Hi, I am a pure mathematician, working in the School of Mathematics and Statistics at UNSW, in Sydney Australia. This blog will touch on various thoughts on mathematics: ideas, patterns, surprises and some hopefully serious discussion on the weaknesses of modern mathematics, which ought to be more widely known and considered. And it will also have other occasional random thoughts, which are, at least in my mind, somewhat mathematical.

This is part of a large scale program I have to reorient mathematics towards a more sensible logical basis. Another aspect of that program is my YouTube channel, called *Insights into Mathematics*, which has more than 500 videos, 27,000 subscribers and 4 million views. The user name is *njwildberger*, so check outย my channelย for a lot more mathematics!

Here is a link to my UNSW web pages:ย My webpages

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JohnThanks for pointing me to this web site. It is an excellent way to share your ideas with like-minded people.

John

njwildberger: tangential thoughtsPost authorHi John, Thanks for the first comment!

Lawson BrouseMr. Wildberger,

I recently “discovered you” and I want to thank you very much your posts, you tube videos, and work have just made my life far more interesting. Thank you, sir.

Lawson Brouse

njwildberger: tangential thoughtsPost authorHi Lawson, Thanks for your kind words!

KasseyProfessor, I want to thank you for the wonderful insight you have given me in mathematics, I’m not a mathematical inclined person, but your YouTube lectures are starting to change my experience with mathematics. – I never thought I would enjoy mathematics. Thank you

njwildberger: tangential thoughtsPost authorI am delighted to hear that. Converting people to this lovely subject is one of my most ardent goals!

CarlosMr. Wildberger, mathematics is just a hobby for me but you poin of view clearly illuminated many paradoxes that i had about infinity. So i am not infinitally grateful but you can set the bound XD. I want to get the book but i want to know if there are any issues about shiping to mexico if i buy from pay pal? Thx once again.

njwildberger: tangential thoughtsPost authorThanks for the comment. As for the book, there is no problem, we ship anywhere if you buy from wildegg.com via paypal.

ubi de feohi, Norman

I am thankful for accidentally bumping into your efforts in teaching math and geometry.

I love your approach, starting from the beginning and moving forward in a very pragmatical and interesting way: relating to things we can picture easily.

I follow the same approach when I teach programming and beginner’s electronics (you may find something searching for “from 0 to C”).

I had major issues in school because my teachers failed to make their subject more interesting than hacking on a Vespa’s engine.

thanks to your videos I’ve been able to pick up math again and now feel fairly comfortable with a lot of things beyond basics.

not long ago I realized that so many concepts in math which I failed to understand in school, are just algorithms and procedures I’ve been implementing in code for the past 25 years without knowing the math counterparts ๐

well, I just wanted to thank you.

I hope I’ll find the time to watch all your videos.

may take a while ๐

cheers,

ubi de feo

njwildberger: tangential thoughtsPost authorHi Ubi, Great to hear that you are picking up with the mathematics again, I am glad my videos have contributed!

Viraj NadkarniHello Prof.Wildberger,

I am a high school student and have been watching your videos since a year and a half.I followed your whole course on Universal Hyperbolic Geometry and linear algebra and am following the differential geometry one now.I had and am still having interest in geometry as a career and have been reading calculus books but when I started your youtube courses I have become a “worshipper” of rational trigonometry.I heartily agree with you that this kind of geometry is indeed the geometry of the future.Your courses and work in mathematics are excellent and your way of teaching is beautiful.Your posts on the real way of delivering mathematics education should recieve implementation as much as they are recieving an audience!

Thank you very much sir for uploading those courses on youtube!! I hope they really draw a greater worldwide crowd for developing the newly emerging geometry!!

Regards,

Viraj Nadkarni

njwildberger: tangential thoughtsPost authorHi Viraj, Thanks for the nice comment and support. You are doing very well to have been able to follow those series still as a high school student! Continue, and you will be able to develop into a thoughtful mathematician.

Omarhey Prof.Wildberger, about three or four weeks ago I bumped into your youtube channel, and watched a couple of your videos, and they are amazing! I liked them a lot ! thank you for your great effort, and I wonder when you will post the third lecture of “the rotation problem and Hamilton’s discovery”, also if you could make a video about “brachistochrone problem” and it’s solution by Newton and Bernoulli that would be awesome, keep up the good work,

I am enthusiastic high school student.

njwildberger: tangential thoughtsPost authorHi, All four lectures on the rotation problem and Hamilton’s discovery are now up at my channel Insights into Mathematics on YouTube (user njwildberger). You might like to watch the lecture MathHistory14: Mechanics and Curves for info on the brachistochrone problem.

Omarthank you prof, I’ve found it in the suggestions bar, but I noticed that part III and IV aren’t included in your (famous math problems) playlist.

VictorHi Prof. Wildberger, i am a college student from Brazil (Mathematics) and your ideas about the problems with angles and trigonometry are the same as mine. When i found out about the rational trigonometry of yours i was really excited and i’ve been watching many of your videos recently to help me out. I love your work, but i can’t found a way to get your book “Divine Proportions”, is there a way that you can help me? I want to learn more about it and present to the mathematicians here in Brazil. Thanks.

Victor.

njwildberger: tangential thoughtsPost authorHi Victor, Nice to hear from you, glad to hear you like RT. Information about my book is at the publishers: wildegg.com but for some reason the Payment system there is not working currently (something I need to fix but i am currently too busy!) So I suggest you get the book from Amazon.com.

And Spread the word!

VictorSoon as a get my hands on your book, sir, i will !

Keep up the good work, your videos are helping me a lot, i am learning so much with you, thanks!

Jud ImhoffHello Professor,

Just want to join the group here in expressing gratitude for your efforts on the YouTube videos. I intend to watch them all and hope you make more.

I like your rejection of infinite sets and questioning of real numbers. It is interesting that Mathematicians haven’t yet been able to clearly understand and communicate the true nature of the continuum and reconcile what I believe is the natural occurrence of the ‘concept of infinity’ in Mathematics. Is it too complicated for Humans to get their minds around? Is there a different type of number perhaps an asymptotic numbers? Can rational numbers also be regarded as such in a certain context?

The ‘Real’ Numbers should be called the ‘Ideal’ Numbers and the Rational Numbers the Real Numbers because the Rational Numbers are the numbers we are in ‘reality’ constrained too as we try ever so hard to approximate the ‘Ideal’. Of course ฯ doesn’t exist because a perfect ‘Ideal’ circle doesn’t exist. Nor for that matter a perfect point, line, length, area etc.

But this only means that the concept of infinity is embedded in our Mathematics, so how do we explain it? Don’t we need too?

Anyway, I’ll continue to watch your videos learn and scribble in my notepad. Wishing there were more great educators like yourself making similar outstanding contributions.

Thanks, Jud Imhoff

njwildberger: tangential thoughtsPost authorThanks for the nice comments, Jud. There are some serious challenges in getting over our addiction to `infinite sets’ and `real numbers’. But we can do it!

NaveedDear Professor Wildberger,

I am fascinated by the first half of your first lecture on Differential Geometry (which is all I have seen so far); you cleared up confusions I have had about conic sections for years, despite having studied and come across them over and over again! Also, shouldn’t quadrance be a standard term in regular use!

Can I ask you for a huge favour: Could you please post lecture notes and/or a set of problem sets to go with your Differential Geometry youtube lectures if you have them, or point me to a suitable resource?

Many Thanks,

Naveed

njwildberger: tangential thoughtsPost authorHi Naveed, I will perhaps try to post some lecture notes that my students took for the course. It will be a few weeks at least though. When I post them, it will probably be at wildegg.com.

NaveedThanks a lot! I look forward to it.

Thiago CarvalhoHi Mr. Wildberger! I came across your blog today whilst searching for new mathematical ideas. I started studying mathematics about 2 years ago after watching documentaries about the universe including visionaries such as Seth Lloyd, Leonard Susskind, Edward Witten and Nick Bostrom. I’m looking forward to seeing more of your videos and participating in some of your discussions. I also have a Youtube channel and a maths website.

njwildberger: tangential thoughtsPost authorHi Thiago, Nice to hear from you, and good luck with your channel.

ByronI recently came across Dr. Wildbergers lectures while searching out some of the info on triangle centers. He has a very high quality lecture style. Good pace, very easy to understand, brilliant, etc. I am now on a mission to watch every video on his YouTube channel and eagerly look forward to learning and enjoying your presentations. A heart felt thanks to Dr. Norman Wildberger for his enormous contribution to humanity.

njwildberger: tangential thoughtsPost authorNice to hear from you. It’s always a thrill when I hear that someone is excited about watching many of my videos. I hope you enjoy them!

AlexHello Dr. Wildberger,

Firstly, I would like to thank you for the tremendous contribution you have brought me and many other in making mathematics a bit more accessible to those of us with less of a natural orientation towards the subject, through your wonderful youtube channel; it has definitely made a number of concepts clearer for me and has also opened my mind to new ideas, such as the problems with real numbers and infinity.

Lately I have been trying to get my head around the Fourier Transform and its applications, mainly regarding various filtering operations (images in particular), as it I have intersected with the problem in both academic and personal endeavors. Although there are many implementations of various filtering operations available, I do not like using something that I donโt understand and Iโm struggling with that part so far. I know that this is less of a pure mathematics subject and more inclined towards applied mathematics or engineering, but if you would enjoy covering this subject (and if time allows of course) I would really appreciate to see your approach on in (and I donโt think I would be the only one).

All the best,

Alex

Martijn van EedenHi mr Wildberger,

Thanks for the inspiring lectures.

About your aversion for the “Real” numbers because they assume infinite actions.. (forgive me for my choice of words, but the point should be clear I guess) I completely agree. However… I think that the “Rational” family misses some essential values like pi, square roots, etc. Wouldn’t it be a good idea to introduce a new family which is defined by all values that we can geometrically construct with simple Euclidean tools: the “Constructibles” for example..

If I am wrong, please tell me why.. I would be eternally thankful.

Kind Regards,

Martijn

njwildberger: tangential thoughtsPost authorHi, It would certainly be worthwhile to develop such a more limited extension of rational numbers. The challenge is: how to do this in a logical and consistent fashion? It turns out to be much more difficult than one might suppose.

Martijn van EedenMaybe, … and this is certainly an uncristalized thought …., the weakness in the first place is the way we tend to write down values in a decimal system. If we would write down values as their true value, independent of the cipher-system, then the values like pi or SQR2 would not be infinite rows of decimals, but just one beautiful “symbol or mini drawing”, as would any rational number too.

So if indeed all true values , “constructible” values, are the values that we can construct with geometric tools, then we obtain all rational numbers and the “true extension to the rational numbers” as you could call it by applying geometry itself to describe numeric values.

So only using integers and operators, we could write down any true value exactly. I will try to work this thought out to a more concrete and practical one..

By the way, the interval story to avoid rational numbers in the “Real number” (minus the Rationals..!) is in my opinion not even a way to describe all missing “Real” numbers, because if you allow infinite repetitions of that process, then you are only describing infinitely small intervals and no single values. If you allow something left, then you allow it also on the right.. right?

njwildberger: tangential thoughtsPost authorOne of the key challenges to make things work logically is that either you have a single canonical way of representing a `number’, such as we do for natural numbers in the Hindu-Arabic system, or you have a well-defined range of different allowable symbols, with a clear notion of equality that allows you to tell if two symbols represent the same `number’, as we do for rational numbers.

Martijn van EedenIn MF128 you are talking about the volume of polyhedrons, the Robbin’s formula and the Bellows conjecture etc. and probably the culinary inventor of tagliatella… who was of course in his time doing research on different shapes and volumes of pasta.. In the previous lectures you specifically mentioned all the time that the points of the polygons had to lay on a circle in order for the theorems to be valid. Shouldn’t then also not the points of the polyhedrons be on a sphere? Of course the classic polyhedrons with regular shapes are already compliant with that, but the irregular cube that you show halfway could be.. and could not be… it is not determined, at least not as far as I could get from your lecture..

Again thanks for your efforts to disclose all this wisdom to the world.

njwildberger: tangential thoughtsPost authorWe started off with Brahmagupta’s formula for a cyclic quadrilateral, but ultimately we are interested in general shapes. The point from a foundational point of view is that all the really good formulas are naturally in terms of quadrances, not distances, so they hold algebraically in considerably wider generality.

Martijn van EedenDear mr Wildberger,

When I was a student some of my teachers loved me, and some of them could skin me alive, because I always questioned them and hit the nail on the head where it hurts the most. I hope that you will not skin me alive when I point out a “mistake” in your reasoning. (With all due respect of course, maybe the word mistake is to harsh, but as a pure scientist we should be hurt by a true statement…).

Lets go for example to the point in MF131 where you want to claim that you can not prove Ptolemy’s theorem by calculating โ(4/5) x โ(882/221) + โ(162/65) x โ(4/17) =? โ(144/85) x โ(50/13) to prove that ab + cd =? pq. I can perfectly do that without taking any square root of infinite complexity: try it yourself please: the numbers all cancel out before taking a square root and you will end up with the perfectly do-able: check if โ49 + โ9 =? 10.. Yes: 7 + 3 = 10!!!! so.. since you are urging us to carefully go through each step.. and I do… therefor I can tell you this.

So I would like to suggest a revising in the direction you are going:

Lets not avoid the existence of square roots. They do exist, only we can not determine their exact value, in other words, the root exist, but their exact decimal value does not exist!!

We can indeed not exactly determine the length of a line segment in a plane, yes, true, but we can still work with it because we can perform exact operations on them so that the result is rational again, and therefor exact..

Another thought is that as you also said yourself a line and a circle that intersect don’t have necessarily common point, because only rational points exist…., that implies that the lines are not continues, but “quantized”. The level of quantization would be infinite so maybe infinity is unavoidable unfortunately.., so. The fact that we can not exactly give the value of the coordinates in a decimal cipher system due to infinite row of decimals, is more due to our cipher system than the question whether the two meet. There meeting point could be described by equations.

So again, yes let us be precise and exact and logical. But we don’t need to avoid things that exist, we just need to learn how to work with them in an exact way.

Kind Regards,

Martijn.

Martijn van EedenHere is a thought about an idea that has been discussed in your lectures.

First three Euclidean axioms:

1 : you can draw a straight line from any point to any other point.

2: you can extend a finite straight line continuously in a straight line.

3: a circle (in the plane) is all the points that are at a given distance from a given point (center).

So it already follows that straight lines are continuous and circles are not!

The statement of Mr. Wildberger that a unit circle and a line y=x do not meet at point [1/โ2,1/โ2], since this is no rational point, is therefor true since the circle has no valid point there, and does not even exist there!

But this also implies that if the circle is not continuous, then its curve is not “infinitely smooth” and therefor the area enclosed by the circle is at the edges purely bound by rational coordinates that according to Euclid connect with straight continuous finite lines.

That means that the enclosed area of a circle is a rational value and not a “irrational” value through ฯ. Of course we have to go very deep here, but it is an essential point.

And it also means that the line y=x does pass through [1/โ2,1/โ2], since it is continuous. We simply cant call it a point, … we cant stop there …, since there is no point there. So lines fill in the “gaps” jumping instantly between two adjacent rational points.