Category Archives: Books

Bertrand Russell, Mathematicians, Pop Books

Fermat's Last Theorem

One of the first "Popular Maths" books I decided to pick up was Fermat's Last Theorem by Simon Singh. In the book, he expertly tells the story of Andrew Wiles and how he (practically) single-handedly solved an age-old mathematics problem. I won't go into any more detail than that because you can read a lot about the story on the web (but really don't, just buy the book).

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Part of what I really love about this book is that it goes into a lot of background detail, a lot of history. It builds up your understanding of why mathematics is the way it is today, and exactly why Fermat's Last Theorem was such a prestigious problem to solve.

One section that really sticks in my mind features a certain man called Bertrand Russell. Now, the more I read about modern mathematics the more his name crops up. Right at the beginning of the 20th Century, Bertrand Russell's research in logic appeared to show that mathematics was flawed. Again, I won't write down the details, I'd only end up copying Simon Singh's own words (buy the book!!!).

Although one passage did stand out in particular. Simon Singh mentioned that mathematicians obviously questioned Russell's work, and then goes on to quote Russell's response:

'But,' you might say, 'none of this shakes my belief that 2 and 2 are 4.' You are quite right, except in marginal cases - and it is only in marginal cases that you are doubtful whether a certain animal is a dog or a certain length is less than a metre. Two must be two of something, and the proposition '2 and 2 are 4' is useless unless it can be applied. Two dogs and two dogs are certainly four dogs, but cases arise in which you are doubtful whether two of them are dogs. 'Well at any rate there are four animals,' you might say. But there are microorganisms concerning which it is doubtful whether they are animals or plants. 'Well, then living organisms,' you say. But there are things of which it is doubtful whether they are living or not. You will be driven into say: 'Two entities and two entities are four entities.' When you have told me what you mean by 'entity', we will resume the argument.

A brilliant read (buy the book).

Books, Proofs, Text Books

How To Study For A Mathematics Degree

After my third module, completed in June 2005, I realised that although I enjoyed the module immensely, I still wasn't entirely happy with my ideas behind how to present proofs. Apparently this is a common problem. The shift from normal arithmetic to proofs is a jump that a lot of students find difficult.

I was so bothered by it I decided to post a query on the Open University forums about further reading. Part of my post read:

"...[I] consistently struggle with all kinds of proofs. I either start off completely incorrectly, or have no idea where to even begin."

The response was amazing, everyone (including more advanced students) agreed with me on the difficulty proofs and the shift of mindset required, and I now have a sizeable reading list to work through!

One of the books suggested in the forum was How To Study For A Mathematics Degree by Lara Alcock. I've tagged this post as being about a text book, but only because it's certainly more of a text book than a pop(ular) book. Although pitched at students that have just finished their A-levels and are about to start at university for the first time, I thought I may benefit from picking it up.

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I came away with some good notes as to how to adjust my learning slightly, and she talked a lot about the basics of proofs and how to acclimatise yourself to them. One of the parts that I feel was key was the realisation that I can try to construct premises and conclusions in terms of definitions. "Starting with the definitions" seems so obvious in retrospect, but when shown something so unfamiliar it can be difficult to align your brain so as to give yourself a solid starting point. Outlining all the definitions you might need gives you that solid starting point. I'll be keeping all this in mind as I start through my first text book...

Another section I benefited from was a section on how to reading mathematics. Or rather, how to benefit whilst reading mathematics. After spending three years of reading mathematics and feeling I've so far done a pretty good job, but Lara introduced some thoughtful comments on note-taking and comprehension. Again... I'm looking forward to applying this when sifting through my next text book.

As good as the book was, I must say I didn't benefit much from the second part on "Study Skills", but this section was only about 80 pages long. The section covers what "university life" is like, and how best to be a student in such an environment. If you've already been to university it'll be of limited use, but if you're straight out of your A-levels it could make for quite an insightful read.

All in all I feel I took away from it what I wanted to take away; that is a base understand of how to approach proofs in a more structured way. At the very least, I now have a starting point!

Analysis, Books, Maths, Text Books

The Maths Club

After my first year, I had a look at the clubs and societies you could join as a member of the OU. I thought it would help me feel a little less like I was learning so remotely. Sure enough, there was a mathematics club, M500.

A nice little bonus that you get for being a member is a discount for the bi-annual revision sessions that are held by the mathematics faculty at the OU. So when I started my last module, I signed up for a revision weekend that was arranged to be held about a month before my exam.

It was great meeting other students. With all that remote learning, it was great to be reminded that you weren't alone. We even had nightly trips to the pub! Just like a physical university!!!

Towards the end of the weekend, one of my fellow students, studying the same module as me, mentioned a book that he'd recently ordered.  It was written by David Brannan, a man apparently partially responsible for the syllabus of our first double-credit module in our second stage, and wrote the book to reflect the content of that module. My classmate thought he buy it to give him a bit of an edge when he finally started the monster-module. In our class he gave everyone the details. The book is called A First Course in Mathematical Analysis, and at over 450 pages, it looks like one serious introduction!

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Finding out about this book couldn't have been any more perfect. It means I can start my easy introductory module this year, and read through my new book on analysis so I'm ahead of the game for when my double-credit module starts in 2016! Perfecto.