The ultimate tactics of self-referential systems by Christine Cordula Dantas

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.

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 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.

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 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 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 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