Finite versus “infinite”

There are several approaches to the modern theory of “real numbers”. Unfortunately, none of them makes complete sense. One hundred years ago, there was vigorous discussion about the ambiguities with them and 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 discipline, 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.

Some of the stumbling blocks have been described at length in my Math Foundations series of YouTube videos. In this blog we concentrate on the problems with real numbers and arithmetic with them.

The basic division in mathematics is between the discrete and the continuous. Discrete mathematics studies locally finite collections and patterns, and relies on counting, beginning with the natural numbers 1,2,3,⋯ and then extending to the integers, including 0, as well as -1,-2,-3,⋯, and to rational numbers, or fractions, of the form a/b with a and b integers, subject to the condition that b non-zero and with a particular notion of equality.

Continuous mathematics studies the “continuum” and functions on it, and relies on measurement, which these days involves also “irrational numbers” like √2,√5 and π that the ancients wrestled with, as well as more modern “irrational numbers” such as e and γ arising from integrals and infinite series. But what do these words and objects actually precisely mean and refer to? We should not presume that just because we use a common term or notion familiar from everyday life, that its mathematical meaning has been properly established.

Up to a hundred years ago, the notion of the “continuum” seemed intuitively straightforward, but difficult to pin down precisely. It appeared that we could rely on our intuition of space, following the philosopher I. Kant’s view that somehow humans had an innate understanding of three-dimensional geometry. However with the advent of modern physics, and in particular relativity and quantum mechanics, the true nature of the “continuum” grew increasingly murky: if time is relative and perhaps finite in extent, and space has an inherent graininess which renders it certainly not infinitely divisible, then what exactly are we modelling with our notion of the “infinite number line”?

While engineers and scientists work primarily with finite decimal numbers in an approximate sense, “real numbers” as infinite decimals are idealized objects which attempt to extend the explicit finite but approximate numbers of engineers into a domain where infinite processes can be ostensibly be exactly evaluated. To make this magic work, mathematicians invoke a notion of “equivalence classes of Cauchy sequences of rational numbers”, or as “Dedekind cuts”.

Each view has different difficulties, but always there is the crucial problem of discussing infinite objects without sufficient regard to how to specify them. I have discussed the serious logical difficulties at length around video 80-105 in the Math Foundations series.

For example the video Inconvenient truths with sqrt(2) has generated a lot of discussion. However not everyone approves of casting doubt on the orthodoxy: the video has more than 1000 likes, but also 316 dislikes. I doubt if I am saying anything in this video which is actually incorrect though — you can judge for yourselves.

Let’s return to the safe side of things. A finite sequence such as s = 1,5,9 may be described in quite different ways, for example as the “increasing sequence of possible last digits in an odd integer square”, or as the “sequence of numbers less than 10 which are congruent to 1 modulo 4”, or as the “sequence of digits occurring in the 246-th prime after removing repetitions”. 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 a sequence somehow implied by the expression “m = 3,5,7,⋯” the situation changes dramatically. It is never possible to explicitly list “all the elements” of an infinite sequence. Instead we are forced to rely on a rule generating the sequence to specify it. In this case perhaps: m is the list of all odd numbers starting with 3, or perhaps: m is the list of all odd primes. Without such a rule, a definition like “m = 3,5,7,⋯” is really rather meaningless.

We can say the words “infinite sequence”, but what are we actually explicitly talking about??

To a computer scientist, an “infinite sequence” is modelled by a computer program, churning out number after number perhaps onto a hard drive, or in former years onto a long tape. At any given point in time, there are only finite many outputs. As long as you keep supplying more memory, or tape, and electricity the process in principle never stops, but in practice will run short of resources and either grind forwards ever more and more slowly (the next output will take two years, hang on just a while!) or just come to a grinding halt when power or memory is inevitably exhausted.

So in this case 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 an essential difficulty with “infinite processes”: the program that generates a given “infinite sequence”: is always far from unique. There is no escape from this inescapable fact, and it colours all meaningful aspects of dealing with “infinity”.

A finite set such as {2 4 6 8} can also be described in many ways, but ultimately it too is only properly specified by showing all its elements. In this case order is not important, so that for example the elements might be scattered over a page. Finite sets whose elements cannot be explicitly shown have not been specified, though we might agree that they have been described.

An example might be: let S be the set of all odd perfect numbers less than 10^{100000}. [A perfect number, like 6 and 28, is the sum of those of its divisors less than itself, i.e. 6=1+2+3 and 28=1+2+4+7+14.] Such a description of S does not deserve to be called a specification of the set, at least not with our current understanding of perfect numbers, which doesn’t even allow us to determine if S is empty or not.

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 strongly 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 “Leprechaun heaven” or “hierarchies of angels” are not generally recognized as proper scientific entities. Infinite sets, angels and Leprechauns 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. I believe it is also closer to the view of modern giants such as H. Poincare and H. Weyl, both of whom were skeptical about our uses of “infinity”.

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 with their essential arithmetic foundation 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, it is possible, perhaps even likely, that we need to understand mathematics in the right way before we will be able to unlock the deeper secrets of the universe.

By reorganizing our subject to be more careful and logical, and by removing dubious axiomatic assumptions and unnecessary philosophizing about “real numbers” and “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 important and ultimately rewarding task of restructuring mathematics properly.

My talk with Daniel Rubin on Real Numbers and the Infinite in Analysis

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.

Welcome to Algebraic Calculus One 2021

Hello Everyone — the Algebraic Calculus One course is now live at

https://www.openlearning.com/courses/algebraic-calculus-one/homepage/?cl=1

and you can find a welcome video at https://www.youtube.com/watch?v=1SFI9QbRpzg

Here is the Welcome to the Course taken from it, and gives some information about it.

Introduction and Welcome

Hi everyone! Welcome to our course. I am Prof Norman Wildberger and I have been teaching at UNSW in Sydney for 30 years, from which I have recently retired, and before that I taught at the University of Toronto for 3 years and Stanford University for 2 years. I am a keen expositor of mathematics on YouTube with my channel Insights into Mathematics and the sister channel Wild Egg Maths, where the videos for this course are posted publically.

I am also the developer of Rational Trigonometry, Chromogeometry, and Universal Hyperbolic Geometry, and am currently putting together exciting series of videos on Solving Polynomial Equations and Exceptional Structures in Mathematics and Physics from Dynamics on Graphs, which you can access by becoming a member on the Wild Egg Maths YouTube channel.

Over my career I have also worked in Harmonic Analysis, Lie group representation theory, Diophantine equations, finite commutative hypergroups, Old Babylonian mathematics (you may have heard about my work with Daniel Mansfield on Plimpton 322 in the international press in 2017), combinatorics, and some mathematical physics, around quantization and star products.

I am hoping this course will open up new pathways for mathematics education and research, and that it will attract a new generation of students to the beauty and power of calculus and geometry!

What is Calculus?

Calculus is a branch of geometry, and also a branch of physics. These two origins of the subject colour almost every part of it.

As part of geometry, calculus is concerned with the areas and tangents of curves. Given a curve in the plane, how do we define and calculate the area determined by that curve, and perhaps other curves or lines? Given a curve in the plane, how do we define and calculate the tangents to that curve at points on the curve, or perhaps near the curve? And what do we mean by a curve in the first place?

As part of physics, calculus is concerned with the motion of bodies, in particular the relationship between the position, velocity and acceleration of bodies. It is motivated by Newton’s laws, which allow us to determine the acceleration on objects from the forces that act on them. The job of calculus is then to determine from that information both velocities and positions.

Both the geometric and physical sides of calculus can be accessed through an applied, real-life point of view emphasizing approximate calculations, or through an abstract, pure point of view, focusing more on exact calculations. We are mostly interested in the theoretical development and logical structure of the subject in this course, but we will be strongly motivated by applied questions, history, and making calculations. We want our theory to support a practical, powerful calculus.

Motivating problems

Here are some fundamental problems that we would like to solve, and that the calculus helps us with.

1. How do you explain the motion of the planets in the night sky?
2. How do you calculate the “area” of a segment of a curve, like a circle, parabola or hyperbola? Does this question even have a precise meaning??
3. How can you determine the trajectory of a particle if you know its starting point and subsequent velocities?
4. What exactly is a “curve” and how can you define and determine a “tangent line” to a given one?
5. How can you find the centre of mass of a planar lamina?
6. What exactly is a “function”, and how can you find maxima and minima of a given one?

Early pioneers of Calculus

Calculus is usually attributed to Leibniz and Newton. These pioneers played a big role in the story. But many important aspects go back to the ancient Greeks, notably Eudoxus and Archimedes, and evidence is emerging that the Babylonians understood some aspects of it.

In modern times, the development of calculus is hard to separate from the emergence of analytic geometry in the 17th century, largely due to Fermat and Descartes. The great Bernoulli family from Switzerland made huge contributions, as did the towering figures of 18th and early 19th century mathematics: Euler, Lagrange, Laplace and Gauss, but in fact there were many others too!

A concrete computational approach

We can, and will, learn much from all of them in crafting this algebraic approach to a traditionally analytic subject. We will add modern geometric developments, notably the power of linear algebra and projective geometry. And never far from sight will be our most important modern allies: calculators and computers! Our concrete computational approach avoids the many contentious aspects of calculus that have dogged it historically.

That means we avoid any mention of completed infinite processes. We are never going to say: “and now let’s do the following infinite number of operations”. We are going to stick with algebraic operations that can be followed step by step by a computer. Limits accordingly play a very limited role in the Algebraic Calculus. We work with rational numbers. Integration comes before differentiation. Discrete situations are generally studied as motivation and prior training for continuous versions.

These are major departures from modern thinking, which considers “calculus” and “infinite processes” to be almost synonymous terms. It means we are not going to consider irrational numbers as exact numbers on the same footing as rational numbers. For us there is a world of difference between 21/7 and “$$\pi$$”. In fact we will not assume that you already know what “$$\pi$$” or “$$\sqrt2$$” or “$$e$$” already are—in fact we try to avoid them altogether. We will not work with “infinite decimals” until we find a finite, concrete, explicit way of introducing and working with “them”.

We will be very careful even about using familiar words like area, function, curve, sequence and number. Our view is that these terms need to be defined rather precisely before they can be accurately used. Until we come up with precise definitions, which are always signified by bold font, you can safely assume that we are adopting a casual, everyday, informal usage of terminology. And thinking about how to make things more precise!

Organization and thanks

The Course has 10 Chapters, which are accessible via the left navigation panel, starting with The Affine Plane. Each Chapter consists of four Modules, and each Module is always divided into four sections called Videos and Notes, Worked Problems, Homework Exercises and Links, Definitions, Notation.

You should expect to spend most of your time going through the Worked Problems and the Homework Questions carefully. The latter are graded for difficulty E (easy), M (medium), H (hard) with the occasional C (challenge or research problem). The average participant should probably expect to spend a minimum of 20 hours on each Chapter, but some of you will need to spend considerably more. That is OK — the more time you put in, the more you will get out of the course.

Comments, questions, discussion and submissions are encouraged. This is your course, so please contribute to making it exciting for everyone!

A big thanks to Dr Anna Tomskova who has worked very hard in helping me put together many of the Problems and Questions,  contributed to the nice diagrams, and worked through many of the calculations.

This calculus course will be very different from any other one that you are likely to meet. I hope you enjoy it, and that you make it your own. And I look forward to a lot of interaction and questions and exciting developments. There’s much to learn!!

All the best,

N J Wildberger

Sydney July 2021

New Members Section on Wild Egg Maths and an exciting new direction for mathematics research/exploration

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! Channel Trailer (2021) for Insights into Mathematics (YouTube) 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. There are also more advanced courses on Differential Geometry and Algebraic Topology. And there’s also a quick and more elementary course which introduces Probability and Statistics. 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. Six: An elementary course in Pure Mathematics meant for a very broad audience This month I am starting an experiment: developing a mostly rigorous course in Pure Mathematics meant for a very general audience of lay people, with only a high school background of mathematics. This course is called Six — not the musical! — because it is all about the magic and mystery of the number six, as encoded by the objects 1,2,3,4,5 and 6. The number 6 plays a distinguished role in mathematics, being the third triangular number, the smallest product of two distinct primes, a factorial, a number intimately connected with each of the Platonic solids, the order of the smallest non-commutative group, the size of the unique symmetric group which has an outer automorphism, the number of points on a conic in Pascal’s theorem, and the number of points on a line involved in the multi-ratio of projective geometry. There are actually quite a few additional occurrences of the special number 6 in mathematics! But a lot of these topics are rather advanced. Our aim is to start with an unfocussed explorative approach with just basic objects and different ways of organizing them. Hopefully as we proceed we will be able to touch base with some of the topics above and others too. The prerequisites are just some basic arithmetic, and a willingness to listen carefully and work through patterns in a systematic organized way. And hopefully an interest in mathematics to start with, but we will be developing on that, and with luck will be able to shed light on why pure mathematics is such a beautiful and remarkable area of study. The videos for the course will be in playlist Six on the YouTube channel Wild Egg mathematics courses . At some further point we will look towards creating an online course at Open Learning. Why not join us? I guarantee you will learn some interesting things. Archimedes’ parabolic area formula for cubics! I try to post a new mathematics video once a week, either at my original YouTube site Insights into Mathematics, or my sister channel Wild Egg mathematics courses. This weekend’s post is particularly interesting I think, because it represents also the first “publication” of this material, albeit in an unusual format –YouTube instead of a paper in an established mathematics journal. Here is the video that presents this new result, at Wild Egg mathematics courses. The video description contains the following: ******************** The very first and arguably most important calculation in Calculus was Archimedes’ determination of the slice area of a parabola in terms of the area of a suitably inscribed triangle, involving the ratio 4/3. Remarkably, Archimedes’ formula extends to the cubic case once we identify the right class of cubic curves. These are the de Casteljau Bezier cubic curves with an additional Archimedean property, characterized either by the nature of the point at infinity on the curve, or alternatively by the geometry of the quadrilateral of control points. This is a very pleasant situation, and shows the power of the Algebraic Calculus to not only explain current theories more carefully and correctly, but also to discover novel results and open new directions. I should have mentioned in the video that this Archimedean situation covers also the special case of a cubic function of one variable, that is a curve with equation y=a+bx+cx^2+dx^3. ************************* Posting research directly to YouTube, or some other web place, is quite an important development I believe. Here I am foregoing the usual refereeing process and uploading the material to the world, or in practice anyone interested in it who can find it. Should academics be allowed to do this? On the one hand the work has not been peer reviewed, but these days peer review is often problematic, with most papers in pure mathematics almost certainly not being reviewed carefully and critically. This is not due to laziness or negligence, rather it is a necessary consequence of the increasing specialization and complexity of the subject. Most reviewers do not have the several weeks, or months, that it would typically require to delve into the details of a longish and complicated paper. It is understandable that on average they only skim the results and try to selectively check accessible proofs. On the other hand, this new process completely sidesteps the usual gatekeepers of knowledge, namely editors and referees. Journals are often oriented to certain points of view or orthodoxies, unstated yet omnipresent. Perhaps they are entering a new phase when they will have to share relevance with the wiki processes of people deciding directly which content creators they value and trust. In the meantime, I hope you enjoy the idea of a two thousand three hundred year old calculus result being extended to the next level! Plimpton 322 and Babylonian exact sexagesimal trigonometry Last week on Aug 24 Daniel Mansfield and I published the paper “Plimpton 322 is Babylonian exact sexagesimal trigonometry” in Historia Mathematica online. The paper has had a huge media response, partly due to the excellent press release created for us by Deb Smith from the Faculty of Science, UNSW Sydney, and partly by the lovely video put together by Brad Hall at UNSWTV with Daniel presenting an overview of our discovery that Plimpton 322 (P322), the world’s most famous Old Babylonian (OB) clay tablet, is actually the world’s first trigonometric table, and also the world’s most exact trigonometric table!! These are remarkable claims that are sure to raise eyebrows not just in the historical community, but also in the mathematical one. How can an unknown scribe, writing almost 4000 years ago with a cuneiform wedge on a small clay tablet, possibly have understood trigonometry not only before anyone else, but in a fashion quite different from anything since (at least, everything before my book on Rational Trigonometry published in 2005)? Could it really be that this ancient form of ratio-based trigonometry, which completely avoids all mention of angles, actually contains a more profound understanding of this fundamental subject than all those hundreds of subsequent tables? Might it be that we are on the verge of a major shift in our understanding of how to teach trigonometry to high school students by incorporating this new/ very old understanding? And could it be that the powerful sexagesimal system that the ancient Sumerians first devised and that is essential to the understanding of P322 holds powerful advantages for modern computing? And of course: do we need to seriously re-evaluate the role of OB mathematics in the history of the subject? How many other important mathematics that is currently credited to the Greeks actually is due to the much earlier cultures of the Sumerians/Akkadians/Babylonians and/or the Egyptians? These are fascinating questions that we hope will be among those discussed as the result of our work. But we do hope that people debate these and other important issues after at least having looked at our paper in some detail. Unfortunately some serious historical academics, as well as at least one science journalist, have leapt to negative conclusions without giving our paper a serious reading. Eleanor and Evelyn: here is the link to the paper again — please have a go at digesting our arguments, which we have spent two years carefully crafting, and which we are confident will change your orientation to this tablet: Plimpton 322 is far more than a teaching aid for teachers to cook up quadratic problems for their students. It is a work of undisputed genius which required a deep understanding of the trigonometry of a right triangle, and took a huge amount of effort to compile. Anyway, I anticipate quite a few more posts on this fascinating development. Let’s indenture our children! The modern economic and political machinery that we have in place is great, isn’t it? It allows us to contemplate actions and arrangements that would have been impossible a few short generations ago. In particular, we are now in the fortunate position of being able to gradually, gently, kindly indenture our children. That means that we commit them to a life of servitude and economic obediance. All completely legally, and in such a subtle fashion that they may only be dimly aware of it. This happy arrangement ensures that the generous life styles that we have voted upon ourselves can continue well into our long and satisfying retirement years; that we can look forward to an efficient health care system oriented towards our greying needs; and that eventually our children can pass on the ever increasing cumulative burden of debt that our parents started, onto our grandkids. We have now several highly effective strategies to ensure our offsprings’ indebtedness. The first idea is both simple and foolproof, and revolves around the key idea of government bonds and debt. We borrow money from rich people to fund the development of our life styles, and promise to pay those rich people back at a higher rate of return than they otherwise would get. That way we get to party now, guarantee ourselves generous pensions once we retire, and ensure a lavish health care system is in place once we start to get decrepit. And the remarkable beauty is: we don’t have to pay for it — our children will! And of course the rich people are happy too, as they get even richer from the scheme. This would be not such a clever idea if it was done on an individual basis, since it is hard to get your children to agree to taking on the personal debts that you have accumulated over a lifetime. Instead, we would find ourselves after some years paying through the nose for indulgences past. But when we do this at a societal level, we get to stretch the life of these bond debts out from our generation to the next one, and crucially we can just issue more bonds to service the debts from our old ones! So we never actually have to pay the piper, but just get to pass the increasing mess on to our kids. The other very cool strategy that is now in place worldwide is to jack up the price of real estate everywhere, so that young people have to enslave themselves to purchase a place to live. We do this by first of all crucially restricting supply: governments have careful “zoning laws” in place that ensure that empty land, even if it is in abundance, can not be accessed for housing. We also ensure that ever more and more people are squeezed into a few mega-cities, where the obvious restrictions on land availability ensure that prices will ever only go upwards. And we orient the tax structures to favour “investors” (i.e. older people) to allow them to speculate advantageously, ensuring that young people who actually want to live in houses or flats to raise families have to juggle two jobs a piece to manage it. And finally we count on pliant governments to maintain our interests, so that if anything comes along to threaten our real estate bubbles, they quickly enact first home buyers loans, or reductions on stamp duty etc to heat up flagging demand and keep prices on the move upwards. Every government knows that whoever is in power when the bubble breaks will be in the electoral wilderness for a generation afterwards, such is the power of our greying voting bloc. The tax system and superannuation laws are set up to advantage senior citizens. Younger people pay for older people’s retirement. This used to be, in agrarian times, a societal convention that was more or less understood: grandma and grandpa were given a room at the back of the house, made sure to be given enough food, and were tended when sick. Now we have managed to hardwire something rather more insidious into the system: we want our original residences with all their accumulated junk into extreme old age, we want mobile health care as well as dialysis machines, we want travel reductions for the elderly, we want a good range of senior cruises and holidays, and we want tax breaks at every opportunity. Then there’s university or college education. That used to be free, or almost free, when the baby boomers were going through the system. But now we have decided that students need to pay for a good part of their higher education, and clearly we can just keep jacking up the prices, forcing them into greater and greater debt before they have even landed their first jobs. Someone has to pay for my retirement, and why should it be me? Increasingly young people are starting to wake up to the shoddy deal that we have dealt them. But there is little use in complaining, since with a demographic as large as us baby boomers, democracy is on our side. Our children can console themselves by the realization that in the fullness of time, they too can pass on the accumulated debts to their children, and by the possibility that when we pass on, the family house will go to them—at least whatever is left of it after the reverse mortgage we took out to finance that half year in Tuscany. Australia is one of the world’s most economically advantaged countries, with 25 years of continued “economic growth”. And we have a mountain of debt, both public (government of different levels), business and private (all those expensive homes). Have a look at the Australian Debt Clock at http://www.australiandebtclock.com.au/. The estimate is a total of about 6 trillion AUS$ of debt, which works out to every man woman and child owing, on average, around

$(6 x 10^12) / (25 x 10^6) =$ 24 x 10^4 =\$ 240,000.

That includes, of course, those young children just coming into the world. But of course the pain is not spread equally, since a lot of people (often rich retirees) hold a good amount of that debt, and so benefit from the larger problem afflicting our society as a whole.

It is a sad situation. Perhaps we will see the day when kids are automatically born into slavery, to look forward to a life of working their way out of it. Perhaps that day is already here?

Upcoming talk on the Goldbach Conjecture

Some exciting news, I will next month be giving a talk which, amongst other things, will resolve the Goldbach Conjecture. That is a rather famous conjecture in Number Theory that asserts that every even number can be written as the sum of two primes.

The talk will be in the Pure Mathematics Colloquium on November 8 2016 at the University of New South Wales, Sydney (UNSW), probably at 3 pm. (Note the change in date from a previous announcement!)

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Speaker: A/Prof N J Wildberger (UNSW)

Title: Primes, Complexity and Computation: How Big Number theory resolves the Goldbach Conjecture

Abstract: The Goldbach Conjecture states that every even number greater than 2 can be written as the sum of two primes, and it is one of the most famous unsolved problems in number theory. In this lecture, we look at the problem from the novel point of view of Big Number theory – the investigation of large numbers exceeding the computational capacity of our computers, starting from Ackermann’s and Goodstein’s hyperoperations, to the presenter’s successor-limit hierarchy which parallels ordinal set theory.

This will involve a journey to a distant, seldom visited corner of number theory that impinges very directly on the Goldbach conjecture, and also on quite a few other open problems. Along the way we will meet some seriously big numbers, and pass by vast tracts of dark numbers. We will also bump into philosophical questions about the true nature of natural numbers—and the arithmetic that is possible with them.
We’ll begin with a review of prime numbers and their distribution, notably the Prime Number Theorem of Hadamard and de la Vallee Poussin. Then we look at how complexity interacts with primality and factorization, and present simple but basic results on the compression of complexity. These ideas allow us to slice through the Gordian knot and resolve the Goldbach Conjecture: using common sense, an Aristotelian view on the foundations of mathematics as espoused by James Franklin and his school, and back of the envelope calculations.

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This lecture will be live streamed on YouTube at
So anyone from around the world who is interested can watch if they like.  Hope you all will be able to join us for this fun, invigorating, and enlightening event! If you are in Sydney on the day, and can head over the UNSW for the event, we will be delighted to see you there.