“It Hurts my Brain” — Wrong! Thinking is Not Hard, Thinking is Beautiful

Can we all please get beyond the myth that “thinking is hard”! This guy from Veritasium means well, but regurgitates the myth: How Should We Teach Science? (2veritasium, March 2017) Thinking is not hard because of the brain energy it takes. That is utter crap. What is likely more realistic psychologically is that people do not take time and quiet space to reflect and meditate. Deep thinking is more like meditation, and it is energizing and relaxing. So this old myth needs replacing I think. Thinking deeply while distracting yourself with trivia is really hard, because of the cognitive load on working memory. It seems hard because when your working memory gets overloaded you cannot retain ideas, and it appears like you get stupid and this leads to frustration and anxiety, and that does have physiological effects that mimic a type of mental pain.

But humans have invented ways to get around this. One is called WRITING. You sit down meditate, allow thoughts to flood your working memory, and when you get an insight or an overload you write them down, then later review, organize and structure your thoughts. In this way deep thinking is easy and enjoyable. Making thinking hard so that it seems to hurt your brain is a choice. You have chosen to buy into the myth when you try to concentrate on deep thinking while allowing yourself to be distracted by life’s trivia and absurdities. Unfortunately, few schools teach the proper art of thinking.

The Arcania of Arkani

It is not often you get to disagree with a genius. But if you read enough or attend enough lectures sooner or later some genius is going to say or write something that you can see is evidently false, or perhaps (being a bit more modest) you might think is merely intuitively false. So the other day I see this lecture by Nima Arkani-Hamed with the intriguing title “The Morality of Fundamental Physics”. It is a really good lecture, I recommend every young scientist watch it. (The “Arcane” my title alludes to, by the way, is a good thing, look up the word!) It will give you a wonderful sense of the culture of science and a feeling that science is one of the great ennobling endeavours of humanity. The way Arkani-Hamed describes the pursuit of science also gives you comfort as a scientist if you ever think you are not earning enough money in your job, or feel like you are “not getting ahead” — you should simply not care! — because doing science is a huge privilege, it is a reward unto itself, and little in life can ever be as rewarding as making a truly insightful scientific discovery or observation. No one can pay me enough money to ever take away that sort of excitement and privilege, and no amount of money can purchase you the brain power and wisdom to achieve such accomplishments.  And one of the greatest overwhelming thrills you can get in any field of human endeavour is firstly the hint that you are near to turning arcane knowledge into scientific truth, and secondly when you actually succeed in this.

First, let me be deflationary about my contrariness. There is not a lot about fundamental physics that one can honestly disagree with Arkani-Hamed about on an intellectual level, at least not with violent assertions of falsehood.  Nevertheless, fundamental physics is rife enough with mysteries that you can always find some point of disagreement between theoretical physicists on the foundational questions. Does spacetime really exist or is it an emergent phenomenon? Did the known universe start with a period of inflation? Are quantum fields fundamental or are superstrings real?

When you disagree on such things you are not truly having a physics disagreement, because these are areas where physics currently has no answers, so provided you are not arguing illogically or counter to known experimental facts, then there is a wide open field for healthy debate and genuine friendly disagreement.

Then there are deeper questions that perhaps physics, or science and mathematics in general, will never be able to answer. These are questions like: Is our universe Everettian? Do we live in an eternal inflation scenario Multiverse? Did all reality begin from a quantum fluctuation, and, if so, what the heck was there to fluctuate if there was literally nothing to begin with? Or can equations force themselves into existence from some platonic reality merely by brute force of their compelling beauty or structural coherence? Is pure information enough to instantiate a physical reality (the so-called “It from Bit” meme.

Some people disagree on whether such questions are amenable to experiment and hence science. The Everettian question may some day become scientific. But currently it is not, even though people like David Deutsch seem to think it is (a disagreement I would have with Deutsch). While some of the “deeper ” questions turn out to be stupid, like the “It from Bit” and “Equations bringing themselves to life” ideas. However, they are still wonderful creative ideas anyway, in some sense, since they put our universe into contrast with a dull mechanistic cosmos that looks just like a boring jigsaw puzzle.

The fact our universe is governed (at least approximately) by equations that have an internal consistency, coherence and even elegance and beauty (subjective though those terms may be) is a compelling reason for thinking there is something inevitable about the appearance of a universe like ours. But that is always just an emotion, a feeling of being part of something larger and transcendent, and we should not mistake such emotions for truth. By the same token mystics should not go around mistaking mystical experiences for proof of the existence of God or spirits. That sort of thinking is dangerously naïve and in fact anti-intellectual and incompatible with science. And if there is one truth I have learned over my lifetime, it is that whatever truth science eventually establishes, and whatever truths religions teach us about spiritual reality, wherever these great domains of human thought overlap they must agree, otherwise one or the other is wrong. In other words, whatever truth there is in religion, it must agree with science, at least eventually. If it contradicts known science it must be superstition. And if science contravenes the moral principles of religion it is wrong.

Religion can perhaps be best thought of in this way:  it guides us to knowledge of what is right and wrong, not necessarily what is true and false. For the latter we have science. So these two great systems of human civilization go together like the two wings of a bird, or as in another analogy, like the two pillars of Justice, (1) reward, (2) punishment. For example, nuclear weapons are truths of our reality, but they are wrong. Science gives us the truth about the existence and potential for destruction of nuclear weapons, but it is religion which tells us they are morally wrong to have been fashioned and brought into existence, so it is not that we cannot, but just that we should not.

Back to the questions of fundamental physics: regrettably, people like to think these questions have some grit because they allow one to disbelieve in a God. But that’s not a good excuse for intellectual laziness. You have to have some sort of logical foundation for any argument. This often begins with an unproven assumption about reality. It does not matter where you start, so much, but you have to start somewhere and then be consistent, otherwise as elementary logic shows you would end up being able to prove (and disprove) anything at all. If you start with a world of pure information, then posit that spacetime grows out of it, then (a) you need to supply the mechanism of this “growth”, and (b) you also need some explanation for the existence of the world of pure information in the first place.

Then if you are going to argue for a theory that “all arises from a vacuum quantum fluctuation”, you have a similar scenario, where you have not actually explained the universe at all, you have just pushed back the existence question to something more elemental, the vacuum state. But a quantum vacuum is not a literal “Nothingness”, in fact is is quite a complicated sort of thing, and has to involve a pre-existing spacetime or some other substrate that supports the existence of quantum fields.

Further debate along these lines is for another forum. Today I wanted to get back to Nima Arkani-Hamed’s notions of morality in fundamental physics and then take issue with some private beliefs people like Arkani-Hamed seem to profess, which I think betray a kind of inconsistent (I might even dare say “immoral”) thinking.

Yes, there is a Morality in Science

Arkani-Hamed talks mostly about fundamental physics. But he veers off topic in places and even brings in analogies with morality in music, specifically in lectures by the great composer Leonard Bernstein, there are concepts in the way Bernstein describes the beauty and “inevitability” of passages in great music like Beethoven’s Fifth Symphony. Bernstein even gets close to saying that after the first four notes of the symphony almost the entire composition could be thought of as following as an inevitable consequence of logic and musical harmony and aesthetics. I do not think this is flippant hyperbole either, though it is somewhat exaggerated. The cartoon idea of Beethoven’s music following inevitable laws of aesthetics has an awful lot in common with the equally cartoon notion of the laws of physics having, in some sense, their own beauty and harmony such that it is hard to imagine any other set of laws and principles, once you start from the basic foundations.

I should also mention that some linguists would take umbrage at Arkani-Hamed’s use of the word “moral”.  Really, most of what he lectures about is aesthetics, not morality.  But I am happy to warp the meaning of the word “moral” just to go along with the style of Nima’s lecture.  Still, you do get a sense from his lecture, that the pursuit of scientific truth does have a very close analogy to moral behaviour in other domains of society.  So I think he is not totally talking about aesthetics, even though I think the analogy with Beethoven’s music is almost pure aesthetics and has little to do with morality.   OK, those niggles aside, let’s review some of Arkani’Hamed’s lecture highlights.

The way Arkani-Hamed tells the story, there are ways of thinking about science that are not just “correct”, but more than correct, the best ways of thinking seem somehow “right”, whereby he means “right” in the moral sense. He gives some examples of how one can explain a phenomenon (e.g., the apparent forwards pivoting of a helium balloon suspended inside a boxed car) where there are many good explanations that are all correct (air pressure effects, etc) but where often there is a better deeper more morally correct way of reasoning (Einstein’s principle of equivalence — gravity is indistinguishable from acceleration, so the balloon has to “fall down”).

philsci_immoral_helim_balloon

It really is entertaining, so please try watching the video. And I think Arkani-Hamed makes a good point. There are “right” ways of thinking in science, and “correct but wrong ways”. I guess, unlike human behaviour the scientifically “wrong” ways are not actually spiritually morally “bad”, as in “sinful”. But there is a case to be made that intellectually the “wrong” ways of thinking (read, “lazy thinking ways”) are in a sense kind of “sinful”. Not that we in science always sin in this sense of using correct but not awesomely deep explanations.  I bet most scientists which they always could think in the morally good (deep) ways! Life would be so much better if we could. And no one would probably wish to think otherwise. It is part of the cultural heritage of science that people like Einstein (and at times Feynman, and others) knew of the morally good ways of thinking about physics, and were experts at finding such ways of thinking.

Usually, in brief moments of delight, most scientists will experience fleeting moments of being able to see the morally good ways of scientific thinking and explanation. But the default way of doing science is immoral, by in large, because it takes a tremendous amount of patience and almost mystical insight, to be able to always see the world of physics in the morally correct light — that is, in the deepest most meaningful ways — and it takes great courage too, because, as Arkani-Hamed points out, it takes a lot more time and contemplation to find the deeper morally “better” ways of thinking, and in the rush to advance one’s career and publish research, these morally superior ways of thinking often get by-passed and short-circuited. Einstein was one of the few physicists of the last century who actually managed, a lot of his time, to be patient and courageous enough to at least try to find the morally good explanations.

This leads to two wonderful quotations Arkani-Hamed offers, one from Einstein, and the other from a lesser known figure of twentieth century science, the mathematician Alexander Gröthendieck — who was probably an even deeper thinker than Einstein.

The years of anxious searching in the dark, with their intense longing, their intense alternations of confidence and exhaustion and the final emergence into the light—only those who have experienced it can understand it.
— Albert Einstein, describing some of the intellectual struggle and patience needed to discover the General Theory of Relativity.

“The … analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months—when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!

“A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration … the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it … yet it finally surrounds the resistant substance.”
— Alexander Gröthendieck, describing the process of grasping for mathematical truths.

Beautiful and foreboding — I have never heard of the mathematical unknown likened to a “hard marl” (sandstone) before!

So far all is good. There are many other little highlights in Arkani-Hamed’s lecture, and I should not write about them all, it is much better to hear them explained by the master.

So what is there to disagree with?

The Morally Correct Thinking in Science is Open-Minded

There are a number of characteristics of “morally correct” reasoning in science, or an “intellectually right way of doing things”. Arkani-Hamed seems to list most of the important things:

  • Trust: trust that there is a universal, invariant, human-independent and impersonal (objective) truth to natural laws.
  • Honesty: with others (no fraud) but also more importantly you need to be honest with yourself if you want to do good science.
  • Humility: who you are is irrelevant, only the content of your ideas is important.
  • Wisdom: we never pretend we have the whole truth, there is always uncertainty.
  • Perseverance: lack of certainty is not an excuse for laziness, we have to try our hardest to get to the truth, no matter how difficult the path.
  • Tolerance: it is extremely important to entertain alternative and dissenting ideas and to keep an open mind.
  • Justice: you cannot afford to be tolerant of dishonest or ill-formed ideas. It is indeed vitally important to be harshly judgemental of dishonest and intellectually lazy ideas. Moreover, one of the hallmarks of a great physicist is often said to be the ability to quickly check and to prove one’s own ideas to be wrong as soon as possible.

In this list I have inserted in bold the corresponding spiritual attributes that Professor Nima does not identify. But I think they are important to explicitly state. Because they provide a Rosetta Stone of sorts for translating the narrow scientific modes of behaviour into border domains of human life.

I think that’s a good list. There is, however, one hugely important morally correct way of doing science that Arkani-Hamed misses, and even fails to gloss over or hint at. Can you guess what it is?

Maybe it is telling of the impoverishment in science education, the cold objective dispassionate retelling of facts, in our society that I think not many scientists will even think of his one, but I do not excuse Arkani-Hamed for leaving it off his list, since in many ways it is the most important moral stance in all of science!

It is,

  • Love: the most important driver and motive for doing science, especially in the face of adversity or criticism, is a passion and desire for truth, a true love of science, a love of ideas, an aesthetic appreciation of the beauty and power of morally good ideas and explanations.

Well ok, I will concede this is perhaps implicit in Arkani-Hamed’s lecture, but I still cannot give him 10 out of 10 on his assignment because he should have made it most explicit, and highlighted it in bold colours.

One could point out many instances of scientists failing at these minimal scientific moral imperatives. Most scientists go through periods of denial, believing vainly in a pet theory and failing to be honest to themselves about the weaknesses of their ideas. There is also a vast cult of personality in science that determines a lot of funding allocation, academic appointments, favouritism, and general low level research corruption.

The point of Arkani-Hamed’s remarks is not that the morally good behaviours are how science is actually conducted in the everyday world, but rather it is how good science should be conducted and that from historical experience the “good behaviours” do seem to be rewarded with the best and brightest break-throughs in deep understanding. And I think Arkani-Hamed is right about this. It is amazing (or perhaps, to the point, not so amazing!) how many Nobel Laureates are “humble” in the above sense of putting greater stock in their ideas and not in their personal authority. Ideas win Nobel Prizes, not personalities.

So what’s the problem?

The problem is that while expounding on these simplistic and no-doubt elegant philosophical and aesthetic themes, he manages to intersperse his commentary with the claim, “… by the way, I am an atheist”.

OK, I know what you are probably thinking, “what’s the problem?” Normally I would not care what someone thinks regarding theism, atheism, polytheism, or any other “-ism”. People are entitled to their opinions, and all power to them. But as a scientist I have to believe there are fundamental truths about reality, and about a possible reality beyond what we perceive. There must even be truths about a potential reality beyond what we know, and maybe even beyond what we can possibly ever know.

Now some of these putative “truths” may turn out to be negative results. There may not be anything beyond physical reality. But if so, that’s a truth we should not hereby now and forever commit to believing. We should at least be open-minded to the possibility this outcome is false, and that the truth is rather that there is a reality beyond physical universe.  Remember, open-mindedness was one of Arkani-Hamed’s prime “good behaviours” for doing science.

The discipline of Physics, by the way, has very little to teach us about such truths. Physics deals with physical reality, by definition, and it is an extraordinary disappointment to hear competent, and even “great”, physicists expound their “learned” opinions on theism or atheism and non-existence of anything beyond physical universes. These otherwise great thinkers are guilty of over-reaching hubris, in my humble opinion, and it depresses me somewhat. Even Feynman had such hubris, yet he managed expertly to cloak it in the garment of humility, “who am I to speculate on metaphysics,” is something he might have said (I paraphrase the great man). Yet by clearly and incontrovertibly stating “I do not believe in God” one is in fact making an extremely bold metaphysical statement. It is almost as if these great scientists had never heard of the concept of agnosticism, and somehow seem to be using the word “atheism” as a synonym. But no educated person would make such a gross etymological mistake. So it just leaves me perplexed and dispirited to hear so many claims of “I am atheist” coming from the scientific establishment.

Part of me wants to just dismiss such assertions or pretend that these people are not true scientists. But that’s not my call to make.  Nevertheless, for me, a true scientist almost has to be agnostic. There seems very little other defensible position.

How on earth would any physicist ever know such things (as non-existence of other realms) are true as articles of belief? They cannot! Yet it is astounding how many physicists will commit quite strongly to atheism, and even belittle and laugh at scientists who believe otherwise. It is a strong form of intellectual dishonesty and corruption of moral thinking to have such closed-minded views about the nature of reality.

So I would dare to suggest that people like Nima Arkani-Hamed, who show such remarkable gifts and talents in scientific thinking and such awesome skill in analytical problem solving, can have the intellectual weakness to profess any version of atheism whatsoever. I find it very sad and disheartening to hear such strident claims of atheism among people I would otherwise admire as intellectual giants.

Yet I would never want to overtly act to “convert” anyone to my views. I think the process of independent search for truth is an important principle. People need to learn to find things out on their own, read widely, listen to alternatives, and weigh the evidence and logical arguments in the balance of reason and enlightened belief, and even then, once arriving at a believed truth, one should still question and consider that one’s beliefs can be over-turned in the light of new evidence or new arguments.  Nima’s principle of humility, “we should never pretend we have the certain truth”.

Is Atheism Just Banal Closed-Mindedness?

The scientifically open-mind is really no different to the spiritually open-mind other than in orientation of topics of thought. Having an open-mind does not mean one has to be non-committal about everything. You cannot truly function well in science or in society without some grounded beliefs, even if you regard them all as provisional. Indeed, contrary to the cold-hearted objectivist view of science, I think most real people, whether they admit it or not (or lie to themselves perhaps) they surely practise their science with an idea of a “truth” in mind that they wish to confirm. The fact that they must conduct their science publicly with the Popperrian stances of “we only postulate things that can be falsified” is beside the point. It is perfectly acceptable to conduct publicly Popperian science while privately having a rich metaphysical view of the cosmos that includes all sorts of crazy, and sometimes true, beliefs about the way things are in deep reality.

Here’s the thing I think needs some emphasis: even if you regard your atheism as “merely provisional” this is still an unscientific attitude! Why? Well, because questions of higher reality beyond the physical are not in the province of science, not by any philosophical imperative, but just by plain definition. So science is by definition agnostic as regards the transcendent and metaphysical. Whatever exists beyond physics is neither here nor there for science. Now many self-proclaimed scientists regard this fact about definitions as good enough reason for believing firmly in atheism. My point is that this is nonsense and is a betrayal of scientific morals (morals, that is, in the sense of Arkani-Hamed — the good ways of thinking that lead to deeper insights). The only defensible logical and morally good way of reasoning from a purely scientific world view is that one should be at the basest level of philosophy positive in ontology and minimalist in negativity, and agnostic about God and spiritual reality. It is closed-minded and therefore, I would argue, counter to Arkani-Hamed’s principles of morals in physics, to be a committed atheist.

This is in contrast to being negative about ontology and positively minimalist, which I think is the most mistaken form of philosophy or metaphysics adopted by a majority of scientists, or sceptics, or atheists.  The stance of positive minimalism, or  ontological negativity, adopts, as unproven assumption, a position that whatever is not currently needed, or not currently observed, doe snot in fact exist.  Or to use a crude sound-bite, such philosophy is just plain closed-mindedness.  A harsh cartoon version of which is, “what I cannot understand or comprehend I will assume cannot exist”.   This may be unfair in some instances, but I think it is a fairly reasonable caricature of general atheistic thought.   I think is a lot fairer than the often given argument against religion which points to corruptions in religious practice as a good reason to not believe in God.  There is of course absolutely no causal or logical connection to be made between human corruptions and the existence or non-existence of a putative God.

In my final analysis of Arkani-Hamed’s lecture, I have ended up not worrying too much about the fact he considers himself an atheist. I have to conclude he is a wee bit self-deluded, (like most of his similarly minded colleagues no doubt, yet, of course, they might ultimately be correct, and I might be wrong, my contention is that the way they are thinking is morally wrong, in precisely the sense Arkani-Hamed outlines, even if their conclusions are closer to the truth than mine).

Admittedly, I cannot watch the segments in his lecture where he expresses the beautiful ideas of universality and “correct ways of explaining things” without a profound sense of the divine beyond our reach and understanding. Sure, it is sad that folks like Arkani-Hamed cannot infer from such beauty that there is maybe (even if only possibly) some truth to some small part of the teachings of the great religions. But to me, the ideas expressed in his lecture are so wonderful and awe-inspiring, and yet so simple and obvious, they give me hope that many people, like Professor Nima himself, will someday appreciate the view that maybe there is some Cause behind all things, even if we can hardly ever hope to fully understand it.

My belief has always been that science is our path to such understanding, because through the laws of nature that we, as a civilization, uncover, we can see the wisdom and beauty of creation, and no longer need to think that it was all some gigantic accident or experiment in some mad scientists super-computer. Some think such wishy-washy metaphysics has no place in the modern world. After all, we’ve grown accustomed to the prevalence of evil in our world, and tragedy, and suffering, and surely if any divine Being was responsible then this would be a complete and utter moral paradox. To me though, this is a a profound misunderstanding of the nature of physical reality. The laws of physics give us freedom to grow and evolve. Without the suffering and death there would be no growth, no exercise of moral aesthetics, and arguably no beauty. Beauty only stands out when contrasted with ugliness and tragedy. There is a Yin and Yang to these aspects of aesthetics and misery and bliss. But the other side of this is a moral imperative to do our utmost to relieve suffering, to reduce poverty to nothing, to develop an ever more perfect world. For then greater beauty will stand out against the backdrop of something we create that is quite beautiful in itself.

Besides, it is just as equally wishy-washy to think the universe is basically accidental and has no creative impulse.  People would complain either way.  My positive outlook is that as long as there is suffering and pain in this world, it makes sense to at least imagine there is purpose in it all.  How miserable to adopt Steven Wienberg’s outlook that the noble pursuit of science merely “lifts up above farce to at least the grace of tragedy”.  That’s a terribly pessimistic negative sort of world view.  Again, he might be right that there is no grand purpose or cosmic design, but the way he reasons to that conclusion seems, to me, to be morally poor (again, strictly, if you like, in the Arkani-Hamed morality of physics conception).

There seems, to me, to be no end to the pursuit of perfections. And given that, there will always be relative ugliness and suffering. The suffering of people in the distant future might seem like luxurious paradise to us in the present. That’s how I view things.

The Fine Tuning that Would “Turn You Religious”

Arkani-Hamed mentions another thing that I respectfully take a slight exception to — this is in a separate lecture at a Philosophy of Cosmology conference —  in a talk, “Spacetime, Quantum Mechanics and the Multiverse”.  Referring to the amazing coincidence that our universe has just the right cosmological constant to avoid space being empty and devoid of matter, and just the right Higgs boson mass to allow atoms heavier than hydrogen to form stably, is often, Arkani-Hamed points out, given as a kind of anthropic argument (or quasi-explanation) for our universe.  The idea is that we see (measure) such parameters for our universe precisely, and really only, because if the parameters were not this way then we would not be around to measure them!  Everyone can understand this reasoning.  But it stinks!   And off course it is not an explanation, such anthropic reasoning reduces to mere observation.  Such reasonings are simple banal brute facts about our existence.  But there is a setting in metaphysics where such reasoning might be the only explanation, as awful as it smells.  That is, if our meta-verse is governed by something like Eternal Inflation, (or even by something more ontologically radical like Max Tegmark’s “Mathematical Multiverse”) whereby every possible universe is at some place or some meta-time, actually realised by inflationary big-bangs (or mathematical consequences in Tegmark’s picture) then it is really boring that we exist in this universe, since no matter how infinitesimally unlikely the vacuum state of our universe is, within the combinatorial possibilities of all possible inflationary universe bubbles (or all possible consistent mathematical abstract realities) there is, in these super-cosmic world views, absolutely nothing to prevent our infinitesimally (“zero probability measure”) universe from eventually coming into being from some amazingly unlikely big-bang bubble.

In a true multiverse scenario we thus get no really deep explanations, just observations.  “The universe is this way because if it were not we would not be around to observe it.”  The observation becomes the explanation.  A profoundly unsatisfying end to physics!   Moreover, such infinite possibilities and infinitesimal probabilities make standard probability theory almost impossible to use to compute anything remotely plausible about multiverse scenarios with any confidence (although this has not stopped some from publishing computations about such probabilities).

After discussing these issues, which Arkani-Hamed thinks are the two most glaring fine-tuning or “naturalness” problems facing modern physics, he then says something which at first seems reasonable and straight-forward, yet which to my ears also seemed a little enigmatic.  To avoid getting it wrong let me transcribe what he says verbatim:

We know enough about physics now to be able to figure out what universes would look like if we changed the constants.  … It’s just an interesting fact that the observed value of the cosmological constant and the observed value of the Higgs mass are close to these dangerous places. These are these two fine-tuning problems, and if I make the cosmological constant more natural the universe is empty, if I make the Higgs more natural the universe is devoid of atoms. If there was a unique underlying vacuum, if there was no anthropic explanation at all, these numbers came out of some underlying formula with pi’s and e’s, and golden ratios, and zeta functions and stuff like that in them, then [all this fine tuning] would be just a remarkably curious fact.… just a very interesting  coincidence that the numbers came out this way.  If this happened, by the way, I would start becoming religious.  Because this would be our existence hard-wired into the DNA of the universe, at the level of the mathematical ultimate formulas.

So that’s the thing that clanged in my ears.  Why do people need something “miraculous” in order to justify a sense of religiosity?  I think this is a silly and profound misunderstanding about the true nature of religion.  Unfortunately I cannot allow myself the space to write about this at length, so I will try to condense a little of what I mean in what will follow.  First though, let’s complete the airing,  for in the next breath Arkani-Hamed says,

On the other hand from the point of view of thinking about the multiverse, and thinking that perhaps a component of these things have an anthropic explanation, then of course it is not a coincidence, that’s were you’d expect it to be, and we are vastly less hard-wired into the laws of nature.

So I want to say a couple of things about all this fine-tuning and anthropomorphic explanation stuff.  The first is that it does not really matter, for a sense of religiosity, if we are occupying a tiny infinitesimal region of the multiverse, or a vast space of mathematically determined inevitable universes.  In fact, the Multiverse, in itself, can be considered miraculous.  Just as miraculous as a putative formulaically inevitable cosmos.   Not because we exist to observe it all, since that after-all is the chief banality of anthropic explanations, they are boring!  But miraculous because a multiverse exists in the first place that harbours all of us, including the infinitely many possible doppelgängers of our universe and subtle and wilder variations thereupon.  I think many scientists are careless in such attitudes when they appear to dismiss reality as “inevitable”.  Nothing really, ultimately, is inevitable.  Even a formulaic universe has an origin in the deep underlying mathematical structure that somehow makes it irresistible for the unseen motive forces of metaphysics to have given birth to It’s reality.

No scientific “explanation” can ever push back further than the principles of mathematical inevitability.  Yet, there is always something further to say about origins of reality .  There is always something proto-mathematical beyond.  And probably something even more primeval beyond that, and so on, ad infinitum, or if you prefer a non-infinite causal regression then something un-caused must, in some atemporal sense, pre-exist everything.  Yet scientists routinely dismiss or ignore such metaphysics.  Which is why, I suspect, they fail to see the ever-present miracles about our known state of reality.  Almost any kind of reality where there is a consciousness that can think and imagine the mysteries of it’s own existence, is a reality that has astounding miraculousness to it.  The fact science seeks to slowly pull back the veils that shroud these mysteries does not diminish the beauty and profundity of it all, and in fact, as we have seen science unfold with it’s explanations for phenomena, it almost always seems elegant and simple, yet amazingly complex in consequences, such that if one truly appreciates it all, then there is no need whatsoever to look for fine-tuning coincidences or formulaic inevitabilities to cultivate a natural and deep sense of religiosity.

I should pause and define loosely what I mean by “religiosity”.  I mean nothing too much more than what Einstein often articulated: a sense of our existence, our universe, being only a small part of something beyond our present understanding, a sense that maybe there is something more transcendent than our corner of the cosmos.  No grand design is in mind here, no grand picture or theory of creation, just a sense of wonder and enlightenment at the beauty inherent in the natural world and in our expanding conscious sphere which interprets the great book of nature. (OK, so this is rather more poetic than what you might hope for, but I will not apologise for that.   I think something gets lost if you remove the poetry from definitions of things like spirituality or religion.  I think this is because if there really is meaning in such notions, they must have aspects that do ultimately lie beyond the reach of science, and so poetry is one of the few vehicles of communication that can point to the intended meanings, because differential equations or numerics will not suffice.)

OK, so maybe Arkani-Hamed is not completely nuts in thinking there is this scenario whereby he would contemplate becoming “religious” in the Einsteinian sense.  And really, no where in this essay am I seriously disagreeing with the Professor.  I just think that perhaps if scientists like Arkani-Hamed thought a little deeper about things, and did not have such materialistic lenses shading their inner vision, perhaps they would be able to see that miracles are not necessary for a deep and profound sense of religiosity or spiritual understanding or appreciation of our cosmos.

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Just to be clear and “on the record”, my own personal view is that there must surely be something beyond physical reality. I am, for instance, a believer in the platonic view of mathematics: which is that humans, and mathematicians from other sentient civilizations which may exist throughout the cosmos, gain their mathematical understanding through a kind of discovery of eternal truths about realms of axiomatics and principles of numbers and geometry and deeper abstractions, none of which exist in any temporal pre-existing sense within our physical world. Mathematical theorems are thus not brought into being by human minds. They are ideas that exist independently of any physical universe. Furthermore, I happen to believe in something I would call “The Absolute Infinite”. I do not know what this is precisely, I just have an aesthetic sense of It, and It is something that might also be thought of as the source of all things, some kind of universal uncaused cause of all things. But to me, these are not scientific beliefs. They are personal beliefs about a greater reality that I have gleaned from many sources over the years. Yet, amazingly perhaps, physics and mathematics have been one of my prime sources for such beliefs.

The fact I cannot understand such a concept (as the Absolute Infinite) should not give me any pause to wonder if it truly exists or not. And I feel no less mature or more infantile for having such beliefs. If anything I pity the intellectually impoverished souls who cannot be open to such beliefs and speculations. I might point out that speculation is not a bad thing either, without speculative ideas where would science be? Stuck with pre-Copernican Ptolemy cosmology or pre-Eratosthenes physics I imagine, for speculation was needed to invent gizmos like telescopes and to wonder about how to measure the diameter of the Earth using just the shadow of a tall tower in Alexandria.

To imagine something greater than ourselves is always going to be difficult, and to truly understand such a greater reality is perhaps canonically impossible. So we aught not let such smallness of our minds debar us from truth. It is thus a struggle to keep an open-mind about metaphysics, but I think it is morally correct to do so and to resist the weak temptation to give in to philosophical negativism and minimalism about the worlds that potentially exist beyond ours.

Strangely, many self-professing atheists think they can imagine we live in a super Multiverse. I would ask them how they can believe in such a prolific cosmos and yet not also accept the potential existences beyond the physical? And not even “actual existence” just simply “potential existence”. I would then point out that as long as there is admitted potential reality and plausible truth to things beyond the physical, you cannot honestly commit to any brand of atheism. To my mind, even my most open-mind, this form of atheism would seem terribly dishonest and self-deceiving.

Exactly how physics and mathematics could inform my spiritual beliefs is hard to explain in a few words. Maybe sometime later there is an essay to be written on this topic. For now, all I will say is that like Nima Arkani-Hamed, I have a deep sense of the “correctness” of certain ways of thinking about physics, and sometimes mathematics too (although mathematics is less constrained). And similar senses of aesthetics draw me in like the unveiling of a Beethoven symphony to an almost inevitable realisation of some version of truth to the reality of worlds beyond the physical, worlds where infinite numbers reside, where the mind can explore unrestrained by bones and flesh and need for food or water.  In such worlds greater beauty than on Earth resides.


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Giving Your Equations a Nice Bath & Scrub

There’s a good book for beginning computer programmers I recently came across.  All young kids wanting to write code professionally should check out Robert Martin’s book, “Clean Code: A Handbook of Agile Software Craftsmanship”  (Ideally get your kids to read this before the more advanced “Design Patterns” books.)

But is there such a guide for writing clean mathematics?

I could ask around on Mathforums or Quora, but instead here I will suggest some of my own tips for such a guide volume.  What gave me this spark to write a wee blog about this was a couple of awesome “finds”.  The first was Professor Tadashi Tokieda’s Numberphile clips and his AIMS Lectures on Topology and Geometry (all available on YouTube).  Tokieda plugs a couple of “good reads”, and this was the second treasure: V.I. Arnold’s lectures on Abel’s Theorem, which were typed up by his student V.B. Alekseev, “Abel’s Theorem in Problems and Solutions”, which is available in abridged format (minus solutions) in a translation by Julian Gilbey here: “Abels’ Theorem Through Problems“.

Tadashi lecturing in South Africa.

Tadashi lecturing in South Africa. Clearer than Feynman?

Tokieda’s lectures and Arnold’s exposition style are perfect examples of “clean mathematics”.  What do I mean by this?

Firstly, what I absolutely do not mean is Bourbaki style rigour and logical precision.  That’s not clean mathematics.  Because the more precision and rigour you demand the more dense and less comprehensible it all becomes to the point where it becomes unreadable and hence useless.

I mean mathematics that is challenging for the mind (so interesting) and yet clear and understandable and visualizable.  That last aspect is crucial.  If I cannot visualise an abstract idea then it has not been explained well and I have not understood it deeply.  We can only easily visualize 2D examples or 3D if we struggle.  So how are higher dimensional ideas visualised?  Tokieda shows there is no need.  You can use the algebra perfectly well for higher dimensional examples, but always give the idea in 2D or 3D.

It’s amazing that 3D seems sufficient for most expositions.  With a low dimension example most of the essence of the general N dimensional cases can be explained in pictures.   Perhaps this is due to 3D being the most awkward dimension?  It’s just a pity we do not have native 4D vision centres in our brain (we actually do, it’s called memory, but it sadly does not lead to full 4D optical feature recognition).

Dr Tokieda tells you how good pictures can be good proofs.  The mass of more confusing algebra a good picture can replace is startling (if you are used to heavy symbolic algebra).  I would also add that Sir Roger Penrose and John Baez are to experts who make a lot of use of pictorial algebra, and that sort of stuff is every bit as rigorous as symbolic algebra, and I would argue even more-so.  How’s that?  The pictorial algebra is less prone to mistake and misinterpretation, precisely because our brains are wired to receive information visually without the language symbol filters.  Thus whenever you choose instead to write proofs using formal symbolics you are reducing your writing down to less rigour, because it is easier to make mistakes and have your proof misread.

So now, in homage to Robert Martin’s programming style guide, here are some analogous sample chapter or section headings for a hypothetical book on writing clean mathematics.

Keep formal (numbered) definitions to a minimum

Whenever you need a formal definition you have failed the simplicity test.  A definition means you have not found a natural way to express or name a concept.  That’s really all definitions are, they set up names for concepts.

Occasionally advanced mathematics requires defining non-intuitive concepts, and these will require a formal approach, precisely because they are non-intuitive.  But otherwise, name objects and relations clearly and put the keywords in old, and then you can avoid cluttering up chapters with formal boring looking definition breaks.  The definitions should, if at all possible, flow naturally and be embedded in natural language paragraphs.

Do not write symbolic algebra when a picture will suffice

Most mathematicians have major hang-ups about providing misleading visual illustrations.  So my advice is do not make them misleading!  But you should use picture proofs anyway, whenever possible, just make sure they capture the essence and are generalisable to higher dimensions.  It is amazing how often this is possible.  If you doubt me, then just watch Tadashi Tokieda’s lectures linked to above.

Pro mathematicians often will think pictures are weak.  But the reality is the opposite.  Pictures are powerful.  Pictures should not sacrifice rigour.  It is the strong mathematician who can make their ideas so clear and pristine that a minimalistic picture will suffice to explain an idea of great abstract generality.  Mathematicians need to follow the physicists credo of using inference, one specific well-chosen example can suffice as an exemplar case covering infinitely many general cases.  The hard thing is choosing a good example.  It is an art.  A lot of mathematician writers seem to fail at this art, or not even try.

You do not have to use picture in your research if you do not get much from them, but in your expositions, in your writing for the public, failing to use pictures is a disservice to your readers.

The problem with popular mathematics books is not the density of equations, it is the lack of pictures.  If for every equation you have a couple of nice illustrative pictures, then there would be no such thing as “too many equations” even for a lay readership.  The same rule should apply to academic mathematics writing, with perhaps an reasonable allowance for a slightly higher symbol to picture ratio, because academically you might need to fill in a few gaps for rigour.

Rigour does not imply completeness

Mathematics should be rigorous, but not tediously so.  When gaps do not reduce clarity then you can avoid excessive equations.  Just write what the reader needs, do not fill in every gap for them.  And whenever a gap can be filled with a picture, use the picture rather than more lines of symbolic algebra.  So you do not need ruthless completeness.  Just provide enough for rigour to be inferred.

Novel writers know this.  If they set out to describe scenes completely they would ever get past chapter one. Probably not even past paragraph one.  And giving the reader too much information destroys the operation of their inner imagination and leads to the reader disconnecting from the story.

For every theorem provide many examples

The Definition to Theorem ratio should be low, for every couple of definitions there should be a bundle of nice theorems, otherwise the information content of your definitions has been poor.  More  definitions than theorems means you’ve spent more of your words naming stuff not using stuff.  Likewise the Theorem to Example ratio should be lo.  More theorems than examples means you’ve cheated the student by showing them lot of abstract ideas with no practical use.  So show them plenty of practical uses so they do not feel cheated.

Write lucidly and for entertainment

This is related to the next heading which is to write with a story narrative.  On a finer level, every sentence should be clear, use plain language, and minimum jargon.  Mathematics text should be every bit as descriptive and captivating as a great novel.  If you fail in writing like a good journalist or novelist then you have failed to write clean mathematics.  Good mathematics should entertain the aficionado.  It does not have to be set like a literal murder mystery with so many pop culture references and allusions that you lose all the technical content.  But for a mathematically literate reader you should be giving them some sense of build-up in tension and then resolution.  Dangle some food in front of them and lead them to water.  People who pick up a mathematics book are not looking for sex, crime and drama, nor even for comedy, but you should give them elements of such things inside the mathematics.  Teasers like why we are doing this, what will it be used for, how it relates to physics or other sciences, these are your sex and crime and drama.  And for humour you can use mathematical characters, stories of real mathematicians.  It might not be funny, but there is always a way to amuse an interested reader, so find those ways.

Write with a Vision

I think a lot of mathematical texts are dry ad suffer because they present “too close to research”.  What a good mathematical writer should aim for is the essence of any kind of writing, which is to narrate a story.  Psychology tells us this is how average human beings best receive and remember information.  So in mathematics you need a grand vision of where you are going.  If instead you just want to write about your research, then do the rest of us a favour and keep it off the bookshelves!

If you want to tell a story about your research then tell the full story, some history, some drama in how you stumbled, but then found a way through the forest of abstractions, and how you triumphed in the end.

The problem with a lot of mathematics monographs is that they aim for comprehensive coverage of a topic.  But that’s a bad style guide.  Instead they should aim to provide tools to solve a class of problems.  And the narrative is how to get from scratch up to the tools needed to solve the basic problem and then a little more.  With lots of dangling temptations along the way.  The motivation then is the main problem to be solved, which is talked about up front, as a carrot, not left as an obscure mystery one must read the entire book through to find.  Murder mysteries start with the murder first, not last.

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That’s enough for now. I should add to this list of guides later. I should follow my own advice too.

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Eternal Rediscovery

I have a post prepared to upload in a bit that will announce a possible hiatus from this WordPress blog. The reason is just that I found a cool book I want to try to absorb, The Princeton Companion to Mathematics by Gowers, Barrow-Green and Leader. Doubtless I will not be able to absorb it all in one go, so I will likely return to blogging periodically. But there is also teaching and research to conduct, so this book will slow me down. The rest of this post is a light weight brain-dump of some things that have been floating around in my head.

Recently, while watching a lecture on topology I was reminded that a huge percentage of the writings of Archimedes were lost in the siege of Alexandria. The Archimedean solids were rediscovered by Johannes Kepler, and we all know what he was capable of! Inspiring Isaac Newton is not a bad epitaph to have for one’s life.

The general point about rediscovery is a beautiful thing. Mathematics, more than other sciences, has this quality whereby a young student can take time to investigate previously established mathematics but then take breaks from it to rediscover theorems for themselves. How many children have rediscovered Pythagoras’ theorem, or the Golden Ratio, or Euler’s Formula, or any number of other simple theorems in mathematics?

Most textbooks rely on this quality. It is also why most “Exercises” in science books are largely theoretical. Even in biology and sociology. They are basically all mathematical, because you cannot expect a child to go out and purchase a laboratory set-up to rediscover experimental results. So much textbook teaching is mathematical for this reason.

I am going to digress momentarily, but will get back to the education theme later in this article.

The entire cosmos itself has sometimes been likened to an eternal rediscovery. The theory of Eternal Inflation postulates that our universe is just one bubble in a near endless ocean of baby and grandparent and all manner of other universes. Although, recently, Alexander Vilenkin and Audrey Mithani found that a wide class of inflationary cosmological models are unstable, meaning that could not have arisen from a pre-existing seed. There had to be a concept of an initial seed. This kind of destroys the “eternal” in eternal inflation. Here’s a Discover magazine account: What Came Before the Big Bang? — Cosmologist Alexander Vilenkin believes the Big Bang wasn’t a one-off event”. Or you can click this link to hear Vilenkin explain his ideas himself: FQXi: Did the Universe Have a Beginning? Vilenkin seems to be having a rather golden period of originality over the past decade or so, I regularly come across his work.

If you like the idea of inflationary cosmology you do not have to worry too much though. You still get the result that infinitely many worlds could bubble out of an initial inflationary seed.

Below is my cartoon rendition of eternal inflation in the realm of human thought:
cosmol_primordial_thoughtcloud_field

Oh to be a bubble thoughtoverse of the Wittenesque variety.

Quantum Fluctuations — Nothing Cannot Fluctuate

One thing I really get a bee in my bonnet about are the endless recountings in the popular literature about the beginning of the universe is the naïve idea that no one needs to explain the origin of the Big Bang and inflatons because “vacuum quantum fluctuations can produce a universe out of nothing”. This sort of pseudo-scientific argument is so annoying. It is a cancerous argument that plagues modern cosmology. And even a smart person like Vilenkin suffers from this disease. Here I quote him from a quote in another article on the PBS NOVA website::

Vilenkin has no problem with the universe having a beginning. “I think it’s possible for the universe to spontaneously appear from nothing in a natural way,” he said. The key there lies again in quantum physics—even nothingness fluctuates, a fact seen with so-called virtual particles that scientists have seen pop in and out of existence, and the birth of the universe may have occurred in a similar manner.
Source: http://www.pbs.org/wgbh/nova/blogs/physics/2012/06/in-the-beginning/

At least you have to credit Vilenkin with the brains to have said it is only “possible”. But even that caveat is fairly weaselly. My contention is that out of nothing you cannot get anything, not even a quantum fluctuation. People seem to forget quantum field theory is a background-dependent theory, it requires a pre-existing spacetime. There is no “natural way” to get a quantum fluctuation out of nothing. I just wish people would stop insisting on this sort of non-explanation for the Big Bang. If you start with not even spacetime then you really cannot get anything, especially not something as loaded with stuff as an inflaton field. So one day in the future I hope we will live in a universe where such stupid arguments are nonexistent nothingness, or maybe only vacuum fluctuations inside the mouths of idiots.

There are other types of fundamental theories, background-free theories, where spacetime is an emergent phenomenon. And proponents of those theories can get kind of proud about having a model inside their theories for a type of eternal inflation. Since their spacetimes are not necessarily pre-existing, they can say they can get quantum fluctuations in the pre-spacetime stuff, which can seed a Big Bang. That would fit with Vilenkin’s ideas, but without the silly illogical need to postulate a fluctuation out of nothingness. But this sort of pseudo-science is even more insidious. Just because they do not start with a presumption of a spacetime does not mean they can posit quantum fluctuations in the structure they start with. I mean they can posit this, but it is still not an explanation for the origins of the universe. They still are using some kind of structure to get things started.

Probably still worse are folks who go around flippantly saying that the laws of physics (the correct ones, when or if we discover them) “will be so compelling they will assert their own existence”. This is basically an argument saying, “This thing here is so beautiful it would be a crime if it did not exist, in fact it must exist since it is so beautiful, if no one had created it then it would have created itself.” There really is nothing different about those two statements. It is so unscientific it makes me sick when I hear such statements touted as scientific philosophy. These ideas go beyond thought mutation and into a realm of lunacy.

I think the cause of these thought cancers is the immature fight in society between science and religion. These are tensions in society that need not exist, yet we all understand why they exist. Because people are idiots. People are idiots where their own beliefs are concerned, by in large, even myself. But you can train yourself to be less of an idiot by studying both sciences and religions and appreciating what each mode of human thought can bring to the benefit of society. These are not competing belief systems. They are compatible. But so many believers in religion are falsely following corrupted teachings, they veer into the domain of science blindly, thinking their beliefs are the trump cards. That is such a wrong and foolish view, because everyone with a fair and balanced mind knows the essence of spirituality is a subjective view-point about the world, one deals with one’s inner consciousness. And so there is no room in such a belief system for imposing one’s own beliefs onto others, and especially not imposing them on an entire domain of objective investigation like science. And, on the other hand, many scientists are irrationally anti-religious and go out of their way to try and show a “God” idea is not needed in philosophy. But in doing so they are also stepping outside their domain of expertise. If there is some kind of omnipotent creator of all things, It certainly could not be comprehended by finite minds. It is also probably not going to be amenable to empirical measurement and analysis. I do not know why so many scientists are so virulently anti-religious. Sure, I can understand why they oppose current religious institutions, we all should, they are mostly thoroughly corrupt. But the pure abstract idea of religion and ethics and spirituality is totally 100% compatible with a scientific worldview. Anyone who thinks otherwise is wrong! (Joke!)

Also, I do not favour inflationary theory for other reasons. There is no good theoretical justification for the inflaton field other than the theory of inflation prediction of the homogeneity and isotropy of the CMB. You’d like a good theory to have more than one trick! You know. Like how gravity explains both the orbits of planets and the way an apple falls to the Earth from a tree. With inflatons you have this quantum field that is theorised to exist for one and only one reason, to explain homogeneity and isotropy in the Big Bang. And don’t forget, the theory of inflation does not explain the reason the Big Bang happened, it does not explain its own existence. If the inflaton had observable consequences in other areas of physics I would be a lot more predisposed to taking it seriously. And to be fair, maybe the inflaton will show up in future experiments. Most fundamental particles and theoretical constructs began life as a one-trick sort of necessity. Most develop to be a touch more universal and will eventually arise in many aspects of physics. So I hope, for the sake of the fans of cosmic inflation, that the inflaton field does have other testable consequences in physics.

In case you think that is an unreasonable criticism, there are precedents for fundamental theories having a kind of mathematically built-in explanation. String theorists, for instance, often appeal to the internal consistency of string theory as a rationale for its claim as a fundamental theory of physics. I do not know if this really flies with mathematicians, but the string physicists seem convinced. In any case, to my knowledge the inflation does not have this sort of quality, it is not a necessary ingredient for explaining observed phenomena in our universe. It does have a massive head start on being a candidate sole explanation for the isotropy and homogeneity of the CMB, but so far that race has not yet been completely run. (Or if it has then I am writing out of ignorance, but … you know … you can forgive me for that.)

Anyway, back to mathematics and education.

You have to love the eternal rediscovery built-in to mathematics. It is what makes mathematics eternally interesting to each generation of students. But as a teacher you have to train the nerdy children to not bother reading everything. Apart from the fact there is too much to read, they should be given the opportunity to read a little then investigate a lot, and try to deduce old results for themselves as if they were fresh seeds and buds on a plant. Giving students a chance to catch old water as if it were fresh dewdrops of rain is a beautiful thing. The mind that sees a problem afresh is blessed, even if the problem has been solved centuries ago. The new mind encountering the ancient problem is potentially rediscovering grains of truth in the cosmos, and is connecting spiritually to past and future intellectual civilisations. And for students of science, the theoretical studies offer exactly the same eternal rediscovery opportunities. Do not deny them a chance to rediscover theory in your science classes. Do not teach them theory. Teach them some theoretical underpinnings, but then let them explore before giving the game away.
With so much emphasis these days on educational accountability and standardised tests there is a danger of not giving children these opportunities to learn and discover things for themselves. I recently heard an Intelligence2 “Intelligence Squared” debate on academic testing. One crazy women from the UK government was arguing that testing, testing, and more testing — “relentless testing” were her words — was vital and necessary and provably increased student achievement.

Yes, practising tests will improve test scores, but it is not the only way to improve test scores. And relentless testing will improve student gains in all manner of mindless jobs out there is society that are drill-like and amount to going through routine work, like tests. But there is less evidence that relentless testing improves imagination and creativity.

Let’s face it though. Some jobs and areas of life require mindlessly repetitive tasks. Even computer programming has modes where for hours the normally creative programmer will be doing repetitive but possibly intellectually demanding chores. So we should not agitate and jump up and down wildly proclaiming tests and exams are evil. (I have done that in the past.)

Yet I am far more inclined towards the educational philosophy of the likes of Sir Ken Robinson, Neil Postman, and Alfie Kohn.

My current attitude towards tests and exams is the following:

  1. Tests are incredibly useful for me with large class sizes (120+ students), because I get a good overview of how effective the course is for most students, as well as a good look at the tails. Here I am using the fact test scores (for well designed tests) do correlate well with student academic aptitudes.
  2. My use of tests is mostly formative, not summative. Tests give me a valuable way of improving the course resources and learning styles.
  3. Tests and exams suck as tools for assessing students because they do not assess everything there is to know about a student’s learning. Tests and exams correlate well with academic aptitudes, but not well with other soft skills.
  4. Grading in general is a bad practise. Students know when they have done well or not. They do not need to be told. At schools if parents want to know they should learn to ask their children how school is going, and students should be trained to be honest, since life tends to work out better that way.
  5. Relentless testing is deleterious to the less academically gifted students. There is a long tail in academic aptitude, and the students in this tail will often benefit from a kinder and more caring mode of learning. You do not have to be soft and woolly about this, it is a hard core educational psychology result: if you want the best for all students you need to treat them all as individuals. For some tests are great, terrific! For others tests and exams are positively harmful. You want to try and figure out who is who, at least if you are lucky to have small class sizes.
  6. For large class sizes, like at a university, do still treat all students individually. You can easily do this by offering a buffet of learning resources and modes. Do not, whatever you do, provide a single-mode style of lecture+homework+exam course. That is ancient technology, medieval. You have the Internet, use it! Gather vast numbers of resources of all different manners of approach to your subject you are teaching, then do not teach it! Let your students find their own way through all the material. This will slow down a lot of students — the ones who have been indoctrinated and trained to do only what they are told — but if you persist and insist they navigate your course themselves then they should learn deeper as a result.

Solving the “do what I am told” problem is in fact the very first job of an educator in my opinion. (For a long time I suffered from lack of a good teacher in this regard myself. I wanted to please, so I did what I was told, it seemed simple enough. But … Oh crap, … the day I found out this was holding me back, I was furious. I was about 18 at the time. Still hopelessly naïve and ill-informed about real learning.) If you achieve nothing else with a student, transitioning them from being an unquestioning sponge (or oily duck — take your pick) to being self-motivated and self-directed in their learning is the most valuable lesson you can ever give them. So give them it.

So I use a lot of tests. But not for grading. For grading I rely more on student journal portfolios. All the weekly homework sets are quizzes though, so you could criticise the fact I still use these for grading. As a percentage though, the Journals are more heavily weighted (usually 40% of the course grade). There are some downsides to all this.

  • It is fairly well established in research that grading using journals or subjective criteria is prone to bias. So unless you anonymise student work, you have a bias you need to deal with somehow before handing out final grades.
  • Grading weekly journals, even anonymously, takes a lot of time, about 15 to 20 times the hours that grading summative exams takes. So that’s a huge time commitment. So you have to use it wisely by giving very good quality early feedback to students on their journals.
  • I still haven’t found out how to test the methods easily. I would like to know quantitatively how much more effective journal portfolios are compared to exam based assessments. I am not a specialist education researcher, and I research and write a about a lot of other things, so this is taking me time to get around to answering.

I have not solved the grading problem, for now it is required by the university, so legally I have to assign grades. One subversive thing I am following up on is to refuse to submit singular grades. As a person with a physicists world-view I believe strongly in the role of sound measurement practice, and we all know a single letter grade is not a fair reflection on a student’s attainment. At a minimum a spread of grades should be given to each student, or better, a three-point summary, LQ, Median, UQ. Numerical scaled grades can then be converted into a fairer letter grade range. And GPA scores can also be given as a central measure and a spread measure.

I can imagine many students will have a large to moderate assessment spread, and so it is important to give them this measure, one in a few hundred students might statistically get very low grades by pure chance, when their potential is a lot higher. I am currently looking into research on this.

OK, so in summary: even though institutions require a lot of tests you can go around the tests and still given students a fair grade while not sacrificing the true learning opportunities that come from the principle of eternal rediscovery. Eternal rediscovery is such an important idea that I want to write an academic paper about it and present at a few conferences to get people thinking about the idea. No one will disagree with it. Some may want to refine and adjust the ideas. Some may want concrete realizations and examples. The real question is, will they go away and truly inculcate it into their teaching practices?

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Primacks’ Premium Simulations

After spending a week debating with myself about various Many Worlds philosophy issues  and other quantum cosmology questions, today I saw Joel Primack’s presentation at the Philosophy of Cosmology International Conference, on the topic of Cosmological Structure Formation. And so for a change I was speechless.

Thus I doubt I can write much that illumines Primack’s talk better than if I tell you just to go and watch it.

He, and colleagues, have run supercomputer simulations of gravitating dark matter in our universe. From their public website Bolshoi Cosmological Simulations they note: “The simulations took 6 million cpu hours to run on the Pleiades supercomputer — recently ranked as seventh fastest of the world’s top 500 supercomputers — at NASA Ames Research Center.”

To get straight to all the videos from the Bolshoi simulation go here (hipacc.ucsc.edu/Bolshoi/Movies.html).

cosmos_BolshoiSim_MD_cluster01_gas_sn320

MD4 Gas density distribution of the most massive galaxy cluster (cluster 001) in a high resolution resimulation, x-y-projection. (Kristin Riebe, from the Bolshoi Cosmological Simulations.)

The filamentous structure formation is awesome to behold. At times they look like living cellular structures in the movies that Primack has produced. Only the time steps in his simulations are probably about 1 million year steps. for example, on simulation is called the Bolshio-Planck Cosmological Simulation — Merger Tree of a Large Halo. If I am reading this page correctly these simulations visualize 10 billion Sun sized halos.  The unit they say they resolve is “1010 Msun halos”. Astronomers will often use a symbol M to represent a unit of one solar mass (equal to our Sun’s mass). But I have never seen that unit “M halo” used before, so I’m just guessing it means the finest structure resolvable in their movie still images would be maybe a Sun-sized object, or a solar system sized bunch of stuff. This is dark matter they are visualizing, so the stars and planets we can see just get completely obscured in these simulations (since the star-like matter is less than a few percent of the mass).

True to my word, that’s all I will write for now about this piece of beauty. I need to get my speech back.

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Oh, but I do just want to hasten to say the image above I pasted in there is NOTHING compared to the movies of the simulations. You gotta watch the Bolshoi Cosmology movies to see the beauty!

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Frankanscifisense

Here’s a v. quick post:  have you been dying to see an intelligent SciFi movie or series?  They are are few and far between right?!  One I am waiting for on DVD is The Martian (2015), I’ve heard ok reviews and the book it was based upon had very good reviews and listening to interviews with the author, Andy Weir, it seems like a quality piece of hard scifi that had some sound engineering physics thought behind it.   Hard to know whether to read the book or watch the film. Film is faster!  Life is short!  Therefore watch the film and sadly miss the book?   Too many mathematics texts to read anyway, so the film it is [sigh]!

If I’m not feeling wide awake enough for a mathematics or physics lecture during my lunch break, I might try a bit of scifi TV or read a science blog article, or sometimes find a good movie to dip into.

And I do mean “dip into”.  I eat fairly quickly, and not too huge helpings, so it’s all over in 15 minutes.  And that’s about as much of a movie I can watch in one session.  Heading out the the theatre is a rare event these days, and besides that, I like to watch a good movie in comparative solitude.

So every purple moon I might find an intelligent SciFi movie.  But I will start watching and get nervous that any moment the story will sensationalize and lapse into horribly saccharine, physically implausible unreality.  You cannot even begin writing a critique of the SciFi genre because 99% of what the film industry turns out is utter crap.  That might seem too harsh, the SFX are vastly better than in days of old, but the stories are the critical component of any good film or book.  And it is the plot, the dialogue, and the whole story structure that really sucks in just about every recent Scifi  film I have seen in the past decade or more.   (Hold on now, I am getting to a good recommendation.)

The problem I think is that the improvements in SFX have outpaced improvements in screenplays.  Older screenplays could be just as good or a lot better than modern scripts because the focus in the old days had to be on stories because the SFX totally sucked.  Take Star Trek as an example.  The modern Star Trek stories have a lot more fancy CGI and the screenplays use a lot more modern science ideas, so they seem pretty cool compared to the camp TV series.  Similar comments could be made about Doctor Who, another generation spanning SciFi series.  But if you analyse them a little more deeply, and think about the dialogue and the psychology, not a lot has really improved.  The dialogue in Start Trek Into the Darkness (2013) was fairly childish.  Whenever a cool science point could be made the pseudo-science explanations lapsed perhaps into even worse quasi-science than the dialogues from the original TV series.  They just use a few more modern science buzz-words.  The actual meat of the scifi science explanations is often a lot worse.  The logic is a lot worse, the liberties taken with reality more extreme. (Recall the “photon torpedo”? … OMG, … let’s not even go there!) The Star Trek franchise should be consulting the chap who wrote the Science of Star Trek books, or Michio Kaku, who can rhapsodize endlessly about plausible scifi science.

I could write a long essay on this, but I won’t.

Can I then get to my recommendation?

Sure dude.  Just hang on one more minute though.   The thing is, I suspect, what makes a really good scifi story is one that dials back the fantasy and aims for a lot of hard realism.  So something like the “near future” genre is always promising, but using plausible and reasonable extrapolations of current science.  Especially stories that obey the principles of conservation of energy, momentum, and the second law of thermodynamics.  Those are perhaps the most blatantly violated principles of science that bad SciFi movies in particular routinely abuse.  My point is that if you discipline your story to obey just these three principles then you will be constraining your plot.  Such constraints are beautiful things.  It forces the other human aspects of your story to be more powerful and it helps make the audience more involved and engaged, even if the average audience member is not aware of the principles.  (I lose count of the number of CGI-mediated violations of conservation of momentum in crashes and fight scenes.  Each instance just makes me more and more nauseous.  even fairly serious film makers like Peter Jackson, routinely violate conservation of momentum — both linear and rotational — in their CGI spectaculars.)

So when someone makes a SciFi film that does not even begin to worry about spectacular CGI, then I am extremely interested.  So here is the recommendation:  go and grab a copy of Robot and Frank (2012).

Robot_and_Frank_movie poster

A movie with no CGI pretensions, and a nice premise on the face of it.

I have only seen the first 15 minutes, so I am still nervous the plot will get derailed later by unrealistic physics or computer science.  But I think this is one film I can happily watch to the end based on the story premise.  Give it a go.

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I guess it is possible the artificial intelligence postulates in this movie will degenerate into implausibility, but over the next week of lunch breaks I’ll risk it. 🙂

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A Beautiful Folding — and the Rise of Transformers

If you want to treat your brain then try watching the MIT lectures by Professor Erik Demaine over at 6.849: Geometric Folding Algorithms: Linkages, Origami, Polyhedra (Fall 2010). Not sure if that was the most recent year his course was offered, but I’m sure you can find the latest version. I will not update this post or any links in any of my blogs, so as always, just Google the key words and you are bound to find what I’m pointing you at.

origami_Erik.Demaine_1

Among many cool results, the two prompting me to write this brief post were:

  1. The universality result that there is a crease pattern from which any modular cuboid polyhedron can be folded.
  2. The self-folding paper construction: a crease pattern can be folded in any way by electrical current stimulation. So we have Origamistless origami.

Ergo: the age of Transformers is upon us! Hahahaha!

Too bad artificial consciousness is not a paper fold.

origami_Erik_Demaine_et_al_2005_cropped

I dunno man. … you see Demaine and his Dad with huge smiles on their faces, glass-blowing. folding cured crease patterns and chatting with John Conway and other legends, and you have to almost cry at the beauty of it all. So much life, so much joy, such intense devotion to art and science.

origami_01_Demaine_0264_Earthtone_Series.original

Oh yeah, … how many mathematicians have their work on permanent collection at MOMA?

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Why Eat When You Can Watch YouTube Instead?

Imagine retiring and making a living reviewing mathematics and science videos on YouTube. Could a computer do this job? This Weeks Finds in Mathematical Physics VDO’s. Seems to have a suitable i-gener dopey ring to it.

Well, if AI ever can respond emotionally to VDO content then perhaps there is no long term future in such an occupation, but for now I’d feel secure in such a retirement occupation if there were donations from readers. Not sure if I would be adding much value with such a service, but sometimes I daydream about some kind of semi-ideal existence. The problem with the idea is that you cannot truly be passionately involved in science or mathematics — to a level that would really add terrific value as a reviewer — unless you are also of the mind that gets captivated by puzzles and wants to explore them.

Because once you start launching an extension of an investigation suggested by a cool lecture or seminar, then you have a time sink. That’s ok though, you would probably simply add your investigations to the VDO review blog.

As for the need …? It would be a conceit to imagine anyone would be interested in a review article. Why not just click on the link that was recommended? Perhaps you have to read a bit of the blog of the person doing the recommendations, just so you feel they have a worthwhile opinion, so you don’t waste your time waiting for the ads and intro of a YouTube clip to get going only to find it is rubbish. But beyond this, I think there is a minor need for good VDO reviews. Maybe not quite yet, but perhaps soon there will be enough awesome science content on the Web that simply using a Google search will not get all the best videos onto the front hits page. So a reputable website with a reliably good quality list would be nice.

A few such lists already proliferate. So maybe my retirement plan is flawed. But there is still the hope that some creative insights could be added to the review, making them worth someone’s time to browse. Then after a few years at this your lists get long and so extended they become unreadable and useless, a list is needed for your list. The tyranny of obsession. When one is truly obsessed it becomes ironically impossible to interest others in your obsession. Then frustrated in not gaining converts, and ever increasingly being convinced of the virtues of one’s obsession, one finds it ever implausible that other people cannot be interested, one eventually then grows mad from the cognitive dissonance, and transcends into existence as an xkcd comic frame.

What I really want is for such brilliant quality science videos that it makes me forget about eating, and feeds my brain through sheer emotional charge. I’ve watched perhaps less than a half dozen such videos in my life so far, perhaps fewer. I will say that apart from Mr Feynman, there is a very nerdy but lunch-forgetting, series of lectures recorded at the Perimeter Institute by guest lecturer Carl Bender (PIRSA:C11025 – 11/12 PSI – Mathematical Physics ).

Actually those lectures gave me such an intellectual hard-on it had the reverse effect. I started making tuna and avocado salad grand sandwiches on whole grain with two quadruple shot latté’s accompanied by dark Whittaker’s dark chocolate and roasted cashew nuts, as my mid-morning brunch endangering my keyboard as I watched Bender gives his lecture’s in the privacy of my study. Tickets were free for this entertainment. Brilliant!

Mr Bender might be a superior educator to Mr Feynman. Feynman wins on entertainment value perhaps, but Bender gets ahead in terms of practical use.

*      *       *

All that is a bit of a long introduction to a plug for one particular video find. “Douglas Hofstadter — Feuerbach’s Theorem: A Beautiful Theorem Deserves a Beautiful Proof” (www.youtube.com/watch?v=4Up22XOEVKc) It’s not viral or mind-blowing. It’s just a simple pleasure to find one like this to watch while eating lunch.

DouglasHofstadter_IsoscelesTriangleTheorem

Hofstadter shows isosceles triangle theorem “proof from The Book” (q.v. Erdos).

I wish Doug Hofstadter had a personal secretary who went around everywhere he speaks and videotaped the talks and lectures. Imagine all the university lectures he has given that have been lost for posterity because he lived in an era before ubiquitous video production. Oh yeah, sure, there will be more Hofstadter’s and Feynman’s in the future. One day even an Isaac Newton level dude or dude-ess will appear and all their talks will be recorded, maybe even their “brain waves” (you know what I mean).

(Is “dude” genderless???)

I was one of the rare theoretical physics, or mathematics major, students in my generation who actually took a course on Euclidean geometry. Most people (who are inclined to think about it) probably think Euclidean geometry was a bit of elementary mathematics in high school, mostly done as part of trigonometry. It’s sad if that’s true. For one thing, it is really cool to get immersed in Euclidean geometry and then slowly realise that when the lessons catch up to the 19th century math we begin to feel like something is uneasy, then we get Lobachevsky and Bolyai and then Gauss and Riemann and when General Relativity finally emerges it is like entering an Alice in Wonderland world.

This “astonishment and wonder” effect actually occurs even when you already know about Einstein’s spacetime and general relativity. There is just something special about studying a good well-paced course of Euclidean geometry with a good historical flavour in addition to the philosophical rigour.

I forget the lecturer’s name for the course I took at Victoria University of Wellington, New Zealand. All I recall was the weird association that struck me as bizarre, that the guy was a philosophy professor. The mathematics department at the time seemed too elite to bother with Euclidean geometry. Mathematics would start only with differential geometry and topology. Euclid was beneath them. (That may no longer be true, or it may be worse, but whatever the case, I am thankful to the VUW Philosophy department.)

So what’s so great about Hofstadter’s Feuerbach theorem lecture?

Just go see for yourself. It’s cool.

Hofstadter has his MacBook desktop exposed, with Geometer’s SketchPad showing some interactive demo’s of basic Euclidean geometry proofs. There are many little highlights: “good theorems deserve good names”; the remote triangle π sum theorems and variants, the “Andrew Wiles called out by a high school kid” anecdote. Another is the proof of the Isosceles Triangle Theorem — the philosophy dude who taught the VUW geometry course did not mention this one, so it was actually new and fresh for me.

That’s pretty awesome isn’t it? That you can find something very elementary and yet new and fresh and brilliant in such a well known century old subject. It’s a great lesson for educators. No subject need ever get stale. There are always creative new ways to present old knowledge. When the good educator finds new wyas to present old topics they are actually adding value and in some sense presenting a new thing, an original new idea, meta to the old idea perhaps, but still new. In my mind this is one reason why GOFAI will never replace a great teacher.

The point is, I think you can add to human experiences by teaching old topics that anyone can just find on Wikipedia or elsewhere, by adding new angles, new ways to express the same ideas. Furthermore, I see this as a useful and creative endeavour. It is a great service to investigate prior knowledge but present it in new crisper or more artistic fashion. Most importantly, I want the school teachers who teach my children, and your children, to understand this, and to not get bogged down by any existing curriculum or style of teaching.

In act one should go further, and teach the teachers to down-right ignore the pre-existing curricula. There is little value in syllabus’ and curricula , or standardized education models. At least when compared to the power of fresh approaches and creative or never-before-seen experiences in learning, compared to such innovations traditional school instruction is perhaps less than valuable, it might even be value-subtracting, if that’s possible! Why might it be “value subtracting”? One reason is that what already is available on the Internet is at most children’s fingertips, at least in the tech-enabled regions of the planet. And whatever is already at one’s fingertips is largely a waste of time trying to re-learn or learn through some inefficient school teacher’s bumbling lessons interrupted by the classroom distractions of other kids.

So teachers! Hear me! In your classroom forget about all the received knowledge and dry textbooks. Teach something new and fascinating or do not teach at all! Give them a book of puzzles rather than a textbook. If you have nothing creative to add then give your students an Internet connection and refer them to Wikipedia. That’s the least you can do for them, and it will at least not harm them.

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Well gosh, I know I had some other things to write about this little VDO of Hofstadter’s, but I seem to have forgotten my original point.

(BTW, Geometer’s Sketchpad is Non-Free software, so I’m not giving you the link! Try Geogebra instead.)

Life advice for Today from OneOverEpsilon

Watch math lectures for lunch, not LOL Cats or Hollywood movies.

There are enough great sciency-math lectures out there now for great entertainment for many years worth of lunchtimes.

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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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