# Infinity: religion for pure mathematicians

Here is a quote from the online Encyclopedia Britannica:

The Bohemian mathematician Bernard Bolzano (1781–1848) formulated an argument for the infinitude of the class of all possible thoughts. If T is a thought, let T* stand for the notion “T is a thought.” T and T* are in turn distinct thoughts, so that, starting with any single thought T, one can obtain an endless sequence of possible thoughts: T, T*, T**, T***, and so on. Some view this as evidence that the Absolute is infinite.

Bolzano was one of the founders of modern analysis, and with Cantor and Dedekind, initiated the at-the-time controversial idea that the `infinite’ was not just a way of indirectly speaking about processes that are unbounded, or without end, but actually a concrete object or objects that mathematics could manipulate and build on, in parallel with finite, more traditional objects.

A multitude which is larger than any finite multitude, i.e., a multitude with the property that every finite set [of members of the kind in question] is only a part of it, I will call an infinite multitude. (B. Bolzano)

Accordingly I distinguish an eternal uncreated infinity or absolutum which is due to God and his attributes, and a created infinity or transfinitum, which has to be used wherever in the created nature an actual infinity has to be noticed, for example, with respect to, according to my firm conviction, the actually infinite number of created individuals, in the universe as well as on our earth and, most probably, even in every arbitrarily small extended piece of space. (G. Cantor)

One proof is based on the notion of God. First, from the highest perfection of God, we infer the possibility of the creation of the transfinite, then, from his all-grace and splendor, we infer the necessity that the creation of the transfinite in fact has happened. (G. Cantor)

The numbers are a free creation of human mind. (R. Dedekind )

I hope some of these quotes strike you as little more than religious doggerel. Is this what you, a critical thinking person, really want to buy into??

From the initial set-up by Bolzano, Cantor and Dedekind, the twentieth century has gone on to enshrine the existence of `infinity’ as a fundamental aspect of the mathematical world. Mathematical objects, even simple ones such as lines and circles , are defined in terms of “infinite sets of points”. Fundamental concepts of calculus, such as continuity, the derivative and the integral, rest on the idea of “completing infinite processes” and/or “performing an infinite number of tasks”. Almost all higher and more sophisticated notions from algebraic geometry, differential geometry, algebraic topology, and of course analysis rest on a bedrock foundation of infinite this and infinite that.

This is all religion my friends. It is what we get when we abandon the true path of clarity and precise thinking in order to invoke into existence that which we would like to be true. We want our integrals, infinite sums, infinite products, evaluations of transcendental functions to converge to “real numbers”, and if belief in infinity is what it takes, then that’s what we have collectively agreed to, back somewhere in the 20th century.

What would mathematics be like if we accepted it as it really is? Without wishful thinking, imprecise definitions and reliance on belief systems?

What would pure mathematics be like if it actually lined up with what our computers can do, rather than with what we can talk about?

Let’s take a deep breath, shake away the cobwebs of collective thought, and engage with mathematics as it really is. Down with infinity!

Or somewhat less spectacularly: Up with proper definitions!

# The truth about polynomial factorization

In yesterday’s blog, called The Fundamental Dream of Mathematics, I started to explain why modern mathematics is occupying a Polyanna land of wishful dreaming, with its over-reliance on the FTA — the cherished, but incorrect, idea that any non-constant polynomial p(x) has a complex zero, that is there is a complex number z satisfying p(z)=0. As a consequence, we all happily believe that every degree n polynomial has a factorization, over the complex numbers.

What a sad piece of delusional nonsense this is: a twelve year old ought to be able to see that we are trying to pull the wool over our collective eyes here. All what is required is to open up our computers and see what really happens!

Let’s start by looking at a polynomial which actually is a product of linear factors. This is easy to cook up, just by expanding out a product of chosen linear factors:

p(x)=(x-3)(x+1)(x+5)(x+11)= x⁴+14x³+20x²-158x-165.

No-one can deny that this polynomial does have exactly four zeroes, and they are x=3,-1,-5, and -11. Corresponding to each of these zeroes, there is indeed, just as Descartes taught us, a linear factor. If I don’t tell my computer where the polynomial is coming from, and just ask it to factor p(x)=x⁴+14x³+20x²-158x-165, then it will immediately inform me that

x⁴+14x³+20x²-158x-165= (x+11)(x-3)(x+5)(x+1).

Now let’s modify things just a tad. Let’s change that last coefficient of -165 to -166. So now if I ask my computer to factor q(x)=x⁴+14x³+20x²-158x-166, then it will very quickly tell me that

x⁴+14x³+20x²-158x-166= -158x+20x²+14x³+x⁴-166.

This is the kind of thing that it does when it cannot factor something. Did I tell you that my computer is very, very good at factoring polynomial expressions? I have supreme confidence in its abilities here.

But wait a minute you say, clearly your computer is deluded Norman! We know this polynomial factors, because all polynomials do. That is what the Fundamental Theorem of Algebra asserts, and it must be right, because everyone says so. Why don’t you find the zeroes first?

Okay, let’s see what happens if we do that: if I press solve numeric after the equation

x⁴+14x³+20x²-158x-166=0

the computer tells me that: Solution is: {[x=-1. 006254],[x=-4. 994786],[x=-11. 00119],[x=3. 002230]}

But these are not true zeroes, as we saw in yesterday’s blog, they are only approximate zeroes. True zeroes for this polynomial in fact do not exist.

True zeroes for this polynomial in fact do not exist.

True zeroes for this polynomial in fact do not exist.

Probably I will have to repeat this kind of mantra another few hundred times before it registers in the collective consciousness of my fellow pure mathematicians!

Let us check if we do get factorization: we ask the computer to expand

(x+1.006254)(x+4.994786)(x+11.00119)(x-3.002230)

and it does so to get

x⁴+14.0x³+20. 00000x²- 158.0x-166.0.

Hurray! We have factored our polynomial successfully! [NOT]

Here is the snag: the coefficients are given as decimal numbers, not integers. That means there is the possibility of round-off. Let me up the default number of digits shown in calculations from 7 to 20 and redo that expansion. This time, I get

x⁴+14.0x³+19. 999999656 344x²- 158. 000005080 13515776x-166. 000016519 11547081.

Sadly we see that the factorization was a mirage. Ladies and Gentlemen: a polynomial that does not factor into linear factors: q(x)=x⁴+14x³+20x²-158x-166.

Here is the true story of rational polynomial factorization: polynomials which factor into linear factors are easy to generate, but if you write down a random polynomial with rational coefficients of higher degree, the chances of it being of this kind are minimal. There is a hierarchy of factorizability of polynomials of degree n, whose levels correspond to partitions of n. For example if n=4, then there are five partitions of 4, namely 4, 3+1, 2+2, 2+1 and 1+1+1+1. Each of these corresponds to a type of factorizability for a degree four polynomial.

Here are example polynomials that fit into each of these kinds:

x⁴+14x³+20x²-158x-166=(x⁴+14x³+20x²-158x-166)

x⁴+14x³+21x²-147x-165= (x³+3x²-12x-15)(x+11)

x⁴+14x³+18x²-172x-216= (x²-2x-4)(x²+16x+54)

x⁴+14x³+19x²-156x-162= (16x+x²+54)(x-3)(x+1)

x⁴+14x³+20x²-158x-165= (x+11)(x-3)(x+5)(x+1).

So the world of polynomial factorizability is much richer than we pretend. But that also means that the theory of diagonalization of matrices is much richer than we pretend. The theory of eigenvalues and eigenvectors of matrices is much richer than we pretend. And many other things besides.

In distant times, astronomers believed that all celestial bodies moved around on a fixed celestial sphere centered on the earth. What a naive, convenient picture this was. In a thousand years from now, that is the way people will be thinking about our view of algebra: a simple-minded story, useful in its way, that ultimately just didn’t correspond to the way things really are.

# The Fundamental Dream of Algebra

According to modern pure mathematics, there is a basic fact about polynomials called “The Fundamental Theorem of Algebra (FTA)”. It asserts, in perhaps its simplest form, that if p(x) is a non-constant polynomial, then there is a complex number z which has the property that p(z)=0 . So every non-constant polynomial equation p(x)=0 has at least one solution. [NOT!]

When we combine this with Descartes’ Factor Theorem, we can supposedly deduce that a degree n polynomial can be factored into exactly n linear factors. [NOT!]

What a useful and crucially important theorem this is! It finds immediate application to integration, allowing us to integrate rational functions by factoring the denominators and using partial fractions to reduce all such to a few canonical forms. In linear algebra, the characteristic polynomial of a matrix has zeroes which are the eigenvalues. In differential equations or difference equations, these eigenvalues allow us to write down solutions. [NOT!]

Unfortunately, the theorem is a mirage. It is not really true. The current belief and dependence on the “Fundamental Theorem of Algebra” is a monumental act of collective wishful dreaming. In fact the result is not even correctly formulated to begin with. It is a false and idealized shadow of a much more complex, subtle and (ultimately) beautiful result.

Wait a minute Norman! Do we not have a completely explicit and clear formulation of the FTA? Are there not watertight proofs? Surely we have lots of concrete and explicit verifications of the theorem!?

Sadly the answers/responses are no in each case. The theorem is not correctly stated, because it requires a prior theory of complex numbers, which in turn require a prior theory of real numbers, and there is no such theory currently in existence, as you can get a pretty clear idea from by opening up a random collection of Calculus or Analysis texts. Or you could go through the fine-toothed comb dismantling of the subject in my MathFoundations YouTube series. And there are, contrary to popular belief, no watertight proofs. All current arguments rest on continuity assumptions that are not supported by proper theories, but ultimately only by appeals to physical intuition. And when we look at explicit examples, it quickly becomes obvious that the FTA is not at all true!

This issue confounded both Euler and Gauss–both struggled to give proofs, and both were defeated. Gauss returned over and over to the problem, each time trying to be more convincing, to be tighter, to overcome the subtle assumptions that can so easily creep into these arguments. He was not in fact successful.

Every few years some one comes up with a new “proof”, often unaware that the real difficulty is in framing the statement precisely in the first place! Unless you have a rock solid foundation for “real numbers”, all attempts to establish this result in its current form are ultimately doomed.

Most undergraduates learn from an early age to accept this theorem. Suspiciously, they are rarely presented with a proper proof. Sometime in a complex analysis course, if they get that far, they are exposed to a rather indirect argument involving Cauchy’s theory of integration. Are they convinced by this? I hope not: why should something so elementary have such a complicated proof? How do we know that some crucial circularity is not built into the whole game, if we only get around to “proving” a result in third year university that we have been happily assuming in all prior courses for years earlier?

And what about explicit examples? Is this not the way to sort out the wheat from the chaff?  Yes it is, and all we need to do is open our eyes clearly and look beyond our wishful dreaming to see things as they really are, not the way we would like them to be in our alternative Polyanna land of modern pure mathematics!

We write down a polynomial equation of some low degree. Let’s say to be explicit x^5-2x^2+x-7=0. Now I open my computer’s mathematical software package (in my case that is Scientific Workplace, which uses MuPad as its computational engine).

This program is no slouch– it has innumerably many times performed what are essentially miracles of computation for me. It can factor polynomials in several variables of high degree with hundreds, indeed thousands of terms. I have just asked it to factor the randomly typed number 523452545354624677876876875744666. It took about 3 seconds for it to determine that the prime factorization is

2 x 3 x 191 x 2033466852397 x 224623712 574400693 .

So let’s see what happens if I ask it to solve

p(x)=x^5-2x^2+x-7=0.

Actually that depends what kind of solving I ask for. If I ask for a numeric solution, it gives

{[x=-1. 166+1. 097i],[x=-1. 166-1. 097i],[x=0.3654-1. 254i],[x=0.3654+1. 254i],[x=1. 601]}.

Indeed we get 5 “solutions”, four consisting of two complex conjugate pairs, and one “real solution”. The reason we only get 4 digit accuracy is because of the current settings. Suppose I up the significant digits to 7. In that case, solve numeric returns

{[x=-1. 166064+1. 09674i],[x=-1. 166064-1. 09674i],[x=0.3654393-1. 253958i],[x=0.3654393+1. 253958i],[x=1. 601249]}.

Are these really solutions? No they are NOT. They are rational (in fact finite decimal) numbers which are approximate solutions, but they are not solutions. Let us be absolutely clear about this. Here for example is

p(-1. 166064+1. 09674i)=4. 164963586 717619097 5×10⁻⁶+9. 468394675 149678997 9×10⁻⁶ i

where I have upped the significant digits to 20, the maximum that is allowed. Do we get zero? Clearly we do NOT.

How about if I ask for solve exact solutions? In that case, my computer asserts that

Solution is: ρ₁ where ρ₁ is a root of Z-2Z²+Z⁵-7.

In other words, the computer is not able to find true exact solutions to this equation. The computer knows something that most modern pure mathematicians seem to be unaware of. And what is that? Come closer, and I will whisper the harsh truth to you..

This .. equation .. does .. not .. have .. exact .. solutions.

# Maths for Humans: Linear, Quadratic and Inverse Relations

The final week of my Future Learn MOOC called: Maths for Humans: Linear, Quadratic and Inverse Relations is about to go live tomorrow. This feels like all I have been doing for the last three or four months, and I am very glad that it is now coming to a close.

A MOOC is a Massive Open Online Course. This is free online education, run potentially on a large scale. Future Learn is a relatively new MOOC platform run by the Open University in the UK, and they have vast experience with distance education over many decades. Maths for Humans is a course that Daniel Mansfield and I have put together at UNSW, supported by UNSW Learning and Teaching Funds from UNSW L&T.

Currently we have about 8500 people registered, but as is typical in such MOOCs only some fraction of that are actively learning — perhaps around a third or so. Which is still a decent number.

And why is a pure mathematician who is re-configuring modern geometry, and also trying to steer the Ship of Mathematics to safer, more placid waters, spending his energies this way? Well one reason is that I have got some grant funding for this, and so some teaching relief. So actually I have not been teaching this semester, because I also have another big project going on to revamp some of our first year tutorials, which I must tell you about some other time.

I happen to think mathematics is far and away the most interesting subject. I reckon a lot of people would be both pleased and enriched to have an opportunity to learn more mathematics in a systematic, structured, and thought-out way. Courses like the one Daniel and I have put together are exactly in this direction. And also they should help high school students and teachers, which is always a good thing from my point of view.

But you might be surprised that, in addition, I actually learn a lot of important things by trying to figure out how best to present material to people who are not necessarily very advanced in mathematics. That motivates a lot of my YouTube channel too. It turns out that having to explain something to someone who perhaps really has no idea about the subject forces me to think hard about what the essence of the matter is, what the key examples are, what to say and what not to say. And invariably I learn something.

What did I learn putting this MOOC together? A lot. I learnt about power laws in biology, about allometry, the study of scaling in biology, about Zipf’s law, thought some more about Benford’s law (which I have mused on from time to time), and reviewed some basic supply and demand kind of elementary economics that I had more or less forgotten. I had a chance to review lots of things that officially I know, but that it is good to solidify. And I also learnt that the bowhead whale is the longest living mammal, with a record lifespan of 211 years.

And I believe that I also learnt the right way to think about the quadratic formula. Let me share with you what I would like to call al Khwarizmi’s identity:

ax^2+bx+c=a(x+b/(2a))^2+(4ac-b^2)/4a

This is the heart of the matter as far as I am now concerned. The usual quadratic formula is just a sloppy consequence that results if one is cavalier about taking “square roots”, which I hope none of you are any more. Geometrically this formula allows us to identify the vertex of the parabola ax^2+bx+c. It is this identity, I’ll bet, that students will learn when they study quadratic equations, one thousand years from now.

The course is lasting only another week, but if you register before the end, then the course contents will be available to you after it closes. So check it out! This link ought to work:

Some thanks: Laura Griffin has a been a big help as our project manager, and Iman Irannejad has done a cracker job with the videos. Ruslan Ibragimov has been splendid with technical assistance, and Joshua Capel and Galina Levitina have both been a big help running the course. Thanks to them, and to the folks at UNSW L&T who have supported the project from afar.

# The ArXiV versus ResearchGate

Lately it seems that ResearchGate is getting more and more popular. This is a website where academics can post their research—both published and unpublished work (subject to copyright restrictions)—keep up with their area of interest, and access other researcher’s work easily. It also provides a bit of a biographical aspect: it feels like a mini Facebook for research academics.

The general public will find it difficult to post things on ResearchGate. This ensures a certain level of trustworthiness and confidence in the materials that are posted. However even if you are not a research academic, you can find and download materials easily.

In mathematics, the ArXiV has for some time now been the default repository for posting preprints. This is a very valuable site that has allowed a lot more visibility to papers which has not been published. It is a way for researchers to stake a claim on work they have done before it is officially published—this can be valuable considering that the waiting time to get published in journals is often more than a year and realistically more like 2 years and up.

The ArXiV has a policy of accepting LaTeX submissions only. This means that a fair amount of expertise with LaTeX is required, and for articles that have a lot of diagrams, the process of uploading them can, or at least used to be, rather onerous. About 3 or 4 years back I was struggling trying to upload some dozens of images that accompanied a paper on Hyperbolic Geometry that I was trying to post to the ArXiV. After several long and painful attempts at the uploading proved unsuccessful, I contacted the ArXiV admin and asked if I could just upload the pdf file. They rather shortly refused, and I found that I consequently lost interest in the ArXiV as a place for my papers. Indeed I always struggled understanding how the ArXiV could justify demanding mathematicians hand over their LaTeX documents, which can easily be copied, cut and pasted rather than the pdfs, which are more high level and secure (at least to an ordinary user such as myself). I suppose I read their explanations, but wasn’t convinced.

So I am rather grateful that ResearchGate allows me to easily upload the pdfs of my papers,  catalogues them conveniently for me, and allows the general public to find these papers, read them online, and download them. This means I no longer have to worry about organizing a list of publications on my own UNSW website–believe me, this is a great convenience.

Goodbye journals! Hello ResearchGate!

Actually I am kidding a bit here: I do still plan on submitting papers regularly to KoG, the Croatian Society for Geometry and Graphics Journal which has to be the physically most attractive journal in pure mathematics, and a few others. But it really makes one think, doesn’t it? I am now in the position of writing papers, publishing them myself on ResearchGate, and letting the world decide whether the papers are worth reading. Brave new times.

Here is the link to my ResearchGate list of publications:

https://www.researchgate.net/profile/Norman_Wildberger/publications

and here is the link to my ArXiV publications:

http://arxiv.org/find/math/1/au:+Wildberger_N/0/1/0/all/0/1

Mathematics journals publish papers by mathematicians on their research discoveries, or sometimes expository works that overview an area. They have traditionally been the main repositories of innovation and development in mathematics—the advance guard of mathematical knowledge—with books being the support troops that bring up the rear and consolidate the progress made.

Mathematics journals used to be published by Academies or Societies, often with a strong regional affiliation. Crelle’s journal was an important journal initiated by a single individual. These days most journals are associated with a Mathematical Society and run as a service to the community, or are commercial enterprises run for a profit.

Modern technology, in particular the internet, is putting the entire enterprise under investigation. Will the mathematics journal in its present form survive? Recently the ArXiV and other online repositories mean that people can upload research papers before they are officially published, and indeed can circumvent the entire publishing process if they so choose. What happens when everything is online and no-one really reads the bound journals? What will be the role of other means of relaying information, in particular video?

Can commercial publishers maintain the (seemingly) exorbitant prices that they have historically been able to charge the main institutional (library) buyers of their wares? How are the roles of editor, referee, typesetter, distributor going to change?

Before the onslaught of the internet, we had already seen a major shift in the world of journals with Donald Knuth’s celebrated creation of TeX, and then LaTeX, which allows authors to typeset their mathematical papers themselves. I use Scientific Workplace to access LaTeX, and it is easily the most useful toolbox that I use as a professional mathematician (I will have to sing its praises in a separate blog sometime). So us mathematicians no longer need typists or typesetters, and in fact we don’t really need the printing and distribution capabilities of traditional journals, since papers only really get published in one place these days, namely the internet, and if someone wants a printed copy, they print it out themselves.

What about editors and referees? Don’t they still perform a valuable function? Yes certainly they do, but their roles are also looking increasingly unclear. Is the refereeing system really working? There are ranges of opinion. As gatekeepers to prestige and success, editors and to a lesser extent referees have been able to shape and direct the course of mathematics. Now increasingly there is a sense that mathematics will go where it will, without supervision, and with powerful people having less of a handle on promoting one area at the expense of others.

But I believe that the real game changer will be coming from video. Already if people want to know something in depth, they are as likely to look it up on YouTube as find a printed explanation. The reality is that we can learn most efficiently when someone who knows what they are talking about explains it to us in a visual way at a level that is appropriate to us. Increasingly mathematicians are going to start to realize that if they really want to engage with a wide audience, and promote their ideas in the broadest possible way, they will have to augment their paper productions with video explanations.

In fact if done well, a video presentation can easily become the primary resource, with an associated pdf as a secondary resource for those who want to pore over the details. Most of us don’t want to pore over the details. So we are going to see an increasing number of academics thinking not so much about their paper CV, but rather about their video CV.

I have already made this transition some years ago. It is just so much more satisfying to explain a good idea by making a video about it, without the onerous obligation of getting every word just right and jumping through all the publishing hoops. Every word doesn’t really have to be right. The ideas have to be right, and they have to be visually interesting and engaging.

And what about the importance of having someone else to `approve’ my ideas for publication? Bah. For me, this is completely unnecessary. I am perfectly capable of deciding myself whether my work is worthy of publishing. If I make a mistake and someone points it out, I can change the video. If someone discovers a better way of working through some point, they can make a comment or publish their own video. And ultimately it will be the public, the viewers, that decide whether videos are worth watching, not some anonymous referee.

Want an example? I am working on revolutionizing hyperbolic geometry. I am slowly doing that with a series of traditional papers in excellent journals. But I am also doing that on YouTube, right here:

# Some fundamental formulas from metrical algebraic geometry (seminar)

This coming Tuesday in the Pure Maths Seminar at UNSW I will be giving a talk: Here are the details, in case you are in Sydney and are interested. The talk is at the School of Mathematics and Statistics, UNSW, Kensington campus, building the Red Centre, which is up the main walkway from Anzac Parade.

________________________________________________________________
Speaker:   Norman Wildberger (UNSW)
Title:         Some fundamental formulas from metrical algebraic geometry
When:       12:00 Tuesday, 9 June 2015
Where:      RC-4082, Red Centre, UNSW

Abstract:  At the heart of metrical algebraic geometry there are hierarchies of beautiful algebraic formulas, starting in one dimension and then working their way up both in dimension and complexity. These  can be viewed as generalizations of formulas of Archimedes, Ptolemy, Brahmagupta, Bretschneider, von Staudt and others, as well as classical results from Euclidean geometry.

One of the interesting aspects of our approach to these formulas, based on Rational Trigonometry, is the pursuit of generality: using general fields and arbitrary quadratic forms. Interesting connections emerge with the theory of special functions, differential geometry and operator analysis.

Most of these formulas would be challenging to find without the use of a computer: this is one reason why they have laid hidden for so long. We’ll see that there are many unanswered questions that invite exploration.

The talk will be accessible to undergraduates.

# Online tutorials for first year mathematics

The first teaching semester of 2015 has just finished here at UNSW. It has been very busy for me, which is a convenient excuse for not having blogged much lately. What’s been happening?

Teaching-wise I got a Learning and Teaching Grant from the University to put together new Online Tutorials for our Higher First Year Maths course MATH1141, which combines Algebra and Calculus. With the fantastic help of Daniel Mansfield, and also Jonathan Kress, we have been busily designing interesting questions that our best first year students can attempt on-line using Maple TA. That is a system devised by the clever people at Maple, based in Waterloo, Canada, which allows mathematical computation in the context of online delivery of course questions.

This means we can coerce, I mean motivate, students to actually do homework! How good is that? Up till now we, along with fellow educators around the world, plead and cajole our students to work on their own going over suggested exercises to practise the skills we are teaching them. But not altogether surprisingly, self-interest intrudes, and students often find themselves with something better to do, in other words something for which marks are assigned.

Our Online tutorials are based on videos to problem solutions that we made that show students how to solve questions from our Problem Notes.

Hopefully they watch the videos, and then answer a suite of five questions, ranging in difficulty from easy to more challenging, on that general topic. With the possibility of adding images, diagrams, lots of good looking and well-laid out mathematics, we can make our questions rather nifty. Here is an example of a typical question, which prompts students to discover the rational parametrization of the hyperbola x^2-y^2=1.

One of the exciting aspects of this is that we have realized we can get our students reading through proofs, and filling in blanks to complete them. This way they get the benefit of walking through a well presented argument and following that logical development—surely a key preliminary to getting them to create coherent proofs themselves.

And of course, there is the flexibility of the online platform that works in their favour, so they can choose the optimal time to do the assignments, which curiously seems to be an hour before Sunday midnight, just before the tutorials close.:)

Next session we are motoring on with more of the same in MATH1241, and next year we hope to roll this out for our big first year course (over 2000 students). Exciting, brave new times for teaching.

# My future glorious life!

I am so excited! Here drinking my morning green tea, i have just been visited by an image of St. Peter, who proclaimed to me that all my personal dreams are about to come true, provided i can write them down on this blog in the next half an hour!! Wow. So please excuse any spelling mistakes: I am about to restructure my life, and I need to do it fast, since i can’t type that quickly, but I need be careful, because I am not allowed to revise anything I write!

First of all, I am going to live a healthy life till I’m 100. Unfortunately he said I could only dictate what happens to me, not anyone else, so I can only hope the rest of my family is okay. And why not beyond a hundred? He said I couldn’t actually change my physical characteristics, so I reckon that beyond 100, frailty is too much of an issue. But I’m wasting time explaining these things to you!

Clearly I want a nice house for us, one that we own. Real estate prices are totally ridiculous here in Sydney, what with…oh I don’t need to explain why, it’s too long a story, suffice to say that for only \$2 million I can get a beautiful three bedroom house in a good neighbourhood not far from the Uni of NSW. So that’s on the list. A new car, of course, perhaps a luxury vehicle, a Lexus say, since I am very happy with our old Camry. Another \$2 million in the bank for a comfortable retirement ahead (not too soon, since I don’t want to get bored) would also be sweet. I can picture it now, a lovely new house, luxury car, money stashed away for retirement travelling and golfing.

Speaking of golf, how about a life membership at one of the nice golf courses close to where I live here. And please St. Peter, throw in a good set of clubs and some nice golf attire. Speaking of clothes, I haven’t redone my wardrobe in a serious fashion for many years: how about a gift voucher for a few thousand bucks at a local department store. I`ll want my theory of Rational Trigonometry to suddenly be understood by the academy, ensuring a promotion at work to full Professor; heh why not one of those Scientia Professorships that pay twice as much with only half the work? That should about wrap it up. But I still have another ten minutes to go. Am I being too modest here? What will my wife think when she finds out I had a blank cheque and came back with only a modest haul. And in a few years that Lexus will be a used car… Shouldn’t I be more ambitious?

And what if some friends come to visit? Let’s upgrade to a 5 bedroom house in Double Bay for \$5 million. Let’s add a top of the range BMW, and–you only live once—a spanking new red Ferrari. Actually I don’t care much about cars, but surely I can get interested once these are parked in my triple garage. The rich and famous who live there are probably a bit stand-offish, so I`ll ask that I become an internationally famous mathematician: Rational Trigonometry will be such an educational breakthrough that I will be in hot demand on the lecture circuit, girls will ask for my autograph in the street, and pollies will invite me to their lavish dinner parties. Russell Crowe and James Packer will become my good buddies, and drag me to those tedious football games, where I can sit in the high class enclosures sipping champers. Speaking of, I need a wine cellar, and say 2000 bottles of the good stuff. Probably I want a library too, and a tennis court—let’s make that house a \$25 million mansion at Piper’s Point instead.

But then all my new buddies will have holiday weekenders and villas in Tuscany. I want a holiday weekender too, say in Leura (I think I only have about 5 minutes left!!), and a villa in Tuscany. And a helicopter, and a horse riding ranch, say out near wherever James has his. And a private jet. Wait! I will need a lot more dough to pay for all the staff, maintenance and taxes that all this stuff requires. I want \$100 million in cash. No, I want \$10 billion in stock. Why limit myself? Why should Bill Gates outdo me in my future philanthropy? I want \$100 billion in cash, bonds and stock. And–

What? It appears that St. Peter has been timing me, and I just ran over my thirty minutes. [Insert sad face here]. The deal is void. Whoops. Better not tell my wife about this.

But to tell the truth, it was all starting to sound like a major headache, if you know what I mean. Now I am back to the essential mystery of my life, and I better get off to work; I have a 9 am meeting.

# The infinitely real delusion, and my recent debate with James Franklin

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.