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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Chain Mile Evil

When I discovered the works of China Mieville, at first through his fabulous piston-driven horrifically gnarly Perdido Street Station, I starting getting pangs of desire to start writing fiction again. Actually “Perdido” is not really horrific. It is gross, sickening, ugly, brutal and yet intricately beautiful. Even the worst of the “monsters” are beautifully described by Mieville, by which I mean his terrifying Slake Moths who feed from and drain psyches.

(Incidentally, there is a creature, called a Teller, who does something similar in Doctor Who, Season 8, episode “Time Heist“. Only it is not as avante garde a destroyer as the Slake Moth. But the Teller does melt brains! Which offers some graphic horromusement, or is it horritainment? You gotta think though, that a protagonist who renders your nonphysical psyche into an empty nothingness is much more existentially horrific. The Slake Moth sucks your soul out, your personal identity and subjective consciousness becomes the empty set.)

The Weaver - 1

A nice ethereal depiction of The Weaver, from Perdido Street Station.

A Quick Quiz

There are more sickening creatures besides the Slake Moths. But try playing a guessing game with my mind, to peer into my psyche, to see if you can tell which other monsters I am speaking of, you might be surprised which ones I am referring to.

Not his daemons. I liked the daemons. They had strong self-preservation instincts and cunning, and so would not be drawn into battle against the Slake Moths.

Not the Handlingers either. Although they were bizarre and not pleasant to read about while having lunch. The same goes for the Khepri sex and the barrage of images Mieville infects the readers mind with when describing the hapless remade criminals, sentenced to bouts of biothaumaturgical grafting and xeonomorphing and heterotyping or their body parts.

Not Mr Motley either. Motley is a cool character. Evil for sure. Ugly for certain. But partly a victim of his time and era in the fictional world of Mieville’s imagination. Mr Motley is not really crazy evil like a Bin Laden or a Ghengis Khan or Hitler or Charles Manson or Pol Pot. Nah man! Motley is merely a banal evil entity, a product of his environment, like Bill Gates or Steve Jobs!! Hahahah! Seriously! Or, … well, maybe I exaggerate. Motley is perhaps closer in characters from nonfiction to, say, someone like a total dickhead like Donald Trump (maybe? Is he really evil or just a douchebag?) or one of those corporate CEO’s from corrupt organizations in the military-industrial complex, like a Union Carbide executive or a Blackwater CEO or Halliburton CEO, one of those high-ups who profit off war, government sanctioned killing and genocide and human misery.

Slake Moth - 1

Hard to find a good drawing of a Slake Moth. How can one capture their essential horror? This one is not too bad.

Do a Bit of Weaving Mr

Not the Weaver either, goddamm! I love the Weaver. Most awesome character in sifi I have come across in decades. Strike that. Most awesome character in scifi eveeeerrrr!

“Snip, snap, the gleaming metal blades sharpen the world weave and I cut the dross and flotsam and remake the  dimensions gleaming and shiny, pretty to the eye and fit template to the mind who delights. I will warp and weave and splice the sentient scenery of a million eyes swooning on the silver and coloured diffractions of the manifold glistening brightnesses. The Grimnebulin creature I will pluck! And send to slithery blistering lair of the gloomy drapers of the weave unreality who make so tortured and unpatterned havoc. We must cut from the fabric! No delightful strand remains whence those spineless wing-ed ones wreak their sloth over the yarn we have made nice.”

Or something like that! Gotta love the Weaver.

The Weaver - 2

This sketch of The Weaver is a good start, but misses out the scissory aesthetic sine qua non of the Weaver.

But there is so much that is (willfully and deliberately artistically) flawed on the ontologies of Bas-Lag (the world of Perdido Street Station) that the novel became like a typical movie for me that I wanted to remake and reinvent. But I cannot. I do not possess the linguistic thaumaturgy.

So I do not wish to write anything like Perdido. What this has inspired me to dedicate some time towards is something far more removed and ethereal. For I think there is, in the real world, as much frantic and incandescently enlightened art and science and natural wonder that surpasses everything in the supercharged fantasy world of China Mieville’s Bas-Lag. But you have to dig deep into this actual world of ours to find it and make it appear more than mundane to the eyes of those who are not aware.

The Weaver - 3

A fairly literal Weaver. The real magic horror of The Weaver is his speech, not his capricious dismembering of creatures for pure aesthetic motives.

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Answer to the Quiz

The most horrific monsters in Perdido Street Station were,

  • Vermishank — the scheming academic who wanted to culture the Slake Moths for military weaponry.
  • Mayor Bentham Rudgutter — for the same reasons Vermishank is a horror.
  • David Serachin — formerly one of Issac’s scientist friends, but who betrayed Lin and Isaac to the authorities. Betrayal is the worst horrors, or one of the worst besides rape and murder.

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Ender’s Ants

Just saw Ben Kingsley playing the Maori warrior Mazer Rackham in the movie Ender’s Game (2013). Kingsley nails the kiwi accent like no other non-Kiwi actor I have seen in decades or perhaps forever. Felt moved to blog a couple of unrelated notes.

Formic Queen

Depiction of the alien Formic Queen.

Military Strategy when Fighting Aliens

The scenario: you have to fight a race of aliens so ancient and unknown that you really have no real insight whatsover into their psychology. In Ender’s Game the Formics are creatures insectoid-like, similar to giant ants.  Naturally enough their psychology would be very weird and hard to understand without close up anthropological/sociological/entomological study.

The human’s training Ender and his colleagues are running massively complex computer simulations of a possible battle with the Formics.  Do you see the big flaw?

If the human’s programming the simulations do not understand the alien psychology, then the training simulations are not going to be very useful when the real war breaks out. They might get lucky. But the premise in Ender’s Game is that the Formics battle formation patterns and movements look chaotic and unpredicatable, apart from the high energy signature emanating from the Queen.

What should they do?

Why can’t script-writers get these story details right?

A Solution — To Be Random

Maybe it ruins the later plot, but the thing to do when faced with indecision about an opponents strategy is to act  randomly yourself.

What I would love to see in the movie is a scene where Ender Wiggins is taken aside and tutored on what exactly the purpose is for the simulation games. The idea should be clearly explained to Wiggins that the simulation games are only useful for training combat readiness, troop cooperation, and execution skill. It cannot teach strategy, because the games will not be acting like the real enemy, the Formics. The simulation games were design by human’s so they will not likely have many scenarios built-in that will realistically simulate fighting actual Formic armies.

So it should be explained to Ender that when facing the real battle he will need to almost forget about any simulation game strategies, and instead think and adapt in real-time. He should be told to act more unpredictably, and to not guess about Formic battle patterns based in the game simulations, but rather to do so based upon what he gleans in the moment of battle.

This is only a brief adjustment to the movie plot, it would take a one minute scene of dialogue between Ender and General Graff, but it would make the movie so much better for a geek like me to watch and enjoy and forgive all the other terrible Hollywoodisms the film commits.

In the end it turns out this way, since Ender is sufficiently unorthodox and imaginative that he ends up doing something that was unthinkable for the adult military commanders.  So I’m not moaning and bitching.  I just think it is more awesome when a script delivers high-brow intellectualization of the characters motives and thinking.  Somehow it makes a fantasy story more immersive and believable when the dialogue and plot have a lot of detailed intellectualizations and binding logic and scentific principles applied rigrously.  For the vast majority of scifi flicks one has to give up scientific realism at so many points in a plot that injecting some really hard science inwhere possible is a very cool thing to do for the thinking audiences out here.

Totally Unpredictable but Not Perfectly Random

As a general meta-strategic point, this is your real strategy when you truly have close to zero insight into your enemy’s psychology and motive.  You would want to be completely unpredictable at times when it pays o have your opponent guessing your next actions, and yet not behave hopelessly chaotic. (Although note that Ender’s Game works as a gripping story partly because Ender Wiggins does end up understanding one of the Formic civilisation’s motives, which is their desperate need to find a world with abundant water.)

So how does one achieve unpredictability without resorting to random incoherence?

This is the bit I write for all my mathematics and science students. Readers who do not know the answer can follow along.  So the idea is that you will draw up a table (or some kid of database) which presents all the possibly non-stupid strategies available.  (An example of a stupid strategy might be the one where you murder your own army, or kill your entire species to allow the aliens to win.  So you make sure none of those types of options sneak into your database! You also would probably want to purge any options that are just silly, like  the ones where the aliens are attacking and you do absolutely nothing not even try to communicate with them or try to show them you are peaceful.)

For the sake of utter simplicity let’s sketch this on a graph, suppose your database of options boils down to choosing a number from a scale, say strategies 1 to 100.  Each strategy is weighted by a frequency which is your and your war counsel’s expert opinions on the probability or likely chance of success of each strategy.

Decison options can be weighted then chosen randomly but biased towards the more favourable options.

Example tableau of strategies you can choose from unpredictably using biases in favour of the more likely successful options.

You see the point is that these strategies are not equally favoured.  So you use a random number to choose one strategy at a particular time when a decision has to be made.  But the random number selects among these strategies in a biased way so that strategy 25 above is half as likely to be chosen by your random number generator as is strategy 30.  The strategies like option 75 here are more likely to be chosen. You get the idea yes?

This tableau will of course shift dynamically as your enemy or opponents shift their strategies.

The point is that no one can predict your decision, you have the ultimate element of surprise which is that you yourself do not even know which decision you will make.  you only have the odds of the various possibilities.  So you can say, “I am more likely to adopt strategy 75 than I am to issue the decisions for strategy 92.”

This is the answer to how you can be totally unpredictable and yet smart and not foolishly random.  You use random numbers to select among possible options in a biased manner.  You bias the outcome towards the options that have higher chance of success.  But there is still a small chance you will choose a low chance of success option.

But when would your tableau ever prevent you with a low success probability option?  Surely these would be purged from the database?  Well, not always.  This would happen, for instance, if the game or battle was at a crucial stage and all the information suggested th enemy was in a stage of trying to predict your next move.  In such circumstances it can pay off to choose a low probability of success option because your enemy will be unlikely to guess this is what you will do.  This sort of option pays off only when the enemy will suffer if they do not take steps to defend themselves against your unlikely decision.

A good example of such a scenario might be the suicide bomber option, or a chess pawn option, where you sacrifice your resources to make it appear like you are giving away an advantage, which hurts your side, but f the enemy does not see your reason for such a sacrifice they will expose themselves to a more lethal attack on your next move.  In chess you might even let your opponent capture your most powerful piece, your Queen, because perhaps if they do choose to capture our Queen you can in three or four decisive moves checkmate them.  They were blind to the secret of your sacrifice.

OK this is where my students have had their lesson and other readers can rejoin.

Would Ender Really Have Tried to “Talk”?

“But OK, that’s fine at first blush, but real military strategy has to more incisive, a good commander has to be willing to be accountable and therefore cannot just roll a dice”, you might think.  That’s true of course and there are many subtleties to fighting a war, and I am totally ignorant of most of them.  You cannot hope, for example, to wage a successful war using a textbook on game theory.

[Spoiler alerts coming!  Skip this next paragraph if you like.]

Here I must confess that although I like high quality hard scifi, I chose to watch the film before reading Scott-Card’s novel.  I enjoy movies this way.  It’s horrible to sit through a movie and constantly complain about the missed adaptation opportunities and unfaithfulness of the script.  So I ended up enjoying Ender.  I must also confess I started writing this post before watching the final battle where Ender commits genocide  [spoiler alert!] while believing himself to be playing merely a final simulation.

Ender’s Game raises this wonderful philosophical issue.  Many in fact.  [spoiler alert!] One of which is, can there ever be a threat to an entire civilisation great enough to morally justify destroying preemptively an entire planet and all life on it?

My point is that the decision to try to communicate with the alien Formic’s was a noble and moral decision that arguably Scott-Card [spoiler alert!]  wants us to believe would have happened had Ender been allowed to face the final confrontation knowing it was not a game.  (Or maybe the General would have then pulled the plug on Wiggins and put someone else in command at the last second?)

But if you were in Ender Wiggin’s shoes and were lazy and just simply choosing from a bunch of preprogrammed stratgey options this diplomatic option might not ever get selected.  So in addition to trying to behave unpredictably so that your opponent cannot easily plan to defend against your moves, one thing you might always want to do is to think freely and outside the scope of what the authorities allow.

Dammit all!  [spoiler alert!] You know I cried when I found out Ender’s last game was for real.  I really internalised the movie and let the raw emotions overcome my senses.  Even if the acting in the film was a bit atrocious, one can let in and filter just the essential emotions and intellectual content.  It make a movie worht watching when you do this artistic suspension of disbelief.   I thought of the ral world horror committed by the second Bush administration in Iraq, and the lying generals and the hapless Colin Powell who seemed to be selecting decisions from a badly written textbook full of errors.  But mostly I thought of poor fcitonal Ender Wiggins and the sickening thoughts that must have ben racing through his mind.

Of all things in the movie, perhaps the most unreal was that his character actually still has the emotional strength to argue his case philosophically with General Graff and Mazer, only minutes after the horrific realisation.  How does one avoid such things?  Such traps for the unwary?

Play all your games ethically!   Even the trashy VDO games, even if they have the most awesome hires graphics.  Refuse even to begin playing the immoral ones that have no option for morality.

Try diplomacy more!  Even if it could wipe out your entire planet!

That would be my thought for the day.


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