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Darius M: on 3/10/16 at 21:37pm UTC, wrote Hi, I think we speak about the same thing from different perspective: ...

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Peter Punin: on 6/12/15 at 10:23am UTC, wrote Dear Christine, Thank you very very much for your kind words. You know, I...

Christine Dantas: on 6/11/15 at 10:27am UTC, wrote Dear Peter, I am sorry that your essay is not among the winners. It is...

Peter Punin: on 6/11/15 at 6:01am UTC, wrote Dear Christine Many congratulations for FQXi contest. It is true that my...

Christine Dantas: on 6/5/15 at 20:50pm UTC, wrote Dear Sara Imari Walker, Thanks. Yes, self-reference is seen as fundamental...

Christine Dantas: on 6/5/15 at 20:48pm UTC, wrote Dear Sylvia Wenmackers, Thank you for your comment, and I wish you good...

Christine Dantas: on 6/5/15 at 20:47pm UTC, wrote Dear Cristinel, Thank you and I wish you good luck as well. Best, ...


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To build the ultimate artificial mimics of real life systems, we may need to use quantum memory.

October 18, 2017

CATEGORY: Trick or Truth Essay Contest (2015) [back]
TOPIC: The ultimate tactics of self-referential systems by Christine Cordula Dantas [refresh]
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Author Christine Cordula Dantas wrote on Feb. 28, 2015 @ 01:18 GMT
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 .

Download Essay PDF File

Sujatha Jagannathan wrote on Feb. 28, 2015 @ 07:32 GMT
Your work moreover relates the recommended common views in comparison to the work of mine.

Great Sir!

Best Regards,

Miss. Sujatha Jagannathan

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Conrad Dale Johnson wrote on Mar. 1, 2015 @ 16:32 GMT

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.

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Author Christine Cordula Dantas replied on Mar. 2, 2015 @ 10:26 GMT

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,


Koorosh Shahdaei wrote on Mar. 3, 2015 @ 13:45 GMT
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


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Author Christine Cordula Dantas replied on Mar. 4, 2015 @ 21:02 GMT
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.



Peter Martin Punin wrote on Mar. 4, 2015 @ 20:28 GMT
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,...

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Author Christine Cordula Dantas replied on Mar. 4, 2015 @ 21:16 GMT
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,


Peter Martin Punin wrote on Mar. 5, 2015 @ 17:36 GMT
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


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Author Christine Cordula Dantas replied on Mar. 6, 2015 @ 21:48 GMT
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).



Andrei Kirilyuk wrote on Mar. 6, 2015 @ 20:02 GMT
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.

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Author Christine Cordula Dantas replied on Mar. 6, 2015 @ 21:41 GMT
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!


George Gantz wrote on Mar. 10, 2015 @ 03:21 GMT
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

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Author Christine Cordula Dantas replied on Mar. 12, 2015 @ 12:38 GMT
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.



Michael James Goodband wrote on Mar. 12, 2015 @ 12:17 GMT
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.


Michael Goodband

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Author Christine Cordula Dantas replied on Mar. 12, 2015 @ 12:27 GMT
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,


Joe Fisher wrote on Mar. 13, 2015 @ 14:09 GMT
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

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Author Christine Cordula Dantas replied on Mar. 13, 2015 @ 15:28 GMT
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.



Joe Fisher replied on Mar. 14, 2015 @ 19:36 GMT
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

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Peter Martin Punin wrote on Mar. 13, 2015 @ 17:37 GMT
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.

There are complementarities.

Best regards


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Author Christine Cordula Dantas replied on Mar. 13, 2015 @ 21:06 GMT
Dear Peter,

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

Best wishes,


Sylvain Poirier wrote on Mar. 18, 2015 @ 11:32 GMT
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...

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Author Christine Cordula Dantas wrote on Mar. 18, 2015 @ 13:06 GMT
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.



Sylvain Poirier replied on Mar. 19, 2015 @ 12:31 GMT
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...

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Author Christine Cordula Dantas replied on Mar. 19, 2015 @ 15:10 GMT
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.



Christian Corda wrote on Mar. 20, 2015 @ 14:38 GMT
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.

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Author Christine Cordula Dantas wrote on Mar. 20, 2015 @ 15:57 GMT
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!


Christian Corda replied on Mar. 20, 2015 @ 16:06 GMT
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.

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Author Christine Cordula Dantas replied on Mar. 22, 2015 @ 14:13 GMT
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...

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Joe Fisher wrote on Mar. 21, 2015 @ 20:04 GMT
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.


Joe Fisher

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Author Christine Cordula Dantas replied on Mar. 21, 2015 @ 20:59 GMT
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.


Thomas Howard Ray wrote on Mar. 30, 2015 @ 05:38 GMT

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,


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Author Christine Cordula Dantas wrote on Mar. 30, 2015 @ 23:07 GMT
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,


James Lee Hoover wrote on Mar. 31, 2015 @ 21:38 GMT

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.


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Author Christine Cordula Dantas replied on Mar. 31, 2015 @ 23:00 GMT
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!



James Lee Hoover replied on Apr. 13, 2015 @ 17:15 GMT

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.


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Author Christine Cordula Dantas replied on Apr. 14, 2015 @ 22:51 GMT
Dear James,

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



Michel Planat wrote on Apr. 3, 2015 @ 14:30 GMT
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.



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Author Christine Cordula Dantas replied on Apr. 3, 2015 @ 16:52 GMT
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,


Joe Fisher wrote on Apr. 6, 2015 @ 15:37 GMT
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

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Christophe Tournayre wrote on Apr. 14, 2015 @ 12:42 GMT
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?



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Author Christine Cordula Dantas wrote on Apr. 14, 2015 @ 22:49 GMT
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?



Christophe Tournayre replied on Apr. 16, 2015 @ 13:31 GMT
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!



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Author Christine Cordula Dantas replied on Apr. 16, 2015 @ 13:41 GMT
I've read it many years ago. It's one of my favorite books.



Cristinel Stoica wrote on Apr. 21, 2015 @ 11:48 GMT
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

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Author Christine Cordula Dantas replied on Jun. 5, 2015 @ 20:47 GMT
Dear Cristinel,

Thank you and I wish you good luck as well.



Member Sylvia Wenmackers wrote on Apr. 21, 2015 @ 20:23 GMT
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

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Author Christine Cordula Dantas replied on Jun. 5, 2015 @ 20:48 GMT
Dear Sylvia Wenmackers,

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


Member Sara Imari Walker wrote on Apr. 22, 2015 @ 22:43 GMT
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?



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Author Christine Cordula Dantas replied on Jun. 5, 2015 @ 20:50 GMT
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).


Peter Martin Punin wrote on Jun. 11, 2015 @ 06:01 GMT
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


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Author Christine Cordula Dantas wrote on Jun. 11, 2015 @ 10:27 GMT
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,


Peter Martin Punin wrote on Jun. 12, 2015 @ 10:23 GMT
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

Congratulations again,

Best regards,


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Author Christine Cordula Dantas wrote on Jun. 12, 2015 @ 10:58 GMT
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?


Darius M wrote on Mar. 10, 2016 @ 21:37 GMT

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

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