The outline of a story, mathematics was most likely discovered.

Some argue about mathematics and if it is discovered or invented. I’ve held an opinion for some time that mathematical things are there to be discovered, we humans invent language and notation to describe it.

When it comes to mathematics and to explore the idea of discovery and invention we need to look around use for evidence. Case one, back to the time of Newton and Leibniz and Calculus. Today we use the notation of Leibniz to do those calculations.

The notation that both used was different and the attempt was to achieve the same measurement albeit through slightly different paths. This was about measuring the area under a curve and tangents from that curve.

From this we could claim that the area under a curve was to give itself up and surrender to our tools even if the tool sets were different.

Let’s look back further in time, lets see what we can say about numbers and that most basic of skills, counting.

Different social groupings around the planet learned to count and created number systems to deal with their needs.

Some used base 60 such as the Babylonians and some groups in South America. The decimal system was used by many other groups, perhaps something to do with hands and fingers. What we know of zero and at least one version of the decimal system was that it was used for religious purposes, that story is from the Indus valley and surrounding regions.

Some people stop counting after a few single digits and then consider all other groups of numbers to be many. Some don’t stop and get to concepts like infinity representing it as a symbol.

We should pause a moment, think of the primes, 1, 2, 3, 5, 7, stopping there, there are infinitely many primes, a fact which is known since the time of Euclid circa 300 BCE.

However before we get bogged down in humans having the skills to count, how about the cicadas, those insects stay underground for for a prime number of years, seven years and then arise for six weeks and where they congregate and make so much noise that people tend to leave the areas they live in while they are around. It should be fair to say that cicadas know how to count in some way, even if they don’t use numerals as we do. They have evolved to do this thing.

Let us now head back to the numbers, people have ways to manipulate them, what people have done and some have become quite skilled with is to restate our discoveries and add to our understanding and abilities by refining the language we use.

This can be shown by the conversion of word problems into more symbolic conversations we have today, this http://jwilson.coe.uga.edu/EMAT6680Fa10/Shields/shields%20iu/BIT%20WP.htm gives a nice example of the conversion of words into symbols.

While this is no rigorous proof, it is the start of a story I’m interested in exploring at a later date.

Reading Mathematics for Fun

The reason to explore is captured in the title of R.P. Feynman’s Book: The pleasure of finding things out.

The things to explore are things that capture our thoughts and surprise us. Be that as it may, humans have evolved a reward system so that when objects strike us as pretty, perhaps a better way of saying that, is symmetric.

When in the gamut of human experience does the ability to describe things happen?

To answer this question we perhaps need first to have looked for a language to describe the world around us.

Formal description that is nouns, dogs, cats, trees, numbers, patterns all needed for the purposes of description to have a way of codify them to an acceptable level of depth.

On patterns and numbers we have abstracted relationships of things to or perhaps better said between each other.

In order to facility this exploration we have defined the knowledge of philosophy to this subject to be named mathematics.

To capture these things we humans have divided up some portions of the universe for the purpose of discussion called domains at least in the English language.

Insider the domain of mathematics there are many discrete headings;

arithmetic,

geometry,

trigonometry,

graphy theory,

proof theory,

statistics,

game theory,

manifolds,

calculus,

algebra,

topology,

logic,

set theory,

real analysis,

it also includes the study of ideas which as the infinite, the infinitesimal and many other topics, all of which fall within the domain.

Now to address the question I began with, but didn’t state as yet.

I’m asked now and again what is it I study in Mathematics, I wanted to answer this but then I decided that the word study is wrong, perhaps I don’t study, I explore small regions of some of these things and eventually perhaps I am at best a very curious onlooker.

On Learning Something New

The following quote is attributed to Richard Feynman: “We are trying to prove ourselves wrong as quickly as possible, because only in that way can we find progress.”

To paraphrase this, if you want to prove something first you should try to disprove it. When you have tried everything, then perhaps, perhaps you have a new idea. It is possible that don’t have enough information, to prove yourself wrong.

When I heard this on a podcast today, I had listened to a few other podcasts about mathematics. It was either on SGU November, or Science(ish) the episode about Pi.

When I consider knowledge I tend to visualise it like wet paint on a canvas flowing but not covering all the space. Different colours representing different depths of the material. We start off with a smattering of something and then we continue to cover the canvas or move to another one and leave this idea partially known, or learning a little.

This year to date, I have painted very slowly and very sporadically for reasons not entirely within my control.

A bit of light reading

Amir D. Aczel’s book Finding Zero: A Mathematician’s Odyssey to Uncover the Origins of Numbers

I quite enjoyed this book, the story of one man and his search for the origin of zero. It turns into a trek across south east Asia, where he crosses the barrier from Hindu Arabic numbers to something a little stranger. It appears that while in the west numbers were used for commerce, in the east they had a lot to do with mysticism.

The book follows the questions of the author as a child from the first time the questions occurred to when where somewhat answered as an adult. The story has the author travelling around the world focusing on what used to be part of the French colonial empire that they called Indochine.

The author sadly passed away in the year of publication, 2015. One of the mathematicians he references Alexander Grotendhieck died in late 2014. What isn’t referenced as it wasn’t widely known at that time is at the end of his life  Grotendhieck relented on the retraction that is mentioned in the book.

Due to the inclusion of Grothendhieck the group Bourbaki gets a mention in passing.

Three photos from a little journey

During the summer we went for a little break to the Ardmore in Waterford.

These are three photos that remind me of those warmer days.

 

pump

This little beauty is a hand bellows for a firew. It is in a tiny rural pub on the outskirts of Ardmore and it still works over a hundred years since it was made.

half-penny

This cast of a half penny adorns the wall of a small dining area to the rear of The Spire Cafe, in Lismore. As I remember predecimal coins this was a bit of a surprise.

However this cafe gave me even more of a surprise, when I went inside I found the meter below on a shelf and they let me take it down and place it on a table to take a photo.

In a former life I used to use analogue meters on a regular basis.

My father had one from this manufacturer when I was a very young. That was when we had lead in solder.

meter

Sometimes you should look at the back of a thing.

backmeter

Kings Of The Wyld

Kings Of The Wyld

I finished it. If you like fantasy all action books, you could to a lot worse than this. 5/5

My usual reading in fiction are more Space Opera, this is outright fantasy.  It doesn’t suffer with an over complicated story. I enjoyed it from beginning to end.

It follows some good plot writing toolchains and this is a good thing.

 

If you need to spend some hours reading fiction, go on, treat yourself it’s great.

 

#kingsofthewyld #fiction #fantasy

 

My Favourite Mathematics Book

It is a hard a question to pick one book  to represent my favourite book. This challenge was set by person who started to organise a group of people to read mathematics books in a book club of sorts. What follows is what I wrote for them on that subject.

I’m going to exclude standard text books as they are used to practice the ideas we are playing with.
I’m going to exclude history of mathematics books, while I read them there are parts of history where I get excited, usually by thinking about how the person being discussed got to the point where they did whatever thing it was that is documented. For instance, Ian Stewart’s very nice ‘Taming the infinite the story of mathematics’, starts with the wonderful line, ‘Mathematics did not spring into being fully formed. It grew from the cumulative efforts of many people, from many cultures, who spoke many languages.’ Knowing this means that I can’t know how or why that happened to be, I can, I suppose, guess but I’ll not be certain without evidence, and then at this remove that evidence is not viewed as it was at the time, we can’t know what went on in the Babylonians minds. Frank Sweetz wrote a book titled Capitalism and Arithmetic, the new math of the 15th Century. We also know that people approximately counted seasons and years from the Mayans to the Celts through the legacy of stones in various trigonometric designs. So I will abandon these books for their distance of the journey.
This thinking has me reflecting on three of my favorite books, in the series about people, in no particular order:
Siobhan Roberts excellent book on John Conway, Genius At Play. Levels of Infinity by Hermann Weyl a collection of his essays including one on Emmy Noether, which is reason alone enough to read the book.
Lastly in a category all on it’s own Lymm Gladwell’s excellent Mathematics and Art, which if you want to characterise it you might say it is a coffee table book, it is also a nice history of mathematics but not in the style of the general history books which tend to be very much aimed at course work. It is beautifully illustrated and just a joy to dip into now and again.
This however leaves whole subsections of mathematics alone, it causes me some grief not to mention the Princeton Companion to Mathematics, edited by Timothy Gowers, or Mathematics From The Birth of Numbers, by Jan Gullberg both of which are tomes and well worth the investment. In this short survey of my shelves I’ve left out a few brilliant things, anything by the popularising group Keith Devlin, Ian Stewart, Marcus Du Sautoy.
Now comes the reason I can can’t name a single book, there is a series that takes a bit of work at times, be that as it may, you can skip the hard parts now and again and still get significant gain from working though this little set of books. It is called The Best Writing in Mathematics and it has been an annual release since 2010 edited by Pitici they are a joy in and of themselves and cover a broad swathe of the subject.
With all that in mind I think you might see why I have problems picking a single book.