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Josh Daniels: on 2/23/17 at 20:36pm UTC, wrote That's super nice idea ad good way to explain such things not only to...

Ian Durham: on 6/12/15 at 15:15pm UTC, wrote Thank you Georgina!

Georgina Woodward: on 6/12/15 at 4:59am UTC, wrote Congratulations on your prize. Yours is the most memorable of the essays I...

Ian Durham: on 5/5/15 at 17:01pm UTC, wrote Oh, hey, now there's an intriguing question!

Kevin Knuth: on 5/5/15 at 4:26am UTC, wrote Hi Ian I really enjoyed your essay! It was very brave as such a dialogue...

Ian Durham: on 4/23/15 at 22:18pm UTC, wrote Hi Michel, Thank you for citing that work! I look forward to hearing about...

Ian Durham: on 4/23/15 at 22:15pm UTC, wrote Hi Mauro! Part of me completely agrees with you that physics needs to be...

Ian Durham: on 4/23/15 at 22:09pm UTC, wrote Hi Sara, Thanks for the comments. I think you raise a very important...


Georgina Woodward: "Joe, sensory products are what is seen. Illumination matters because it..." in The Sudoku Universe, Why...

fally jonash: "It is a well-maintained site where people can learn about various topics. I..." in A Wonderful Outcome

fally jonash: "Your article is very interesting and fantastic, at the same time the theme..." in In Search Of Other Earths

Joe Fisher: "Dear Georgina, The (INVISIBLE) “sensory products” you keep writing..." in The Sudoku Universe, Why...

Joe Fisher: "Dear Georgina, I failed to mention that although conventional chess game..." in The Complexity Conundrum

Joe Fisher: "Dear Steve Agnew, On December 7, 2017, I have emailed : “Dear..." in The Complexity Conundrum

Lena Smith: "All HP printers carry a unique qualities. if the product is still under..." in Conjuring a Neutron Star...

Lena Smith: "All HP printers carry a unique qualities. if the product is still under..." in Conjuring a Neutron Star...

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The Complexity Conundrum
Resolving the black hole firewall paradox—by calculating what a real astronaut would compute at the black hole's edge.

Quantum Dream Time
Defining a ‘quantum clock’ and a 'quantum ruler' could help those attempting to unify physics—and solve the mystery of vanishing time.

Our Place in the Multiverse
Calculating the odds that intelligent observers arise in parallel universes—and working out what they might see.

Sounding the Drums to Listen for Gravity’s Effect on Quantum Phenomena
A bench-top experiment could test the notion that gravity breaks delicate quantum superpositions.

Watching the Observers
Accounting for quantum fuzziness could help us measure space and time—and the cosmos—more accurately.

December 11, 2017

CATEGORY: Trick or Truth Essay Contest (2015) [back]
TOPIC: The Raven and the Writing Desk by Ian Durham [refresh]
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Author Ian Durham wrote on Mar. 11, 2015 @ 20:38 GMT
Essay Abstract

In this essay, I use a dialogue between characters from Lewis Carroll’s Alice’s Adventures in Wonderland to discuss the relationship of mathematics to physical reality. In it, I propose that there are two realities: representational and tangible. Mathematics belongs to the former. We can reconcile the two by taking Eddington’s stance that the universe is nothing more than our description of it.

Author Bio

Ian Durham is Professor of Physics at Saint Anselm College where he sometimes ventures — unharmed! — into the Mathematics Department. They even let him serve as Acting Chair of Mathematics once. He holds a PhD in mathematical physics from the University of St. Andrews (Scotland) where he and his wife once danced with Will & Kate. His alter ego, Cyrus Bohm, helped promote FQXi’s recent video contest.

Download Essay PDF File

George Gantz wrote on Mar. 12, 2015 @ 17:09 GMT
Ian -

Oh, what a delightful respite from the drudgery of all these essays! Thank you.

One quibble - in the answer to "So are there as many integers as there are rational numbers?" Hatter concludes there are multiple infinities - but since we can enumerate the rational numbers (aleph null) that is not quite correct. We need to include the irrational numbers to reach aleph 1.

Otherwise an excellent play on the concepts of tangible, representational, descriptive and actual. It's all real don't you think?

Cheers - George Gantz

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Author Ian Durham replied on Mar. 12, 2015 @ 17:29 GMT
Hello George,

Thank you for the kind words! And you are correct. I goofed a bit in talking about the aleph numbers. Hopefully the primary point shines through anyway.



Author Ian Durham replied on Mar. 13, 2015 @ 02:41 GMT
So you've gotten me thinking. I am aware that rational numbers and integers have the same cardinality, but I'd be curious to see a proof (I'm sure there are many). Certainly from an intuitionist standpoint, they shouldn't have the same cardinality. Is there a number theorist lurking on this board who can perhaps point out a good proof of that?

Member Tejinder Pal Singh replied on Mar. 14, 2015 @ 17:34 GMT
Hello Ian,

You might like the nice demo of Cantor's proof given here.



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Ed Unverricht wrote on Mar. 14, 2015 @ 16:42 GMT
Dear Ian Durham,

I guess no one told you, but the Queen of Hearts has spoken. There are 3.14159 names on the list today and to anyone who disputes that "off with their heads"!

Regards, Ed Unverricht

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Mohammed M. Khalil wrote on Mar. 14, 2015 @ 22:45 GMT
Dear Prof. Durham,

Thank you for this delightful dialogue. I really enjoyed it.

I would like to invite you to read my essay; it also deals with the relation between mathematics and physical reality.



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Georgina Woodward wrote on Mar. 17, 2015 @ 08:05 GMT
Hi Ian, this essay is really lovely. I think tangible mathematician, not the representational pseudonym Lewis Carroll would approve- If he was alive, Which he isn't....

I'm beginning to feel like I'm one of your characters.

Is the universe tangible? I think the visible universe is representational. Formed from received data but not out there touchable as the data takes time to arrive and a universe in motion continues in motion always becoming.So not as seen.

We can assume there is a tangible universe but it is only potentially so- as we cannot reach it to touch it, our probes only just leaving our solar system. So a bit like the potential infinity that might be counted but can't actually -unless you are a fictional talking dormouse! I wonder whether 'concrete' rather than 'tangible' may better capture the sense of being in some way more actual than abstract or representational without the necessity of being touchable.

I love, Quote: "Hatter: After all, just because I’m a character in a dialogue doesn’t make me any less a part of the universe." Which gets us thinking about how different kinds of reality relate to each other. Certainly the ink on the page or pixels on the screen coding the dialogue are tangible but they are also representational and the characters represented almost 'come to life'and almost seem tangible but are despite wishful thinking only as tangibly real as Lewis Carroll deceased.

Great fun, best of luck. Georgina

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Author Ian Durham replied on Mar. 19, 2015 @ 00:08 GMT
Thank you for the kind words, Georgina. I am glad you enjoyed the essay! Indeed, how do different realities relate to one another? That is an intriguing question.

Eckard Blumschein wrote on Mar. 18, 2015 @ 06:03 GMT
Dear Ian Durham,

If you need a convincing number theorist I recommend Salviati. Of course, he as well as Euclid are presently kept for heretical or at least outdated.

Not just you might hopefully agree on that any cardinality in excess of the plausible distinction between an unbounded but discrete grid of numbers (a_0) and the endlessly divisible continuum (a_1) has not proved useful in science.

Do you think I'll stick with tangible reality? Instead I prefer conjecturing reality as something that fits to confirmed by experience and reasoning self-consistent basic relationships like causality.


Eckard Blumschein

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Lorraine Ford wrote on Mar. 18, 2015 @ 13:36 GMT
Hi Ian,

This is a very enjoyable, easy to read essay. But re

"HATTER: Neither is there anything particularly tangible about 'two' or 'ten. ' They are abstract concepts."


"MARCH HARE: And mathematics is representational? HATTER: Precisely. . .MARCH HARE: No. I simply don't buy it. I'll stick with tangible reality, thank you very much" :

If numbers are representational, i.e. if numbers represent physical reality, then you can't really say that numbers are "abstract concepts". What "tangible reality" does a number represent?

I contend in my essay (Reality is MORE than what Maths can Represent) that numbers MUST represent fundamental physical structures. And, though it's seemingly not a complete solution to the number "problem", I contend in my essay that a natural or a rational number must represent a ratio: a "relationship between information categories" where the category in effect cancels out.



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Author Ian Durham replied on Mar. 19, 2015 @ 00:21 GMT
Hi Lorraine,

The number 3 can represent many tangible things: 3 donuts, 3 tortoises, 3 coins. In fact that is precisely the point of the abstraction. Russell, extending the ideas of Frege, essentially says that numbers are similarity classes. So when we say "there are three coins in my pocket" we are asserting that the objects in my pocket share some (possibly vague) similarity. We could just as easily say "there are three things in my pocket" and those three things might be a coin, a candy wrapper, and a key. While they are not terribly similar, they do share the common fact that they are made of ordinary matter and so we may classify them as such.

Conversely, I can't say "there are three airs in my pocket" (referring to the air we breathe). I can't "count" air in the sense of that sentence. I could count air _molecules_, but not "the air". As Russell points out, these ideas are deeply connected to the language we use to express them.


Anonymous replied on Mar. 19, 2015 @ 08:18 GMT
But Ian!

Re "3 donuts, 3 tortoises, 3 coins": You are talking about subjective structures/relationships that exist in your brain, which you can represent by the written symbol "3" or the spoken word "three" etc. There is nothing abstract about what exists in your brain, which you subjectively experience.

The point I was trying to make is: what is the reality behind quantity in FUNDAMENTAL physical reality?!! What "tangible reality" does a number represent? Surely, we've got to stop always looking to an abstract platonic realm to solve every difficulty?

I contend in my essay (Reality is MORE than what Maths can Represent) that numbers MUST represent fundamental physical relationships. I contend in my essay that a natural or a rational number must represent a ratio: a "relationship between information categories" where the category in effect cancels out.



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Lorraine Ford replied on Mar. 19, 2015 @ 08:26 GMT
Above post was from me, Lorraine Ford.

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Member Rick Searle wrote on Mar. 19, 2015 @ 01:32 GMT
I must say, Ian, that the only essay I enjoyed as much as yours in this contest was the one by Tommaso Bolognesi,and have voted accordingly. Though I am still confused as to how all these mathematical madness gives us such accurate models of the world.

Please take the time to read and vote on my essay:

Best of luck!

Rick Searle

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Author Ian Durham replied on Mar. 19, 2015 @ 01:35 GMT
Thanks Rick! I think we're all confused about that. :-)

Eckard Blumschein replied on Mar. 19, 2015 @ 02:50 GMT
Rick and Ian,

"Mathematical madness" and "we're all confused about that. :-)"? Sounds like emotions of freshmen students who were confronted with a shut up and calculate attitude even in the simple case of using the definitely not mysterious complex calculus.

Maybe, Wigner intended bringing irrationality into matters that seemingly evade common sense? I noticed that he almost adored J. v. Neumann who on his part admired v. Békésy for his held for more genuine nobility. Irrationality is a basis for belief. Suppressed doubts in the correctness of their believe might have affected G. Cantor, Hausdorff, Gödel, Grotendieck and others. Cantor died in a madhouse, others behaved otherwise abnormal.

Incidentally, to those who don't understand my hint, in a fictitious dialog Salviati was used to utter Galileo Galilei's still compelling reasoning: "The relations smaller than, equal to, and larger than are not applicable on infinite quantities, only on finite ones".


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Michel Planat wrote on Mar. 19, 2015 @ 09:16 GMT
Dear Ian,

Good essay. It shows the distinction between maths (the representational) and physics (the tangible) at an elementary level. Can you comment on the quantum reality? If one agrees that quantum measurements are contextual the quantum world is tangible, isn'it?

The cardinality of the Monster Group M (big but not infinite) is representational at the moment but it can become tangible (will be effectively counted with the computer), the cardinality of most sporadic groups became tangible.

All the best,


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Author Ian Durham replied on Mar. 21, 2015 @ 02:04 GMT
Hello Michel! Thank you for your comments. And your essay is on my list to read (I noticed that you wrote a dialogue as well). I absolutely believe that the quantum world is tangible. In my opinion, if it weren't, then nothing would be tangible.

James Lee Hoover wrote on Mar. 28, 2015 @ 06:35 GMT

Very fun and imaginative dramatization. Math is a representational reality which means it must be peer reviewed by others (like BICEP2) to make sure it really represents tangible reality. Eddington's observation did help prove Einstein's theory but at the same time Eddington mistrusted mathematical derivations from relativity theory to explain "degenerate stars."

Thanks for sharing an imaginative creation that helps clarify roles of math and physics.

My "Connections" tries to show the relationships of math, the mind, and physics in new discoveries in quantum biology, LHC and DNA.


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James Lee Hoover replied on Apr. 13, 2015 @ 17:40 GMT

Time grows short, so I am revisiting essays I’ve read to assure I’ve rated them. I find that I rated yours on 3/28, rating it as one I could immediately relate to. I hope you get a chance to look at mine :


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Martin Seltsam wrote on Mar. 29, 2015 @ 22:53 GMT
Dear Professor Durham,

I am just a humble student but noticed a similarity of one moral your excellent story conveys to the main thesis presented in my little opera "Map = Territory" where I ponder the possibility of an actual merger of the description and the described in fundamental physics.

Your message "the universe is nothing more than our description of it" reminded me of my beginner's take on the subject and I would very much appreciate an opinion of an accomplished professional like you (if some time can be found for that). I would be honoured by your feedback and advice.

With deep respect and best wishes,


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Jonathan Khanlian wrote on Apr. 1, 2015 @ 20:05 GMT
Hi Ian,

I hope this isn't a re-post. I thought I commented on your essay but when I checked back to today I didn't see it. (Un)Luckily I had it saved in a word document...

I totally agree with what you were trying to say via your dialogue… or maybe I completely disagree… or… Anyway, at the very least I liked the dialogue (reminded me of Gödel-Escher-Bach) and think the thoughts you provoked with it definitely relate to my Digital Physics movie essay. I think we’re still trying to reach a consensus on infinity, probabilities, and mathematics in general, so I see little hope for achieving a Grand Unified Theory in physicists before some of these fundamental ideas are better addressed.

I also tried to have a little fun with the essay format, so hopefully it makes it an easy read. I’d love for you to comment on my thoughts regarding real numbers in my essay. I think you might also like taking a crack at a few of the questions I have at the end of my essay. Here’s a sample of one that might interest you:

How quickly could a tape be processed through a Turing machine and is this constraint physical or informational in nature?

Thanks again for making an interesting and entertaining read.


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Joe Fisher wrote on Apr. 9, 2015 @ 16:07 GMT
Dear Ian,

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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Louis Hirsch Kauffman wrote on Apr. 10, 2015 @ 07:00 GMT
Dear Ian,

Lovely! Here is a quote from G. H Hardy in his "Mathematician's Apology":

(p. 70) " A chair or a star is not in the least like what it seems to be; the more we think of it, the fuzzier its outlines become in the haze of sensation which surrounds it; but '2' or '317' ha nothing to do with sensation, and its properties stand out more clearly the more closely we scrutinize it. ... 317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way."

For Hardy, mathematics was more tangible than physical reality.


Lou Kauffman

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Alma Ionescu wrote on Apr. 18, 2015 @ 16:20 GMT
Dear Ian,

I am particularly amused by and happy to have read you essay. I must say that I wanted to make mine a dialogue as well, but then reconsidered as I thought it might be deemed as not matching with the contest; I'm glad at least one person wouldn't have thought that ;) I decided in the end to just go with a light and funny writing tone.

I like the symmetry of your essay. It begins and ends with the same action, somewhat like a Turing machine caught in an infinite loop. It must mean it's meant to be read and enjoyed by thinking beings. In this context, the eternal 6 o'clock made me consider what would happen if the March Hare would put on the table such a machine rummaging through an uncomputable calculation. I enjoyed very much how you are working your way to the argument of the representational and tangible realities and found very satisfying in this respect (and hilarious) the Hatter's remark that "A character in a dialogue has chosen tangible reality. How comforting."

Profound and packed in dry humor. Great read! Wish you the best of luck in the competition and I'm rating your work accordingly, in the hope that it will make a difference.

Warm regards,


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Jeffrey Michael Schmitz wrote on Apr. 20, 2015 @ 21:04 GMT
Dear Ian,

As others have commented, this is a wonderful choice of style. I wish you could of included one or two more topics, but this was fun and clear.

All the best,


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Jeffrey Michael Schmitz replied on Apr. 22, 2015 @ 00:16 GMT

I am replying to my own thread, because I realized it was incomplete.

In writing class (which you clearly do not need)we learned "show do not tell". This essay was showing, not telling. Is the idea of something the thing itself? Is the name of an idea the idea? Is an equation of the universe, the universe (at this point I clearly went beyond the text)? You let us make these decisions, which I do not know if I like or hate you for putting us in such a position.

All the best,


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Author Ian Durham wrote on Apr. 21, 2015 @ 21:05 GMT
Thank you all for the kind comments! I have been busy with work and have not been able to find the time to respond to each of you individually, but I appreciate the thoughts! I am very glad indeed that I was able to inject a bit of humor without taking away from the depth of the subject. I do wish I had had more space to elaborate on some points, but perhaps I will do so in the future elsewhere. Thank you all again!

Michel Planat wrote on Apr. 22, 2015 @ 06:07 GMT
Dear Ian,

Thank you for going to my essay. I expect that my essay will not be red as a tree but as a surface with punctures, it is non local in some sense. I also hope a big picture is emerging, some points are ongoing research (as those pointed out in the abstract), some technical aspects may not be familiar to quantum physicists (e.g. modular forms and characters).

If you go to reference [17] just appearing in QIP, you can see that I cite a work of yours on the "order theoretic quantification of contextuality", meanwhile I also found another measure of geometrical contextuality that I am currently working on. A mathematician would say the Langlands program but I stay closer to physics.



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Author Ian Durham replied on Apr. 23, 2015 @ 22:18 GMT
Hi Michel,

Thank you for citing that work! I look forward to hearing about your additional measure of geometrical contextuality when it is finished.

Indeed, I am not all that familiar with modular forms and characters. But that is one of the beautiful points of your essay. It introduces some really terrific ideas to the physics community.



Member Sara Imari Walker wrote on Apr. 22, 2015 @ 16:24 GMT
Dear Ian,

Thanks for the very enjoyable essay. The distinction between representation and tangible realities is well articulated. One of the oddities of this dichotomy seems to be that some physical systems can characterize what may (or may not) be tangible using representation. So it seems these two modes of description are not entirely independent but that the representational influences the tangible and the tangible constrains what can be represented. You seem to suggest a starker line between the two, so I am curious on your thoughts where these two domains interact. In particular, my view is that addressing this aspect is essential to addressing why math seems to work for describing physical reality.



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Author Ian Durham replied on Apr. 23, 2015 @ 22:09 GMT
Hi Sara,

Thanks for the comments. I think you raise a very important issue: where *is* the line between tangible and representational reality? Is the line, if it exists, well-defined or fuzzy? And if we err on the side of caution, should we err on the tangible side or the representational side?

I'm torn on this issue, to be honest. I'm not a pure operationalist, but I'm not even sure operationalism, at least as it is practiced today, is even purely tangible. I do firmly believe that physics should be about the physical, but I also believe that well-founded theories are as broad as possible in their explanatory nature. So, to give you an example, if you've ever watched Susskind's videos on what you need to know about modern physics, he starts out talking about Newton's laws. Whenever I teach introductory material, I start with energy and momentum because Newton's laws are easily derived from conservation of momentum and energy. When I made this point to a friend of mine, he argued that introducing Newton's laws (and general laws of motion) first, made more sense from an operationalist view. So, in that sense, I am not an operationalist.

On the other hand, as I said, I do think we need to default to physical reality. Feynman said something about this once. In fact Carl Caves said something to me once many years ago to this effect: we're physicists and we're doing physics so we should default on the side of physics.

So, in short, I unfortunately don't have a neat and clean answer to your question. I think the line I painted in the essay is quite stark for that particular example. It may be that there is a stark line in *every* example, but that there is no single, unifying way of defining that line such that we can "know" it in a given case (if that makes any sense).



Member Giacomo Mauro D\'Ariano wrote on Apr. 22, 2015 @ 21:31 GMT
Dear Ian,

very nice essay. I still need to read it in full detail to appreciate all the sides of the conversation. For the moment let me say that I appreciate the choice of Lewis Carrol, which, btw, raised the famous logical paradox that proves that even the "modus ponens" must be formalized. If logic itself must be formalized, than you would agree with me that Physics must be formalized, otherwise would not be logical, whence not logical falsifiable, whence not scientific. As you know, this is the point raised in my essay.

Hope to see you soon, and enjoy a thorough conversation about this crucial point about physical science.

My best regards


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Author Ian Durham replied on Apr. 23, 2015 @ 22:15 GMT
Hi Mauro!

Part of me completely agrees with you that physics needs to be formalized if it is built on logic (which it partially is). But another part of me says that it is ultimately built on what we can measure (see my reply to Sara above). In other words, it still has to explain what we see in a lab. It's not clear that we can formalize everything we see. Certainly people have tried ever since Hilbert's original proposal (and, to some extent, Newton and others in the 17th century tried as well, before physics took a more experimental turn). It seems telling to me that no one has fully succeeded in this endeavor yet, though of course that doesn't mean it can't be done.

Hope to see you soon! Let me know if you will be in Boston at some point.



Member Kevin H Knuth wrote on May. 5, 2015 @ 04:26 GMT
Hi Ian

I really enjoyed your essay!

It was very brave as such a dialogue is extremely challenging. You have to maintain a direction to the conversation leading the reader to thoughts without explicitly stating them, and all the while make the conversation lively, engaging and natural. Its a daunting challenge, and I think that you did a great job!

I enjoyed how the dialogue began and ended with the two attempting to shove the Doormouse into the teacup.

I wonder, did the Doormouse actually count to infinity in the same way that Achilles caught up to and passed the Tortoise?



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Author Ian Durham wrote on May. 5, 2015 @ 17:01 GMT
Oh, hey, now there's an intriguing question!

Georgina Woodward wrote on Jun. 12, 2015 @ 04:59 GMT
Congratulations on your prize. Yours is the most memorable of the essays I read and deserves recognition among the winners. Well done, kind regards Georgina

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Author Ian Durham replied on Jun. 12, 2015 @ 15:15 GMT
Thank you Georgina!

Josh Daniels wrote on Feb. 23, 2017 @ 20:36 GMT
That's super nice idea ad good way to explain such things not only to children, but to grown up people as well and that's one of the reasons why you're going to one of my favourite writers. I'm pretty sure, you'll be successful in every field you'll choose, but I'll be happy, if you become a writer of my writing team. You can check review up and let me know about your decision, I'll be waiting for it anyway.

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