And the math will set you free by Cristinel Stoica

Dear Sylvia,

Thank you for the comments. You raise interesting questions.

One point you make is about my words "That line contains your entire life". You thoughtfully reply "After all, my life is not a text". Well, could you please tell me the part of your life which is not a text? I don't intend to be too curious, rather to ask you a trick question, since if you can tell this, you will make it into a text :) A quick reading of my essay may leave the reader with the feeling that I choose to ignore consciousness for example, but as I wrote, "I don't claim we can explain consciousness, with or without mathematics." What I wrote about this may clarify what I mean: "However, any feeling we may have, there are neural correlates associated to it, and hence, physical correlates. And these physical correlates are in the domain of known physics, which is strongly mathematized." About your excellent remark that you can choose to narrate the same life differently, and make it closer to one number or another, I fully agree, but I can't see why would this be a problem. As you know, there is a bijection between the points on a segment and those in a square, or the entire space, but that bijection can't be continuous. I understand that your remark reveals that representing everything on a segment is counterintuitive, and I like it. I just wanted to make a point regarding whether math is discovered or invented.

Another point you make is "could there really be an isomorphism between the universe and a mathematical structure? In my opinion, at best we can find an isomorphism between _the structure of_ the universe and a mathematical structure." I agree, isomorphism is between structures. For example, there are more isomorphisms between the set of real numbers and other structures. The real line is isomorphic with a square, if we refer to the category of sets, with a line if we refer to the category of topological spaces, with a totally ordered set if we refer to the order, with a vector space, with a metric space, with a group, semigroup, ring, field, etc., it all depends on the structure we are interested in. As I explained in the essay, the structure is captured in the relations, all relations that can be described by propositions. So there's nothing that can be left outside the structure, if we take into account all the true propositions about the world. Saying "ascribing a structure to the universe leaves a lot of degrees of freedom" is not necessarily true, I mean, of course it is true if we leave outside some of the truths. About the substance, I don't know what you mean by this. Is it something that has effects? Then its properties are captured in the structure. If you think that there are properties of the substance that don't have effects to the structure, then I have nothing to say about it, and anything we would say would be out of our possibilities of verifications. I think the text in a book is what makes a book, and not the paper or the electronic memory used to keep a copy of that text.

It was a deep pleasure to talk with you, and I wish your essay will do well, since I loved it.

Best wishes,

Cristi

Dear Sylvia,

Thank you for the answer. I think I understand what you mean, but I don't think I made you understand what I mean.

"If I tell you an episode of my life, my life itself will not turn into words." Sure it will not. And even if you give me the complete list of episodes, your life will not turn into words. What I mean is that its structure will be captured in those words. So, going back to my words "That line contains your entire life", I should have said perhaps "That line contains a complete description of your entire life". The main subject of my essay was about the relation between mathematics and physics, and I was careful to mention that I don't aim to explain unphysical stuff, perhaps consciousness being an example.

You give a good example, that of a color. But you answer it yourself, you can reproduce it on a computer (and that information can be expressed if we wish as words, or as a point on a segment). Your worry that there will be not enough room for hardware in the universe to simulate the color is a new element you add. But the proof of principle remains: the possibility to describe that particular color exists. I am talking about the possibility of a description, not a real implementation of it, not of a hardware. I don't think that, in order to prove it, one should effectively build that description. For instance, the number π exists even if its decimals are nowhere written completely, and even if the entire universe is not enough to write it.

"On your view, can you destroy all copies of a book (including our memories of reading it etc.) without destroying the text?" If you destroy all writings containing Pythagora's thereom, you will not destroy it, you will destroy the information about it that we have. It will be soon rediscovered. But if you destroy all books by Shakespeare, I don't expect that we will rewrite them soon. They will still be in that segment, but we will lose the address to retrieve them. Think for example at a computer. When you normally delete a file, you delete a reference to it, but the information remains on the hard drive, and can be recovered by special software, until you overwrite it with other information. When you lose the address, the file is not lost. A home is not lost when you lose the address. But you lose your access to it. In the case of a book, the address is the book.

But when we talk about life, the things are different... I don't think that a complete description of your life values at least 0.000...01% of your life. You may think that this is a contradiction: on the one hand I claim that everything is isomorphic to a mathematical structure, on the other hand, I don't reject the possibility that consciousness or life is more than this. Maybe this is not obvious in my essay of this year (where I focused on the relation between mathematics and the physical world), but perhaps the one for last year is more on this topic.

Best wishes,

Cristi

Dear Anshu and Tejinder,

Thank you for the kind comments.

"The duck: for us, a description of the duck is different from the duck; thus we said that mathematics lacks in material substance. Would you agree?"

I don't really know what to answer, since I really know neither the nature of mathematics nor that of material substance.

If we call material substance the thing from which things are made, what is that thing? Things are made of atoms, which seem hollow, and made of protons and neutrons, which seem hollow, which are made of quarks. Which are what? and why can't they have independent existence? Are particles waves of some stuff, or waves of probability? Are they fields of what? Can we know about these things more than mathematical abstractions? Are they more than clicks in detectors? You see, I am very confused about all this, and I must admit I don't know what is the substance making all this stuff. I don't want to sound like an ancient oriental wise pondering about the illusion of everything but nothingness, nor like a post modernist cubist surrealist, I just really don't know. What we know are operations from which we infer relations. We know propositions of the form "if we do this we obtain that outcome, with that probability". And we know that in the world there are some regularities, which look like the regularities we find in mathematical structures and nowhere else. So maybe these two things, matter and mathematics, are just one thing, the thread that connects these regularities. Call it the substance making the world, call it mathematical structure, in both cases is just a thread connecting the regularities. So if we say that mathematics lives on a material support, or that what we call material substance is in fact a collection of relations, or of propositions which are true about those relations, or mathematics, would it be a difference? And if there is a substance upon which the regularities are imposed in a way which looks like mathematics, then how can two so different things be so intimatelly connected, if they are not one and the same?

So I agree that "a description of the duck is different from the duck", but we don't have ducks, we only have descriptions of them. We are talking here about two descriptions of ducks. If the descriptions are identical and there is no way to check how the real ducks are (or how real the ducks are), can they be different?

"And if pushed, what stance would you take: Platonism, or non-Platonism? :-)"

None. Should I pick one? If the Platonist view distinguishes between the ideal world and our world, which is just an approximate pale shadow of the former, I clearly am not satisfied with it. Why have an ideal world of shapes and live in the cave? What use would be for that ideal world? And, as you said, how can we know about it, other than by extra-sensorial perceptions, which I don't even know what can be? If non-Platonism means that mathematics is just a secretion of the thinking matter, then again I am not satisfied, because the very source of that secretion is subject to mathematical laws. To avoid circularity, I feel forced to admit identity between the two.

But if I admit the identity of the two, this looks like a mathematical universe hypothesis, or mathematical monism. In this case, what brings the mathematical structure into existence? I don't know, but whatever we would consider to be the reality (including material substance), we face the same question: what brings it into existence? I don't have an answer for this, but I would rather have a single answer about the existence of the two ducks which are one, than two answers about what brings into existence two so different ducks, and another answer about what makes them so similar. Maybe all that there is is just mind, which contemplates an infinite diversity of propositions, which all arise from the principle of explosion (this essay page 9), and selects from these subworlds which are logically consistent, creating by this all mathematical structures, hence all possible worlds. Maybe. But I don't know :)

Best wishes,

Cristi

Hi Bill,

Thank you for the kind comments.

You ask "If the universe is isomorphic to math, then why we are unable to analytically solve any of our PDEs in physics?"

Why would we be able to analytically solve any of our PDEs in physics? You seem to state it as if it would be an inevitable consequence of the hypothesis that the universe is isomorphic to a mathematical structure.

"if "A Supreme Something" had ordered me to design a physical world--and to do so in way isomorphic to mathematics--I'd like to think that I could have concocted a physical setup far more computationally efficacious than the one we now find ourselves in!"

Why would you do it computationally efficacious? And could you do it like this, and in the same time allow the complexity we observe and we need to exist?

Anyway, our current mathematical models of the physical world are very good approximations. Put it conversely, the universe seems to be able to approximate efficiently our mathematical models, which are indeed not so computationally efficacious. So even if the universe would not be isomorphic to math, it seems to be doing so well the job of a mathematical structure, including these computations.

Thanks again for these interesting questions, and good luck in the contest!

Best wishes,

Cristi

Hi Jon,

Thank you for the kind and interesting comments. I am not sure if all mathematics can be used to explain the universe. Maybe there are parts that don't have correspondent in the universe, although they may have in other universes. Also, some mathematical theories are based on opposite axioms. For example, Euclidean and non-Euclidean geometries contradict one another when it comes about parallel lines. But there is a way to incorporate contradiction and use it as a fecundity principle to create mathematical universes, including ours (see this essay page 9). I look forward to read your essay, in this brief time that remains.

Best wishes,

Cristi

Thanks, Christine, and congratulations to you too, for your beautiful winning essay!

Best wishes,

Cristi

Dear Steve,

Thank you very much for your kind and interesting comments.

Best regards,

Cristi