I’ve been going over a couple of ideas for some blogs recently, but haven’t quite had the time or depth to write them all out. I had a look through my ideas, and realised that the reason I haven’t posted them is that I don’t think I can explain all the concepts very simply and easily, and so I choose to not write them up…so I’m going to stop doing that. Very simply, from now on I’m going to give broader explanations of ideas. If you have any questions, feel free to leave a comment and I’ll get back to you.
The first thing I’d like to do is address a question – or perhaps a concept – that most people have in regards to mathematics.
Most people think that they are not logical enough to do mathematics. This puts them off ever even looking at the subject, because they don’t think they’ll understand it.
This view point is note entirely correct. Whilst there is a lot of logic in mathematics and it is certainly helpful to be able to think logically, it is not all there is to it. Mathematics is – by necessity – a very creative science. To stretch mathematics to new areas, you need to be creative in your approach. To attempt a proof of something that no one else has done, you need to be creative.
This being said, there are rules and logical steps that you follow – just like everywhere else. Music is governed by certain laws of physics. Art is governed by visual laws. Poetry and literature are governed by laws of the medium it is written in – even poets like e.e. cummings ascribe to certain laws, albeit they enjoy breaking most of them. Mathematics is the same.
A few people I have told this to have been sceptical of my views. They remember their mathematics at school as being very strict and formal. Very few were privileged enough to be shown the joys of creating new mathematics, or taught to think rather then remember. To people who view mathematics as being a logic and rule based science with no deviations, I suggest you look up some recreational math authors (or contact me for some who I’ve read).
In my mind, creativity is crucial to development in mathematics. You can’t do new maths without creating, you can’t find anything new without creating.
A small example of this: unsolved number theory problems.
Now, most of these problems are very easy to state (for example, the collatz conjecture, which I’ve written about in a previous post). The majority deal with finding out if a certain pattern satisfies every number – this requires us to look at the infinite. How exactly do you approach the infinite without some creativity?
You can’t start counting and eventually get there – infinity is easily viewed as the biggest number you can think of, plus 1. You can’t ever reach it. Your only way of examining it is to simplify it, or to creep up on it and surprise it.
Here’s a devious proof that shows this principle:
assume you want to show that there is a set of consecutive composite numbers of any size.
(so the number n is not prime, nor is n+1, n+2,…for the next s numbers.)
The numbers 8,9,10 show that there exists a set of 3 consecutive numbers. 25,26,27,28 is a set of 4.
But how do you show that there is a set of any integer?
We need to deal with infinity here. We need to find a way to show that there is a set for any number, no matter how big the number is.
here’s the trick: if you take a number n, and you times it by n-1,n-2…2 then the number you get is called the factorial of n, and is shown as n!. so n!=n*(n-1)*(n-2)*…*2
now this is obviously composite. We can also show that n!+2 is composite as n! has a factor of 2, as does 2, so it is composite. We can do this for any factor of n! – so for all numbers from 2 to n-1, we can add it n! and get a composite number. Assume we choose k. then n!=n*(n-1)*…*k*…*2, and n!+k=k( (n*(n-1)*…*(k+1)*(k-1)*…*2+1), so k is a factor.
this means that all numbers from n!+2 up to n!+n are composite. So to find a set of size s, take the s numbers after (s+1)!+1 – they’ll all be composite. (quick question, are there any sets of size 2? Why?)
Thus we have a way of finding any amount of consecutive composite numbers – all the way up to infinity.
It is not at all obvious in the above proof that you need to use a factorial – that required some creativity to come up with. This is why there are still so many unsolved number theory problems – computers can not calculate all the numbers from 1 to infinity, and so we need to use human ingenuity to find shortcuts to infinity.
I am not saying here that computers will not be able to prove number theory problems. There is a role for computers in proofs. Sometimes we can reduce a problem down to a set of numbers – so if it satisfies these numbers, then it satisfies all numbers. This is the way the four colour theorem in Graph theory was solved. I wouldn’t be surprised if the Collatz conjecture is similar. but I believe that no computer will ever be able to take a conjecture and prove it from scratch. I may be mistaken, I do think I recall a book that said that Turing proved that computers won’t be able to find solutions to all problems, which leaves the possibility that some problems will be proved by computers, but my current knowledge does not know of any that have been done by a computer.
Now I’m the sort of person who enjoys both logic and creativity. And I find that maths satisfies both of these loves. I enjoy looking at unsolved problems and attempting to think my way through a new approach. So here are a few that I have been looking at lately/ enjoy thinking about.
there is a separate blog post about this, but to reiterate: take any number n. if it’s odd, multiply it by 3, and add 1. If it’s even, divide it by 2. Repeat for the new number. Keep doing that, and eventually you’ll reach 1 – or at least, that’s the conjecture (which means it hasn’t been proved).
This problem has been around for around 70 years ( I think), and has some interesting properties that I’ve seen – but there’ll be more on that when I have enough to post!
Goldbach’s conjecture states that there are an infinite number of “twin primes” where p and p+2 are prime. A few examples are 3,5 ; 101, 103 etc. This one is lots of fun to play with if you enjoy working out primes, and you can start finding links/ideas almost immediately.
The abc conjecture.
Take three numbers (a,b,c) that have no common factors, with a+b=c. let d be the product of the distinct prime factors of a,b,c.
The conjecture states that d is very rarely much less than c.
This conjecture is quite difficult for a non-mathematician to get their head round at first. It has some interesting and important consequences.
The 196 algorithm
Take any 2 digit number, reverse it, and add the two numbers. Repeat this until the number reached is a palindrome.
196 is the smallest number for which no palindrome has been found. There is no proof yet that this algorithm never gives a palindrome for 196.
163+361=524+425=949 which is a palindrome.
These are only a few. There are a lot more, and that’s just in number theory.
Which brings me to another thought: different branches of mathematics.
Something that not many people seem to understand is how diverse mathematics is. They seem to assume that if you have a degree in mathematics, you know all mathematics….this is definitely not true!
The easiest way to see this is to look at it like this:
Assume that, a long time ago, there was a village in the middle of nowhere with a very detailed language. Now a huge fight erupted, and because of the fight, many people in the village decided to leave and start new villages. These people all formed their own groups which then left the village and headed off in their own direction, never to travel back to the original village.
now over the following thousands of years, this happened many times. Occasionally, splinter groups from one village would join up with another splinter group, forming a new village, but people never travelled to an existing village.
Through this timeline, little changes in the language would become apparent to an outsider. Each village would start forming new words for new ideas or ‘bastardizing’ old words for old ideas or things. Eventually (assuming there had been no outside influence on the language) there would be many different languages, but each language would be similar to each other. An outsider coming in would probably have to learn the basics of the original language, and, from there, have a chance to communicate with the different villagers. But there would be big difficulties. Certain topics and words would be recognisable, but others would be completely alien. The outsider may hear a strange word, and start to investigate it, only to find out it means exactly the same as a completely different word in a different village. Or the outsider may hear a very familiar word, but be shocked to discover it means something completely different.
Mathematics is slightly like this. Certain ideas are the same in all branches, and there are always overlaps. To jump from one branch to another is not always a simple process. Sometimes it requires learning a whole new language, at other times it requires learning a new translation. Sometimes, there are branches that act as an in-between, which can make the jump easier.
At times, mathematicians have found interesting and amazing links when looking at supposedly separate branches. An example of this is complex numbers. When they were first introduced, Riemann began to examine them. He started looking at what happened when he applied certain functions to them. He looked at what happened when he put them into the zeta function…and discovered an amazing link to prime numbers, in the completely separate branch of number theory!
Collaboration amongst mathematicians is always encouraged, but only recently have mathematicians began to collaborate with non-mathematicians. The results have pretty impressive –quantum physicists have discovered a link to the work of Riemann, economists and mathematicians discovered the ideas of game theory which became important to cryptography and computer scientists started to overlap with logicians, set theorists and graph theorists. There have also been interesting and wonderful advances in medicine, psychology and the more ‘physical’ sciences when applied mathematicians have looked at their problems.
We are coming into an age where collaboration is an email away. Mathematicians are finding more and more ways to help their fellow man. Whilst some mathematicians may not enjoy the ‘corrupting of the pure fields of abstract math’ to aid in other subjects, the benefit to all is obvious.