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
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
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
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?
Best,
Christine
Dear James,
Yes, time runs... And my reading list is late. Hope to be able to read yours and others! :/
Best,
Christine
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!
Regards,
Christophe
I've read it many years ago. It's one of my favorite books.
Regards,
Christine
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!
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
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?
Best,
Sara
Dear Cristinel,
Thank you and I wish you good luck as well.
Best,
Christine
