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.
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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.
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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