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
The ultimate tactics of self-referential systems by Christine Cordula Dantas
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 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.
Best,
Christine
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
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.
Regards,
Christine
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.
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!
Christine
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.
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.
Respectfully,
Joe Fisher
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.
Christine
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 the last grade has been universally attained, two chief kinds of men. The one kind is composed of those who have not yet elevated themselves to the full feeling of their freedom and absolute independence, who are merely conscious of themselves in the representation of outward things. These men have only a desultory consciousness, linked together with the outward objects, and put together out of their manifoldness. They receive a picture of their Self only from the Things, as from a mirror; for their own sake they cannot renounce their faith in the independence of those things, since they exist only together with these things. Whatever they are they have become through the outer World. Whosoever is only a production of the Things will never view himself in any other manner; and he is perfectly correct, so long as he speaks merely for himself and for those like him. The principle of the dogmatist is: Faith in the things, for their own sake; hence, mediated Faith in their own desultory self, as simply the result of the Things.
But whosoever becomes conscious of his self-existence and independence from all outward things--and this men can only become by making something of themselves, through their own Self, independently of all outward things--needs no longer the Things as supports of his Self, and cannot use them, because they annihilate his independence and turn it into an empty appearance. The Ego which he possesses, and which interests him, destroys that Faith in the Things; he believes in his independence, from inclination, and [seizes] it with affection. His Faith in himself is immediate.
--Fichte,
Introduction to Fichte's Science of Knowledge
http://en.m.wikisource.org/wiki/Introduction_to_Fichte%27s_Science_of_Knowledge
8 days later
Christine,
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,
Tom
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,
Christine
Christine,
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.
Jim
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!
Best,
Christine
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.
Best,
Michel
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,
Christine
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
7 days later
Christine,
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.
Jim
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?
Regards,
Christophe