Perhaps, some day, you will attend one of my research seminars! I give these now and then, to let my colleagues here at UNSW, or at conferences, know the results of my latest exciting mathematical researches and breakthroughs. Over the years I have talked about group representations, Lie theory, hypergroups, special functions, number theory, combinatorics, and mathematical physics, but these days it is usually some aspect of geometry or rational trigonometry—whatever I have been working hard at for the last year or so.

Having given such talks dozens of times over the past few decades, I am now just starting to suspect, somewhat painfully, that most of them have probably been quite boring. Not to me of course!—I find them singularly interesting, especially when full of the adrenalin that talking for an hour in front of a room full (or half full, or one quarter full) of highly intelligent people gives you.

What made me suspicious? Nothing obvious, just a few subtle tell-tale signs, a hint here or there. People sleeping during the lecture. Audible snores. Some colleagues deeply engaged in test marking, others seeming to meditate quietly with their eyes closed, yet others studying the cloud formations out the window. The drooping eyelids followed by the dropping head, and then the automatic jerk as the body regains consciousness before hitting the desk. The awkward silence at question time, the nominal polite question from the seminar organizer or a friend in the audience.

So because one shouldn’t jump to conclusions quickly, I now make discreet observations of members of the audience at other pure mathematics seminars. And I am pleased to report that with some key exceptions, the phenomenon seems to be almost universal. Not only are my seminars boring, but in fact most pure mathematical seminars are boring—judging solely by the audience attention and reaction.

This is surely a conundrum, seeing as pure mathematics has to be one of the most fascinating areas of human endeavour! How can we explain it?

To answer this question, I have submitted myself to intense psychological self-examination in the interests of Science. The results are not pretty, and don’t cast me and my fellows in a glowing light. This is reality journalism, self-confession and science reporting, all rolled into one.

The need to impress When I give a lecture to my professional colleagues, I pretend I am interested in informing and entertaining them. In reality my motives are much more nefarious and self-centered: I want to convince them that I have not been twiddling my thumbs for the last year, that I deserve to get more research money, that I ought to be promoted, and that I am generally not the moron I appear to myself most of the time. To do this is no easy task, but I have a well-trodden path to follow.

The key is to make my talk as technical and difficult to understand as possible. If the listeners can’t absorb and follow my seminar, they won’t suspect it is mostly uninteresting, and ultimately rather trivial (the key result boils down to setting a derivative to zero, or solving a quadratic equation, or something equally mundane). I formulate the most general version of everything, give the most specialized and convoluted examples, and make sure that the theory gets dressed up as something much more subtle and difficult than it really is.

Keep expository stuff down to a minimum Since most of my colleagues aren’t familiar at all with the particular areas I investigate, they would probably benefit most by an entertaining, expository, and wide ranging overview of the area. They would like to see the gems in the subject, the really beautiful arguments, the most important and useful results, the surprising connections with adjacent disciplines. But giving them what they want would be like dousing water on my all-important reputation. Most of the really interesting things in my area have been established long ago, perhaps by Euler, Sophus Lie, Felix Klein or Hermann Weyl. How is explaining their lovely insights going to enhance my reputation, increase my prospects for promotion, or improve my chances of getting one of those obscenely rich Australian Research Council grants?

Rising up in the cult of complexity Modern pure mathematics gets a bit insular, and so it becomes really challenging to compare the relative importance of different people’s work. Is my theory of Modular cuspidal cohomology of the functorial duals of p-adic proto-sheafs on a transcendental delta ring more interesting than your theory of Simplicial foliations of the pseudo-twisted maximal operator on the spinor bundle of a perverse quantum monoid? Who’s to say?

What ultimately counts is what we can get our colleagues to believe about the depth and importance of our research fumblings, how many papers in prestigious (i.e. unreadable) journals we publish, and how big and influential our circle of citation/conference-buddies becomes. This is a zero-sum game, my friend, and  the complexity and incomprehensibility of my seminars is a key tool to impress the Dickens out of you and my colleagues. Academic self-interest must prevail, and so I am happy to say that my next seminar will be…deep, profound and extremely important! In other words, boring.

Modern mathematics is enormously complicated and sophisticated. It takes some courage, and perhaps some foolishness, to dare to suggest that behind the fancy theories lie serious logical gaps, and indeed error. But this is the unfortunate reality. Around the corner, however, is a new and more beautiful mathematics, a more honest mathematics, in which everything makes complete sense! It is my job to give people glimpses of this better, more logical alternative, and to empower young people especially to not be afraid to question the status-quo and the dubious thinking that currently holds sway over the subject. My MathFoundations series of videos will investigate these problems in a systematic way; let me here at least briefly outline some of the problems, so you can get an initial idea, and so that perhaps some of you will start to think more seriously about these important issues. I will be saying a lot more about these topics in future posts.

The notion of rigour in mathematics is a difficult one to pin down. Certain historical periods accepted notions or arguments that later were deemed insufficiently precise, or even incorrect, but this often became clearer only once a more accurate way of thinking emerged. A familiar illustration is the geometry of Euclid’s Elements, which for most of the last two thousand years was considered the model for logical presentation of mathematics. Only in the nineteenth century did it become acknowledged that Euclid’s definitions of point and line were imprecise, that he implicitly used rigid motions for proofs without defining them, that intersections of circles were taken for granted, that notions of betweenness were used without being supported by corresponding definitions, that arguments by pictures were implicitly used, and that most of the three-dimensional parts of the geometry were logically unsubstantiated. In each of these cases it became possible to talk about alternative ways of thinking, due to non-Euclidean geometries, linear algebra, and the idea of geometry over finite fields. Einstein’s theory no doubt played a big role in loosening people’s conviction that Euclidean geometry was somehow God-given.

The foundations of trigonometry are also suspect as soon as one inquires carefully into the nature of an angle—a difficult concept that Euclid purposefully avoided. It requires either the notion of arc-length or area contained by a curve, and both of these require calculus. The usual pastiche of trigonometric relations depend logically on a prior theory of analysis; a point that even most undergraduates never really properly see. Indeed the very notion of a curve was problematic for seventeenth and eighteenth century mathematicians, and even to this day it is not straightforward. For example, one of the supposedly basic results about curves is the Jordan curve theorem: a simple closed curve in the plane separates the plane into two regions; but it is the rare undergraduate who can even state this result correctly, least of all prove it.

There are even surprising and serious logical gaps with first year calculus. The foundations of the “real number line” are notoriously weak, with continued confusions as to the nature of the basic objects and the operations on them. Attempts at trying to define “real numbers” in the way applied mathematicians and physicists would prefer—as decimal expansions—run into the serious problems of how to define the basic operations, and prove the usual laws or arithmetic. [Try to define multiplication between two infinite decimals, and then prove that this law is associative!] The approaches using equivalence classes of Cauchy sequences, or Dedekind cuts, suffer from an inability to identify when two “real numbers” are the same, and purposefully side-step the crucial issue of how we actually specify these objects in practice. Dedekind cuts in particular are virtually picking oneself up with one’s own boot straps, with a notable poverty of examples. The continued fractions approach, while in many ways the most enlightened path, suffers also from difficulties. The result of these ambiguities is a kind of fantasy arithmetic of real numbers, a thought-experiment floating above and beyond the reach of concrete examples and computations. Which is why the computer scientists have such a headache trying to encode these “real numbers” and their arithmetic on our computers.

The serious problems with the continuum are reflected by an attendant state of denial by our first year Calculus texts, which try to bluff their way through these difficulties by either pretending that the foundations have been laid out properly elsewhere, can be replaced by some suitable belief system dressed up using “axiomatics”, or can be glossed over by appeals to authority. The lack of examples and illustrative computations is illuminating. A challenge to those pure mathematicians who object to these claims: can you show us some explicit first year examples of arithmetic with real numbers??

The Fundamental Theorem of Algebra, a key result in undergraduate mathematics, that a polynomial of degree n has a zero in the complex plane, is almost never proved properly. While it ostensibly appears to be `proved’ in complex analysis courses, it is doubtful that this is convincing to students: after all, by the time one has studied complex analytic functions to the point of being able to apply Liouville’s theorem, who can say for sure whether one has not already used the very result one is ostensibly proving, perhaps implicitly? In fact complex analysis as laid out in undergraduate courses is very much open to criticism, and not just because of the nebulous situation with `real numbers’. Yet this crucial result (FTA) is used all the time to simplify arguments.

Closely connected with all of this is Cantor’s theory of `infinite sets’ and its current acceptance by the majority as the foundation of mathematics. The essential problem that ultimately overwhelmed Cantor is still with us: what exactly is an “infinite set”? For a long time now it has been well-known that Cantor’s initial “definition” of an infinite set was far too vague; consideration of the “set of all sets”, or the “set of all groups” or the “set of all topological spaces” are fraught with difficulty and indeed paradox. The modern attitude is to slyly substitute some other terms like “class” or “family” or “category” when possible contradictions might arise. Hopefully fellow citizens will have the decency to not raise the question of what exactly these words mean! If everyone plays along, there is no problem, right?

Other weaknesses of modern analysis arise with issues of constructability and specification. What do we actually mean when we say “Let G be a Lie group”, or “Consider the space of all analytic functions on the circle” or “Now take the nth homology group”?? Terminology is important: I have never seen a proper discussion of what the words let, consider or take actually mean in pure mathematics, despite their universal usage. Difficulties with terminology also affect the core set-up: the modern mathematician likes to frame her subject in terms not only of sets but also of functions. The latter term is almost as problematic as the former.

What precisely is a “function”? Okay, the usual definition is something like “a rule that inputs one kind of object and outputs a possibly different kind of object”. But this passes the buck from defining the term “function” to defining the term “rule”. Are we thinking about a computer program here? If so, what kind of program? What language and syntax? What conventions about how to specify a program, and how does one tell if my program defines the same “function” as your program??

The modern analyst likes to go further, and also talk about “arbitrary functions”, allowing not only those that can be described in some concrete way by an arithmetical expression or a computer program, but also all those “functions which are not of this form”. What exactly this means, if anything, is highly debatable. The lack of clear examples that can be brought to bear on such a discussion is a hint that we are chatting here about something other than mathematics. Surely a distinction ought to be drawn between “functions” which one can concretely specify and “functions” which one can only talk about. Even better would be to cease discussion about the latter entirely, or at least relegate them to philosophy!

The theoretical use of limits in calculus is generally lax. This despite all the huffing and puffing with epsilons and deltas, whose seeming precision obscures the more devious sleights of hand, of which there are many. For example, while care is often used to `prove’ the Intermediate Value Theorem (which is obvious to any engineer or physicist), the use of `limit’ in the usual definition of the Riemann integral is almost a complete cheat. Have a look at your calculus book carefully in this section, and see what I mean! Most first year students are blissfully unaware of the vast logical gaps in their courses. Most mathematicians do not go out of their way to point these out.

Of course there is much more to be said about these issues. All of them will be addressed in my MathFoundations YouTube series, but I think it useful to also begin a discussion of them here in this blog. There is another, more beautiful, mathematics waiting to be discovered, but first before we can properly see it, we need to clean out the cobwebs that currently obstruct our vision.

We live in interesting times; a good thing—so far at least. One of the momentous waves of change which is just now starting to roll over universities and academics around the world is a whole new online way of learning, accessible from essentially anywhere, for free. This will have a deep and profound effect on academic life. The Australian recently ran an article on this development, featuring my friend and colleague Chris Tisdell (Google: Tisdell, seismic shift, education to access the article). 

Increasingly you can log onto YouTube, or iTunes U, or other repositories, and start learning about anything you want. While in many areas the offerings are still in a scattered and embryonic form, the amount of material and resources is increasing exponentially, and the process seems clearly irreversible. More organized courses called MOOCS are using platforms such as Coursera, EdX, Udacity and others to train tens of thousands of students (how successfully is still a question). Other platforms are being established as you read this.

No amount of feet dragging by academics, textbook publishers, college administrators, and other entrenched interests will likely stop this trend. The reality is that universities as sole repositories of high-end knowledge and learning is coming to an end. Academics like myself will have to adapt or be prepared to go the way of the harness and carriage-makers a hundred years ago with the advent of the motor car. The lesson is clear: change, or be made irrelevant.

Right now, I have about four tutorials a week in first year Linear Algebra or Calculus. I find myself saying the same things as the lecturer in the next room, and that I have said dozens of times in the past. The same scenario is repeated with little variation in thousands of colleges around the world. This is an unsustainable situation, much as perhaps we would like it to continue, as our jobs largely depend on it.  The reality is that having thousands of essentially identical first year tutorials/classes around the world on, say, “the derivatives of inverse hyperbolic functions”, or “how to apply the normal distribution” is increasingly a situation approaching its use-by date. Clearly it is vastly more logical and practical for a few people to develop the lessons really well, and put them on YouTube for anyone, anywhere to watch whenever they feel like it. Once that happens, and students can access easily the information they need, thousands of academic jobs almost immediately become redundant.

The teaching role of universities, especially for large popular subjects, will inevitably change from providing primarily learning content to providing primarily assessment, support and certification. People will pay to get a certificate of achievement. They will no longer be so willing to pay to get instruction that they can easily get for free online. No doubt there is a social aspect of going to university; meeting other young people, playing cards or soccer during lunch hours, and chatting to your university lecturers. Attending a class can be a positive experience. But it can also be rather lukewarm: some college level lecturers are not stellar teachers, have ordinary communication skills and little real training in education. Once the choice between a mediocre live lecture and a high production video with powerful graphics and an entertaining dynamic expositor is available, I think we all know where most students will go. The core idea that universities and colleges primarily provide instruction, and rather high priced instruction at that, has to change.

Many of my academic colleagues will, quite understandably, be upset at this development. My own efforts at posting lots of mathematics videos online at YouTube (my channel is called Insights into Mathematics, user: njwildberger, check it out!), along with those of Chris Tisdell (his channel is called Understand Mathematics, user: DrChris Tisdell, check it out!) are seen by some of our colleagues as competitive with the traditional lecture format. But the reality is that the changes that are coming are made inevitable by the technology at hand; the question is only whether one is willing to embrace them and move forward on the train, or stand still like deer in the headlights. We see ourselves as potential bridges to the future: establishing UNSW as a key contributor in providing quality mathematics instruction to the world, along with more established and well-funded players like MIT, Stanford etc.

For centuries universities have been elite institutions catering to the sons of the rich to ready them for positions of power and privilege. In the twentieth century that scenario gradually expanded, allowing first women and then more and more middle class and even working class students into the college and university framework. While this has been a great contributor to the rise in equality in the Western world, still most of the rest of the world was excluded from the process, as the high-end educational institutions were concentrated in mostly well-to-do Western countries. The current technology supports a massive expansion of knowledge into the third world, as well as empowering ordinary people, young and old, rich or poor, to learn, learn, learn, as long as they want to! It will be one of the really big game-changers in the brave new world of tomorrow. Education is a killer application for the internet.

Every so often my thoughts turn to the curious economic situation we have built for ourselves here in the Western world, over the last century or two. The greatest building and economic boom in history has been hurtling along for some time now, but there are some serious issues that are starting to become difficult to ignore, even though we might like to. Most of these revolve around a four letter word which is the cure to all evils to some, and the cause of all evils to others: debt.

Around two hundred years ago, governments in Denmark and Germany started raising money for building projects by issuing bonds: asking people for loans which would be paid back over a long time period (say 30 years); which paid a generous rate of interest; and which were backed by the buildings themselves. Since then, the bond market has blossomed into a huge industry with many variants, but the basic principle remains constant: the public/government borrows money from rich citizens to spend on needy projects, and promises to pay back, with interest, sometime in the future. The rich citizens win–they get a sure and safe return on their money, often with a better return than they could otherwise expect from other investments. The public wins: they get shiny new buildings, roads, airports, armies, hospitals or schools, and generous pensions for civil servants. The governments win: they get credit for providing essential services, a happy populace, and money to spend promoting re-election. It seems like a win-win situation: but are there really no losers?

Unfortunately, there often are. They are called children, grand-children, and great grand-children. Boiled down to its essentials, the basic equation is this: we borrow from our children to support a lifestyle we otherwise could not afford. The beauty in this arrangement, from the point of view of the generation making it, is that the approval, or even the awareness, of the future generations who will pay for it is not required! With the the modern financial, legal and government set-up, we are able to simply legislate monies into existence. This is a modern kind of financial sleight of hand, as if from the Ministry of Magic in Harry Potter world. Need an extra few billion dollars? Wave the legislative wand, and lo!, it is there.

In time, the borrowing inevitably starts to escalate: it is all too easy. We find ourselves always requiring more money, each time to cover not just our bigger and bolder plans, but also the interest charges of our previous loans, which grow more and more onerous. Once we get used to the idea of an inevitable “national debt”, we don’t resist when the politicians jack it up yet a little bit more—in our best interests, naturally.

After a while, successive generations become simultaneously victims and beneficiaries of the system, which comes more and more to resemble a Ponzi scheme (first-in get paid, last-in lose). Each new generation, as they become conscious of the state of affairs, looks backwards with annoyance at the previous cohort for having gotten them into the mess, and then looks ahead at ways of passing the buck to the next lot. Government debt getting into stratospheric levels? Let’s just sell off a bit more of the public domain; our roads, our power systems, our hospitals, our airports, our public lands. Still not enough cash to support all the lavish entitlements and retirement plans that we all so richly deserve? Borrow more money from our ever-more-wealthy wealthiest citizens, or perhaps the Asians, or just print some more.

With the baby boomer demographic bulge moving into retirement, and life expectancies still on the rise, many commentators are now starting to realize that the pressures on the current generation of young people to keep the system going is shortly going to become acute, perhaps even unmanageable without serious alterations to expectancies.

Governments are raising retirement ages, pension plans are, in many places, starting to look wobbly (my own UniSuper superannuation scheme here in Australia has started making rumblings about not enough cash to meet future committments), and while the mainstream media maintains their reluctance to acknowledge and assess the problem fairly, more and more discontent is registering in alternate forums on the internet. The prospect looms of future action by younger people to rein in the lavish retirement packages that the baby boomers have lined up for themselves. What legislation can give, legislation can take away.

In the university sector here in Australia, I joined a few years too late to enter the “old superannuation scheme”, which overly generously allows my older colleagues to have retired in their late fifties with essentially full salaries as pensions. My cohort will have to work well into their sixties to receive possibly only 1/2 to 2/3 of our salaries. Does this seem fair? Well not to me, but then again to an already-retired person it is probably quite acceptable. And future generations? What kind of weight will my daughter’s demographic group have to collectively bear? It’s a question.

My conferences in Novosibirsk and Baska (on the isle of Krk in Croatia) are over and I am in Zagreb for a couple of nights before my flight home. Novosibirsk is in the middle of Russia, the largest city in Siberia and third largest in Russia itself. It straddles the Ob river, one of the longest in the world, and was built largely to support the Trans-Siberian railway bridge built more than a century ago across the river.

A typical walk in the Academic town between Institutes

The Geometry conference was held in a satellite suburb called something like “Academic Town” in Russian: a sprawling complex of Scientific Institutes set in a forest, with pleasant walking paths through the birch and pines. The northern forest reminds me of Canada, and so I enjoyed the walks I took, several with my new young friend Maxim, a student of Alexander Mednykh, the mathematician who had invited me.

The Ob sea: man made reservoir outside Novosibirsk

Maxim showed me the Botanical Garden and local Russian Orthodox church, all in wood, set amongst the trees, as well as the Ob sea/reservoir, an impressive man-made lake which is hard to see across it is so big.

Sasha Mednykh and me in downtown Novosibirsk

Alexander and his son Ilya took me on a nice walking tour of Novosibirsk itself, showing me the theatre, train station, some interesting squares, and even a semi deserted fair with a Ferris wheel we took a ride on, to get a good view of the Ob and the longest subway bridge in the world over it.

Main street in Academic town, Novosibirsk

I stayed in a pleasant student dorm which had a few floors acting as a hotel: there were not a lot of accommodation options. The weather was lovely, end of summer warmth and blue skies.

The conference itself was fun, particularly the dinner with lots of toasts, of which I gave one! I met some interesting people; one of my neighbours was from distant Yakutsk, a name familiar from years playing Risk, but of which I know rather little, and he told me about life there. Evidently the permafrost spews out slabs of ice from the ground erratically in places, even in the middle of the summer!

I also had some good mathematical discussions, one in particular with I. K. Sabitov from Moscow, who had proven the lovely theorem that given a polytope in three dimensional space, there is an algebraic equation satisfied by the volume whose coefficients depend only on the quadrances (squared lengths) of the edges. However for complicated polytopes this polynomial can be exceedingly large. (For a tetrahedron the relation goes back to Tartaglia, and it was also discovered by Euler.)

A view from Novosibirsk

One small aspect of Russian life perhaps worth retelling: in the small minibus that Maxim and I took to go into the city, passengers pass up their fares to the driver hand to hand via other passengers, even from the back, till it gets to the driver. He dispenses the various lots of change and it snakes its way back, each person taking the amount due him/her, passing the rest on. Can I imagine this happening in a big city in Australia or America? Not really, to tell the truth. It says something about the level of trust people have with strangers. Perhaps an aspect of the `quality of life’ that doesn’t get as much attention as it should.

Downtown Novosibirsk: The center of Russia

Alexander Dimitryovich, as he is known by his students and colleagues (Russians use the first two names as a sign of respect, and the second name is that of the father) has a small office which is quite often full of students, so there is almost no room to turn around. There was a nice feeling of working together there, and I had some chance to exchange some interesting thoughts about hyperbolic geometry, a subject I have developed a deep interest in. (My Rational Trigonometry extends to the hyperbolic setting, and gives a rather new, and of course exciting, view of this classical subject.) I was happy that Alexander Dimitryovich was willing to entertain using my algebraic reformulations to make some computations.

So… I am eager to go back to Siberia. Next time I will perhaps try to stay a bit longer, and get further out of town, perhaps to the Altai mountains which are not so far away.

Mathematics throughout its history has wrestled with a major schism: between the discrete and the continuous. In the earliest times this was the difference between arithmetic and geometry.

Arithmetic ultimately comes down to the natural numbers 1,2,3,4,5,6…. These are so fundamental and familiar that most ordinary folk don’t see much point in `defining’ them. But us pure mathematicians like to ponder such things, and it is fair to say that the issue is still open to further insights coming from programmers and computer science. One way or the other natural numbers are symbols that we write down to help us count; the number of apples in a bushel, children around a campfire, stars in the sky.

Geometry, on the other hand, ultimately comes down to points and lines. It is not so easy to say exactly what these are. In the 19th century mathematicians started to acknowledge that the bible of geometry—Euclid’s Elements—didn’t deal adequately with this issue. Things were okay as long as you just assumed you knew about points, lines, and the plane; and accepted various physically obvious properties they satisfied.

Fortunately Descartes and Fermat some 300 years earlier had constructed a framework—the Cartesian plane—which allowed geometry to become subservient to arithmetic: a point is an ordered pair [x,y] of numbers, and a line is an equation of the form ax+by=c. This was a wildly successful conceptual leap. It allowed algebraic techniques to bear on higher order curves, like conics or cubics, gave a straightforward and uniform treatment of many geometrical problems, and led to the development of the calculus.

But there was a heavy price to be paid for this arithmetization, which was mostly unacknowleged for centuries. The precise and logical arithmetical form of geometry which Descartes’ system gives us has curious aspects that diverge from our everyday physical experience. No longer do two circles which pass through each others centers meet. We cannot guarantee that a line passing through the center of a circle meets that circle.

This explains, I believe, why Euclid shied away from an arithmetization of his geometry: he knew that standard geometrical constructions yielded `irrational’ numbers whose arithmetic he did not understand. The Greeks’ numerical system was cumbersome compared to our Hindu-Arabic system, they had no good notation for algebra, and they considered geometry more fundamental. So ultimately Euclid choose to consider a `line’ as a primitive object which need not be defined, and carefully avoided using distances and angles as the main metrical measurements. For him, logical purity trumped practical considerations.

Modern geometry has steered away from the concern and esteem for rationality of the ancient Greeks. The trigonometry we ostensibly teach in high school texts is logically half-hearted and involves a hefty amount of cheating; at the research level we have resorted to simply walking away from this challenge. Euclid would be appalled at the sad state of affairs in modern geometry, and would find it inexplicable that the majority of educated people have almost no understanding of this beautiful subject!

The problems with irrationals have been around for two and a half thousand years, and are still with us, whether we like it or not, acknowledge it or not. Deep at the bottom of modern mathematics lies a gnarled and warted toad: the lack of a true understanding of the continuum. Many (but not all!) modern mathematicians will view this statement with skepticism. We like to believe we understand the continuum—in the context of `real numbers’—and have faith that the definitions involving Dedekind cuts, Cauchy sequences, or just axiomatic assumptions, deal adequately with the problems. Unfortunately, they do not.

In my opinion, the continuum is actually much, much more complicated than mathematicians think. Our current view of the continuum is analogous to the simple-minded model of the heavens that ancient, and not so ancient, peoples had: that we live surrounded by a large celestial sphere on which the stars are pinned, and on which the sun, moon and planets move. For better or worse, the true celestial story is vastly richer and indeed more interesting than this, and so it is with the continuum.

Modern mathematics has accepted a confusion which has spread its poisoned tentacles into almost every aspect of the subject. By accepting the logically dubious, we come to accept also that some parts of mathematics are just inherently vague and obscure—that logic has its limits, and beyond that is a kind of no-mans land of convenient but arbitrary assumptions. Mathematics loses its certainty, and descends into shades of grey. This shrugging away the bounds of careful reasoning at the research level also naturally affects the integrity of mathematics education.

The reader will want some initial evidence to support these statements. Look in any modern Calculus textbook in the introductory section which purports to establish, or review, the fundamental properties of `real numbers’. Almost all resort to waffling or unwarranted assumptions, with a few honest exceptions that admit to the lack of proper foundations. Then consider how the modern computer programming community deals with `real numbers’. What you find is that they don’t, because they can’t; the rigour of their machines interferes with wishful thinking. Instead, the programmers work with floating point representations or rational number computations, which are light years away from working with `real numbers’.

So let me put some of my cards on the table: I propose to shine a clear light on this whole issue, to expose the logical confusion that underpins modern analysis, and most importantly, to provide a way forward that allows one to envision a time when mathematics will be logical and true, without waffling and the ugliness that naturally follows it around.

In my MathFoundations YouTube series at http://www.youtube.com/course?list=EC5A714C94D40392AB&feature=plcp I will tackle the detailed mathematical aspects of this campaign. In this blog I hope to provide some overall framing and discussion of both the mathematical and the sociological aspects of this unfortunate delusion—a delusion that has got its stranglehold on mathematics education, as much as research level mathematics, since the beginning of the twentieth century. We need to move into a bigger, brighter, more honest space.

I am flying on Monday to Novosibirsk in Siberia, and then a week later to Krk, an island in Croatia. The occasions are two interesting geometry conferences, where I will meet old friends, make new ones, and talk about my recent work. Travel is one of the joys of my life, and even a long plane flight is only half a burden, knowing that it ends in—a new place! It’s a chance to step out of the groove, in a small way, for a while.

The life of a research mathematician, if the truth be told (and I want to do that), is outwardly not very exciting. The routine is piled on thick; calculus and linear algebra lectures don’t change much from year to year, writing papers is rather boring and tedious, and the stable, conservative environment of most universities means that one’s working life is like a sheltered cocoon. Jobs are hard to come by, so career mobility for most of us is limited, and we get pretty familiar with our stomping grounds after a few decades.

The inward life, however, is a totally different story. Mathematics present us with such a rich and fascinating framework of ideas, concepts and challenges that I can think of no work that would give me even half the satisfaction I get from being a research mathematician. I would not trade my job for any other, not in law, politics, business or medicine. The search for patterns, the hunt for a key equation, the challenge in proving what our intuition tells us loud and clear must be true…until we find out a week later that our intuition was wrong and the story goes in a quite different direction! What a happy life to be able to think about interesting things; I am very grateful, and one of the purposes of this little blog is to share some of the rich ideas I get to think about, with you all.

At both conferences I will be talking about the Rational Trigonometry of a Tetrahedron,  albeit in somewhat different forms. You are all familiar with ordinary trigonometry (the word means the study of triangle measurement); with those angles and lengths, cosines and sines, and a whole raft of complicated formulas. It’s useful, but not altogether pretty.

Hopefully you have heard that there is now a new and better way of tackling the whole subject, called Rational Trigonometry, discovered about 10 years ago by yours truly. This story of a completely new way of thinking about a very classical subject is rather interesting; as you might expect such a bold departure from tradition doesn’t occur very often in elementary mathematics. Is it really better? Does it make computation simpler? Does it lead to a lot of beautiful new mathematics? Yes, yes and yes. And the crucial question: is it what we ought to be teaching our young people in high schools?? Definitely yes!

I will be telling you more about this discovery of mine, and the reaction that it has gotten from colleagues etc. in due time. An interesting consequence is that new doors have been opened to me by this understanding; vistas and trails that lie before me now, undreamt of a decade ago. The possibility of a new approach to the entire subject is starting to emerge as the fog slowly settles—a more careful, honest, logical and beautiful mathematics, more closely aligned to computer science.

The consideration of “revolution” in the context of mathematics is to many practitioners unlikely and even heretical. The safe confines of academia enclose an even more secure installation of pure research mathematics, where orthodoxy, accepted practice and authority largely rule.

Perhaps my whole life has been building up to the realization that, even in mathematics, there are true paths, and false paths, and paths in-between, and that ultimately only I can decide which is which—for myself. If you allow me, I propose to take you on some little mathematical journeys, and show you new possibilities for thinking. Then you too can decide what is true, what is false, and what is in-between—for yourselves.

Thought I’d give you a window on my week to come, since it is early Monday morning—so you can get an idea of what mathematicians do. Today is my big teaching day, I have four tutorials, all on first year calculus. I am handing back the tests the students wrote last week, and will try to convince them they must make more effort writing coherently and expressing their mathematical thoughts more carefully. Our first year students often don’t have a lot of practice in writing mathematics. Sometime during the day I will go for a swim at the Uni pool to make four hours of teaching go smoother.

Tomorrow I will chair a meeting of our Learning and Teaching Committee, we need to get going on mentoring new tutors, thinking about a L&T seminar for later in the term, and putting one of our colleagues forward for a teaching prize. I will also talk with one of my graduate students, Nguyen Le, on how we can construct a planar triangle over the rational numbers that has incenters in both blue (Euclidean), red and green (relativistic) geometries simultaneously. She has been working on triangle geometry, and we would also like to start writing a paper on some extensions of classical results involving Gergonne, Nagel, Incenter points etc.

I will need to schedule a list of mentors to oversee our new casual tutors (the School of Mathematics and Statistics employs outside help to also tutor some first year classes).

On Wednesday and Thursday I have three hours of my Algebraic Topology class. I need to have marked the problem sets before that—something else to do on Tuesday—hand them back, and give lectures on a review of free groups and non-commutative groups, and also on applications of winding and turning numbers (for example the ham sandwich theorem!) Nguyen will video one of those classes for me, and at some point, probably the weekend, I will edit it and post it on my YouTube channel (user njwildberger).

Friday I don’t have classes but I want to spend a good chunk of time with my Masters student Sharanjit Paddam who is working on rational views of classical tesselations of the hyperbolic plane. I am heading overseas the following week, to Novosibirsk (Russia) and Krk (Croatia) for two geometry conferences, will be gone three weeks, and want to make sure his project is on track.

I also need to spend some time preparing the talks I will give at these conferences. But what I also would like to do, in fact what I really want to do, this week is: think about the group structure on a cubic curve (one step up from the conic sections like ellipses, parabolas and hyperbolas) in terms of ternary forms. I have this reformulation of algebraic geometry kicking around in the back of my mind, which starts by reconsidering the basic nature of an algebraic curve; not as a set of points, but rather as… well something both more complicated but simpler to deal with. I`ll explain this at some later point when I understand it better, but suffice it to say for now that I know there are interesting newish algebraic objects which I call `pods’ floating around under the noses of classical algebraic geometers. I have been putting together Geometer’s Sketchpad worksheets that allow me to explore cubics and play around with group and pod structures.

The other thing that has been on my mind is the Cayley transform, something that I thought quite a lot about when I was a graduate student, and post-doc at Stanford. I reckon that we need a more algebraic theory of Lie groups, and I have known for a long time, although I haven’t written anything on it, that the key is replacing the exponential map with the Cayley transform, suitably interpreted.

Jack Hall did a Master’s project on related issues with me some years ago. He has finished a PhD at Stanford under Ravi Vakil, who curiously was also briefly a student of mine: he sat in on my first year algebra course at the University of Toronto when he was a precocious high school student. Jack will be taking up a position in Canberra next year, it will be nice to have him as part of the mathematics scene.

Anyway, the Cayley transform is a lovely thing, ought to be more widely appreciated and used. I would like to extend Rational Trigonometry to the Lie group setting, a big project of course, but something that I like to contemplate now and then.

I have also been feeling guilty about not posting any more videos in my WildLinAlg series for quite some time. If I can make time, I might put together a new video on the linear algebra of cubic splines. Not sure if I will get around to that though.

So that is my week, coming up! Lot’s of fun, and interesting.

The London Olympics are over, what a success they have been. Many Australians however are pondering the performance of our athletes, as we only placed 10th.

Only placed 10th?? By whose count?

The current rankings, which are published by most of the international media, rank countries by order of Gold medals won. How can this be at all sensible or fair, since it ignores silver and bronze medals entirely? If you are going to give out prizes, those prizes ought to be worth something.

There is another popular ranking system which we also see; which tallies the total number of medals won. This is also patently simplistic. Why should a bronze count as much as a gold?

The uncritical acceptance of these two simple-minded scoring systems around the world reflects an astonishing mathematical naivety. Surely we can do better in the 21st century! For a country like Britain to put on such a complex, dazzling show is inconsistent with its media performing just a cursory back-of-the-envelope calculation to determine rankings.

More than a hundred years ago, a fairer ranking system whose proposed by the English press: each gold is worth 5, each silver is worth 3 and each bronze is worth 1. So using a little bit of multiplication and addition, we get a much clearer and more equitable picture of how different countries performed—in total. Of course one can argue about the weightings, but I personally think these are quite sensible.

With such a weighted ranking system, all medal winners contribute to their countries standing, but on a scale that is reflective of the different levels of achievement. If we had adopted this, silver and bronze medallists would be happier knowing that their win has contributed to the national account, and we would see fewer dejected  athletes having just placed second or third in the world!

My calculations of the Weighted Ranks for the London Olympics are available at http://www.maths.unsw.edu.au/news/2012-08/weighted-olympic-rankings-london-2012-n-j-wildberger. Australia places 8th, not 10th; while Great Britain gets beaten into third place by Russia—perhaps this is the reason the British press don’t promote this fairer system this year!;— Spain goes from 21st to 14th, and Canada goes from 35th place to 22nd.

Congratulations to all our athletes for their excellent 8th place in a highly competitive meet. Let’s hope that the Olympics Committee can consider instituting this weighted ranking system as the gold standard for Rio, or that at least the international press can think about adopting it. A little mathematics can go a long way!

On Monday, after my early 9 am class, I drove up to Sydney’s North Shore all the way to Wahroonga. I was slated to give a talk at Knox Grammar School to 70 or 80 Heads of Independent High School Maths Departments from around NSW, on the subject of “National Curriculum Issues and opportunities for revitalizing geometrical thinking in the classroom”, which I admit is a rather long-winded title.

I had been invited by Joshua Harnwell, a teacher there, who I had met in an earlier Board of Studies meeting, also about these ghastly new mathematics curricula proposed by ACARA, the national group entrusted with coming up with a syllabus for maths education around the country.

Knox Grammar is a lovely old-money private school set in the affluent suburbs of Sydney. There are a lot of such around, quite a difference from Canada where I grew up, where almost everyone except close friends of the Duke of Buckingham (or some such) just went to the nearest local high school, which was invariably a public school. Meaning it was free, and open to all who lived in the area. Although Australia prides itself on its egalitarianism, there is in some curious way quite a lot more class differentiation here than in Canada where I was raised, although perhaps I am just getting older and wiser to such things. In particular there are so many private schools in Oz that the public school system languishes a bit for funds, and there is an unhealthy divide educationally. Growing up where I did, my default view is that governments have an obligation to level the educational playing field as much as possible.

Not to say anything against Knox, a lovely school; and they put on a really fine lunch for us in a high-ceilinged glass-enclosed foyer, with a fine view of the spacious grounds. I had pleasant conversation with some high school heads over our salmon, talking about the merits of GeoGebra and other dynamic software packages.

After lunch I gave my talk, recorded for posterity since I had brought my trusty Sony video camera with me. So in case you weren’t there, and you are interested in the topics, which are quite important from my point of view, you can find the video and pdf of the talk at http://www.maths.unsw.edu.au/news/2012-08/national-curriculum-talk-norman-wildberger-knox-grammar.

One of the side points I made was that we need to rethink, or rather the media ought to rethink, the ranking system used to score the Olympics. Turns out that was related to some of the geometry I talked about, and maybe next time I’ll tell you about that.

In any case I enjoyed the opportunity to talk to high school teachers about something important; it feels good to get out of the ivory tower of academia every so often.