A few days ago I had an online conversation with Dr Daniel Rubin who is a mathematician living in the US and who works in analysis, geometry and approximation theory. The topic was one close to my heart: Daniel wanted to hear of my objections to the status quo concerning the foundations of modern analysis: namely my rejection of “real number arithmetic” and why I don’t accept “completed infinite processes”. And naturally he wanted to do his best to rebut them.
Here is a link to our chat:
It is certainly encouraging to see that some analysts are willing to engage with the uncomfortable idea that their discipline might actually be in serious logical difficulties. Most of us are reluctant to accept that something we have been working on for years and years might actually be wrong. I applaud Daniel for the courage to engage with these important ideas, and to consider how they fit, or don’t fit, into his current view on analysis.
When we learn pure mathematics, there are many things that we at first don’t understand, perhaps because they are obscure, or perhaps because we are not smart enough — it is easy not to be sure which. Our usual reaction to that is: let me try to accept the things which are cloudy, and hopefully with further learning things will become clearer. This is a reasonable approach to tackling such a difficult subject. However it does require us to put aside our natural skepticism, and accept what the more established figures are telling us at critical points in the theoretical development, even if we imagine this is only temporary.
A good example is: “analysis is built from axiomatic set theory.” In other words the foundations of “infinite sets” and so the basic logical structure of the “arithmetic of real numbers” is a consequence of work of logicians, and can be taken for granted without much further inquiry. Or to put it less politely: it is not the job of an analyst to work out clearly the foundations of the subject; this is something that can be outsourced.
In this fashion dubious logical sleights of hand can creep into an area, transmitted from generation to generation and strengthened with each repeat. Young academics in pure mathematics are under a lot of pressure to publish to obtain a foothold in the academic ladder. This means they do not often have time to mull over those knotty foundational questions that might have been bugging them secretly at the backs of their minds. They probably don’t spend a lot of time on the history of these problems, many of which go back centuries, and in former times engaged the interest of many prominent mathematicians.
Later in their career, if our young PhD has been lucky enough to score an academic job, they might be in a position to go back over these core problems and think them through more carefully. But even then there is often not a lot of “academic reward” in doing so: their fellows are not particularly interested in endeavors that are critical of the orthodoxy — pure mathematics is quite different in this regard than science or even applied mathematics!
And journals are uniformly not keen on publishing papers on foundational issues, especially ones which challenge accepted beliefs. As pure mathematics rests on a premise of logical correctness, any questioning of that is seen as subversive to the entire discipline.
But maybe some serious consideration and debate of the underlying logical structure is just what the discipline really needs.
I certainly enjoyed our conversation and I think there are valuable points in it. I hope you enjoy it, and look forward to another public YouTube discussion with Daniel.
I am recently retired from 30 years at the University of New South Wales (UNSW) Sydney. But I don’t plan on giving up on mathematics explanation and discovery any time soon — it is just too much fun, and exciting.
But to cement this new direction, I have decided to embark on an additional, quite different directions of explanation — to chart a course in mathematics exploration for the general viewer, offering you a road map to get into a wide range of interesting topics in pure mathematics that you can investigate also on your own — after some orientation on my part.
The first topic is particularly exciting — it is a series on Solving Polynomial Equations. You will all know that the standard extension of the quadratic formula to cubic equations involves complicated expressions with cube and square roots, that the quartic equation is even more complicated, and that this method breaks down, at least partially in the quintic and higher cases. Galois theory was designed partly to try to understand the obstructions to writing down formulas for zeros of higher degree polynomials in terms of radicals.
But since I don’t believe in irrational quantities except in an applied, approximate sense, these “solutions by radicals” are intrinsically suspect for me. Now I am going to show you an exciting alternative, which actually meshes closer to what physicists and engineers do to solve equations — using power series and rational extensions of them in the coefficients of the given equations.
With this rather dramatic shift in point of view, I claim that an entirely new landscape emerges, which remarkably connects with a rich hierarchy of combinatorial objects related to Catalan numbers and their generalizations. We will meet binary and ternary trees, polygonal subdivisions, Dyck paths, standard tableaux, and make lots of contact with many interesting entries in the Online Encyclopedia of Integer Sequences.
You might be surprised. Could it be that we will be able to solve the general polynomial equation with this major new point of view!?
To access this exciting series, please JOIN our Members section on my YouTube channel Wild Egg Maths. See for example this informational video:
For a minimal amount (around $5 / month) you will have a rich stream of interesting videos to watch. We are going to be delving into lots of other topics too — from graph theory to projective geometry to a new world of convexity to triangle geometry in hyperbolic geometry. There will also be quite a few advising videos on how to do research as an amateur or as a graduate student.
The videos will be informal, hands- on and will encourage you to participate. I look forward to having you join us!
Hi everyone, I’m Norman Wildberger, a soon-to-be retired professor of mathematics at UNSW in Sydney Australia, and I want to tell you about this channel which will introduce you to a wide variety of mathematical topics with a novel slant. The content is aimed at a very broad audience from everyday people with an interest in maths to graduate students working on a PhD in the subject. A link to this introductory video is given below, so you will be able to find quickly any of the playlists that I describe.
I believe that mathematics should be completely clear and straightforward, and that ideally a beginner should be able to navigate through one of the many branches of the subject, one step at a time, supported by lots of explicit examples and concrete computations, with the logical structure visible at all times.
That means however that I no longer buy the standard religion of “real numbers”, which are anchored in an arithmetic reliant on infinite processes. It’s not possible to add up an infinite number of things, so why do we pretend that we can?
I also don’t believe in the “hierarchies of infinite sets” that supposedly form the foundation for modern mathematics, following Cantor. It’s not possible to exhibit a “set” with an infinite number of elements, so why do we pretend that we can?
The pure mathematical community depends on these and other fancies to support a range of “theories” that appear pleasant but are not actually corresponding to reality, and “theorems” which are not logically correct. Measure theory is a good example –this is a subject in which the majority of “results” are without computational substantiation. And the Fundamental theorem of Algebra is a good example of a result which is in direct contradiction to direct experience: how do you factor x^7+x-2 into linear and quadratic factors? Answer: you can’t do this exactly — only approximately.
By removing ourselves from the seductive but false dreamings of modern pure mathematics, we open our eyes to a more computational, logical and attractive mathematics –where everything is above board, where computations actually finish in finite time, where examples can be laid out completely, and where we acknowledge the proper distinction between the exact and the only approximate. This is a pure mathematics which is closer to applied mathematics, and more likely to be able to support it. It also gives us many new insights, more precise definitions, and theorems which are actually …correct.
In this channel, we explore the beginnings of such an exciting new way of learning and doing and teaching mathematics. I present you with topics that are developed and explored in a sequence of YouTube videos, usually from rather elementary beginnings. These topics are organized in Playlists, so you can work your way through them sequentially and strengthen your understanding slowly and steadily.
The History of Maths series is great for high school teachers and anyone with a general interest in mathematics — so much of the subject makes more sense when viewed in a historical context. There is also a playlist on Ancient Mathematics and another on Old Babylonian mathematics. The latter topic is close to my heart — a paper in Historia Mathematica a few years ago with Daniel Mansfield on Plimpton 322 generated international coverage in hundreds of newspapers, including the New York Times.
Wild Trig is an introduction to Rational Trigonometry — a more general and algebraic view of trig that allows much more extensive and quicker calculation for many problems and that opens the door to many new theoretical possibilities, such as chromogeometry! This is based on my book: Divine Proportions: Rational Trigonometry to Universal Geometry.
Famous Math Problems discusses a wide range of —famous math problems, some of them with novel solutions!
Wild Lin Alg A and the follow up Wild Lin Alg B is a first year undergraduate course in Linear Algebra, from largely a geometric point of view.
The most extensive series is the MathFoundations series, which comes in parts MathFoundationsA (videos 1-79), MathFoundationsB (videos 80-149) and MathFoundationsC (videos 150-present). This series examines so many important topics in the subject. The most recent videos for example give a new treatment of the Algebra of Boole, transcending the more usual Boolean Algebra (which is not really what Boole intended) and open the door for simpler logic gate analysis by engineers.
The most elementary series is: Elementary Math (K-6) Explained which is for parents and teachers of primary school students, and will give you tools to understand the important mathematical skills and concepts their children need to learn. In this direction, there is also a course on Math Terminology for Incoming Uni Students meant for people from a non- English speaking background.
Universal Hyperbolic Geometry is a more advanced series on geometry which will give you an exciting new completely algebraic way to understand the hyperbolic geometry of Gauss, Lobachevsky and Bolyai, and which connects more naturally with relativistic physics. There are hundreds of new theorems here, many very beautiful. I will be developing this a lot more in the coming years.
So this is a large amount of content that is consistently oriented towards avoiding infinite processes and arguments which are not supportable by explicit computation. It is a new kind of mathematics. If you work through some of this, your mathematical understanding will deepen, you will see connections that were invisible, and your appreciation for the logical beauty of the subject will continue to grow. Mathematics is surely the richest intellectual discipline, and I want to empower more people, young and old to experience it directly, to learn lots of fascinating things, to be challenged, and to explore on your own. For those of you aspiring to do some research on your own, there will be plenty of new directions to think about!
My understanding is very different from my fellow mathematicians. So why do I have such a unique perspective? One reason is that I have simply worked in lots of areas of mathematics.
I have done work in number theory, developing the most powerful general algorithm for solving large Diophantine equations, and unravelling the algebraic structure of Gaussian periods. I’ve done work on Pell’s equation –basically discovered the simplest explanation of why solutions are always possible.
I have worked in Lie group harmonic analysis, solving the Horn conjecture (with A. H. Dooley and J. Repka) on eigenvalues of sums of Hermitian matrices. I’ve initiated the moment map of a Lie group representation and found a geometric Fourier transform which explains *-products on coadjoint orbits of compact Lie groups. The wrapping map introduced with A. H. Dooley gives a broad explanation for the effectiveness of A. A. Kirillov’s orbit theory.
In work with D. Arnal I’ve introduced quasi-standard Young tableux, building from my geometric “diamond” construction of the irreps of SU(3), which is of considerable interest to physicists. I have also given combinatorial constructions of G2 and the simply laced Lie algebras, excluding E8.
In 2005 I wrote a book which introduces Rational Trigonometry, and then extended that to a complete rewrite of hyperbolic geometry. This gives a large scale revision of Euclidean and non-Euclidean metrical geometries. With this I have further discovered a remarkable three-fold symmetry in planar geometry called chromogeometry.
I have developed the theory of finite signed hypergroups, which are probabilistic versions of finite groups, and developed a duality theory for them, somewhat like Poyntriagin duality for abelian groups, and also applied ideas of entropy to them.
For the last five years I have been developing the Algebraic Calculus, which is a coherent approach to Calculus which avoids real numbers and infinite processes, and is correspondingly more general and often gives new insights. Videos for this can be found at the sister channel Wild Egg mathematics courses, while the course itself is on openlearning.
If you are interested in learning more about my research at the more advanced level, there is a Playlist on this channel of Math Seminars, and also a smaller one on Research Snapshots, which I hope to enlarge in the future.
I have a Vice Chancellor’s award at UNSW for teaching excellence and have been very involved in the development of online tutorials for mathematics courses there.
In summary, my aim is to put this wealth of research and teaching experience to work in framing a more fruitful path for mathematics education, and opening up a more solid approach to pure mathematics research, connected more strongly to computational reality. Come along and join me on an exciting journey to explore new and better foundations and directions for 21st century pure mathematics! Once we face the music and see things as they really are, not just how we want them to be, there is much to do.
In the last fifteen years or so, I have become increasingly disenchanted with the way modern mathematics deals with, or rather doesn’t deal with, the serious logical problems which beset the subject. These difficulties arise from a misunderstanding of the nature of `infinite sets’ and `the continuum’, and then extend further in many directions.
`Infinite sets’ are propped up, according to the standard dogma, by certain axiomatics, which lift the burden of having to actually define properly what we are talking about, and prove the various theorems that we would like to have true. What a joke these ZFC axiomatics are. The entire situation is ironic to the extreme: in fact Cantor’s Set Theory was vigorously opposed by most prominent mathematicians during his day, and then collapsed in a catastrophic heap at the beginning of the 20th century due to the discovery of irrefutable paradoxes. And now, fast forward a hundred years later: not only has Set Theory been resurrected, essentially with no new ideas—most of the key concepts go back to Cantor or Turing, and are just endlessly recycled—but now most of us believe that this befuddled and imprecisely laid out subject is actually the correct foundation for the rest of mathematics! This is little short of incredible. I feel I have woken from a dream, while most of my colleagues are still blissfully dozing.
And our notion of the continuum is currently modelled by the so-called ‘real numbers’, which in fact are far removed from most sensible people’s notions of reality. These phoney real numbers that most of my colleagues pretend to deal with on a daily basis are in fact hazy and undefined creations that frolic and shimmer in a fantasy underworld deep beneath the computational precisions of our computers, ready to alleviate us from the dull chore of striving for precise computations, and incorporating correct error bounds when we can obtain only approximations.
We are talking about irrational numbers here; numbers whose names even lay people are familiar with, such as sqrt(2), and pi, and Euler’s number e.
Supposedly there are myriads of other ones, given by various arcane procedures, formulas and properties. The actual theory and arithmetic of such real numbers is never laid out completely correctly; rather we find brief ‘summaries’ of the wished-for properties that these creatures have, properties that ensure that theoretically many standard computational problems have solutions, even if our computers can in fact not find them.
Ask a modern pure mathematician to make the computation pi+e for you, and see what kind of bemused look you get. Is not the answer the same as the question? Is this not how we all do `real number arithmetic’??
The belief in `real numbers’ supports a false mathematical dream-world where almost everything has a solution; a Polyanna fantasy land which can be conjured up by words but not written down on paper. (Of course the computer scientist or applied mathematician or scientist knows that in reality all meaningful computations occur with rational numbers or floating point decimals).
What a boon it is to live in the `infinitely real’ dreamscape of the modern pure mathematician! To conjure up `constructions’ and ` computations’ these days we need only scribble words, phrases and descriptions together. This is why so many of the ‘best’ journals are filled with page after page of what might be generously called `mathematical prose’. See my submission `Let H be a load of hogwash’ to get a feeling for this language of modern mathematics that the journals encourage.
Most pure mathematicians feel little obligation to address the claims of logical weakness. Objections such as mine may be safely ignored. Unlike scientists, we don’t feel the obligation to step up to the plate and respond rationally to criticism, as it clearly cannot be correct: since the majority rules! As long as we all play along, and ignore the increasingly obvious gaps between what our computers can do and what we are claiming, everyone can pretend that things are merry.
But could the tide be turning? A little while ago, James Franklin and I had a public debate (quite civilized and friendly I would add) in the Pure Maths Seminar in the School of Mathematics and Statistics UNSW, and lo and behold– the room was filled to capacity, people were huddled at the doors from outside trying to hear what was said, and my heresies were not met with a barrage of hoots, tomatoes and derision.
Judging from the many comments, it is no longer such a one-sided debate as it was a few decades ago. I reckon that young people’s comfort and trust in computers has a lot to do with it. What is it really, if you can’t get your computer to model it?? Only a fantasy.
You can join the revolution, too. Don’t be so accepting of everything you are told. Ask for explicit examples and concrete computations. Be suspicious of appeals to authority, or the well worn method of swamping with jargon. And of course, watch as many of my videos as you can, for a slow but steady introduction to: a more sensible world of pure mathematics.
Perhaps the forces of confusion and orthodoxy will soon be on the back foot.
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 effectively support the crucial practice of writing. Learning slips into a lower gear. Such an approach cannot be the future of mathematics education. Tens of thousands of students going through the motions? They will find something more worthwhile to do with their time, like just watching YouTube maths videos!
But a slight rethinking of the enterprise, together with some common sense, can perhaps orient us in a more profitable direction. An education system ought to make enough money to at least fractionally 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!
njwildberger: tangential thoughts 