Dear Amtheamatics — The Silver Saxphones Say I Should Abuse You

That’s a slight corruption of the lyric from Bob Dylan’s “I Want You“. I was doodling around with some mathematics when Dylan’s song came up on my playlist.  Although it is a song about relationships it was speaking to me about mathematics and science this day.

Math Girls v1 cover

Math Girls series, volume 2, by Hiroshi Yuki, cover.

I really do want to abuse mathematics. I’d like to get it to work for me in the craziest ways. I’d like to write a novel about some advanced unforeseen mathematical theorems and investigations. To do so would require inventing some impossible mathematics. If this is to be done then the result would likely not be true mathematics, in that it would have little or no connection to future theorems and results in mathematical sciences.

The point of the novel would be to illuminate literature with a glimpse of the wondrous dream-world that mathematical minds tend to swim about in most days. So my novel would not need to be mathematically accurate. Just highly realistic. Inspirational without being 100% plausible. But plausible enough that a layperson or even many professional mathematicians, would not be able to tell the difference. Is this sort of semi-realism possible?

Surely it’s possible. The question is can I write such stuff!

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In the Head of a Symbolist

A lot of serious mathematicians would probably baulk against my project. “Why the heck wold you want to do fictional mathematics when real mathematics is so much more exciting?”, scream the grey saxaphones of the soulless.

Actually I do not have a great response to that question. Because real mathematical investigation is exhilarating. But I do have a weak reply. Partly, (and here most people might sympathise with me) doing real mathematics is bloody hard work. 90% of the time you have a problem to solve and cannot see your way to the solution. 9% of the time the solution seems clear but getting to the end of it seems like a marathon race or akin to sitting through 100 hours of parliamentary debates and select committee meetings. It’s not always like this, but when the only solution to a puzzle seems to be to grind away on some repetitive search task and tedious run-of-the-mill calculation, then the parliamentary analogy can seem subjectively appropriate.

That’s the non-glorious side of mathematics that most people experience from school. However, I’d like to put together a novel that presents the other mostly hidden glorious side of mathematics.

A mathematician will get a question stuck in their head. They might (in the past) go to a library to find the answer, or (these days) Google for the answer. 80% of the time they probably find someone has already answered the question. The other 20% of the time there is tremendous excitement in finding an unanswered question. It is tremendously exciting because it is so rare to find a good unanswered and unasked question. Although there are infinitely many unanswered questions and only finitely many answered questions, it does paradoxically seem very hard to find a good unanswered question. For a mathematician or scientist they are like gold. (This precludes the many asked questions that remain unanswered, since they have already been asked they are not the same kind of gold, more like silver or bronze.)

So if one is lucky there is no answer and no one has asked the question before. This is exciting and dangerous. It is dangerous because then the question will haunt the mathematician. Sometimes to the end of their life.

But such an event is also the fire of life. It can drive your mind like nothing else. Even cliché’s like “better than sex” do not even apply. It goes beyond cliché, and must, as with some religious experiences, “be experienced”.

In fact I would ague that genuine mathematical insight is a spiritual experience. And I am fully prepared to defend this thesis. One day I might even do so for real. It is an important idea that our modern western civilisation tends to discount as anti-intellectual and un-rigorous. But I think this is an unfair judgement and to paraphrase Kurt Gödel (one of the preeminent mathematical logicians of the twentieth century, and certainly the most famous), “a prejudice of our times“.

Yes. I think if one is really committed to investigating mathematics, whether one cares to admit it or not, one is engaged in a spiritual pursuit. It is certainly possible to be engaged with this spiritual discipline and yet deny vociferously that it is spiritual. If you do not believe in spiritual reality then naturally even when you are exercising spiritual impulses you will deny it. Almost everyone does this at some in point in life. You find yourself acting altruistically yet deny this is your motive. Someone tells you that you are acting selfishly or prejudicially and yet you deny it, but objectively there there can be no denial.

I have read (but not interviewed) a few mathematicians who strongly believe the exercise of mathematics is nothing more than manipulating symbols on paper or in one’s mind using certain rules. These rules are what we refer to as “mathematics”. They are wrong. They may be correct that this is what they truly believe. They may also be correct that in some societies and circles of acquaintances this definition of “what it means to be mathematics” is exactly such cold unemotional symbol manipulation.

But I can justify with a high degree of rigour that there is an alternative definition of “Mathematics” (yes, with a capital “M” for Mphasis) that goes far beyond the impoverished thinking of a symbol manipulator. Gödel knew this also.

My project is to take this higher plane spiritual view of Mathematics and put it into a novel that anyone can read and appreciate. It would not be to popularise mathematics. But my hope it would give a reader a sense of renewed wonder at the world. The human mind can go places without hallucinogenic drugs that most people never get to see. And these places can be amazing and awesome, scary and beautiful, captivating and sometimes almost horrific and frightening in their depth and complexity. Breathtaking and rejuvenating, sometimes deadening black & white in repetitiveness and then bursting with colours beyond the physical spectrum of anyone’s imagination.

Hmmm … that last hyperbolé might capture what I really wish to communicate. You see, one of the truly spiritual wonders of mathematics is that in investigating a challenging problem a mathematician is forced to dream beyond what they can imagine. How is that possible? What happens is that the problem reveals a computation or mini-puzzle that must be solved to answer the original question. Sometimes the solution to this sub-problem is so unexpected and revelatory that the mathematician has to stop and pause for wonderment. It is at once beyond what the mathematician could have imagined, so they check their logic and  …  yes, it is true, there was no mistake in the calculations. So the mathematician is then flipped in consciousness into believing what was previously unimaginable.

In this unfolding there is every hint of a truly spiritual endeavour. The final steps in this process are mechanical and logical, but getting to this point is the spiritual journey. Then having mechanically checked everything is ok the final dawning consciousness of the importance of the result for other branches of mathematics, or for the practical problem at hand, is again nothing short of a spiritual awakening. You do not have to believe or appreciate the spiritual significance. Many mathematicians refuse to and go to pains to avoid emotional responses to their own work. But the spiritual significance is real nonetheless.

It is not an easy thing to recognise either. Such mathematical spiritual realisations are often not “beautiful” in the same way as great art or music. They tend to be austere and elemental in their beauty. A perfect circle is, after all, quite boring. A hand-drawn circle seems to many people to have more “spirit”, especially when it is part of a greater work of art. But a mathematical mind finds more in a perfect circle than the line on paper. They see many, many new and interesting properties, and I am not even going to explain the transcendental number π, that is only one of many beauties in a circle. But if they try to communicate these niceties to the general public then a lot of the mystery seems to be inexplicable, and the beauty vanishes because the medium of communication is too dull.

This is the general problem of mathematical popularization. It is a contradictory endeavour. Mathematics cannot truly be communicated unless one learns the mathematics. So to attempt to popularize mathematics is fraught with impossibilities and paradoxes. You need to simplify concepts for a general audience, and at some point in simplification the essential mathematical mystery can get entirely lost. What remains is a façade, almost empty words that just “have to be believed”.

You know what I mean. When people say,

“Andrew Wiles proved the hundred year old Fermat’s Last Theorem in 1995. Wiles’ work was hundreds of pages of proof and an exposition of diverse fields of mathematics, connecting Modular Forms with Elliptic Equations. Yet Pierre De’Fermat wrote that a proof of his theorem was found that was wonderful but would not fit in the margin of his book.”

Then we are supposed to be impressed right?

We are supposed to be impressed that Fermat had a wonderful proof which remained undiscovered for hundreds of years, and Andrew Wiles worked his butt off finding a proof that was a tour de force and involved mathematical ideas that were completely unknown to Fermat. And we are supposed to be impressed by all of this as if we understood the effort. Well, for sure I was impressed by Wiles’ achievement. And I can even retain some residual amusement that perhaps Fermat had an elegant proof but it was probably flawed.

But to have any insight into the spiritual wonder of Fermat’s Last Theorem is truly difficult to gain, unless one has some inkling of understanding f the meaning of the theorem and the tremendous complexity and intricacy and unifying ideas of Wiles’ proof. At one point in the BBC documentary about Wiles’ efforts Andrew Wiles has a moment where tears well up in his eyes as he remarks, “I will never do anything as important as this again”.

That almost gets to the spirit. It is a beautiful moment. Wiles has this seemingly simultaneous emotion of loss of greatness (“never again”) superposed with triumph (“as important as this”).

My point is that the general audience has to somehow trust that all of this is as awesome as the documentary and commentary suggest. The fact Andrew Wiles is not an actor really helps! But the inner core of emotion can only be guessed at. If I had to try to explain what Wiles was thinking I would take another essay, and even then to get to the heart of the spiritual aspects of Wiles’ work would take Wiles’ own words, and even then much of it would probably be lost in his own prejudices and misconceptions about the philosophy of mathematics, despite his authoritative knowledge of his own proof.

A Japanese Author Who Did Not Abuse Mathematics

Just want to now plug one author who has managed to avoid corrupting mathematics and yet tell an exciting and highly readable story. The novel “Math Girls” and it’s sequels, by Hiroshi Yuki, are best-sellers in Japan, and the first two volumes have recently been translated into English by Tony Gonzalez for Bento Books.

Excertp from Math Girls, p62.

Excerpt from Math Girls, vol.1., by Hiroshi Yuki.

The mathematics in these novels is the real deal. So give them a go. And if you are a high school teacher then I suggest retiring your textbooks, convert them to computer monitor stands, and using these novels instead. The textbooks can be a reference. But for learning, at least for beginning students, give them these novels at first, please! Once inspired then release the textbooks.

Actually don’t do that. After the novels, release the puzzles and curiosities in worksheets and recreational mathematics books. Keep the textbooks accessible but chained up in the reference shelf.

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Oh yeah … why “Amtheamtics” in the title?

That is my most common typing of “mathematics”. The sequence my fingers hit the correct letters on my keyboard permute the letters this way about 60% of the time. My funniest typo is “does not” which 20% of the time comes out as “doe snot”. Another common typo is “student” which 50% of the time becomes “studnet”. Probably my most common typo is “whihc”.

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