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

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

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 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