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”).


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.

*      *       *

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.


Greater Thoughts that Cannot Be Imageoned

Most scientists do not enter their chosen fields because the work is easy. They do their science mainly because it is challenging and rewarding when triumphant. Yet few scientists will ever taste the sweet dew drops of triumph — real world-changing success — in their lifetimes. So it is remarkable perhaps that the small delights in science are sustaining enough for the human soul to warrant persistence and hard endeavour in the face of mostly mediocre results and relatively few cutting edge break-throughs.

Still, I like to think that most scientists get a real kick out of re-discovering results that others before them have already uncovered. I do not think there is any diminution for a true scientist in having been late to a discovery and not having publication priority. In fact I believe this to be universally true for people who are drawn into science for aesthetic reasons, people who just want to get good at science for the fun of it and to better appreciate the beauty in this world. If you are of this kind you likely know exactly what I mean. You could tomorrow stumble upon some theorem proven hundreds of years ego by Gauss or Euler or Brahmagupta and still revel in the sweet taste of insight and understanding.

Going even further, I think such moments of true insight are essential in the flowering of scientific aesthetic sensibilities and the instilling of a love for science in young children, or young at heart adults. “So what?” that you make this discovery a few hundred years later than someone else? They had a birth head start on you! The victory is truly still yours. And “so what?” that you have a few extra giants’ shoulders to stand upon? You also saw through the haze and fog of much more information overload and Internet noise and thought-pollution, so you can savour the moment like the genius you are.

Such moments of private discovery go unrecorded and must surely occur many millions of times more frequently than genuinely new discoveries and break-throughs. Nevertheless, every such transient to invisible moment in human history must also be a little boost to the general happiness and welfare of all of humanity. Although only that one person may feel vibrant from their private moment of insight, their radiance surely influences the microcosm of people around them.

I cannot count how many such moments I have had. They are more than I will probably admit, since I cannot easily admit to any! But I think they occur quite a lot, in very small ways. However, back in the mid 1990’s I had, what I thought, was a truly significant glimpse into the infinite. Sadly it had absolutely nothing to do with my PhD research, so I could only write hurriedly rough notes on recycled printout paper during small hours of the morning when sleep eluded my body. To this day I am still dreaming about the ideas I had back then, and still trying to piece something together to publish. But it is not easy. So I will be trying to leak out a bit of what is in my mind in some of these WordPress pages. Likely what will get written will be very sketchy and denuded of technical detail. But I figure if I put the thoughts out into the Web maybe, somehow, some bright young person will catch them via Internet osmosis of a sort, and take them to a higher level.


There are a lot of threads to knit together, and I hardly know where to start. I have already started writing perhaps half a dozen manuscripts, none finished, most very sketchy. And this current writing is yet another forum I have begun.

The latest bit of reading I was doing gave me a little shove to start this topic anew. It happens from time to time that I return to studying Clifford Geometric Algebra (“GA” for short). The round-about way this happened last week was this:

  • Weary from reading a Complex Analysis book that promised a lot but started to get tedious: so for a light break YouTube search for a physics talk, and find Twistors and Spinors talks by Sir Roger Penrose. (Twistor Theory is heavily based on Complex Analysis so it was a natural search to do after finishing a few chapters of the mathematics book).
  • Find out the Twistor Diagram efforts of Andrew Hodges have influenced Nima Arkani-Hamed and even Ed Witten to obtain new cool results crossing over twistor theory with superstring theory and scattering amplitude calculations (the “Amplituhedron” methods).
  • That stuff is ok to dip into, but it does not really advance my pet project of exploring topological geon theory. So I look for some more light reading and rediscover papers from the Cambridge Geometric Algebra Research Group (Lasenby, Doran, Gull). And start re-reading Gull’s paper on electron paths and tunnelling and the Dirac theory inspired by David Hestene’s work
  • The Gull paper mentions criticisms of the Dirac theory that I had forgotten. In the geometric algebra it is clear that solving the Dirac equation gives not positively charge anti-electrons, but unphysical negative frequency solutions with negative charge and negative mass. So they are not positrons. It’s provoking that the authors claim this problem is not fully resolved by second quantisation, but rather perhaps just gets glossed over? I’m not sure what to think of this. (If the negative frequencies get banished by second quantisation why not just conclude first quantisation is not nature’s real process?)
  • Still, whatever the flaws in Dirac theory, the electron paths paper has tantalising similarities with the Bohm pilot wave theory electron trajectories. And there is also a reference to the Statistical Interpretation of Quantum Mechanics (SIQM) due to Ballentine (and attributed also as Einstein’s preferred interpretation of QM).
  • It gets me thinking again of how GA might be helpful in my problems with topological geons. But I shelve this thought for a bit.
  • Reading Ballentine’s paper is pretty darn interesting. It dates from 1970, but it is super clear and easy to read. I love that in a paper. The gist of it is that an absolute minimalist interpretation of quantum mechanics would drop Copenhagen ideas and view the wave function as more like a description of what could happen in nature, tat is, the wave functions are descriptions of statistical ensembles of identically prepared experiments or systems in nature. (Sure, no two systems are ever prepared in the exact same initial state, but that hardly matters when you are only doing statistics rather than precise deterministic modelling.)
  • So Ballentine was suggesting the wave functions are;
    1. not a complete description of an individual particle, but rather
    2. better thought of as a description of an ensemble of identically prepared states.

This is where I ended up, opening my editor to draft a OneOverEpsilon post.

So here’s the thing I like about the ensemble interpretation and how the geometric algebra reworking of Dirac theory adds to a glimmer of clarity about what might be happening with the deep physics of our universe. For a start the ensemble interpretation is transparently not a complete theoretical framework, since it is a statistical theory it does not pretend to be a theory of reality. Whatever is responsible for the statistical behaviour of quantum systems is still an open question in SIQM. The Bohm-like trajectories that the geometric algebra solutions to the Dirac theory are able to compute as streamline plots are illuminating in this respect, since they seem to clearly show that what the Dirac wave equation is modelling is almost certainly not the behaviour a single particle. (One could guess this from Schrödinger theory as well, but I guess physicists were already lured into believing in the literal wave-particle duality meme well before Bohm was able to influence anyone’s thinking.)

Also, it is possible (I do not really know for sure) that the negative frequency solutions in Dirac theory can be viewed as merely an artifact of the statistical ensemble framework. No single particle acts truly in accordance with the Dirac wave equation. So there is no real reason to get ones pants in a twist about the awful appearance of negative frequencies.

(For those in-the-know: the Dirac theory negative frequency solutions turn out to have particle currents in the reverse spatial direction to their momenta, so that’s not a backwards time propagating anti-particle, it is a forwards in time propagating negative mass particle. That’s a particle that’d fall upwards in a gravitational field if the principle of equivalence holds universally. As an aside note: it is a bit funky that this cannot be tested experimentally since no one can yet clump enough anti-matter together to test which way it accelerates in a gravitational field. But I presume the sign of particle inertial mass can be checked in the lab, and, so far, all massive particles known to science at least are known to have positive inertial mass.)

And as a model of reality the Dirac equation has therefore, certain limitations and flaws. It can get some of the statistics correct for particular experiments, but a statistical model always has limits of applicability. This is neither a defense or a critique of Dirac theory.  My view is that it would be a bit naïve to regard Dirac theory as the theory of electrons, and naïve to think it should have no flaws.  At best such wave-function models are merely a window frame for a particular narrow view out into our universe.  Maybe I am guilty of a bit of sophistry or rhetoric here, but that’s ok for a WordPress blog I think … just puttin’ some ideas “out there”.

Then another interesting confluence is that one of Penrose’s big projects in Twistor theory was to do away with the negative frequency solutions in 2-Spinor theory. And I think, from recall, he succeeded in this some time ago with the extension of twistor space to include the two off-null halves. Now I do not know how this translates into real-valued geometric algebra, but in the papers of Doran, Lasenby and Gull you can find direct translations of twistor objects into geometric algebra over real numbers. So there has to be in there somewhere a translation of Penrose’s development in eliminating the negative frequencies.

So do you feel a new research paper on Dirac theory in the wind just there? Absolutely you should! Please go and write it for me will you? I have my students and daughters’ educations to deal with and do not have the free time to research off-topic too much. So I hope someone picks up on this stuff. Anyway, this is where maybe the GA reworking of Dirac theory can borrow from twistor theory to add a little bit more insight.

There’s another possible confluence with the main unsolved problem in twistor theory. The Twistor theory programme is held back (stalled?) a tad (for 40 years) by the “googly problem” as Penrose whimsically refers to it. The issue is one of trying to find self-dual solutions of Einstein’s vacuum equations (as far as I can tell, I find it hard to fathom twistor theory so I’m not completely sure what the issue is). The “googly problem” stood for 40 years, and in essence is the problem of “finding right-handed interacting massless fields (positive helicity) using the same twistor conventions that give rise to left-handed fields (negative helicity)”. Penrose maybe has a solution dubbed Palatial Twistor Theory which you might be able to read about here: “On the geometry of palatial twistor theory” by Roger Penrose, and also lighter reading here: “Michael Atiya’s Imaginative Mind” by Siobhan Roberts in Quanta Magazine.

If you do not want to read those articles then the synopsis, I think, is that twistor theory has some problematic issues in gravitation theory when it comes to chirality (handedness), which is indeed a problem since obtaining a closer connection between relativity and quantum theory was a prime motive behind the development of twistor theory. So if twistor theory cannot fully handle left and right-handed solutions to Einstein’s equations it might be said to have failed to fulfill one it’s main animating purposes.

So ok, to my mind there might be something the geometric algebra translation of twistor theory can bring to bear on this problem, because general relativity is solved in fairly standard fashion with geometric algebra (that’s because GA is a mathematical framework for doing real space geometry, and handles Lorentzian metrics as simply as Euclidean, not artificially imposed complex analytic structure is required). So if the issues with twistor theory are reworked in geometric algebra then some bright spark should be able to do the job twistor theory was designed do do.

By the way, the great beauty and advantage Penrose sees in twistor theory is the grounding of twistor theory in complex numbers. The Geometric Algebra Research Group have pointed out that this is largely a delusion. It turns out that complex analysis and holomorphic functions are just a sector of full spacetime algebra. Spacetime algebra, and in fact higher dimensional GA, have a concept of monogenic functions which entirely subsume the holomorphic (analytic) functions of 2D complex analysis. Complex numbers are also completely recast for the better as encodings of even sub-algebras of the full Clifford–Geometric Algebra of real space. In other words, by switching languages to geometric algebra the difficulties that arise in twistor theory should (I think) be overcome, or at least clarified.

If you look at the Geometric Algebra Research Group papers you will see how doing quantum mechanics or twistor theory with complex numbers is really a very obscure way to do physics. Using complex analysis and matrix algebra tends to make everything a lot harder to interpret and more obscure. This is because matrix algebra is a type of encoding of geometric algebra, but it is not a favourable encoding, it hides the clear geometric meanings in the expressions of the theory.

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So far all I have described is a breezy re-awakening of some old ideas floating around in my head. I rarely get time these days to sit down and hack these ideas into a reasonable shape. But there are more ideas I will try to write down later that are part of a patch-work that I think is worth exploring. It is perhaps sad that over the years I had lost the nerve to work on topological geon theory. Using spacetime topology to account for most of the strange features of quantum mechanics is however still my number one long term goal in life. Whether it will meet with success is hard to discern, perhaps that is telling: if I had more confidence I would simply abandon my current job and dive recklessly head-first into geon theory.

Before I finish up this post I want to thus outline very, very breezily and incompletely, the basic idea I had for topological geon theory. It is fairly simplistic in many ways. There is however new impetus from the past couple of years developments in the Black Hole firewall paradox debates: the key idea from this literature has been the “ER=EPR” correspondence hypothesis, which is that quantum entanglement (EPR) might be almost entirely explained in terms of spacetime wormholes (ER: Einstein-Rosen bridges). This ignited my interest because back in 1995/96 I had the idea that Planck scale wormholes in spacetime can allow all sorts of strange and gnarly advance causation effects on the quantum (Planckian) space and time scales. It seemed clear to me that such “acausal” dynamics could account for a lot of the weird correlations and superpositions seen in quantum physics, and yet fairly simply so by using pure geometry and topology. It was also clear that if advanced causation (backwards time travel or closed timelike curves) are admitted into physics, even if only at the Planck scale, then you cannot have a complete theory of predictive physics. Yet physics would be deterministic and basically like general relativity in the 4D block universe picture, but with particle physics phenomenology accounted for in topological properties of localised regions of spacetime (topological 4-geons). The idea, roughly speaking, is that fundamental particles are non-trivial topological regions of spacetime.  The idea is that geons are not 3D slices of space, but are (hypothetically) fully 4-dimensional creatures of raw spacetime topology.   Particles are not apart from spacetime. Particles are not “fields that live in spacetime”, no! Particles are part of spacetime.  At least that was the initial idea of Geon Theory.

Wave mechanics, or even quantum field theory, are often perceived to be mysterious because they either have to be interpreted as non-deterministic (when one deals with “wave function collapse”) or as semi-deterministic but incomplete and statistical descriptions of fundamental processes.   When physicists trace back where the source of all this mystery lies they are often led to some version of non-locality. And if you take non-locality at face value it does seem rather mysterious given that all the models of fundamental physical processes involve discrete localised particle exchanges (Feynman diagrams or their stringy counterparts).   One is forced to use tricks like sums over histories to obtain numerical calculations that agree with experiments.  But no one understand why such calculational tricks are needed, and it leads to a plethora of strange interpretations, like Many Worlds Theory, Pilot Waves, and so on.   A lot of these mysteries I think dissolve away when the ultimate source of non-locality is found to be deep non-trivial topology in spacetime which admits closed time-like curves (advanced causation, time travel).  To most physicists such ideas appear nonsensical and outrageous.  With good reason of course, it is very hard to make sense of a model of the world which allows time travel, as decades of scifi movies testify!  But geon theory doe snot propose unconstrained advanced causation (information from the future influences events in the past).   On the contrary, geon theory is fundamentally limited in outrageousness by the assumption the closed time-like curves are restricted to something like the Planck scale.   I should add that this is a wide open field of research.  No one has worked out much at all on the limits and applicability of geon theory.    For any brilliant young physicists or mathematicians this is a fantastic open playground to explore.

The only active researcher I know in this field is Mark Hadley. It seemed amazing to me that after publishing his thesis (also around 1994/95 independently of my own musings) no one seemed to take up his ideas and run with them.  Not even Chris Isham who refereed Hadley’s thesis.  The write-up of Hadley’s thesis in NewScientist seemed to barely cause a micro-ripple in the theoretical physics literature.    I am sure sociologists of science could explain why, but to me, at the time, having already discovered the same ideas, I was perplexed.

To date no one has explicitly spelt out how all of quantum mechanics can be derived from geon theory. Although Hadley I surmise, completed 90% of this project!  The final 10% is incredibly difficult though — it would necessitate deriving something like the Standard Model of particle physics from pure 4D spacetime topology — no easy feat when you consider high dimensional string theory has not really managed the same job despite hundreds of geniuses working on it for over 35 years. My thinking has been that string theory involves a whole lot of ad hockery and “code bloat” to borrow a term from computer science! If string theory was recast in terms of topological geons living as part of spacetime, rather than as separate to spacetime, then I suspect great advances could be made. I really hope someone will see these hints and connections and do something momentous with them.  Maybe some maverick like that surfer dude Garett Lisi might be able to weigh in and provide some fire power?

In the mean time  geometric algebra has so not been applied to geon theory, but GA blends in with these ideas since it seems, to me, to be the natural language for geometric physics. If particle phenomenology boils down to spacetime topology, then the spacetime algebra techniques should find exciting applications.  The obstacle is that so far spacetime algebra has only been developed for physics in spaces with trivial topology.

Another connection is with “combinatorial spacetime” models — the collection of ideas for “building up spacetime” from discrete combinatorial structures (spin foams, or causal networks, causal triangulations, and all that stuff). My thinking is that all these methods are unnecessary, but hint at interesting directions where geometry meets particle physics because (I suspect) such combinatorial structure approaches to quantum gravity are really only gross approximations to the spacetime picture of topological geon theory. It is in the algebra which arises from non-trivial spacetime topology and it’s associated homology that (I suspect) combinatorial spacetime pictures derive their use.

Naturally I think the combinatorial structure approaches are not fundamental. I think topology of spacetime is what is fundamental.

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That probably covers enough of what I wanted to get off my chest for now. There is a lot more to write, but I need time to investigate these things so that I do not get too speculative and vague and vacuously philosophical.

What haunts me most nights when I try to dream up some new ideas to explore for geon theory (and desperately try to find some puzzles I can actually tackle) is not that someone will arrive at the right ideas before me, but simply that I never will get to understand them before I die. I do not want to be first. I just want to get there myself without knowing how anyone else has got to the new revolutionary insights into spacetime physics. I had the thrill of discovering geon theory by myself, independently of Mark Hadley, but now there has been this long hiatus and I am worried no one will forge the bridges from geon theory to particle physics while I am still alive.

I have this plan for what I will do when/if I do hear such news. It is the same method my brother Greg is using with Game of Thrones. He is on a GoT television and social media blackout until the books come out. He’s a G.R.R. Martin purest you see. But he still wants to watch the TV adaptation later on for amusement (the books are waaayyy better! So he says.) It is surprisingly easy to enforce such a blackout. Sports fans will know how. Any follower of All Black Rugby who misses an AB test match knows the skill of doing a media blackout until they get to watch their recording or replay. It’s impossible to watch an AB game if you know the result ahead of time. Rugby is darned exciting, but a 15-aside game has too many stops and starts to warrant sitting through it all when you already know the result. But when you do not know the result the build-up and tension are terrific. I think US Americans have something similar in their version of Football, since American Football has even more stop/start, it would be excruciatingly boring to sit through it all if you knew the result. But strangely intense when you do not know!

So knowing the result of a sports contest ahead of time is more catastrophic than a movie or book plot spoiler. It would be like that if there is a revolution in fundamental physics involving geon theory ideas. But I know I can do a physics news blackout fairly easily now that I am not lecturing in a physics department. And I am easily enough of an extreme introvert to be able to isolate my mind from the main ideas, all I need is a sniff, and I will then be able to work it all out for myself. It’s not like any ordinary friend of mine is going to be able to explain it to me!

If geon theory turns out to have any basis in reality I think the ideas that crack it all open to the light of truth will be among the few great ideas of my generation (the post Superstring generation) that could be imagined. If there are greater ideas I would be happy to know them in time, but with the bonus of not needing a physics news blackout! If it’s a result I could never have imagined then it’d be worth just savouring the triumph of others.


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:

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.

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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var MyStupidStr = “Gadammit! Where’d You Put My Variables!?”;

This WordPress blog keeps morphing from Superheros and SciFi back to philosophy of physics and other topics. So sorry to readers expecting some sort of consistency. This week I’m back with the Oxford University series, Cosmology and Quantum Foundations lectures. Anthony Valentini gives a talk about Hidden Variables in Cosmology.

The basic idea Valentini proposes is that we could be living in a deterministic cosmos, but we are somehow trapped in a region of phase space where quantum indeterminism reigns. In the our present epoch region there are hidden variables but they cannot be observed, not even indirectly, so they have no observable consequences, and so Bell’s Theorem and Kochen-Specker and the rest of the “no-go” theorems associated with quantum logic hold true. Fine, you say, then really you’re saying there effectively are no Hidden Variables (HV) theories that describe our reality? No, says Valetini. The Hidden Variables would be observable if the universe was in a different state, the other phase. How might this happen? And what are the consequences? And is this even remotely plausible?

Last question first: Valentini thinks it is testable using the microwave cosmic background radiation. Which I am highly sceptical about. But more on this later.


The idea of non-equilibrium Hidden Variable theory in cosmology. The early universe violates the Born Rule and hidden variables are not hidden. But the violent history of the universe has erased all pilot wave details and so now we only see non-local hidden variables which is no different from conventional QM. (Apologies for low res image, it was a screenshot.)

How Does it Work?

How it might have happened is that the universe as a whole might have two (at least, maybe more) sorts of regimes, one of which is highly non-equilibrium, extremely low entropy. In this region or phase the Hidden Variables would be apparent and Bell’s Theorems would be violated. In the other type of phase the universe is in equilibrium, high entropy, and Hidden Variables cannot be detected and Bell’s Theorem’s remain true (for QM). Valentini claims early during the Big Bang the universe may have been in the non-equilibrium phase, and so some remnants of this HV physics should exist in the primordial CMB radiation. But you cannot just say this and get hidden variables to be unhidden. There has to be some plausible mechanism behind the phase transition or the “relaxation” process as Valentini describes it.

The idea being that the truly fundamental physics of our universe is not fully observable because the universe has relaxed from non-equilibrium to equilibrium. The statistics in the equilibrium phase get all messed up and HV’s cannot be seen. (You understand that in the hypothetical non-equilibrium phase the HV’s are no longer hidden, they’d be manifest ordinary variables.)

Further Details from de Broglie-Bohm Pilot Wave Theory

Perhaps the most respectable HV theory is the (more or less original) de Broglie-Bohm pilot wave theory. It treats Schrödinger’s wave function as a real potential in a configuration space which somehow guides particles along deterministic trajectories. Sometimes people postulate Schrödinger time evolution plus an additional pilot wave potential. (I’m a bit vague about it since it’s a long time since I read any pilot wave theory.) But to explain all manner of EPR experiments you have to go to extremes and imagine this putative pilot Wave as really an all-pervading information storage device. It has to guide not only trajectories but also orientations of spin and units of electric charge and so forth, basically any quantity that can get entangled between relativistically isolated systems.

This seems like unnecessary ontology to me. Be that as it may, the Valentini proposal is cute and something worth playing around with I think.

So anyway, Valentini shows that if there is indeed an equilibrium ensemble of states for the universe then details of particle trajectories cannot be observed and so the pilot wave is essentially unobservable, and hence a non-local HV theory applies which is compatible with QM and the Bell inequalities.

It’s a neat idea.

My bet would be that more conventional spacetime physics which uses non-trivial topology can do a better job of explaining non-locality than the pilot wave. In particular, I suspect requiring a pilot wave to carry all relevant information about all observables is just too much ontological baggage. Like a lot of speculative physics thought up to try to solve foundational problems, I think the pilot wave is a nice explanatory construct, but it is still a construct, and I think something still more fundamental and elementary can be found to yield the same physics without so many ad hoc assumptions.

To relate this with very different ideas, what the de Broglie-Bohm pilot wave reminds me of is the inflaton field postulated in inflationary Big Bang models. I think the inflaton is a fictional construct. Yet it’s predictive power has been very successful.   My understanding is that instead of an inflaton field you can use fairly conventional and uncontroversial physics to explain inflationary cosmology, for example the Penrose CCC (Conformal Cyclic Cosmology) idea. This is not popular. But it is conservative physics and requires no new assumptions. As far as I can tell CCC “only” requires a long but finite lifetime for electrons, which should eventually decay by very weak processes.  (If I recall correctly,  in the Standard Model the electron does not decay.)  The Borexino experiment in Italy has measured the lower limit on the electron lifetime as longer than 66,000—yottayears, but currently there is no upper limit.

And for the de Broglie-Bohm pilot wave I think the idea can be replaced by spacetime with non-trivial topology, which again is not very trendy or politically correct physics, but it is conservative and conventional and requires no drastic new assumptions.

What Are the Consequences?

I’m not sure what the consequences of cosmic HV’s are for current physics. The main consequence seems to be an altered understanding of the early universe, but nothing dramatic for our current and future condition. In other words, I do not think there is much use for cosmic HV theory.

Philosophically I think there is some importance, since the truth of cosmic HV’s could fill in a lot of gaps in our civilisations understanding of quantum mechanics. It might not be practically useful, but it would be intellectually very satisfying.

Is Their Any Evidence for these Cosmic HV’s?

According to Valentini, supposing at some time in the early Big Bang there was non-equilibrium, hence classical physics more or less, then there should be classical perturbations frozen in the cosmic microwave radiation background from this period. This is due to a well-known result in astrophysics where perturbations on so-called “super Hubble” length scales tend to be frozen — i.e., they will still exist in the CMB.

Technically what Valentini et al., predict is a low-power anomaly at large angles in the spectrum of the CMB. That’s fine and good, but (contrary to what Valentini might hope) it is not evidence of non-equilibrium quantum mechanics with pilot waves. Why not? Simply because a hell of a lot of other things can account for observed low-power anomalies. Still, it’s not all bad — any such evidence would count as Bayesian inference support for pilot wave theory. Such weak evidence abounds in science, and would not count as a major breakthrough, unfortunately (because who doesn’t enjoy a good breakthrough?) I’m sure researchers like Valentini, in any sciences, in such positions of lacking solid evidence for a theory will admit behind closed doors the desultory status of such evidence, but they do not often advertise it as such.

It seems to me so many things can be “explained” by statistical features in the CMB data. I think a lot of theorist might be conveniently ignoring the uncertainties in the CMB data. You cannot just take this data raw and look for patterns and correlations and then claim they support your pet theory. At a minimum you need to use the uncertainties in the CMB data and allow for the fact that your theory is not truly supported by the CMB when alternatives to your pet theory are also compatible with the CMB.

I cannot prove it, but I suspect a lot of researchers are using the CMB data in this way. That is, they can get the correlations they need to support their favourite theory, but if they include uncertainties then the same data would support no correlations. So you get a null inconclusive result overall. I do not believe in HV theories, but I do sincerely wish Valentini well in his search for hard evidence. Getting good support for non-mainstream theories in physics is damn exciting.

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Epilogue — Why HV? Why not MWI? Why not …

At the same conference Max Texmark polls the audience on their favoured interpretations of QM. The very fact people can conduct such polls among smart people is evidence of areal science of scientific anthropology. It’s interesting, right?! The most popular was Undecided=24. Many Worlds=15. Copenhagen=2. Modified dynamics (GRW)=0. Consistent Histories=0. Bohm (HV)=5. Relational=2. Modal=0.

This made me pretty happy. To me, undecidability is the only respectable position one can take at this present juncture in the history of physics. I do understand of course that many physicists are just voting for their favourites. Hardly any would stake their life on the fact that their view is correct. still, it was heart-warming to see so many taking the sane option seriously.

I will sign off for now by noting a similarity between HV and MWI. There’s not really all that much they have in common. But they both ask us to accept some realities well beyond what conservative standard interpretation-free quantum mechanics begs. What I mean by interpretation-free is just minimalism, which in turn is simply whatever modeling you need to actually do quantum mechanics predictions for experiments, that is the minimal stuff you need to explain or account for in any metaphysics interpretations sitting on top of QM. There is, of course, no such interpretation, which is why I can call it interpretation-free. You just go around supposing (or actually not “supposing” but merely “admitting the possibility”) the universe IS this Hilbert space and that our reality IS a cloud of vectors in this space that periodically expands and contracts in consistency with observed measurement data and unitary evolution, so that it all hangs together consistently and a consistent story can be told about the evolution of vectors in this state space that we take as representing our (possibly shared) reality (no need for solipsism).

I will say one nice thing about MWI: it is a clean theory! It requires a hell of a lot more ontology, but in some sense nothing new is added either. The writer who most convinces me I could believe in MWI is David Deutsch. Perhaps logically his ideas are the most coherent. But what holds me back and forces me to be continually agnostic for now (and yes, interpretations of QM debates are a bit quasi-religious, in the bad meaning of religious, not the good) is that I still think people simply have not explored enough normal physics to be able to unequivocally rule out a very ordinary explanation for quantum logic in our universe.

I guess there is something about being human that desires an interpretation more than this minimalism. I am certainly prey to this desire. But I cannot force myself to swallow either HV(Bohm) or MWI. They ask me to accept more ontology than I am prepared to admit into my mind space for now. I do prefer to seek a minimalist leaning theory, but not wholly interpretation-free. Not for the sake of minimalism, but because I think there is some beauty in minimalism akin to the mathematical idea of a Proof from the Book.


Rovelli’s Roll

In a highly watchable talk in the Oxford University lecture mini-series on Cosmology and quantum Foundations Carlo Rovelli gives a lot of persuasive arguments about why the Many Worlds Interpretation is suspect. But he goes fast and furious sometimes. Sometimes constructing strawman arguments (I do not think anyone seriously thinks just literally interpreting mathematics in a given model of physics leads to necessarily great ontological truths, apart from the likes of characters like Tegmark perhaps) but I think generally even these points are well made and interesting to ponder. Rovelli describes his own current opinion as “Everettian” — which means not a traditional Many Worlds interpretation but rather Relative State interpretation.


One observer observing another, screenshot from Carlo Rovelli’s lecture.

There are many key slides in his presentation that I thought worthy of mentioning and which inspired this current post of mine.

In another slide Rovelli puts up a couple of threads, one is,

    • “Why don’t we see superpositions?” — what a silly question! Because in textbook QM we do not see the state, we see eigenvalues. We see where is the position o the electron or it’s momentum, never it’s wavefunction.
    • These (facts) are described by the position in phase space in classical physics; and by points in the spectra of elements of the observable algebra in quantum physics.

Which is cool, but then he riles the zen masters by writing:

  • They can be taken as primary elements, and the quantum formalism built up from them.

First, I should point out this is not erroneous. You can build up a theory from elements that are such primitives as “points in the spectra of elements of the observable algebra”.

But I think this is misleading for purists and philosophers of physics. Just because one approach to calculating expectation values works does not make it’s mathematical elements isomorphic in some sense to elements of physical reality. So I think Rovelli un-does some of his good arguments with such statements. (I’m not the expert Rovelli is, I’m just sayin’ ya know …)

You might counter: “Well, if you are not willing to take your theoretical elements of reality direct from the best mathematical model’s primitives, then where are you going to define your ontology (granting you are wishing to construct a realist interpretation)?”

I would concede, “ok, for now, you can have a favoured realist interpretation based on the primitives of your observables algebra.” But I think you are always going to have to admit this will be temporary, only an “effective interpretation” that is current to our present understandings.

My point is that while this makes for great contemporary physics it does not make for good philosophy (love of both knowledge and truth). The reason is blatant. If all you have is a model for computing amplitudes then there is really only a small probability for hoping this is a dead accurate and “True” picture of the real ontology in our universal physics. You can certainly freely pin your hopes on this chance and see where it leads.

I, for one, think that such an abstraction as an “observable algebra” although nice and concrete and clean, is just too abstract to be wisely taken literally as the basis for a realist interpretation. Again, I’m “just sayin’…”.

There are many more good discussion points in Rovelli’s lecture.

The Wavefunction is a Computational Tool

This meme has always gelled with me. You can map a wavefunction over time, for example, you can visualize an atomic electron’s orbital. But at no single moment in time is the electron ever seen to be smeared out over it’s orbital. To me, as a realist, this means the electron is probably not a wave. But it’s temporal behaviour manifests aspects of wave-like properties. Or to be bold: over time the (non-relativistic) constant energy electron’s state is completely coded as a wave. I will admit in future we might find hard evidence that electron’s truly are waves of some weird spacetime foamy medium, not waves in an abstract mathematical space, but I do not think we are there yet, and I think we will not find this to be so. My guess would be electrons are extended topological geons, perhaps a little more gnarly than superstrings, but less “super”. I think more like solitons of spacetime than embedded strings.

The keyword there for philosophy is “coded”. The wave picture, or if you prefer, the Heisenberg state matrix representation, (either the Schrödinger or Heisenberg mathematical tool will do) is a code for the time evolution of the electron. But in no realist sense can it be identified as the electron.  Moreover, if you are willing to accept the Schrödinger and Heisenberg pictures are equivalent then you have a doubled-up ontology.  To me that’s nonsense if you are also a realist.

Believe it or not though, I’ve read books where this is flatly denied and authors have claimed the electron is the wavefunction. I really cannot subscribe to this. It violates the principle of separation of ontology from theory (let me coin that principle if no one has before!). A model is not the thing being modelled, is another way to put it.

On a related aside note: John Wheeler was being very cheeky or highly provocative in suggesting the “It from Bit” meme. It sounds like a great explanatory concept, but it seems (to me) to lack some unknown extra structure needed to motivate sound belief. Wheeler also talked about “equations written on paper cannot bring themselves into existence” (or something to that effect). But I think “It from Bit” is not very far removed from equations writing themselves into a universe.

EPR is Entanglement with the Future?

That’s not quite an accurate way to encapsulate Rovelli’s take on EPR, but I think it captures the flavour. Rovelli is saying that in a Relational QM interpretation you do not worry about non-locality, because from each observers (the proverbial Alice and Bob at each end of an EPR experiment, or non-human apparatus if you prefer to drop the anthropomorphisms) point of view there is a simple measurement, nothing more. The realisation entanglement was happening only occurs later in the future when the two observers get back together and compare data.

I’m not quite with Rovelli fully on this. And I guess this makes me a non-Everettian. There might be something I’m missing about all this, but I think there is something to explain about the two observers from a “Gods eye” view of the universe at the time each makes their measurements. (Whether God exists is irrelevant, this is pure gedankenexperiment.)  If you are God then you witness effects of entanglement in the measurement outcomes of Alice and Bob.

The recent research surrounding the ER=EPR meme seem to give a fairly sound geometric or geometrodynamic interpretation of EPR as a wormhole connection. So I think Rovelli does not need to invoke anything fancy to explain away EPR entanglment. ER=EPR has, I believe, put the matter of the realist interpretation mechanism of entanglement to rest.

No matter how many professors shout out, “do not attempt to make mental mechanical models of QM, they will fail!” I think ER=EPR defies them at least on it’s own ground. (Ironically, Susskind says just such things in his popular Theoretical Minimum lectures, and yet he was one of the original ER=EPR co-authors!)

What About Superposition: Is Superposition=ER?

I am now going beyond what Rovelli was entertaining.

If you can explain entanglement using wormholes, how about superposition?


ER=EPR depiction from a  nice article by “Splitting Spacetime” Bao, Pollack, and Remmen (2015)


I have not read any good papers about this yet. But I predict someone will put something on the arxiv soon (probably have already since I just haven’t gotten around to searching.) In a hand-waving manner, superpositions are bit like self-entanglement. A slightly harder interpretation might be that at the ends of a wormhole you could get particle duplication or mirror-effects of a sort.

One might even get quite literal and play with the idea that when an electron slips down a minimal wormhole it’s properties get mirrored at each end. Although, “mirror” is not the correct symmetry. I think perhaps just “copied at each end” is better. Cloned at each end? Whatever.

Maybe the electron continually oscillates back and forth between the mouths in some way? Who knows. It does require some kind of traversable ER bridge, or maybe just that when the bridge evaporates in a finite time the electron’s information snaps but to one end, but not both ends. Susskind and Hawking both concur now that there is no black hole information loss right? So surely a little ol’ electron’s information is not going to get lost if it wanders into a minimal ER bridge.

Then measurement or “wave function collapse” is likely a process of collapse of the wormhole. But in snapping the ER bridge the particle property can (somehow) only get restored at one end. Voila! You solve Schrodinger’s Cat’s dilemma.

Oh man! Would I not love t0 write a detailed technical mathematical exposition of all this. Sigh! Someone will probably beat me to it. Meehhh … what do I care, I’m not doing physics for fame or fortune.

Someone will have to eventually worry about stability of minimal ER bridges and the like. Then there are Lorentzian wormholes and closed time-like curves to consider. That Bao, Pollack, Remmen (2015) paper I cited above talks about “no-go” theorems arising from admitting ER bridges, no-go for causality violation and no-go for topology change.  I think what theoretical physics needs is an injecting of going past such no-go theorems.  They have to be “goes”.  Especially topology change.  If topology change implies violation of causality then all the better.  It only needs to have direct consequences at the Planck scale, then it’s not so scary to admit into theory, whatever the mess it might cause for modelling.  The upshot is that at the macroscopic scale I think allowing the “go” for these theorems rather than the “no-go” will reveal a lot of explanatory power, maybe even most of the explanation for the core phenomenon of quantum mechanics.  They mention concerns about violation of causality All of which I think is brilliant. I can see this sort of deep space structure explaining a lot of the current mystery about quantum mechanics, and in a realist interpretation. Awesome! And that I am not “just sayin” — it truly would be justifiably awesome.

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Hmmm … had a lot more to say about Rovelli’s talk. Maybe another day.

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Propelled by Beams of Intellectual Light

One frustrating thing about being a mathematics teacher is the difficulty of conveying to young students the sometimes terrifying giddiness of plunging deep into mathematics. There is an awesome sort of thrilling vertigo associated with trying to understand, and work through, high level mathematics.

The cool thing about mathematics is that it is endlessly capable of providing such a thrill, no matter what your age or talent, no matter what level of ability you already have. There are also many different paths one can explore to get these adrenalin rushes.  Godel’s incompleteness theorems loosely suggest there is no end to the depths and heights of mathematical investigation.  There will always be a need for new distilled crystallized axioms that try to best express our most basic and unquestionable mathematical presumptions.  A possible future might even see multiple parallel universes of mathematics, pure imaginary worlds that can never collide because their alternative fundamental axioms will never be able to be proven to be across-world consistent, and yet which cannot be proven to be inconsistent.

One recent path I took was reading about some recent discoveries from the papers of Srinivasa Ramanujan. Ramanujan’s work is one of the most amazing collections in mathematical history. Not always the most applicable to modern technology (hardly any physicists have ever made use of Ramanujan’s results), but as pure abstract journeys of the mind Ramanujan’s work stands almost unparalleled in history.


Ramanujan’s manuscript. The representations of 1729 as the sum of two cubes appear in the bottom right corner. An equation expressing a near counter example to Fermat’s last theorem appears. Image courtesy Trinity College library. (From: )

The analogy I conjured up was that of climbing Mount Everest without ropes or oxygen. Getting deep into mathematics can be that terrifying. You constantly get the piercing anxiety of, “I will never understand this!” Everyone knows this feeling, because school mathematics is still compulsory in most countries. Everyone hits this barrier at some stage. No matter how good they are with mathematics. People only vary in when they get to such a wall.

People who hit this wall early probably bifurcate: they either haul themselves over the wall with gargantuan heart-pounding effort, and continue on to excel in mathematics or sciences, while the others get daunted by the height and cannot see the other side and give up.


An image I borrowed from blog, it seemed to capture the physical/intellectual interconnections in our minds, plus a sort of infinite mental expanse quite well.  I didn’t want to post a brain explosion image!  That might give the wrong impression.

This is, I know, too simplistic a picture, but I think it captures the psychological impact of mathematical learning for many people. Sometimes you can hit several walls, and maybe only the sixth or seventh seems insurmountable, and so you retire grand mathematics ambitions and turn to maybe history or teaching or a relatively safe branch of applied mathematics. But even going down those seemingly safe avenues there are walls of unimaginable height and beauty that can disturb you.

And this is part of the wonder and beauty of mathematics.

The other imagery that came to my mind was the primal fear the astronaut Chris Hadden spoke about when sitting confined in a Space Shuttle launch cabin and getting the giant or all kicks up the backside when the main rockets ignited and hurled him into orbit. (By the way; Hadden recorded a pitch-perfect cover of David Bowies’ “Major Tom” track while in orbit on the ISS, one Bowie himself described as, “the best cover he had heard.”) Going by recent history, there is only a 1 in 30 to 1 in 40 chance of surviving such rocket launches. So it is a fantastic gamble deciding to be an astronaut. It is amazing people still volunteer, considering robots are almost capable of performing most of the tasks needed in space missions. Why take the risk?

If you talk to Hadden and his colleagues I’m sure they will tell you it is dozens of times over worth the risk. Just watching the Earth slowly rotate underneath in the vastness of black space is something that seems to change the soul.

With surviving great overwhelming terror comes profound spiritual awareness.

The terror can be purely mental, it does not have to be physical. But there is a fascinating connection here in the human brain. Terror and other similar deep emotions like fear and envy, arise in the amphibian primitive centres of our brains, the amygdala and hippocampus, while the impact rises up to higher conscious brain functions and we can sometimes get an experience of an inner world of abstract delight and insight when these primitive regions are stimulated. (I know the mappings of brain regions to psychological states is not as simple as this, so neurologists please do not hassle me about this, q.v.  The Amygdala Is NOT the Brain’s Fear Center, by Joseph DeLoux, Psychology Today, 10 August 2015. The amygdala is more correctly merely a threat-response system, it is not a source of conscious fear, the amygdala merely contributes in small part to a more neocortex driven feeling of fear or fright.)

The flight-or-fight response originates primarily in the amygdala, and it is an unconscious response. The consciousness of being in sheer panic or rage filters up to higher brain regions only after a few seconds or moments, which is neurologically a fairly long time — at least a few dozen or hundred cycles of 40 Hz brain wave activity. But we are eventually consciously aware of our responses. What the conscious systems do with these feelings is then a complex matter. Some people are able to thrive on the fear or rage and go deeper into the rabbit hole. Others rebel and go for safety. So perhaps a whimsical caricatured “difference between” X-Game competitors and Wingsuit flying daredevils and a mathematical genius is only the type of stimulus they fly form or dive into. Get anxiety from heights or open spaces or hanging upside down then you might be more of a mathematician. Get anxiety from an undecipherable maze of symbols on paper that are demanding decryption, and feel ill at the hopelessness of untangling them, then you might be more of a skydiver or rock climber.

So I wonder if the act of doing mathematics has a tremendous amount of associated unconscious neural activity? I wonder if this translates into a thrill and adrenalin rush when some insight is gained and a forbidding intellectual wall appears to crumble and a new revelatory insight into the world of mathematics is unveiled? And I even wonder if this can be addictive?

Whatever your inclination, when you next hear about a mathematical or scientific breakthrough, spare a thought for those who made those endeavours possible. For every breakthrough there are hundreds or thousands of researcher’s who will never get the accolades and awards, but who daily put themselves through the anxiety-ridden turmoil of smashing their minds up against intellectual barriers and paralyzing laser grids of the mind, or who feel constantly like they are falling from infinite heights of mental anguish and never know when the fear will cease. But all they are doing is sitting or pacing around in their laboratory or study wondering desperately where the much needed inspiration will come from to rescue them from the impending calamity of intellectual loss.

When the magical insight arrives, if ever, then the risk of the depression becomes all worth it, because the thrill of insight and discovery in the invisible planes of abstract theory and intellectual monuments is like being driven across the cosmos on beams of light. The journey is an expansion of your mind, it receives new ideas, allows your brain to form new connections, and opens up fields of intellectual inquiry previously barred. The propulsion system is imagination, insight and, dare I say t, some sort of spiritual impulse. If you are a physicalist then I suppose there are neural correlates for all of this — and you may think of it all as non-miraculous if it makes you feel better —, but think deeply and you might realise there is something more. Doing truly insightful mathematics or science does, I think, at the very forefront and apex of human endeavour, bring something new into the physical universe. From where it comes is perhaps unknowable. You might admit, if you have been touched by real inspiration, that perhaps, just possibly, maybe even likely, there is a world of imagination and abstraction beyond our physical reality, perhaps even closer to us that the atoms constituting our body, perhaps more like the essence of our selves, an existence our body and brain are merely borrowing temporarily ‐ or the converse, depending on your point of view.

This is what is so hard to explain to young children and even mature students. To explain the feeling of experience of these intellectual thrills, and to even hope to remotely compare them to physical danger and excitement, is incredibly difficult for a teacher. These are in the realms of “you really have to experience it for yourself.”

To replicate such intellectual adventure in a classroom is one of the prime responsibilities of a teacher. Yet our schools suppress most attempts in these directions, sadly. I call upon call teachers to put away textbooks and exam-preparation sessions, and replace them with adventures into mathematical depths that offer no clear or easy chance of escape. How you motivate such exploration is up to you, all I can say is try it! Just give your students freedom to explore. Then be prepared to catch them with your firm gedanken safety rope when they cry out in terror!


The Man Who Knew Infinity“, by Robert Kanigel, Abacus Books, 1992 (See goodreads.)

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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.


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.


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.


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.

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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” ( It’s not viral or mind-blowing. It’s just a simple pleasure to find one like this to watch while eating lunch.


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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Pressing the Origins

If I were pressed to write less than 700 words on the current state of cosmology and tie it in with infinite number theory and the deplorable state of scientific media communication, then I might write something like this following email to my friend Syko.

Let me first get you “in” on this conversation:

Syko made a comment in an earlier email to the effect that he could agree the origin of the universe can be taken seriously, but … (in his words):

I … rather support the notion of the origin. I struggle with the idea that the universe is infinite. Doesn’t make sense to me. Don’t buy it.

Then I wrote back saying something like:

Things do not need to make sense for them to be real.  There are some wonderful and bizarre levels of “infinite”, in fact far more infinitely many layers of infinity than most ordinary people realise, but since you Syko are not an ordinary person I can reveal some of the panorama for you.  And in any case, you do not get to buy in to Nature, Nature has bought you, and you have no say in this deal!   If it is an infinite universe then it’s infinite and you’ll have to suck that up, if it is finite then it’s finite and we have to live within that.    As Feynman said, “The theory of quantum electrodynamics describes Nature as absurd from the point of view of common sense. And it agrees fully with experiment. So I hope you accept Nature as She is — absurd.”

One thing about the potentially infinite expanse is that it is darn hard to kill off our universe.  How can you destroy it?  In 1998 those crazy astronomers measured the expansion was accelerating.  So gravity will not seem to be crushing us out of existence in a few trillion trillion years, so it’ll just keep expanding forever from what current measurements can predict.

But within that infinite expansion there are amazing things that could happen.

One is a cyclical time cosmology.  You can Google that along with “Penrose” since any discussion of cyclic time without Roger Penrose is probably New Age clap-trap. Penrose is however the real deal.  I WARN YOU, it is pretty awesome stuff!   So Google at your own peril!  hahahaha!
Then Syko replied:

Haha. Nature is not for me to buy, but the theories of mere men are, until proven. The infinite thing is a little to abstract. I get that there was a beginning. That the universe could be expanding. But the idea that it has no end or frontier and that it never ends. Well , I don’t think that’s right. It seems lazy to say it’s infinite.

And that led to my longer email about cosmology, the infinite, and science communication.  Here it is …

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Hey Syko,

I’m curious why you think it is lazy to theorise the universe could have infinite extent in time?  (Note that this does not imply infinite extent in space unless the expansion does not slow down to an asymptotic limit.)

Surely the idea of an infinite time is not really lazy, none more so than theorising time will be finite.  However, stating a theory which begins with such an assumption, either way, finite or infinite, is a lazy approach in a sense (and perhaps that’s what you mean?)  because it is simply assuming a fact that should be provable or falsifiable by other means, either by a less presumptive theory or by observational evidence.

But in any case, the best evidence available to date tells us the universe will expand forever.  This is not a theory.  It is a fact about the dark energy component of the universe together with the theory of general relativity.  It could be wrong, but it’s the best answer we have at present.

These physical “potential infinities” are one thing.  But the really exciting stuff, I think, is in pure mathematics where the transfinite numbers are considered.  It takes some mental effort to wrap your head around the concepts, but there are amazing possibilities involving transfinite number theory being applied to solve hitherto impossible problems in analysis (calculus) including maybe tackling problems that arise in quantum mechanics and general relativity when calculations arrive at irreducibly infinite numerical answers.  The idea before was that the theories had to be wrong or incomplete because they gave infinite numerical answers to fairly basic questions.   But modern mathematic suggests the idea that the infinite number answers might be totally sensible if interpreted according to transfinite arithmetic.  However, this is not all worked out and there is a communication gap between the physicists and the mathematicians.

Ordinal number spiral

When you have time… I am also curious about people who cannot conceive of anything existing prior to the “Big Bang”.   People are fond of saying that the Big Bang arose out of nothingness as a quantum fluctuation.  But this is sheer madness, since quantum fluctuations cannot fluctuate without a pre-existing spacetime in which to fluctuate.  And no one has ever shown how spacetime can fluctuate itself into existence from nothing.   In fact, there is not even a primitive philosophy about how to do it, so the physics has absolutely no hope of actually explaining the existence and origin of the universe.

So I really lose patience and can get quite irate with scientists in the media who get into the public airways and start saying things like how physics has explained the origin f the universe.  It is utter nonsense and gives physics a bad reputation in my opinion.   To give you some idea of the scale of this lunacy, at least in my opinion, I would describe it as analogous to a reputable biologist speaking to the media in all calmness and coolness and seriousness telling them that not only all known diseases, but in fact all future possible diseases from all possible vectors whether they be based either on DNA-based pathogens or non-DNA based life, have all been cured in theory by recent discoveries in the field of quantum medicine.

What physics can do is explain how things evolved after the period of cosmic hyper-inflation which happened after the universe became more than a singularity.  The actual history prior to this is completely mysterious to physics.  We do not even know if there ever was an initial singularity.   People who argue science says otherwise are completely deluded and are impossible to debate and argue rationally with.

That’s my opinion.  I wish people would take my opinion more seriously and read them, and take them to heart, before issuing any public relations announcements on behalf of science.  Of course if people did so and qualified all their claims and speculations correctly no science press conference could possibly last for less than about 30 minutes I imagine.  Hahahaha!

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You know what, I actually prefer Sir Roger’s hand-drawn diagrams:

Penrose CCC diagram

If you are interested in business strategy and economics and marketing, there is a dude Ben Thompson writing a great blog on such matter.  I mention him because he also uses hand-drawn diagrams to amazing effect.  It’s only data viz art, but funky cool.  Check him out here:,  and there is his podcast here:


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