently within the system, by the system. It is a freely and non-formal interpretation of Gödel's theorem for autonomous self-referential systems.

    I hope this helps a little. I will read your essay opportunely.

    Best wishes,

    Christine

    Dear Christine,

    Thank you for the attention-grabbing essay. You mentioned: "... nature is a self-referential system obeying bounds similar to those respected by mathematics...", I partially share your view, as part of math can be unphysical, similarly part of physics can be unmathematical. Why physics and math intersect is because physics is about quantity and quantity fits math well, whatever is not measurable can't simply be about math, mentioning among other things self-consciousness or teleportation in quantum physics etc. These are also reflected in my essay in more details.

    Kind Regards

    Koorosh

      Dear Mrs. Dantas

      An ineradicable intellectual fashion - probably a survival of logical positivism which for long had exercised a real dictatorship on philosophy of science - reduces mathematics to "a kind of language."

      It is a real pleasure for me to discover in your essay some as original as relevant objections to this commonplace.

      Before we talk about your paper, I wish to reiterate my great astonishment to see so many people persist on the reduction of mathematics to "a kind of language", knowing that some rather trivial reasonings are sufficient to shake these certainties.

      Language in the original sense of the term is a cultural phenomenon which for this reason (i) changes when we move from a linguistic community to another and (ii) evolves over time in an at least partly contingent manner. Mathematics, among humanity, is universal, and any evolution of mathematical knowledge - because "mathematics" and "knowledge of mathematics" do not mean the same thing - is never contingent. The emergence of a new paradigm, such as non-Euclidean geometry, always occurs so that the old paradigm becomes a particular case of the new. When new fields appear, such as probabilities at the time of Pascal, they always represent bonds of unity with already existing fields, or even enhance this unity, as it is the case of Cantor's set theory. (Here, obvious links can be established with (i) "insaturation of mathematics" and (ii) "[nature as a] space for realizations to come.") Finally, Frege already said that changing mathematical symbols does not affect the meaning of mathematical propositions - for example, the jargon of Schwartz distributions prefers "Dt" to "df / dt" and "I" to "∫" without changing anything at this level) - whereas contingent changes in the configuration of phonemes regarding a given language would make the latter ineffective on its role in communication.

      In short, even at this basic level, a gap opens between mathematics and "a kind of language", but by going further, epistemological discernment inevitably leads to your starting assumption. If we consider mathematics as a kind of language, it is impossible to attempt a rational explanation of this "unreasonable effectiveness" of mathematics in physics. To try such a rational explanation, we first have agree the idea that mathematics must be something else / more than just language. Your final words mentioning the famous Galileo quote aptly summarize the problem: It is not sufficient that the great book of philosophy (here philosophy of nature) is written in mathematical terms. This book can be adequate if and only if th nature in turn "behaves mathematically", and if this is the case, the reduction of mathematics to a language like any other can not be satisfactory.

      I totally agree with you that mathematics can be used as language since its allows the communication of laws of nature, but this does not explain how or why nature "behaves" in a mathematizable way. For the "unreasonable effectiveness" of mathematics in physics to be rationally explainable, there must be some deep analogy between mathematics and nature.

      Regarding this analogy, your approach in terms of self-reference and insaturation is as interesting as convincing.

      Nature is there because it is there, knowing that it could although not be there. On the other hand, nature is becoming, i.e. "a space for realizations to come", as you say. (If you envisage a longer study, you might perhaps reformulate the latter point in terms of block universe suggested by relativistic constraints. The block universe, certainly controversial, is not inconsistent with "a space for realizations to come", but requires reformulations. Since you are reading Jean CavaillГЁs dans le texte, you manifestly speak French; therefore I recommend you, in this specific context, the french paper of Fabien Besnard "Temps des philosophes, temps des physiciens, temps des mathГ©maticiens" http://fabien.besnard.pagesperso-orange.fr/articles/temps.pdf . Regarding mathematics, please allow me to make a little remark. When you say - with CavaillГЁs - that "Mathematics constitutes a becoming (...)", you risk claiming in spite of you - and despite CavaillГЁs - constructivist views opposing your own self-referential approach of mathematics. It is for this reason that I insisted above on the anti-constructivist differentiation between "mathematics" and "knowledge of mathematics." Our human knowledge of mathematics is obviously becoming ("en devenir" chez CavaillГЁs), but this does not conflict with the conception of mathematics existing per se, timelessly, irreducibly to anything, in short, existing only by self-reference. Certainly, in absolute terms, all approaches claiming objective existence of mathematics are metaphysical theories, nor provable nor refutable. But in my own essay, A Defense of Scientific Platonism without Metaphysical Presuppositions by Peter Martin Punin, I try to show (i) that competing theories of this ("Platonistic") approach are in turn metaphysical theories and (ii) that, compared to its (neither less, nor more metaphysical) competing theories under epistemological criteria such as simplicity, economy of assumptions, consistency and so on, approaches claiming objective existence of mathematics as edifices irreducible to anything are still the most plausible. Here let -me just mention a historical example which, in my opinion, denotes that mathematics exists before being discovered. As you know, Saccheri, arround 1730, tried to prove Euclid's parallel postulate by reductio ad absurdum. He thought that replacing the parallel postulate by one of its both negations would entail a lot of inconsistencies. But these inconsistencies did not occur, could not occur. Despite himself, unwittingly, Saccheri was making non-euclidean geometry, and this almost a century before it was discovered by Bolyai, Lobachevsky and others. Deduce from the previous point that non-euclidean geometry existed before its discovery is perhaps metaphysical. But explaining the necessary failure of Saccheri on the bases of ultra-formalism or constructivism would lead to arguments approaching farfetchedness.

      Let us therefore assume the objective existence of irreducible mathematics. From this perspective, knowledge of mathematics is insaturated, since at any time, something remains to be discovered. However, from an ontological perspective we can not speak of unsaturation of mathematics; the latter exists as such. By contrast,we can mention the "physical" insaturation of mathematics, in the sense that only a small part of mathematics is interpreted by nature. But, on the other hand, nature as "space for realizations to come" tends to invest "physically" insaturated mathematics. Finally, since our knowledge of nature is in turn insaturated, discoveries about nature can favor discoveries in mathematics.

      Please, just tell me even if my interpretation of your final words is correct:

      Laws of nature and mathematics are the same, except that mathematics transcends knowledge areas covered by laws of nature.

      Well, I hope, that my understanding of your essay is not too far from what you mean. Anyway, this reading was actually beneficial for me. I obviously have to apologize for my English, but I am (i) not a native speaker and (ii) always under time pressure, knowing that il faut de tout pour faire un monde. And if you have the time, it would be a pleasure to know your comments on my own contribution My paper is unfortunately not animated by the admirable humor being yours. It may be a matter of personal style and probably also of generation.

      With best regards

      Peter Punin

        Dear Koorosh,

        Thanks for reading my essay and your comments. The bounds that I refer to do not concern the (direct) intersection of physics and mathematics, although this is certainly a point often noted (and I mention it peripherally), but concerns bounds to self-referential systems, which both mathematics and nature seem to obey in different strata. I'm preparing a list of essays to read as time permits, and yours is in my list.

        Best,

        Christine

        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

                        Dear Christine,

                        Thanks for your kind reply with the answer to my point 4). Yes, you are correct: I strongly appreciate "thinking outside the box".

                        Cheers, Ch.

                        Dear Dr.Dantas,

                        I posted a comment at your site that was unnecessarily contemptuous and devoid of the civility all contributors are entitled to. I deeply regret having done so, and I do hope that you can forgive my slurring of your fully deserved reputation.

                        I suspect that I may be suffering a relapse of Asperger's Disorder. While this might explain my distasteful action, it cannot in any way justify it.

                        Respectfully,

                        Joe Fisher

                          Dear Joe,

                          No need for apologies, so please do not worry about it at all. You may place your criticisms to my essay here any time at your discretion and if you think there are connections to your essay, you can also mention them by giving the specific points.

                          Take care and good luck.

                          Christine

                          I would like to add this fine quote by Fichte:

                          The highest interest, and hence the ground of all other interest, is that which we feel for ourselves. Thus with the Philosopher. Not to lose his Self in his argumentation, but to retain and assert it, this is the interest which unconsciously guides all his Thinking. Now, there are two grades of mankind; and in the progress of our race, before the last grade has been universally attained, two chief kinds of men. The one kind is composed of those who have not yet elevated themselves to the full feeling of their freedom and absolute independence, who are merely conscious of themselves in the representation of outward things. These men have only a desultory consciousness, linked together with the outward objects, and put together out of their manifoldness. They receive a picture of their Self only from the Things, as from a mirror; for their own sake they cannot renounce their faith in the independence of those things, since they exist only together with these things. Whatever they are they have become through the outer World. Whosoever is only a production of the Things will never view himself in any other manner; and he is perfectly correct, so long as he speaks merely for himself and for those like him. The principle of the dogmatist is: Faith in the things, for their own sake; hence, mediated Faith in their own desultory self, as simply the result of the Things.

                          But whosoever becomes conscious of his self-existence and independence from all outward things--and this men can only become by making something of themselves, through their own Self, independently of all outward things--needs no longer the Things as supports of his Self, and cannot use them, because they annihilate his independence and turn it into an empty appearance. The Ego which he possesses, and which interests him, destroys that Faith in the Things; he believes in his independence, from inclination, and [seizes] it with affection. His Faith in himself is immediate.

                          --Fichte,

                          Introduction to Fichte's Science of Knowledge

                          http://en.m.wikisource.org/wiki/Introduction_to_Fichte%27s_Science_of_Knowledge

                          8 days later

                          Christine,

                          You are one of the few people I seek out in these essay competitions, because I know you will always have something worthwhile and thought provoking to say.

                          (Have you read Jose Luis Borges's short story, "The Library of Babel?" I would be surprised if you haven't.)

                          Anyway, mathematics, or meta-mathematics, can't define mathematics? I would call that a meta-definition, A complete bounded set of models, after all, would assume a bounded universe -- while our most substantiated physical cosmology (general relativity) is based on what Einstein called "finite and unbounded". Just something to ponder.

                          You deserve my highest rate, and I hope you get a chance to visit my essay.

                          All best,

                          Tom

                          Dear Thomas,

                          Thank you for the kind words... I am flattered! Yes, I have read several books and short stories by Jorge Luis Borges, including Bioy Casares, I enjoy Latin American Fantastic Fiction. Yes, I'll be reading your essay as soon as time allows. Good luck and best wishes,

                          Christine

                          Christine,

                          Interesting concept of a self-referential system, saying we create the world we're trying to forecast. "Math and physical laws are a pure self-metabolism," allowing for it all to grow and reproduce? I don't see adherence to the anthropic principle but isn't it anthropomorphic? Or not?

                          My connections (math, mind, physics) simply accepts math's utilization as a tool of describing the physical world with missteps peer reviewed (BICEP2) and studies of the classical world leading to discovers in quantum biology, etc.

                          I would like to see your thoughts on that.

                          Jim

                            Dear James,

                            Thanks for reading and posting your comment here. I don't think it is anthropomorphic in the sense of placing humans as the most special conscious beings in the Universe, while practicing their mathematics for self-discovery. We would be just an instance of that practice, which is probably occuring elsewhere as other instances, by means that we may -- or may not -- understand yet.

                            Your paper is on my reading list, thanks!

                            Best,

                            Christine

                            Dear Christine,

                            You write "Mathematics represents the ultimate tactics of self-referential systems to mimic themselves", a sentence that summarizes most of your arguments that Nature is a self-referential system and mathematics is the best (and irreducible) way to describe itself. If this is so, Physics is in some sense mathematics: Tegmark's thesis. Also a system of axioms cannot escape itself: more or less Goedel's findings.

                            As you don't provide explicit examples, I have in mind the material in Yanofsky's essay (e.g. Hilbert's Nullstellentsatz), it is not easy for me to follow you very far. I accept that self-referential systems are plethora in nature but it does not mean that they are compatible, they may even be a multiverse of possibilities with possibly different kinds of mathematics.

                            One noticeable point in your text is that the human brain is an instance of nature self-reference. It may be that, oppositely, the human mind cannot avoid self-reference, this is in no way equivalent to the self-referential character of nature and more compatible to Hawking's sentence at the end of my own essay.

                            In summary, although your essay is very well written, I feel not satisfied. Let me know if I misunderstood you.

                            Best,

                            Michel

                              Dear Michel,

                              Thank you for reading my essay and posting your comments. It is perfectly fine to be not satisfied, I certainly am not myself, as what I wrote was just some tentative first steps into my investigation. My points are completely compatible with different kinds of mathematics, in fact I firmly believe that is possible and even quite natural. I need more time in order to write a more detailed comment, though. As for your point on Tegmark, I disagree. See my last answer (point # 1) to Sylvain a few comments above.

                              Best wishes,

                              Christine

                              Dear Dr. Dantas,

                              I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

                              All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

                              Joe Fisher

                              7 days later

                              Christine,

                              Time grows short, so I am revisiting essays I've read to assure I've rated them. I find that I rated yours on 3/31. I hope you get a chance to review mine.

                              Thanks for sharing yours.

                              Jim

                              Dear Christine,

                              I found your essay very interesting. I understand the logic behind self referential systems and the need for an extrinsic reference. However I cannot grasp where is the external / extrinsic reference of mathematics. Would it be possible to help me seeing the extrinsic reference?

                              Regards,

                              Christophe