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RECENT POSTS IN THIS TOPIC

**Thomas Ray**: *on* 4/24/15 at 8:20am UTC, wrote Hi Alma, Just to let you know that I replied to your excellent post on my...

**Rick Searle**: *on* 4/24/15 at 2:51am UTC, wrote Hi Alma, I wanted to let you know that I thought your essay was wonderful...

**Thomas Ray**: *on* 4/23/15 at 20:31pm UTC, wrote Dear Alma, I suspected as much. My own formal education was cut short by...

**Alma Ionescu**: *on* 4/23/15 at 18:42pm UTC, wrote Dear all, Thank you for your heartwarming comments! Should you want to...

**Alma Ionescu**: *on* 4/23/15 at 17:24pm UTC, wrote Dear Mauro, Thank you very much for returning me the visit and for your...

**Alma Ionescu**: *on* 4/23/15 at 17:17pm UTC, wrote Dear Tom, You’ve hit a bull’s eye there :) I would dearly love to...

**Alma Ionescu**: *on* 4/23/15 at 17:08pm UTC, wrote Dear Steve, Thank you for your generous compliments! I am very happy to...

**Giacomo D'Ariano**: *on* 4/23/15 at 1:17am UTC, wrote Dear Alma nicely written essay. I agree on many points, and especially...

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Philosophers, physicists and neuroscientists discuss how our sense of time’s flow might arise through our interactions with external stimuli—despite suggestions from Einstein's relativity that our perception of the passage of time is an illusion.

A devilish new framework of thermodynamics that focuses on how we observe information could help illuminate our understanding of probability and rewrite quantum theory.

An unusual approach to unifying the laws of physics could solve Hawking's black-hole information paradox—and its predicted gravitational "memory effect" could be picked up by LIGO.

Objective reality, and the laws of physics themselves, emerge from our observations, according to a new framework that turns what we think of as fundamental on its head.

The impossibility of building a perfect clock could help explain away microscale weirdness.

FQXi FORUM

May 20, 2019

CATEGORY:
Trick or Truth Essay Contest (2015)
[back]

TOPIC: Trick and Truth: The Natural Effectiveness of Math in Fundamental Science by Alma Ionescu [refresh]

TOPIC: Trick and Truth: The Natural Effectiveness of Math in Fundamental Science by Alma Ionescu [refresh]

We consider the unreasonable effectiveness of math in physical sciences by drawing clear differences between pure math and mathematical physics. We take into consideration the preferred methods of the fields, their historical background and examples from other fields. We build a theory of their similarities and divergences. We present some examples to support the hypothesis that the partial success that we investigate is based on the simplicity and the reducibility of the fields to their component parts, as well as naturalness and proximity to the fundamental parts of reality. Using our current knowledge of the directions and advancements of each field, we try to predict potential future directions.

Female, born 1983, Eastern Europe, life-long learner with a background in IT and business. Some formal math and physics background completed with courses and independent study during the past few years. I keep up to date with current research directions as well as theoretical developments and experimental results. Primary interests in high energy physics-theory, astrophysics, general relativity and quantum cosmology.

Dear Alma,

Your beautiful essay is very fun and entertaining, and in the mean time so profound. You fill a gap and go at the root of the problem, when you discuss*the differences between pure math and mathematical physics*. Indeed, *the simplicity and the reducibility of the fields to their component parts* is responsible for the success of mathematics as a tool to investigate physics, or "Math works perfectly in physics because, as we will show, they are both simple and modular in their own way". As you well pointed out, a simple tool would be limited to the intention which led to its invention, while "when one makes the complex plane, fractals grow in it uninvited, without having been planned, placed there or even wanted, like leaves on a light bulb". This tool takes over and starts living its own life, leading us to new, unexpected destinations, "but when we lose the correct track we have no way to find it again without more new [physical] insights". You are right that we, as species, don't do math for that long time, and "there is still time for math". Despite the fact you mention, that "we haven’t discovered anything fundamentally unexpected for years", I share your optimism, summarized in your conclusion that *there is still time*.

Best wishes,

Cristi Stoica (link to my essay)

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Your beautiful essay is very fun and entertaining, and in the mean time so profound. You fill a gap and go at the root of the problem, when you discuss

Best wishes,

Cristi Stoica (link to my essay)

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Dear Alma Ionescu,

Your writing style is exquisite and you phrase the problem well: "*It would be very nice to know once and for all if math is a human invention. … If it is … we could cross it out of our list of places to search for deeper meaning. If it isn't, and there hidden logical rules to the universe … then we can see it as an intrinsic part of reality*."

I...

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Your writing style is exquisite and you phrase the problem well: "

I...

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Dear Edwin,

Thank you for reading my essay and thank you for your very kind words!

To be honest I was preparing to read your essay anyway. I will probably not be able to read all the essays, but I at least had in intention to read the highest rated works. I will gladly comment on it after your lovely invitation to dialogue.

To respond at least partly to your comment, I think...

view entire post

Thank you for reading my essay and thank you for your very kind words!

To be honest I was preparing to read your essay anyway. I will probably not be able to read all the essays, but I at least had in intention to read the highest rated works. I will gladly comment on it after your lovely invitation to dialogue.

To respond at least partly to your comment, I think...

view entire post

Dear Alma,

I'm very pleased that I could help you realize something enough to state it unequivocally. Sometimes all it takes is a slightly different way of seeing it.

I really like your analogies, and completely agree with your view that math has become "complicated and therefore accessible only to adepts, which is equivalent to worshiping it more than in the times when we...

view entire post

I'm very pleased that I could help you realize something enough to state it unequivocally. Sometimes all it takes is a slightly different way of seeing it.

I really like your analogies, and completely agree with your view that math has become "complicated and therefore accessible only to adepts, which is equivalent to worshiping it more than in the times when we...

view entire post

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Dear Ms. Ionescu,

You asked: “Why is a physical theory less sturdy than the math it’s composed of?”

Because physical theory and mathematics are both absurd abstractions.

Accurate writing has enabled me to perfect a valid description of untangled unified reality: Proof exists that every real astronomer looking through a real telescope has failed to notice that each of the...

view entire post

You asked: “Why is a physical theory less sturdy than the math it’s composed of?”

Because physical theory and mathematics are both absurd abstractions.

Accurate writing has enabled me to perfect a valid description of untangled unified reality: Proof exists that every real astronomer looking through a real telescope has failed to notice that each of the...

view entire post

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Dear Alma,

Congratulations for a very interesting and well written essay. You have an enjoyable writing style, and some of your sentences have a literary feel – almost poetic! I really liked the spherical cow in the void, the light bulb growing leaves, and the list of ailments that befell our biggest mathematicians… I could almost feel their pain! You should start a blog or write a...

view entire post

Congratulations for a very interesting and well written essay. You have an enjoyable writing style, and some of your sentences have a literary feel – almost poetic! I really liked the spherical cow in the void, the light bulb growing leaves, and the list of ailments that befell our biggest mathematicians… I could almost feel their pain! You should start a blog or write a...

view entire post

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Dear Marc,

Thank you for taking the time to read my essay and for your kind words!

I'm not sure I can say I have an optimistic attitude or hold hope. Should there be a theory of everything, I realize it's not even granted that we will ever find it. Unlike winning the lottery (where someone has to win at some point), a TOE is not a guaranteed miracle. It's more that I am somewhat...

view entire post

Thank you for taking the time to read my essay and for your kind words!

I'm not sure I can say I have an optimistic attitude or hold hope. Should there be a theory of everything, I realize it's not even granted that we will ever find it. Unlike winning the lottery (where someone has to win at some point), a TOE is not a guaranteed miracle. It's more that I am somewhat...

view entire post

Dear Alma,

Fun speculations! I like the idea that beings living in a universe based on math that does not support conservation laws would have an appropriate mathematics that would work in their world, and also mathematics that resembles ours, but is only used in philosophy classrooms! It looks like something out of a Borges story, or something that could exist in some remote corner of...

view entire post

Fun speculations! I like the idea that beings living in a universe based on math that does not support conservation laws would have an appropriate mathematics that would work in their world, and also mathematics that resembles ours, but is only used in philosophy classrooms! It looks like something out of a Borges story, or something that could exist in some remote corner of...

view entire post

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Dear Marc,

Right, I forgot to say something; you first mentioned that my writing style is enjoyable. I think I can say I mostly learned English from Terry Pratchett's books, so even if I can't compare my writing to his, I think I retain some of the formulation quirks and comparisons he used to make.

Cheers,

Alma

Right, I forgot to say something; you first mentioned that my writing style is enjoyable. I think I can say I mostly learned English from Terry Pratchett's books, so even if I can't compare my writing to his, I think I retain some of the formulation quirks and comparisons he used to make.

Cheers,

Alma

Alma,

This is a well written and enjoyable essay.

Sadly, it is true that many great scientists and mathematicians have died prematurely. Riemann and Maxwell for example. I understand that Euler blind? My memory may be fuzzy on that one. Emmy Noether did not even leave a textbook:-(

But as you note, even a few 10's of thousands of man-years does produce results. And as long as it is written down, it will be remembered.

You mention quaternions and their use for describing orbits ... You might be interested in my essay.

Best Regards and Good Luck,

Gary Simpson

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This is a well written and enjoyable essay.

Sadly, it is true that many great scientists and mathematicians have died prematurely. Riemann and Maxwell for example. I understand that Euler blind? My memory may be fuzzy on that one. Emmy Noether did not even leave a textbook:-(

But as you note, even a few 10's of thousands of man-years does produce results. And as long as it is written down, it will be remembered.

You mention quaternions and their use for describing orbits ... You might be interested in my essay.

Best Regards and Good Luck,

Gary Simpson

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Thank you, Gary! Your kind words are much appreciated.

Yes indeed, the one that went blind would be Euler. He tried to learn to write without looking while he still retained some sight, but after losing his sight his writing was too unclear to serve any purpose. I used that trick to encourage the readers' feeling of participation, because I thought you would recognize every case. I'm also glad you liked the man-years approximation.

I downloaded your essay and took a look. It's certainly quite dense so it will take me a while to digest it, but it looks appetizing and I will certainly comment on it once I manage to understand it.

Warm regards and best wishes,

Alma

Yes indeed, the one that went blind would be Euler. He tried to learn to write without looking while he still retained some sight, but after losing his sight his writing was too unclear to serve any purpose. I used that trick to encourage the readers' feeling of participation, because I thought you would recognize every case. I'm also glad you liked the man-years approximation.

I downloaded your essay and took a look. It's certainly quite dense so it will take me a while to digest it, but it looks appetizing and I will certainly comment on it once I manage to understand it.

Warm regards and best wishes,

Alma

Dear Alma:

A very nice and well written essay, I really like it! I agree that math can't work on it's own --- that's the biggest problem my "pragmatic physicist" has with Tegmark's mathematical universe. It isn't of much use knowing we live in a mathematical universe if we don't know where we live. We'd still have to go out and figure that out. It didn't really become clear to me though whether you are arguing for or against reductionism, or maybe the middle way in which reductionism works 'in principle' but not 'in practice'?

-- Sophia

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A very nice and well written essay, I really like it! I agree that math can't work on it's own --- that's the biggest problem my "pragmatic physicist" has with Tegmark's mathematical universe. It isn't of much use knowing we live in a mathematical universe if we don't know where we live. We'd still have to go out and figure that out. It didn't really become clear to me though whether you are arguing for or against reductionism, or maybe the middle way in which reductionism works 'in principle' but not 'in practice'?

-- Sophia

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Dear Sophia,

Very nice to hear from you!

Actually that's a very good question because I didn't consider reductionism when I wrote the essay. My thoughts on the topic are not as clear as they should be considering the difficulty of the topic - but to answer your question, I think that maybe I can say that reductionism can work in practice but not in principle. What I mean is that finding descriptions for emergent phenomena may be possible and they would probably come in handy, however it may not be completely satisfactory. To give the best example that can illustrate my case - some day we may find a great description of the human brain and it will have life-saving applications in medicine; that description however may not include qualia and even if it does and we can rationally argue that it's complete, I very much doubt that it can*feel* complete.

Warm regards,

Alma

Very nice to hear from you!

Actually that's a very good question because I didn't consider reductionism when I wrote the essay. My thoughts on the topic are not as clear as they should be considering the difficulty of the topic - but to answer your question, I think that maybe I can say that reductionism can work in practice but not in principle. What I mean is that finding descriptions for emergent phenomena may be possible and they would probably come in handy, however it may not be completely satisfactory. To give the best example that can illustrate my case - some day we may find a great description of the human brain and it will have life-saving applications in medicine; that description however may not include qualia and even if it does and we can rationally argue that it's complete, I very much doubt that it can

Warm regards,

Alma

Alma,

Interesting read. I like your chatty, almost poetic style, like a narrative we want to continue, and I like your Socratic inquisitions.

When you say "We seem to come to a halt momentarily ..", do you mean nothing stunning has occurred, like finding the Higgs through LHC was expected -- but they now expect a larger-mass Higgs is being hidden as they increase energy.

I think scientific approaches can be more directly looking for solutions to mysteries in the classical world like looking into the almost 100% efficiency of photosynthesis and tying it to superposition in the quantum world and quantum biology's look into the European robin's navigation N to S. The math follows.

Enjoyed your essay.

Jim

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Interesting read. I like your chatty, almost poetic style, like a narrative we want to continue, and I like your Socratic inquisitions.

When you say "We seem to come to a halt momentarily ..", do you mean nothing stunning has occurred, like finding the Higgs through LHC was expected -- but they now expect a larger-mass Higgs is being hidden as they increase energy.

I think scientific approaches can be more directly looking for solutions to mysteries in the classical world like looking into the almost 100% efficiency of photosynthesis and tying it to superposition in the quantum world and quantum biology's look into the European robin's navigation N to S. The math follows.

Enjoyed your essay.

Jim

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Dear Jim,

Thank you for reading my essay and for your kind words!

I said that we seem to have come to a halt at the present time because some long sought after principles have failed to show, like SUSY or Higgs' brothers, dark matter signatures in LUX or any other number of different things we were hoping to find, but my main idea was to plead for patience. That we didn't find something in the last years does not mean much because we haven't been searching for long enough to despair.

The examples that you give, new unexpected puzzles that may lead to new conceptions about the world or new valuable applications, are wonderful and help make my point.

Warm regards,

Alma

Thank you for reading my essay and for your kind words!

I said that we seem to have come to a halt at the present time because some long sought after principles have failed to show, like SUSY or Higgs' brothers, dark matter signatures in LUX or any other number of different things we were hoping to find, but my main idea was to plead for patience. That we didn't find something in the last years does not mean much because we haven't been searching for long enough to despair.

The examples that you give, new unexpected puzzles that may lead to new conceptions about the world or new valuable applications, are wonderful and help make my point.

Warm regards,

Alma

Alma,

I revisited your essay and am thankful you checked out mine. I find that I did not rate yours, something I usually do to those I enjoy, so I am rectifying that. I like the bounciness, the hopefulness and the openness of your essay.

Jim

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I revisited your essay and am thankful you checked out mine. I find that I did not rate yours, something I usually do to those I enjoy, so I am rectifying that. I like the bounciness, the hopefulness and the openness of your essay.

Jim

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Dear Alma,

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

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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

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Dear Alma,

thanks for reading my essay and your words. I want to make the unification of science using math very clear.

But I gave the complement back. I also read your essay and it is really great. Much easier to read then my essay (and maybe also easier to uderstand for any reader). I'm glad that the conclusion of our essay are (in principle) the same.

I wish you also the best luck for the contest (for that I gave a high rate).

Torsten

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thanks for reading my essay and your words. I want to make the unification of science using math very clear.

But I gave the complement back. I also read your essay and it is really great. Much easier to read then my essay (and maybe also easier to uderstand for any reader). I'm glad that the conclusion of our essay are (in principle) the same.

I wish you also the best luck for the contest (for that I gave a high rate).

Torsten

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Dear Torsten,

Thank you for taking the time to read my essay! I am very glad that you found our ideas similar. Actually after submitting, by reading other essays, I realized that I must have underestimated the level that was acceptable. I hope that the lightness of the style doesn't take much off the weight of the argument.

Warm regards,

Alma

Thank you for taking the time to read my essay! I am very glad that you found our ideas similar. Actually after submitting, by reading other essays, I realized that I must have underestimated the level that was acceptable. I hope that the lightness of the style doesn't take much off the weight of the argument.

Warm regards,

Alma

Dear Alma,

Excellent analysis in the spirit of Cartesian doubt. Deep, all-encompassing questions on fundamental questions of Mathematics and Physics and the direction of finding the answers to them:

"That our knowledge manifests gaps around emergent phenomena seems to be an indication that we lack some insight of the mathematical description itself, not just of what happens...

view entire post

Excellent analysis in the spirit of Cartesian doubt. Deep, all-encompassing questions on fundamental questions of Mathematics and Physics and the direction of finding the answers to them:

"That our knowledge manifests gaps around emergent phenomena seems to be an indication that we lack some insight of the mathematical description itself, not just of what happens...

view entire post

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Dear Alma,

Don't know for what reason your essay had not caught my eye. Probably because of the number of essays. Better late than never. Certainly a brilliant contribution and will reflect in my rating.

Under the section,**Simplicity and What Follows**, you wrote, "My room and everything in it is made from one type of lepton and two types of quarks... The universe takes a...

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Don't know for what reason your essay had not caught my eye. Probably because of the number of essays. Better late than never. Certainly a brilliant contribution and will reflect in my rating.

Under the section,

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Dear Akinbo,

What I had in mind when I mentioned 'matchsticks' was simplicity. Matchsticks are small objects, completely similar and having the same properties and it takes some imagination to think that one can build things out of such seemingly inappropriate pieces.

That being said, I will add that I read your essay and will be commenting on your page shortly.

Warm regards,

Alma

What I had in mind when I mentioned 'matchsticks' was simplicity. Matchsticks are small objects, completely similar and having the same properties and it takes some imagination to think that one can build things out of such seemingly inappropriate pieces.

That being said, I will add that I read your essay and will be commenting on your page shortly.

Warm regards,

Alma

Alma,

You mention the Langlands program, which is pretty close to what I am looking into. The generalization of the Tanyama-Shimura conjecture into algebraic varieties and categories is one reason why I have been looking into this homotopy approach to the foundations of mathematics.

Your essay was interesting and was free of obvious problems or errors. Mathematics has qualities as...

view entire post

You mention the Langlands program, which is pretty close to what I am looking into. The generalization of the Tanyama-Shimura conjecture into algebraic varieties and categories is one reason why I have been looking into this homotopy approach to the foundations of mathematics.

Your essay was interesting and was free of obvious problems or errors. Mathematics has qualities as...

view entire post

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Dear LC,

Thank you for taking the time to read my essay and for your kind words! It's very attention worthy that you are mentioning the Langlands program as a direction of your interest. I hope you do continue your research in this area and obtain the results you're hoping for. Considering your ideas, I think you'd be obtaining results nothing short than spectacular. Wish you good luck in...

view entire post

Thank you for taking the time to read my essay and for your kind words! It's very attention worthy that you are mentioning the Langlands program as a direction of your interest. I hope you do continue your research in this area and obtain the results you're hoping for. Considering your ideas, I think you'd be obtaining results nothing short than spectacular. Wish you good luck in...

view entire post

Dear Alma,

It was a pleasure reading your well-written and enjoyable essay, with many points to take home, and commonalities with the ideas in our essay. Especially the emphasis that mathematical physics is as good as the physical insight preceding it [what we called conceptual unification].

A noteworthy remark you make concerns thinking of mathematics as an emergent process. Very interesting. Now on the one hand of course elegant theorems follow from axioms. Can one think of this as emergence in the same sense as emergence in physical and biological systems? Does a theorem possess emergent properties which the axioms do not have, and which could not have been anticipated just by staring at the axioms. We suppose yes, as for instance features we see in number theory - properties possessed by a collection of numbers which simply cannot be guessed by looking at individual numbers. Perhaps you make a very important point here. Can emergence in physical systems be in any way linked to the corresponding emergence in the maths used to describe them? We will appreciate any thoughts you might have on this.

Kind regards,

Anshu, Tejinder

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It was a pleasure reading your well-written and enjoyable essay, with many points to take home, and commonalities with the ideas in our essay. Especially the emphasis that mathematical physics is as good as the physical insight preceding it [what we called conceptual unification].

A noteworthy remark you make concerns thinking of mathematics as an emergent process. Very interesting. Now on the one hand of course elegant theorems follow from axioms. Can one think of this as emergence in the same sense as emergence in physical and biological systems? Does a theorem possess emergent properties which the axioms do not have, and which could not have been anticipated just by staring at the axioms. We suppose yes, as for instance features we see in number theory - properties possessed by a collection of numbers which simply cannot be guessed by looking at individual numbers. Perhaps you make a very important point here. Can emergence in physical systems be in any way linked to the corresponding emergence in the maths used to describe them? We will appreciate any thoughts you might have on this.

Kind regards,

Anshu, Tejinder

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Dear gentlemen,

Thank you for this greatly engaging comment! I like your formulation of conceptual unification and with your permission I will keep it.

I too think that theorems have emergent properties. One thing that seems to confirm this conjecture is that we have theorems that are very hard to prove. If all the properties were contained in the number theory we have so far,...

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Thank you for this greatly engaging comment! I like your formulation of conceptual unification and with your permission I will keep it.

I too think that theorems have emergent properties. One thing that seems to confirm this conjecture is that we have theorems that are very hard to prove. If all the properties were contained in the number theory we have so far,...

view entire post

Dear Alma,

Thank you for your comments. It will be great if someone took to investigating common emergence in physical phenomena and the in the mathematics used to describe it. For instance, there are concrete studies which derive quantum theory as an emergent phenomenon, and gravity as an emergent phenomenon, and they use precise mathematics at all levels. It will be interesting to explore...

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Thank you for your comments. It will be great if someone took to investigating common emergence in physical phenomena and the in the mathematics used to describe it. For instance, there are concrete studies which derive quantum theory as an emergent phenomenon, and gravity as an emergent phenomenon, and they use precise mathematics at all levels. It will be interesting to explore...

view entire post

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I don't know to what extent theorems have emergent properties as it might be the things theorem refer to. I have been working on how spacetime is a coarse grained manifestation of large N entanglements, which have a period 8 structure. In this way the basic SLOCC structure is 4-bit or 8-bit even if there are 2^n = N large number of quantum bits. The geometry of spacetime time then comes about from event horizons that are measures of these entanglements. This in part goes in favor of those who say that time is a sort of illusion, but as I think space is a bit of an illusion also.

On this basis we could say that distance measures, say the Pythagorean theorem, is not emergent so much as the underlying constructions of points, lines, planes and so forth. Given that those elements exist, even if they are emergent, then Pythagoras was right as is so much of other mathematics. Of course we are talking about physical emergence, not so much mathematical. Mathematics as an idealized system of models does not seem to my mind emerge as such, but rather the underlying natural system the mathematics models is emergent.

LC

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On this basis we could say that distance measures, say the Pythagorean theorem, is not emergent so much as the underlying constructions of points, lines, planes and so forth. Given that those elements exist, even if they are emergent, then Pythagoras was right as is so much of other mathematics. Of course we are talking about physical emergence, not so much mathematical. Mathematics as an idealized system of models does not seem to my mind emerge as such, but rather the underlying natural system the mathematics models is emergent.

LC

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*I am copying these posts here as they are now hidden.

Dear Anshu, Tejinder,

My sincerest apologies for the confusion! It is very nice of you to say it's understandable and thank you for that :)

Warm regards,

Alma

Dear Anshu, Tejinder,

My sincerest apologies for the confusion! It is very nice of you to say it's understandable and thank you for that :)

Warm regards,

Alma

Dear LC,

That is a very subtle point you make there and I'm not sure I am skilled enough in this domain - I am probably not :) - to be able to bring the discussion further. But at least so as to clarify what I have in mind, I will bring the following example, which I realize is built on a different framework than the one you specified. There exists a mathematical proof for why and how phase transitions happen , but it relies on a lattice treatment (Sinai and Berezin). Until today, there is no treatment that can show the same thing for systems with more degrees of freedom. The emergence here is given by the extra idea that needs to exist in order to complete the proof. This extra idea is what adds insight and moves the pieces of the demonstration together in the right order, so they compose something that was not necessarily and sufficiently implied by the existing theory. That being said, i fully agree that the underlying natural models expressed by mathematics show emergent properties.

Cheers,

Alma

That is a very subtle point you make there and I'm not sure I am skilled enough in this domain - I am probably not :) - to be able to bring the discussion further. But at least so as to clarify what I have in mind, I will bring the following example, which I realize is built on a different framework than the one you specified. There exists a mathematical proof for why and how phase transitions happen , but it relies on a lattice treatment (Sinai and Berezin). Until today, there is no treatment that can show the same thing for systems with more degrees of freedom. The emergence here is given by the extra idea that needs to exist in order to complete the proof. This extra idea is what adds insight and moves the pieces of the demonstration together in the right order, so they compose something that was not necessarily and sufficiently implied by the existing theory. That being said, i fully agree that the underlying natural models expressed by mathematics show emergent properties.

Cheers,

Alma

Dear Alma,

I spent a few hours this morning reading your essay and preparing a non trivial comment for you. You will have a very good comment and appreciation from me by the end of the day. May be this is an instance of distant entanglement between brains at least not a pure coincidence. My thoughts during the two hour fast walk I just had was about rivalry between the two hemispheres that you might be call Phys and Math., a kind of quantum superposition that collapses one side or the other depending on context.

Have a good afternoon.

Michel

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I spent a few hours this morning reading your essay and preparing a non trivial comment for you. You will have a very good comment and appreciation from me by the end of the day. May be this is an instance of distant entanglement between brains at least not a pure coincidence. My thoughts during the two hour fast walk I just had was about rivalry between the two hemispheres that you might be call Phys and Math., a kind of quantum superposition that collapses one side or the other depending on context.

Have a good afternoon.

Michel

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Dear Alma,

Great essay! You have an engaging writing style that is enjoyable and accurately expresses your arguments. I particularly liked the section "Simplicity and What Follows" and "The Shape of Things to Come" and I share your optimism. I also discussed the role of simplicity in my essay; I would like to take your opinion.

All the best,

Mohammed

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Great essay! You have an engaging writing style that is enjoyable and accurately expresses your arguments. I particularly liked the section "Simplicity and What Follows" and "The Shape of Things to Come" and I share your optimism. I also discussed the role of simplicity in my essay; I would like to take your opinion.

All the best,

Mohammed

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Dear Alma,

In simple sentences and well chosen pieces you gave a relevant hint of what relates maths and physics, a proof of maturity that some experts either never had or have lost. Let me quote you "When one makes the (upper-half) complex plane, fractals grow in it uninvited", "Einstein’s cosmological constant and black holes – mere second-handunwanted consequences of his theory – and Higgs’ field were firstly miraculous ideas based on physical intuition, not lucky attempts to search the whole of math in the hope of a chance finding" and "It may very well be a gap in our total knowledge of the world, not of the physical, or biological or mathematical fields in isolation."

I had more time than you to think about the power of mathematical physics and always found this remarkable coincidence between maths and physics in several fields from electronic engineering to quantum information. About 20 years ago, I have been fascinated by the possible connection between cognition and the quantum: I gave a few references at Vincent Douzal blog. At the the same time, I attended a school on Grothendieck's dessin d'enfants that remained "dormant" until very recently.

You are the first to acknowledge me of introducing modular forms in the program! May the moonshine topic is no longer as fashionable as it was, or just too abstract!

I don't want to be too long, but reading you revitalized my forgotten quest of putting our cognitive ability into the picture. I remembered me reading the Nobel winner in Physiology or Medicine: John Eccles "I here express my efforts to understand with deep humility a self, myself, as an experiencing being. I offer it in the hope that we human selves may discover a transforming faith in the meaning and significance of this wonderful adventure that each of us is given on this salubrious Earth of ours, each with our wonderful brain, which is ours to control and use for our memory and enjoyment and creativity and with love for other human selves." --How the Self Controls Its Brain, pp. 180-1 (1994).

I suspect that the gap in our knowledge of the world lies in our neglect of other fields such as those concerning ourselves. I also suspect that maths can be as much effective there with enough imagination.

I wish you all the best and keep ready for further interaction.

Michel

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In simple sentences and well chosen pieces you gave a relevant hint of what relates maths and physics, a proof of maturity that some experts either never had or have lost. Let me quote you "When one makes the (upper-half) complex plane, fractals grow in it uninvited", "Einstein’s cosmological constant and black holes – mere second-handunwanted consequences of his theory – and Higgs’ field were firstly miraculous ideas based on physical intuition, not lucky attempts to search the whole of math in the hope of a chance finding" and "It may very well be a gap in our total knowledge of the world, not of the physical, or biological or mathematical fields in isolation."

I had more time than you to think about the power of mathematical physics and always found this remarkable coincidence between maths and physics in several fields from electronic engineering to quantum information. About 20 years ago, I have been fascinated by the possible connection between cognition and the quantum: I gave a few references at Vincent Douzal blog. At the the same time, I attended a school on Grothendieck's dessin d'enfants that remained "dormant" until very recently.

You are the first to acknowledge me of introducing modular forms in the program! May the moonshine topic is no longer as fashionable as it was, or just too abstract!

I don't want to be too long, but reading you revitalized my forgotten quest of putting our cognitive ability into the picture. I remembered me reading the Nobel winner in Physiology or Medicine: John Eccles "I here express my efforts to understand with deep humility a self, myself, as an experiencing being. I offer it in the hope that we human selves may discover a transforming faith in the meaning and significance of this wonderful adventure that each of us is given on this salubrious Earth of ours, each with our wonderful brain, which is ours to control and use for our memory and enjoyment and creativity and with love for other human selves." --How the Self Controls Its Brain, pp. 180-1 (1994).

I suspect that the gap in our knowledge of the world lies in our neglect of other fields such as those concerning ourselves. I also suspect that maths can be as much effective there with enough imagination.

I wish you all the best and keep ready for further interaction.

Michel

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Dear Michel,

It brings me great joy that a scientist of your caliber has found things to appreciate in my essay. For your comment alone and it was well worth participating into this contest.

To me it is natural to speak about modular forms because in my opinion they*are* the Langlands program, first and foremost. The most famous achievement of the program lies with modular...

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It brings me great joy that a scientist of your caliber has found things to appreciate in my essay. For your comment alone and it was well worth participating into this contest.

To me it is natural to speak about modular forms because in my opinion they

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Dear Alma,

As I told you in my Essay page, I have read your intriguing Essay. You made an excellent and original work. Here are some comments/questions:

1) I see your pretty statement that "the belief that math has an independent existence beyond our daily lives is based on the observation that even a child can intuitively understand math" confirmed every day by my son David, 4 years old, who plays with numbers...

2) Your aphorisms "Physics is the only science that can work with spherical cows in a void", "As opposed to an engine, one can’t fix math", "Mathematical physics is only as good as physical insight.", "We can’t expect math to work on its own", "Physics is simple", "There is still time for math", "You know you’re missing something when there’s just too much you can’t explain" and "Just wait to see our children" are fantastic! For the last one, see my point 1).

3) Your stress we don’t have a quantitative match between theories in pure math and the description of nature. I think we will never have it.

4) Do you think physics is not scale invariant?

5) I agree with you that the Langlands program sounds like good news for physics, but it must be handled very carefully. I know two possibilities to translate an intractable problem into another framework which sometimes generate confusion: the Maldacena conjecture, which, in my opinion, does nor resolve the black hole information paradox as it is often claimed, and the "Einstein frame versus Jordan frame" controversy in astrophysics observations.

Finally, I found the reading of your beautiful Essay very interesting and enjoyable. Thus, I am giving you a deserved highest rate.

I wish you best luck in the contest.

Cheers, Ch.

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As I told you in my Essay page, I have read your intriguing Essay. You made an excellent and original work. Here are some comments/questions:

1) I see your pretty statement that "the belief that math has an independent existence beyond our daily lives is based on the observation that even a child can intuitively understand math" confirmed every day by my son David, 4 years old, who plays with numbers...

2) Your aphorisms "Physics is the only science that can work with spherical cows in a void", "As opposed to an engine, one can’t fix math", "Mathematical physics is only as good as physical insight.", "We can’t expect math to work on its own", "Physics is simple", "There is still time for math", "You know you’re missing something when there’s just too much you can’t explain" and "Just wait to see our children" are fantastic! For the last one, see my point 1).

3) Your stress we don’t have a quantitative match between theories in pure math and the description of nature. I think we will never have it.

4) Do you think physics is not scale invariant?

5) I agree with you that the Langlands program sounds like good news for physics, but it must be handled very carefully. I know two possibilities to translate an intractable problem into another framework which sometimes generate confusion: the Maldacena conjecture, which, in my opinion, does nor resolve the black hole information paradox as it is often claimed, and the "Einstein frame versus Jordan frame" controversy in astrophysics observations.

Finally, I found the reading of your beautiful Essay very interesting and enjoyable. Thus, I am giving you a deserved highest rate.

I wish you best luck in the contest.

Cheers, Ch.

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Dear Christian,

Thank your for reading my essay and for your kind words, this is an honor for me! Allow me to reply to your points:

1) It looks like someone has a little genius physicist in preparation ;) He is lucky to have you as he will learn a lot from you as he is growing up.

2) Thank you! It is perhaps my main point, that regardless of what the past generations did not complete, the next generation is arriving and I trust their ingenuity.

3) I think so too. There is a finite number of laws and seemingly always more math.

4) I believe in scale invariance up to the extent where I would risk saying that a final theory should first guess the ratios and only after that the quantities. I hope I didn't seem to doubt it; if I did, I must have chosen an unfortunate expression.

5) You are perfectly right from my point of view. Progress does not always equal benefits as the atomic bomb history shown. In this case, theories that are only conformally equivalent should be very well understood first and their limits of applicability rigorously established, so as to avoid false promise and dead ends.

Thank you again for your generous comments and wish you the best of luck in the contest and in your research!

Cheers,

Alma

Thank your for reading my essay and for your kind words, this is an honor for me! Allow me to reply to your points:

1) It looks like someone has a little genius physicist in preparation ;) He is lucky to have you as he will learn a lot from you as he is growing up.

2) Thank you! It is perhaps my main point, that regardless of what the past generations did not complete, the next generation is arriving and I trust their ingenuity.

3) I think so too. There is a finite number of laws and seemingly always more math.

4) I believe in scale invariance up to the extent where I would risk saying that a final theory should first guess the ratios and only after that the quantities. I hope I didn't seem to doubt it; if I did, I must have chosen an unfortunate expression.

5) You are perfectly right from my point of view. Progress does not always equal benefits as the atomic bomb history shown. In this case, theories that are only conformally equivalent should be very well understood first and their limits of applicability rigorously established, so as to avoid false promise and dead ends.

Thank you again for your generous comments and wish you the best of luck in the contest and in your research!

Cheers,

Alma

Dear Alma,

It was my pleasure. He is myself who thanks you for your kind words on my little son David. I think to have understood your point on scale invariance. It is nice and sharable.

Cheers, Ch.

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It was my pleasure. He is myself who thanks you for your kind words on my little son David. I think to have understood your point on scale invariance. It is nice and sharable.

Cheers, Ch.

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Dear Alma,

You essay and comments are insightful and you seem to be a charming person. I was also interested in Leifer's essay viewing the whole of knowledge as a scale-free network. Your idea of looking at possible phase transitions is developed in his Ref. [13], Sec. G, p. 63 where you can read that "the critical exponents of the phase transition equal the critical exponents of the infinite-dimensional percolation". On my side, in my Neuroquantology paper quant- ph/0403020, I wrote in the abstract "Time perception is shown to depend on the thermodynamics of a quantum algebra of number and phase operators already proposed for quantum computational tasks, and to evolve according to a Hamiltonian mimicking Fechner's law. The mathematics is Bost and Connes quantum model for prime numbers. The picture that emerges is a unique perception state above a critical temperature and plenty of them allowed below, which are parametrized by the symmetry group for the primitive roots of unity." We recently revisited the BC model in the context of Riemann hypothesis and quantum computation http://iopscience.iop.org/1751-8121/labtalk-article/45421. This is a good sign that a good mathematical theory may have many inequivalent applications.

Today I have in mind to approach the subject of cognition with the tools I am advocating in my essay, it may take a while. I already mentioned that rivalry between the two cerebral hemisphres looks like a qubit.

Thank you very much Alma for the stimulus you are giving me. My very best regards.

Michel

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You essay and comments are insightful and you seem to be a charming person. I was also interested in Leifer's essay viewing the whole of knowledge as a scale-free network. Your idea of looking at possible phase transitions is developed in his Ref. [13], Sec. G, p. 63 where you can read that "the critical exponents of the phase transition equal the critical exponents of the infinite-dimensional percolation". On my side, in my Neuroquantology paper quant- ph/0403020, I wrote in the abstract "Time perception is shown to depend on the thermodynamics of a quantum algebra of number and phase operators already proposed for quantum computational tasks, and to evolve according to a Hamiltonian mimicking Fechner's law. The mathematics is Bost and Connes quantum model for prime numbers. The picture that emerges is a unique perception state above a critical temperature and plenty of them allowed below, which are parametrized by the symmetry group for the primitive roots of unity." We recently revisited the BC model in the context of Riemann hypothesis and quantum computation http://iopscience.iop.org/1751-8121/labtalk-article/45421. This is a good sign that a good mathematical theory may have many inequivalent applications.

Today I have in mind to approach the subject of cognition with the tools I am advocating in my essay, it may take a while. I already mentioned that rivalry between the two cerebral hemisphres looks like a qubit.

Thank you very much Alma for the stimulus you are giving me. My very best regards.

Michel

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Dear Alma,

A beautiful essay in all ways, original, insightful and with perfect English and logic, well argued.

I feel I should reproduce the phrases that I felt hit the target most perfectly;

'We can’t expect math to work on its own'

'You know you’re missing something when there’s just too much you can’t explain.'

'We’ve carefully chosen bits of math that resemble the phenomena we wanted to study'

'In pure math, any inconsistency is shot dead on sight.'

'Math requires us to be very careful when shaping a theory because the slightest false step will bring down the whole edifice that we’ve carefully built by creating a contradiction.'

'when we lose the correct track we have no way to find it again without more new insights.'

No time now, or even need, for detailed questions, time for scoring. i hope you may have time to read mine, which identifies a specific and important case of loosing the correct track but accepting illogicality by being satisfied that maths is enough. I'd love to hear your views.

Very well done, and thank you.

Peter

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A beautiful essay in all ways, original, insightful and with perfect English and logic, well argued.

I feel I should reproduce the phrases that I felt hit the target most perfectly;

'We can’t expect math to work on its own'

'You know you’re missing something when there’s just too much you can’t explain.'

'We’ve carefully chosen bits of math that resemble the phenomena we wanted to study'

'In pure math, any inconsistency is shot dead on sight.'

'Math requires us to be very careful when shaping a theory because the slightest false step will bring down the whole edifice that we’ve carefully built by creating a contradiction.'

'when we lose the correct track we have no way to find it again without more new insights.'

No time now, or even need, for detailed questions, time for scoring. i hope you may have time to read mine, which identifies a specific and important case of loosing the correct track but accepting illogicality by being satisfied that maths is enough. I'd love to hear your views.

Very well done, and thank you.

Peter

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Alma Ionescu wrote on Apr. 18, 2015 @ 15:08 GMT stub

Dear En,

This is nice and to the point. Your writing displays an interesting personal stance and I'm sure you enjoyed the exercise. I think that you're right when you're saying, in the third paragraph, that Wigner's expression should not be taken literally as it was more a metaphorical way of encouraging new lines of thought and...

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Dear En,

This is nice and to the point. Your writing displays an interesting personal stance and I'm sure you enjoyed the exercise. I think that you're right when you're saying, in the third paragraph, that Wigner's expression should not be taken literally as it was more a metaphorical way of encouraging new lines of thought and...

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Hi Alma--

Your essay was a joy to read. There is a wonderful lyrical quality to your writing style. I loved your sayings, especially "You know you're missing something when there's just too much you can't explain". I shall incorporate that one into my daily lexicon.

Quick question, admittedly somewhat off-point: Which philosopher has influenced you the most?

Very best regards and luck in the contest,

Bill.

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Your essay was a joy to read. There is a wonderful lyrical quality to your writing style. I loved your sayings, especially "You know you're missing something when there's just too much you can't explain". I shall incorporate that one into my daily lexicon.

Quick question, admittedly somewhat off-point: Which philosopher has influenced you the most?

Very best regards and luck in the contest,

Bill.

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Dear Bill,

Thank you very much for your kind words! Actually I like that question. As it stands, the philosopher in question would be Weinberg. I am half joking because his name sound German and imposing, but only half because besides Popper, Russell and newer philosophers like Maudlin and Earman I can’t say I enjoy philosophy (to my embarrassment) and none of these four philosophers...

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Thank you very much for your kind words! Actually I like that question. As it stands, the philosopher in question would be Weinberg. I am half joking because his name sound German and imposing, but only half because besides Popper, Russell and newer philosophers like Maudlin and Earman I can’t say I enjoy philosophy (to my embarrassment) and none of these four philosophers...

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Hi Alma--

I admire your preferences for thinkers and philosophers. If you haven't already, I urge to read Jim Holt's book, "Why does the world exist"?. It's a wonderful book. Furthermore, Holt devotes an entire section to Weinberg (chapter 9). Reading Weinberg's views on philosophy, the multiverse, etc., was fascinating.

Best regards,

Bill.

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I admire your preferences for thinkers and philosophers. If you haven't already, I urge to read Jim Holt's book, "Why does the world exist"?. It's a wonderful book. Furthermore, Holt devotes an entire section to Weinberg (chapter 9). Reading Weinberg's views on philosophy, the multiverse, etc., was fascinating.

Best regards,

Bill.

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Alma,

You're a winner. I do not exaggerate when I say that in my opinion, your essay ranks near Wigner's own, in its breadth of knowledge and depth of insight.

A couple of confessions I have to make: I have followed and enjoyed your intelligent commentary in other forums, so I knew I would get to your essay eventually. And the other confession is that I checked on the internet to verify how young you are; I would never have believed it from your mature postings. I do hope your circumstances permit you to go on to higher academic pursuits -- it's obvious that you have the "right stuff."

I didn't know the story of von Neumann and the engine. I do recall another one, though, of his lightning-fast calculating skill -- mathematicians like to try and fool each other with problems that seem complicated, but can be made simple with "tricks". One of them concerns two trains approaching each other, with a fly flying back and forth between the engines -- how far does the fly travel before getting squashed when the trains collide? One can get a quick answer by knowing the short cut of averaging -- otherwise, calculating the series by brute force is long and tedious. It is said that when the problem was put to von Neumann, he gave the solution almost immediately. Asked if he knew the short cut, he looked puzzled and replied, "What could be easier than summing the infinite series?" The Wolfram site has an article on the two-train problem.

Highest marks, and good luck!

Tom

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You're a winner. I do not exaggerate when I say that in my opinion, your essay ranks near Wigner's own, in its breadth of knowledge and depth of insight.

A couple of confessions I have to make: I have followed and enjoyed your intelligent commentary in other forums, so I knew I would get to your essay eventually. And the other confession is that I checked on the internet to verify how young you are; I would never have believed it from your mature postings. I do hope your circumstances permit you to go on to higher academic pursuits -- it's obvious that you have the "right stuff."

I didn't know the story of von Neumann and the engine. I do recall another one, though, of his lightning-fast calculating skill -- mathematicians like to try and fool each other with problems that seem complicated, but can be made simple with "tricks". One of them concerns two trains approaching each other, with a fly flying back and forth between the engines -- how far does the fly travel before getting squashed when the trains collide? One can get a quick answer by knowing the short cut of averaging -- otherwise, calculating the series by brute force is long and tedious. It is said that when the problem was put to von Neumann, he gave the solution almost immediately. Asked if he knew the short cut, he looked puzzled and replied, "What could be easier than summing the infinite series?" The Wolfram site has an article on the two-train problem.

Highest marks, and good luck!

Tom

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Dear Tom,

You’ve hit a bull’s eye there :) I would dearly love to pursue further education but I am not in the appropriate circumstances for that. Hopefully that will change in the future, but my age might become a problem at some point. We’ll see; where there’s life, there’s hope. I am surprised and very flattered by your confession! I too look up people when I am curious so I take it as a sign of genuine interest. Thank you very much for your kind words and for your many compliments!

I am very happy you liked the anecdote on von Neumann, and thank you for the Wolfram page on the problem! I chose Neumann because his life produced the most numerous and funniest anecdotes about scientists or mathematicians (except for the Pauli effect, which is even funnier). If you’d like to amuse yourself with a few more and find out the punch line of the Ford anecdote, visit this page; another rich deposit of Neumann’s memorable sayings is here; I also read more about the legend here.

Thank you again and my best wishes!

Alma

You’ve hit a bull’s eye there :) I would dearly love to pursue further education but I am not in the appropriate circumstances for that. Hopefully that will change in the future, but my age might become a problem at some point. We’ll see; where there’s life, there’s hope. I am surprised and very flattered by your confession! I too look up people when I am curious so I take it as a sign of genuine interest. Thank you very much for your kind words and for your many compliments!

I am very happy you liked the anecdote on von Neumann, and thank you for the Wolfram page on the problem! I chose Neumann because his life produced the most numerous and funniest anecdotes about scientists or mathematicians (except for the Pauli effect, which is even funnier). If you’d like to amuse yourself with a few more and find out the punch line of the Ford anecdote, visit this page; another rich deposit of Neumann’s memorable sayings is here; I also read more about the legend here.

Thank you again and my best wishes!

Alma

Dear Alma,

I suspected as much. My own formal education was cut short by life circumstances, and it's made me an advocate for universally free lifetime education. I don't see how democracy in the world can survive without it.

I don't know how it is in Europe -- in the States, however, the disparity between privileged members of society with full access to educational institutions and financial resources, and the rest of the country (the vast majority) has reached a critical point. The class division has never been greater in my long lifetime.

If there's any justice left in the world, you'll get the chance to fulfill your potential. It would be shame for a person of your fine intellect to be denied. And don't make age the issue!

Thanks for the tip on von Neumann.

All best,

Tom

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I suspected as much. My own formal education was cut short by life circumstances, and it's made me an advocate for universally free lifetime education. I don't see how democracy in the world can survive without it.

I don't know how it is in Europe -- in the States, however, the disparity between privileged members of society with full access to educational institutions and financial resources, and the rest of the country (the vast majority) has reached a critical point. The class division has never been greater in my long lifetime.

If there's any justice left in the world, you'll get the chance to fulfill your potential. It would be shame for a person of your fine intellect to be denied. And don't make age the issue!

Thanks for the tip on von Neumann.

All best,

Tom

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Hi Alma,

Just to let you know that I replied to your excellent post on my page.

Best,

Tom

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Just to let you know that I replied to your excellent post on my page.

Best,

Tom

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Dear Alma,

Your essay is extremely insightful and directly addresses this forum topic. I really enjoyed the development of your conclusions, in particular "mathematical physics is only as good as physical insight." My essay makes the exact same point, and discusses how changing the physical insight subsequently effects the mathematical abstraction, and how changing mathematical representation affects physical explanation. Your examples are excellent and very pertinent, in fact exciting. In particular, your juxtaposition of the Langlands program with Godel is brilliant. The Taniyama–Shimura–Weil conjecture is an incredible connection of different mathematical areas, and I absolutely agree that an equivalent to the Langlands program applies to physics; indeed it is a cornerstone in physical explanation. Schroedinger is perhaps one immediate example, though there are many more, and I'm familiar with some university programs specifically designed to connect different areas of physics for this very reason. As I mentioned, I liked how you juxtaposed this with your discussion of Gödel, and how you distinguished mathematical physics when countering the consequences of incompleteness. My essay too shows how what was considered a limitation can actually be expanded when attempting to physically model undecidability, and thus consider the physical requirements resulting from incompleteness. Finally, your appreciation of the humanity and lives of some of the greats, coupled with your optimism for the future is quite noble. This essay is a great contribution and I give it a 10.

Please check out my essay as well, and we have many things in common and our ideas very well support each other.

Thanks,

Steve Sax

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Your essay is extremely insightful and directly addresses this forum topic. I really enjoyed the development of your conclusions, in particular "mathematical physics is only as good as physical insight." My essay makes the exact same point, and discusses how changing the physical insight subsequently effects the mathematical abstraction, and how changing mathematical representation affects physical explanation. Your examples are excellent and very pertinent, in fact exciting. In particular, your juxtaposition of the Langlands program with Godel is brilliant. The Taniyama–Shimura–Weil conjecture is an incredible connection of different mathematical areas, and I absolutely agree that an equivalent to the Langlands program applies to physics; indeed it is a cornerstone in physical explanation. Schroedinger is perhaps one immediate example, though there are many more, and I'm familiar with some university programs specifically designed to connect different areas of physics for this very reason. As I mentioned, I liked how you juxtaposed this with your discussion of Gödel, and how you distinguished mathematical physics when countering the consequences of incompleteness. My essay too shows how what was considered a limitation can actually be expanded when attempting to physically model undecidability, and thus consider the physical requirements resulting from incompleteness. Finally, your appreciation of the humanity and lives of some of the greats, coupled with your optimism for the future is quite noble. This essay is a great contribution and I give it a 10.

Please check out my essay as well, and we have many things in common and our ideas very well support each other.

Thanks,

Steve Sax

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Dear Steve,

Thank you for your generous compliments! I am very happy to know that you enjoyed my writing and my ideas, especially because, as you mention, our work follows the same direction. Honestly I wasn’t sure that my usage of examples instead of properly formed arguments may be thought of as lacking in rigor but I am glad it is not so. Actually your comment is hinting at the fact that the examples themselves were what most readers enjoyed, and I thank you for that insight! The equivalences I used may form the general idea of a conjecture rather than feel compelling and I was aware that there was some factor of risk when it came to putting together such remote areas of science. I am glad it turned out to be comprehensible and not a waste of time for the reader.

My best wishes,

Alma

Thank you for your generous compliments! I am very happy to know that you enjoyed my writing and my ideas, especially because, as you mention, our work follows the same direction. Honestly I wasn’t sure that my usage of examples instead of properly formed arguments may be thought of as lacking in rigor but I am glad it is not so. Actually your comment is hinting at the fact that the examples themselves were what most readers enjoyed, and I thank you for that insight! The equivalences I used may form the general idea of a conjecture rather than feel compelling and I was aware that there was some factor of risk when it came to putting together such remote areas of science. I am glad it turned out to be comprehensible and not a waste of time for the reader.

My best wishes,

Alma

Dear Alma

nicely written essay. I agree on many points, and especially about complexity, about which you say that “That our knowledge manifests gaps around emergent phenomena seems to be an indication that we lack some insight of the mathematical description itself, not just of what happens physically.” And I agree that that this limits our knowledge in fields as biology, along with phys. and math themselves.

My best wishes for your essay and for you

Mauro

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nicely written essay. I agree on many points, and especially about complexity, about which you say that “That our knowledge manifests gaps around emergent phenomena seems to be an indication that we lack some insight of the mathematical description itself, not just of what happens physically.” And I agree that that this limits our knowledge in fields as biology, along with phys. and math themselves.

My best wishes for your essay and for you

Mauro

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Dear all,

Thank you for your heartwarming comments!

Should you want to remain in touch, you can reach me at alma[dot]ionescu83[at]gmail.com.

Warm regards,

Alma

Thank you for your heartwarming comments!

Should you want to remain in touch, you can reach me at alma[dot]ionescu83[at]gmail.com.

Warm regards,

Alma

Hi Alma,

I wanted to let you know that I thought your essay was wonderful and that I am glad you seem to have done well in the contest.

All the best,

Rick Searle

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I wanted to let you know that I thought your essay was wonderful and that I am glad you seem to have done well in the contest.

All the best,

Rick Searle

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