Let’s talk about a rich and fascinating branch of mathematics called algebraic geometry. The subject has its beginnings with Descartes’ realisation that geometry could be approached algebraically by first introducing coordinates. In this way points become pairs, or triples, of numbers; lines become linear equations; conics become quadratic equations etc., while relations between objects can be encoded and studied purely algebraically.

In this brief note, I want to outline a somewhat radical birds-eye view of the subject, without getting into details. I probably should qualify my expertise here, in that I am not a professional algebraic geometer in the usual sense of the word. Nevertheless I have been studying the subject from a new point of view for more than ten years now, and have arrived at some rather novel understandings of what the subject is about. So what follows is my ten-minute take on algebraic geometry.

The most essential fact of the subject is that it is divided equally into two interlocking areas, the affine theory and the projective theory. The former theory rests on a vector space over a field, the latter theory rests on the associated projective space of lines through the origin. Neither is primary, contrary to popular belief; they are equal partners, and pretty well all aspects of the subject have both an affine and a projective version.

And what field are we working over? Certainly the rational number field is by far the most important, but finite fields are also  interesting, as are various extensions of the rationals,  for example the complex rationals obtained by adjoining a square of -1. But the truest theory is that which applies across the board to all fields (with the notable exception of  fields of characteristic two, which ought not to be called fields!) Note that the usual ‘field of complex numbers’, built on the so called ‘real numbers’, must be avoided at all costs if one aspires to be logically careful; it is a fantasy arena in which almost all our dreams come true, at the cost of abandoning our hold on mathematical reality and diminishing the natural number-theoretical richness of the subject.

Returning to the large-scale organization of the subject, there is a complementary and largely independent subdivision of the subject into various layers depending on the complexity, or degree, of the objects and operations involved.  The main distinction is between the first half–the linear theory, and the second half–the nonlinear theory.

The linear half of algebraic geometry is the more important half, and it goes by another name: linear algebra. This is the study of points, lines, planes and their generalizations and relations. The nonlinear half is itself divided roughly into two halves: the quadratic theory and the non-quadratic theory. The quadratic half is again more important than the non-quadratic half, and is occupied with conics and their associated metrical structures, namely bilinear or quadratic forms.

The non-quadratic half of the nonlinear half is again roughly equally divided into the cubic/quartic half and the higher degree half. Degrees three and four seem to be naturally linked, and support structures that don’t easily generalise to higher degrees. Although one could keep on subdividing, it seems reasonable to lump degrees five and higher into one-eighth of the subject.

I ‘ll try to figure out how to make a table to summarise the situation. But at least you get a sense of the various natural compartments of the subject, at least along the lines of how I see things currently.

What one studies in each of these areas is another important matter of course, but one that seems secondary to me to the basic subdivisions described here. Perhaps this rough guiding framework may provide a simple-minded but helpful orientation to the beginning student.

This semester I have been on Long Service Leave, so I am off the hook for teaching, and can spend more time with my graduate students Ali Alkhaldi and Nguyen Le, do some investigations into hyperbolic geometry and related issues, make more videos, and do some travelling. Ali is in his fourth year of the PhD, and is writing up his work on the parabola in hyperbolic geometry, which is now blossoming into a major re-evaluation of this subject, with dozens of new theorems. Nguyen is in her second year, and is making good progress on various aspects of Euclidean and relativistic triangle geometry, at this point related to the Incenter hierarchy.

Another main pre-occupation in the last few weeks is a large LT (Learning and Teaching) grant that Chris Tisdell, Bruce Henry and I have applied for, in conjunction with other colleagues here at UNSW, and some other universities in Australia. The country’s chief scientist, Ian Chubb, has organized a largish pool of money to be allocated to projects that improve teaching of maths and science in secondary schools in Australia, and our project proposes to address this by creating online professional development courses for high school teachers that teach them more mathematics and science.

Teachers are exposed to lots of in-service development that addresses the educational side of things: how to improve student learning, principles of effective pedagogy, teaching strategies etc. But in mathematics I think the greater problem is that not enough high school maths teachers understand the content of the subject well enough. We hear lots of anecdotal stories of Principals hiring Phys Ed teachers to teach mathematics because of shortages: after a quick 6 week training period the hapless new teacher is expected to inspire and motivate his/her students in a subject many of them already find difficult. Clearly not a very good situation.

Our idea is to make some high quality online courses that lay out the new Australian maths and physics curriculum in an engaging way for Years 11 and 12 teachers. The lectures for these courses would be make freely available on YouTube to anyone. With enough resources, we are hoping to put together videos and other materials with cool graphics, animations, demonstrations etc that will inspire high school teachers (and students too).

In addition to these courses, we hope to also organize a (YouTube) video library of Year 9 and 10 maths subjects, also aimed for teachers, that systematically presents the subject in a careful and fun manner. Let’s hope we get funded! If we do, I will probably be spending more time in the next few years making videos and online courses, and less time doing traditional teaching, which would be a nice change. [As you probably know, I like making YouTube videos!]

On another front, I will be heading overseas soon, visiting fellow geometers at the Universities of Innsbruck and Graz in Austria, and then on to Toronto via Florida to visit my family and friends. I will stop at my old alma mater the University of Toronto to talk maths with my friend Joe Repka and others, and I usually give a talk there.

My daughter Ali loves going to Canada, and she is taking some extra weeks off school to make the trip worthwhile. I will try to give her some personal maths instruction to make up for her lost classes, but that is not always easy! Right now we are talking about Pythagoras’ theorem, and I am trying to get her to see that it is really rather remarkable.  I think that possibility is not emphasized in school: that we are here witnessing a small miracle: make a right triangle, carefully draw three squares on the three sides (graph paper is essential for this), compute the areas of those squares, and then notice that the sum of the smaller two equals the third. And if you try it with a triangle which is not right, it doesn’t work! Isn’t that amazing?

Thought I’d venture into a bit of politics today. The channels of communication and social media that are opening around us, like flowers in spring, invite us to reconsider the role of the individual citizen in decision-making processes, at least in democratic countries. Are we not on the cusp of a transforming technology that allows—not just public debate orchestrated by main stream media—but the possibility of real input by ordinary people in the major policy decisions of our various levels of government?

Please join me today for a short thought experiment. Imagine a government website called mygov.org, where you may navigate to either Federal, State or Municipal government levels. Let’s suppose that we head for the State level. There we find:

1. An overview of the structure of the State government—who are our representatives, what are the major departments, who is in charge of what, contact information and links etc.
2. A summary of the current State Budget, together with summaries of previous budgets and a range of charts showing budget allocations in graphical form over various ranges of years. The citizen (you or I or our next door neighbour) can get a sense for where the State government gets its money (mostly from us of course!) and what it is spending it on. In particular a summary of the current levels of debt are prominently visible.
3. A record of our politicians’ debates in the various houses of government, a current listing of bills being proposed, and written statements from elected representatives as well as public experts on issues of current policy.
4. And of greatest interest! The VOTING BOOTH: an electronic portal that allows voting citizens (via a user name and password) input into the issues of the day. For example: Should we decrease the cost of the Airport train link to encourage tourists to use it, keep it as it is, or hike it to make more money? On this issue we can read pros and cons from various groups which have some expertise or direct involvement, as well as summaries from politicians and civil servants that have an opinion. There are threads of comments for public debate. Costings of the various alternatives from Treasury are there, as well as a legal analysis, if relevant. And at the bottom, you have a chance to vote: by using (say) one to three of your 10 yearly STATE VOTING COUPONS. If this issue is one that you feel strongly about, you can assign some weighted allocation of votes to the issue, and all our votes are combined to determine a public response to the topic.

Obviously this last point needs some mathematical and sociological expertise to set up, and tinkering no doubt will be needed to get it working well. During the first years of this tentative real-time democracy, perhaps the government would be legally obliged to follow the electorate if a 75% majority was in one direction or another, but only encouraged to follow us if the range was 50%-75%. The role of politicians would move subtly towards framing questions and providing balanced and detailed (!) views of the different sides to issues, persuading us by providing facts and careful reasoning, not just cliches and wishful thinkings.

But are we, the rabble, sensible and intelligent enough to hold some of the reins of power?  Are we really interested and willing? My guess to both of these questions is a tentative yes. We would want to ensure some checks and balances.

And what about all those entrenched and vested interests? That’s also a question.

Ever wonder if it might just be advantageous to think differently from those around you? Sure there are lots of cons: maybe you tend to get bored by what you consider inane talk about mindless sporting events, celebrity gossip or international news. While everybody else dreams of four bedroom mansions with fully chrome kitchen appliances and expensive German cars parked out in the driveway, perhaps your thoughts are on growing cacti, or brushing up on 19th century Russian literature, or whether RomeoVoid might ever come out with a new album?

Well, the good news is: being different is actually economically good for you! I propose to expound this (novel??) theory here in this blog: the Nobel prize jury for economics knows where to find me, and I am fully prepared to wait a few years.

Let’s explain the basic idea with a simple model. In the Land of Pi, people like to eat apples and bananas. For various reasons ostensibly connected with supply and demand, bananas are more expensive than apples: in fact one banana costs the same as four apples. Everywhere in the land of Pi, where the citizens are placidly uniform and all more or less think the same pleasant thoughts, everyone agrees that  B=4A.

Want to trade your two bananas for my eight apples? Sure, that’s only fair. But your two bananas for my nine apples? You’ve got to be kidding! I know the true value of things.

In the far distant Land of E however, where apples and bananas are also the two main fruits, a different agricultural rhythm prevails. Perhaps the place is more tropical, and bananas grow more easily, or perhaps the United Apple Consortium has better political and marketting savvy: in any case apples and bananas in this land have equal value: one apple and one banana are worth more or less the same. Young people, old people, rich or poor all realize that  B=A is the state of affairs here.

Now let’s suppose that you, a happy citizen of the Land of Pi, take a long and perilous journey to the Land of E. The Lands are far apart, so essentially no trade takes place between them–a fact that we can deduce from the marked differentials between the values of apples and bananas. You’ve scraped up some Land of E money doing some casual labour, and head off to the store to buy food.

You are going to be in for a bit of a surprise in the fruit section. Instead of bananas being four times as expensive as apples, which all reasonable people know to be the true value, here they are actually equal in price! Let’s make the assumption that for an hours worth of work, you can get roughly the same amount of food as you can in Pi. It means that bananas are much cheaper here than you are used to, while apples are more expensive. Naturally you are going to buy lots of bananas, and you are going to think—this is great, I am getting a lot more food for my hour of labour here! Of course you might get a hankering for apples now and then, but every time you buy one the exorbitant price will annoy you.

Of course this example involves two products that are more or less interchangeable as a food source—you don’t actually need to buy both of them. There are no doubt other qualifications to add before undergrads starting learning Wildberger’s Theory of Relative Values. The Nobel committee will want me to quantify the theory, but I think the basic idea doesn’t really need much mathematical underpinning. If you have a different value system from those around you, some things appear cheaper to you while others are more expensive, so if you can choose you are better off.

Here is a way of turning the situation around to see the argument in a different light: suppose someone opened a new store where all the prices were as usual in the first week, but then in the second week they were all marked up or down in a random fashion. When would you prefer to go shopping? If you didn’t actually have to buy any one particular item, I bet you’d go during the second week.

Here is another example: here in Australia at the beginning of the twenty-first century, I can go to a movie for $10, a play for $50, a symphony concert for $60, an opera for $200, or a Barbra Streisand show for $1200. My estimations of the “true value” of these things? Something like: movie $15, play $30, concert $20, opera $50, and Barbra’s show I won’t say, in case you are a fan.

You can deduce that I rarely see plays, go to concerts, attend operas or pay money to hear famous people sing. Because I am a cultural slob? No, simply because they are worth a lot less to me than the market wants. Movies, on the other hand, are worth more to me than I can get them for. Naturally I see lots of movies, and am happy.

So an interesting psychological ploy now manifests itself. Instead of working harder to make more money to spend on stuff, why don’t you just judiciously re-orient your thinking so that you value things differently than the market? The direction of re-orientation is not really important, the main principle is that the more widely your value lists differ from those of the majority, the better off you are in real terms. Non-conformity is the new economic black!

Suppose that I could hypnotise myself into thinking that movies are much more interesting and worthwhile than I think they already are—each experience worth hundreds of dollars: say on a par with a helicopter ride over the Grand Canyon, or those early morning balloon flights over the Hunter Valley Vineyards. Wouldn’t I be a lucky chap then, managing to snare such bargains for the ridiculously low price of $10 a shot! You get the idea; I’d end up with thousands of dollars of value each month for next to nothing.

On a related note, I would be interested in knowing what people think about how much things are “really worth”.  How much are different kinds of cars worth? How much are holidays to various places worth? How much are dates with different types of women worth? How much is early retirement worth? (One of these questions seems a lot more interesting than the others, don’t you think??)

And of course, how much are apples and bananas really worth?

I’ve just come back from a few weeks in Thailand, where I gave talks at Chulalongkorn University, Thammasat University and Chiang Mai University. I also attended a conference on Geometry and Graphics in Bangkok, and met my mathematician friend Paolo Bertozzini, who is always a pleasure to talk to, full of insights and anecdotes from his long experience in the Land of Smiles. By the way, Thailand is a fascinating and wonderful place, visit if you can!

Travelling gives me time to muse; I can’t always be at the computer (even though on this particular trip I did spend a lot of time working on two papers with my graduate students) and waiting for airplanes or sitting in them gives one time to speculate. A topic that I have been pondering is: to what extent are mathematicians scientists? Or are we actually something else?

Probably I am steering towards the something else. Sure, mathematics and science have a lot in common. Science uses lots of maths first of all in setting up its theories. This used to be much more true of the physical sciences, but increasingly the biological sciences are also becoming more mathematical–or at least some aspects of them are. And some applied mathematicians are pretty close to being physicists, but not really experimental ones. I believe a strong case could be made that most mathematicians do research like scientists: we observe patterns, try to formulate theories to explain them, and then subject those theories to experiments–in our case calculations–to discover that they are probably wrong and need to be modified.

But in my experience this somewhat standard view-point misses an important distinction that needs a historical perspective to appreciate. Mathematics has been around a lot longer than science. The Greeks were doing mathematics at a very high level more than two thousand years ago. Mathematicians have a long sense of history, humbled and awed by the great minds which have preceeded us, of accomplishments in centuries gone by which we can no longer hope to surpass, or even equal.

The scientist thinks, and feels, quite differently. Science really only kicked off about 500 years ago in Europe, when people slowly started thinking thoughts like: how do we really know when something is true? Can belief and truth be separated? Does our desire that the world be a certain way prevent us from seeing it as it really is?

I remind you of the answers to these kinds of questions that people came slowly to appreciate: that the source of all true knowledge is observation: careful, unbiased and thorough. From the observations we make, we formulate theories to explain them. The simplest and most powerful theories take precedence. Finally we examine the implications and predictions of our theories and see if these are born out. If so, we strengthen our faith in our theories but do not become dogmatic about them. We are prepared to be wrong, and to change our minds when confronted with new evidence and explanations.

How much deep knowledge and power resides in the understanding which I have crudely summarized in the previous paragraph! A whole brave new way of thinking, of seeing, of understanding the world. Brave, because we are prepared to face the music, however it may sound. No longer must the heavens dance to a tune of our liking. Maybe we are small, and insignificant, and weak. But we will have the courage to admit it, and to carry on none-the-less in understanding the world, unconcerned if we are no longer at the center of God’s great plan, should such a One exist. And we are not upset if our theories overturn and disprove the thoughts of a previous generation–in fact we welcome such, and strive towards the breakthrough that upturns the applecart.

I believe that modern mathematics has lost its way logically, and that a new and far more interesting mathematics awaits us. I have a fair amount of evidence to support this point of view. Rational trigonometry gives a much simpler and more powerful approach to trigonometry and geometry, making computations easier—but the kicker is that it actually makes logical sense, as opposed to classical trigonometry, the development of which is a logical basketcase! And Universal hyperbolic geometry is likewise a complete logical overhaul of hyperbolic geometry, again replacing pictures and wishful thinkings with simpler and much more careful reasonings. Both of these new developments result in many novel and beautiful theorems.

It has been interesting, and I will admit somewhat (but not overly) disappointing, to see how uninterested my fellow pure mathematicians are in contemplating really new directions of thinking, and how unsure they are in applying their own critical analysis to weigh the evidence, rather than rely on authority and precedence.

The force of habit in people’s thinking weighs heavily on them, the mark of a heavy and bloated subject. How can I inject more scientific thinking amongst my fellow pure mathematicians? How can I make the subject lighter? These are the kinds of thoughts I have been thinking in Thailand.

This is an important topic on which I will have a lot to say. Today, let’s just gently introduce a big stumbling block for modern analysis.

There are several approaches to the current theory of “real numbers”. Unfortunately, none of them makes sense. One hundred years ago, there was vigorous discussion about the difficulties, ambiguities and even paradoxes. The topic was intimately linked with Cantor’s theory of “infinite sets”.

As time went by, the debate subsided, but the difficulties didn’t really go away. A largely unquestioning uniformity has settled on the mathematics community, with most students now only dimly aware of the logical problems with “uncomputable numbers”, “non-measurable functions”, the “Axiom of choice”, hierarchies of “cardinals and ordinals”, and various anomalies and paradoxes that supposedly arise in topology, set theory and measure theory.

A hundred years ago, the notion of the “continuum” appeared intuitively straightforward, but it was difficult to pin down precisely. The Greeks had struggled with irrational numbers, but the decimal number approach of Simon Stevin in the 16th century seemed reasonable, especially considering that in practice the further digits beyond the three dots in pi=3.14159263… are hardly of practical significance. Great mathematicians like Newton, Euler, Gauss, Lagrange and others were always interested in a combination of applied questions (physics and astronomy mostly) along with more theoretical questions. In the later part of the nineteenth century an increasing preoccupation with trying to pin down the fundamentals of analysis led to both more careful definitions but also to the realization that the default view of irrational numbers as infinite decimals was shaky.

However with the advent of relativity theory and quantum mechanics, the concept of the continuum again became murky: if time is relative and perhaps finite in extent, and space has an inherent graininess which is not infinitely divisible, then what exactly are we modelling with our notion of the `infinite number line’?
While engineers and scientists viewed real numbers primarily as “decimals which go on till we don’t care anymore”, nineteenth century mathematicians introduced the ideas of  “equivalence classes of Cauchy sequences of rational numbers”, or as “Dedekind cuts”, or sometimes as “continued fractions”. Each view has different difficulties, but always there is the crucial problem of discussing `infinite objects’ without sufficient regard to how to specify them.

The twentieth century saw an entirely new sleight-of-hand; the introduction of “axiomatics” removed the time-honoured obligation of defining mathematical objects before using them. This was a particularly unfortunate and wrong-headed turn of events that has done much to diminish the respect for rigour in modern mathematics.

A finite sequence such as S=1,1,2,5,14,42,132,429,1430 may be described in many different ways, but ultimately there is only one way to specify such a sequence S completely and unambiguously: by explicitly listing all its elements.

When we make the jump to infinite sequences, such as the sequence of Catalan  numbers C=1,1,2,5,14,42,132,429,1430… (sequence A000108 in Sloane’s Online Encyclopedia of Integer Sequences (OEIS) at http://oeis.org/ ) the situation changes dramatically. It is never possible to explicitly list “all” the elements of such a “sequence”; indeed it is not clear what the word “all” even means, and in fact it is not clear even what the term “infinite sequence” precisely means. (Do you think you have a good definition?)

But assuming for a moment that we have some idea of the terms involved: still we are obliged to admit that in the absence of a complete list of the elements, we can specify the Catalan sequence C essentially only by giving a rule which generates it. A quick look at Sloane’s entry for the Catalan numbers shows some obvious problems: which of the potentially many rules that generate the Catalan numbers are we going to use? How are we going to tell when one rule actually agrees with a seemingly quite different one? Is there some kind of theory of `rules’ that we can apply to give meaning to the generators of a sequence?

If we think in terms of computation, an infinite sequence can also be modelled by a computer program, churning out number after number onto a long tape (or these days your hard drive). At any given point in time, there are only finite many outputs. As long as you keep supplying more tape, electricity, and occasionally additional memory banks, the process continues. The sequence is not to be identified by the `completed output tape’, which is a figment of our imagination, but rather by the computer program that generates it, which is concrete and completely specifiable. However here we come to the same  essential difficulty with infinite processes: the program that generates a given infinite sequence is never unique. There is no escape from this inescapable fact, and it colours all meaningful aspects of dealing with `infinity’. It seems that any proper theory of real numbers presupposes some kind of prior theory of algorithms; what they are, how to specify them, how to tell when two of them are the same.

Unfortunately there is no such theory.

With sets the dichotomy between finite and `infinite’ is much more severe than for sequences, because we do not allow a steady exhibition of the elements through time. It is impossible to exhibit all of the elements of an `infinite set’ at once, so the notion is an ideal one that more properly belongs to philosophy—it can only be approximated within mathematics. The notion of a `completed infinite set’ is contrary to classical thinking; since we can’t actually collect together more than a finite number of elements as a completed totality, why pretend that we can? It is the same reason that `God’ or `the hereafter’ are not generally recognized as proper scientific entities. Both infinite sets, God and the hereafter may very well exist in our universe, but this is a philosophical or religious inquiry, not a mathematical or scientific one.

The idea of `infinity’ as an unattainable ideal that can only be approached by an endless sequence of better and better finite approximations is both humble and ancient, and one I would strongly advocate to those wishing to understand mathematics more deeply. This is the position that Archimedes, Newton, Euler and Gauss would have taken, and it is a view that ought to be seriously reconsidered.

Why is any of this important? The real numbers are where Cantor’s hierarchies of infinities begins, and much of modern set theory rests, so this is an issue with widespread consequences, even within algebra and combinatorics. Secondly the real numbers are the arena where calculus and analysis is developed, so difficulties lead to weakness in the calculus curriculum, confusion with aspects of measure theory, functional analysis and other advanced subjects, and are obstacles in our attempts to understand physics. In my opinion, we need to understand mathematics in the right way before we will be able to unlock the deepest secrets of the universe.

By reorganizing our subject to be more careful and logical, and by removing dubious `axiomatic assumptions’ and unnecessary philosophizing about `infinite sets’, we make it easier for young people to learn, appreciate and contribute. This also strengthens the relationship between mathematics and computing. It is time to acknowledge the orthodoxy that silently frames our discipline. We need to learn from our colleagues in physics and computer science, and begin the slow, challenging but ultimately rewarding task of restructuring mathematics properly.

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.