MOOCs and TOOCs and the role of problem solving in maths education

A quick quiz: which of the following four words doesn’t fit with the others??

Massive/Open/Online/Courses

We are going to muse about MOOCs today, a hot and highly debated topic in higher education circles. Are these ambitious new approaches to delivering free high quality education through online videos and interactive participation over the web going to put traditional universities out of business, or are they just one in a long historical line of hyped technologies that get everyone excited, and then fail to deliver the goods? (Think of the radio, TV, correspondence courses, movies, the tape recorder, the computer; all of which held out some promise for getting us to learn more and learn better, mostly to little avail, although the jury is still out on the computer.)

It’s fun to speculate on future trends, because of the potential—indeed likelihood—0f embarrassment for false predictions. Here is the summary of my argument today: MOOCs in mathematics are destined to fail essentially because the word Massive is intrinsically unrelated to the other words Open, Online and Courses. But, a more refined and grammatically cohesive concept: that of a TOOC, or Targeted Open Online Course, is indeed going to have a very major impact.

When we are teaching mathematics at any level, there are really two halves to the job. The first half is the one that traditionally get’s the lion’s share of attention and work: creating a good syllabus with coherently laid-out content, which is then clearly articulated to the students. The other half, which is almost always short-changed, and sometimes even avoided altogether, is to create a good set of exercises which allow students to practice and develop further their understanding of the material, as well as their problem-solving skills. In my opinion, really effective teaching involves about equal effort towards both halves; again this is rarely done, but when it is, the result usually stands well out above the fray.

Here are some examples of mathematics textbooks in which creating the problem sets probably occupied the authors as much as did the writing of the text: first and foremost Schaum’s Outlines (on pretty well any mathematics subject), which are arguably the most successful maths textbooks of the 20th century, and deservedly so, in my opinion. Then come to mind Spivak’s Calculus, Knuth’s The Art of Computer Programming, Stanley’s Enumerative Combinatorics, and no doubt you can think of others.

Good problems teach us and challenge us at the same time. They are the first and foremost example of Gamification in action. Good problems force us to review what we have learnt, give us a chance to practice mundane skills, but also give us an opportunity to artfully apply these skills in more subtle and refined ways.  They provide examples of connections which the lecture material does not have a chance to cover, they give students a chance to fill in gaps that the lectures may have left. When combined with a good and comprehensive set of solutions, problems are the best way for students to become active in their learning of mathematics, a critically important aspect. When further combined with a skilled tutor/marker who can point out both effective thinking and errors in student’s work, make corrections, and advise on gaps in our understanding, we have a really powerful learning situation.

Here is where the Massive in MOOCs largely kills effective learning. It is the same situation as in most large first year Calculus or Linear Algebra classes around the world. Officially there may be problem sets which students are exhorted to attempt, but in the absence of required work to be handed in and marked, students will inevitably cut down to a minimum the amount of written work they attempt. In the absence of good tutors who can mark and make comments on their written work as they progress through the course, students don’t get the feedback that is so vital for effective learning.

Once you have thousands of students taking your online maths courses, it becomes very challenging to get them to do problem sets and have these marked in a reasonable way. The currently fashionable multiple choice (MC) question and answer formats that people are flocking to can go some small way down this road, but rarely far enough. Students need to be given problems which require more than picking a likely answer from a,b,c or d. They need to define, to compute, to evaluate, to organize, to find a logical structure and to explain it all clearly. This is practice doing mathematics, not going through the motions!

When we are planning an open course for possibly tens of thousands of students from all manner of backgrounds, the possibility to craft really good problems accessible to all diminishes markedly. There is no hope of giving feedback to so many students for their solutions, so all we can aspire to are MC questions that inevitably ride on the surface of things and don’t allow the crucial practice of writing. Learning inevitably takes a nose-dive, there is no other way. Such an approach cannot be the education of the future in mathematics, unless that future be washed out and limpid. Tens of thousands of students going through the motions? I don’t think so. They will find something more worthwhile to do with their time, like just watching YouTube maths videos!

But all is not lost. A slight rethinking of the enterprise, together with some common sense, can save the day. An education system has to make roughly enough money to support itself. People are willing to pay for something if it has value to them, and they tend to work harder at an activity if they have committed to it monetarily. All good technical writing has a well-defined audience in mind. These are almost self-evident truths. What we need is to think about crafting smaller, targeted open online courses, that generate enough income to support some minimal but effective amount of feedback on students’ work on real problem sets. By real I mean: problems that require thinking, computation, explanation.

Can this be done? Yes it can, and it will be the big education game changer, in my humble opinion. We will want to stream people into appropriate courses at the right level. Entry should be limited to those who have enough interest and enthusiasm to fork out some—perhaps minimal, but definitely non-zero!—amount of money, which hopefully can be dependent on the participant’s region; and who can pass some pre-requisite test. Yes, testing for entry is an excellent, indeed necessary, idea that will save a lot of people from wasting their time. Having 300 people from 10,000 pass a course is not a successful outcome. Better to have targeted the course first to those 1000 who were eager and capable. Then you get a lot more satisfaction across the board, from both students and the educators involved.

A major challenge will be how to provide effective feedback for written work. Relying exclusively on MC exercises should be considered an admission of failure here. If and when this challenge is overcome, TOOCs will have the potential to radically transform our higher education landscape!

A radical birds-eye overview of algebraic geometry

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.

Recent LT grant application

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?

 

mygov.org and the looming threat of democracy

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.

The economic value of being different

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?

Are mathematicians scientists?

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.

Difficulties with real numbers

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.