Dear Christophe,

Thanks for reading and for your question.

In an extrinsic view (but see last paragraph below), a self-referential object X would be equipped with a series of maps to itself, the endomaps {a}, and its evolution to another self-referential state, Y, with endomaps {b}, would be carried out by some extrinsic map, say, f: X -> Y, wherein the composition f o a = b o f holds, for all a and b.

But for an intrinsic view, the {a} have a new meaning within the system: a map a_i operates with itself and this must result in a "change of state" within X. That is, the composition a_i o a_j = a_k (not necessarily i=j) results in a new state Y, with self-referential operators {b}, mixed from the original ones {a}, which I suppose to be countably infinite. This requires an "autonomy" for the evolution, which is fed by a combinatorial, potentially non-exhaustive, set of endomaps. This self-characterization acquires an internal meaning as "active", and externally as "autonomous".

On the other hand, the external map f has no meaning inside the self-referential object. So it is difficult (or impossible) to characterize it-- a different mathematics, perhaps?

Best,

Christine

    Dear James,

    Yes, time runs... And my reading list is late. Hope to be able to read yours and others! :/

    Best,

    Christine

    Thanks. You might be interested by the following book: Gödel, Escher, Bach: An Eternal Golden Braid by Douglas R. Hofstadter. Beware, it is long, 800 pages!

    Regards,

    Christophe

    I've read it many years ago. It's one of my favorite books.

    Regards,

    Christine

    5 days later

    Dear Christine,

    You make very interesting points and you express them in an excellent writing style. I enjoyed your unique approach, and I think I understand your point and agree. I like that you go beyond the viewpoint that mathematics is an "ovely-effective language", and see it as being much more than this. It is a good point you are making that no value of 'truth' can be extracted beyond the actual framework that is used, implying the truth of the axioms. You are saying that "an ultimate regression that is not mathematically based is impossible" and I agree because the patterns underlying math cannot be expressed by something that contains no pattern and such an exercise would indeed bring very little benefit or further understanding. I liked very much your conjectures, both irreducibility and insaturation, and also the "Mathematics is an existence condition for autonomous self-referential systems, in particular, the Universe", they clarify the role of mathematics!

    This was a very good read and wish you good luck in the contest!

    Cristi Stoica

      Dear Christine C. Dantas,

      I quite enjoyed the style of your essay.

      Given the title, I was expecting a reference to the work of Wheeler. ;-) Well, you did cite Gödel as a source of inspiration for the second conjecture. To me, it was a nice surprise to see that you cited Cavaillès. (I have only recently read some of his work, because a colleague pointed me towards it, although it was not connected to the current subject.)

      What attracted me in the abstract -the reason I put this essay on my reading list- was that you question the Galilean assumption of mathematics as a language. So, the part I liked best was the ending. I wonder why you did not use that part as the starting point. As the essay was presented now, I didn't find the conjectures entirely convinving: there seem to be some gaps in the argumentation. For instance, you describe mathematics as "overly effective" in general. I agree that we can apply mathematics to anything we like, but this doesn't show that it is always appropriate, effective, ... to do so. In some cases, a lot may be lost in the 'translation' (to stick to the language metaphore) to a mathematical description. Couldn't this be a case of seeing only nails when all we have is a hammer?

      Best wishes,

      Sylvia Wenmackers - Essay Children of the Cosmos

        Dear Christine,

        I enjoyed your essay very much (especially the conjecture nickname's, they are very cute). I am in agreement about many of the aspects in which you connect nature as what is physically manifest and math as what is possible. And, in particular that the connection between the two is fundamental. I've been thinking along similar lines recently (including in my essay here in this contest).

        One thing I will point out is that self-referential systems, while dynamically not reducible to non-self-referential systems can converge on attractor states that look like they are not-self-referential (can be fully explained by local rules). I think this is significant because it suggests that most of what is important in invoking self-reference is to explain how we got to a certain state and not necessarily the state itself.

        For example, I would argue that there are lots of states of the world that can exist - like conscious beings - that are consistent with physical reality that is not-self-referential if you look at a static snapshot, but that you would never be able to explain how they came to exist without it. I think your viewpoint is similar, but you seem to take that self-referential structure itself as what is fundamental, whereas my view is that is an emergent property of universe - is that an accurate interpretation?

        Best,

        Sara

          a month later

          Dear Cristinel,

          Thank you and I wish you good luck as well.

          Best,

          Christine

          Dear Sylvia Wenmackers,

          Thank you for your comment, and I wish you good luck with your essay.

          Best.

          Dear Sara Imari Walker,

          Thanks. Yes, self-reference is seen as fundamental (and actually I see it as related to concurrent processes, a theme of a previous essay of mine).

          Best.

          5 days later

          Dear Christine

          Many congratulations for FQXi contest. It is true that my personal view your test was the most interesting, with that of Alexey and Lev Burov, ex aequo in the ranking. It is a great pleasure to have met you on this occasion.

          Bonne continuation,

          kind regards

          Peter

          Dear Peter,

          I am sorry that your essay is not among the winners. It is extremely insightful and valuable to readers coming from different backgrounds. In fact, I see it as a kind of prize in itself, to have this beautiful text, as well as the comments that follow, as a rigorous reference and source of ideas. Thank you very much for having writting it, as it enriches anyone interested in the subject. Its value is intrinsic and independent of a prize, although it does matter to have one's name among the winners in terms of getting a broader attention, so it is really a pity in this sense.

          Thank you for your kind attention throughout this contest, and I feel humbled that you gave such a deep value to my essay. I enjoyed very much our exchanges.

          Best wishes,

          Christine

          Dear Christine,

          Thank you very very much for your kind words. You know, I participated in this contest for the sole purpose of exchanging ideas and find interesting contacts. In both respects, it was a great success for me. I had some very good rankings (at least two 10 and several 8) systematically followed by massive down-ranking, but frankly, this kind of mentality does not matter.

          I'm really glad I met you on this occasion. It would please me to stay in contact with you. Since you are working on quantum cosmology, I later would like to return to you for your views on the following question: How to conciliate or at least to balance the semi-intuitive cosmic time with the very controversial epistemological status of "time"()? (i) at the quantum level and (ii ) at the macro-quantum interaction? But that's for later; I am currently on another paper that requires a lot of time. If you want, I will send it to you from here several weeks or even later; it's a bit complicated.

          Here is my private e-mail adress

          peter.punin@wanadoo.fr

          Congratulations again,

          Best regards,

          Peter

          Dear Peter,

          Thank you very much, and I'll be very happy to keep contact with you. Yes, there are many fundamental questions on quantum gravity/cosmology that are extremely difficult to address, and the issue of time (at whatever level) is certainly the most perplexing, frustrating, fascinating of all... In case you are interested, I have an essay submitted on the Nature of Time, the first contest organized by the FQXi.

          On the Nature of Time - Or Why Does Nature Abhor Deadlocks?

          http://fqxi.org/community/forum/topic/297

          Best.

          9 months later

          Hi,

          I think we speak about the same thing from different perspective:

          http://fqxi.org/community/forum/topic/2291

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