The Foundations of Mathematics (Part 1)

Hello Reader,

In this post I would like to show you what mathematics actually is. Mathematicians on the whole do try to avoid philosophical questions, but in this particular case, the question required answering for mathematics to progress. To understand why, we must return to the mathematical world of the 20^th century and a legendary German mathematician called David Hilbert.

In the 20^th century, thanks in part to the work of Hilbert, mathematicians were also beginning to ask the same question we are: What is mathematics? How do we know that anything we prove in mathematics is true?

When we prove something in mathematics, we inevitably make some assumptions. For example, in Euclid’s proof of the existence of infinite primes, he assumed the fundamental theorem of arithmetic. Of course, we can then prove the fundamental theorem of arithmetic, but then we make some assumptions about other, even more basic things. As we go further and further down the rabbit hole of provable statements, we eventually come to statements called axioms, which are essentially unprovable statements that are so obviously true that we just assume they are. For example, the axiom of equality states that x = x.

So, in essence, mathematics is the search for logical conclusions from a set of presumed truths called axioms....but, unfortunately, it gets more complicated than that.

It wasn’t until the early 20^th century that mathematicians really began to question the foundations of mathematics. After all, based on the above definition of maths, the entire subject is like a tower of cards, each card depending on others for support all the way down to the base cards (the axioms). But what if these base cards were shaky...the entire tower could come crumbling down. This period became known as the foundational crisis of mathematics.  Mathematicians desperately began to try to clarify the foundations of mathematics, but each attempt seemed to find paradoxes and inconsistencies. Many feared the end of mathematics was nigh.

David Hilbert, who by this time was a renowned and highly respected mathematician, set up the Hilbert Program. The aim of this program to strengthen the foundations of mathematics by creating a finite system of axioms that could be shown to be consistent (they would never contradict on another) and complete (all statements about mathematics could be shown to be true from this set of axioms). Hilbert essentially wanted to create a foundation for the tower of cards that would never crumble or have to be re-laid. Wouldn’t that have been nice...

Unfortunately for Hilbert, another German mathematician by the name of Kurt Godel was about to utterly destroy the Hilbert Program by showing that what it hoped to achieve was in fact impossible. In 1931, Godel proved two fundamental truths about mathematics:

1)      Within any axiomatic system with a finite set of axioms, there exist statements which cannot be proved true or false within the system (ie. No matter how many axioms we have, there will always be unsolvable problems in mathematics)

2)      For any axiomatic system, if the system includes a statement of the system’s own consistency, the system is inconsistent (ie. The axioms of maths cannot be proved to be consistent using maths)

To say this shook the very core of mathematicians around the world would be an understatement. Many mathematicians’ worst fears had been realised, that mathematics could all be somehow untrue. Thankfully, as is often the case with revelations as large as this, the reaction was a little overstated.

The next post will explain what followed Godel’s revelations, how the crisis was solved (actually leading to mathematics rising from the ashes stronger than ever before) and maybe some notes on a truly mind-boggling but relatively uncomplicated, personal favourite topic of mine.

Thanks for reading J

What Is Mathematics?

Hello Reader,

To tie up, for now, this theme of proof, I thought I'd go ahead and show you the man who shook the 20th century mathematical world with the revelation that, no matter what we prove in mathematics, we can never be sure any of it is true :o (shocked face indeed!).

Before I go further, I'd like to give you something to think about. A seemingly simple question.

Fundamentally, What Is Mathematics?

If you have any ideas, please comment with your answer below.

Otherwise, watch this space as my next post will try to answer this question, and introduce you to the foundations of mathematics.

Thanks for reading :)

Proofs and Primes

Hello reader,

Following on from my post on Pythagoras, which turned into a post about proofs, I decided I should give an example of what I meant about how mathematicians can't use 'evidence', only pure logic.

Euclid (yep, him again) came up with this ingenious proof that there are an infinite number of prime numbers (if you don't know what prime numbers are, you have genuinely come to the wrong blog):

Firstly, as some groundwork, you must understand the fundamental theorem of arithmetic which is that every number can be uniquely written as the product of prime factors

For example:

236 = 2 x 2 x 59
113163 = 37 x 59 x 61
4398618 = 2 x 3 x 7 x 104729 (yes, 104729 is prime)

Prime numbers are like elements in chemistry (except there's an infinite amount of them, as we shall shortly prove). They can be multiplied together in different ways to make all the 'compound numbers' that exist, and every compound number can be broken down into its primes. But the primes cannot be 'split' / divided by any number (other than themselves and 1). Now, onto Euclid's proof

Let's first assume that there are in fact a finite number of primes. Lets say the 'set of all primes' is {2, 3, 5, 7, 11, 6th prime, 7th prime, prime8, prime9, p10.....and so on until the largest prime} In that case, we can multiply them all together to get the number P.

So P = p1 x p2 x p3 x p4 x p5..... x largest prime

then let's make a new number one bigger than P:      Q = P + 1

Now, if Q is a prime, then it is a prime existing outside the set which was meant to be the 'set of all primes'. This is a contradiction. Therefore, since we have seen a contradiction, we must assume our beginning assumption was incorrect ie. we assumed that there are a finite number of primes. This must be incorrect.

If Q is not prime, then as the fundamental theorem of arithmetic shows, it must be divisible into prime factors. In other words, it must be divisible by some prime number p. But this prime number p cannot be in the 'set of all primes'; if it were it would divide P and so cannot divide P + 1 (ie Q). Thus, once again, we have shown that there must exist a prime outside the set of all primes, which is a contradiction. Therefore (see above) it is incorrect to believe there are finite primes.

Therefore there are infinite primes.

Thanks for reading, and a special thanks to Zachary for recommending this proof which is a perfect example of what I was hoping to explain :)

Pythagoras' Theorem (Part 2)

Hello reader,

As you may have noticed in my last post, I didn't actually cover that much about Pythagoras' Theorem at all...yeh, sorry about that :p . That said, the theme of rigorous mathematical proof is something I'll return to in the near future, so it wasn't a complete waste of time.

So...Pythagoras' Theorem.

For starters here is a quick, non-rigorous, geometrical proof of Pythagoras' Theorem.

This is not a complete proof on its own however, as an accompanying set of algebra is required.You should be careful when seeing geometrical 'proofs' without an accompanying algebraic solution as these can be faulty (our eyes are not very good tools for mathematics).



(Source of animation: Wikipedia)



Something you may not have thought about, but which is crucially important to Pythagoras' Theorem is that it's converse is also true. This means that given 3 integers a < b < c iff (if and only if) a² + b² = c² then there exists a triangle with side lengths a, b and c with a right angle between sides a and b.

It is therefore natural for people to be interested in finding sets of integers which satisfy the equation a² + b² = c² . Such sets are called 'Pythagorean Triples'. For example, {3, 4, 5} is a Pythagorean triple because 3² + 4² = 5².

A Pythagorean triple generating formula was discovered by Euclid (a post-Socratic, Greek mathematician) and written in his book 'Elements' which is historically one of the most influential books ever written in science. The formula is as follows:

For any two integers m and n:

f (m , n) = {2mn , m² + n² , m² – n² }

is a pythagorean triple (try it out!)

I feel this would be a good time to let you in on a somewhat humbling fact. Almost everything the standard GCSE student knows about mathematics was known over 2000 years ago. We're playing catch up, and we've got a very long way to go.

For those of you who have further interest in Pythagoras' theorem or triples, I would recommend investigating its generalisations into higher dimensions and non-Euclidean geometry. But that will do for now on Pythagoras on this blog.

Thanks for reading :)

Pythagoras' Theorem (Part 1)

Hello reader,

Every student who has ever studied mathematics has come across Pythagoras' theorem. The sum of the squares of two sides equal blah blah blah...

And by using the theorem to solve pages upon pages of monotonous questions, most students learn the formula as just that...a formula to solve questions on maths exams. But it's so much MORE than that!

Pythagoras came up with his famous theorem in some time around 500BC. Does it not strike you as odd then that it has not been disproved yet? For example, around the same time, biologists thought the human body was made of clay and earth, and physicists thought the earth was the centre of the universe (and would do for another 2000 years). Yet, what mathematicians were discovering then is considered as true today as in 500BC and will be considered true forever more. So what is it about mathematical theorems which gives them such longevity?

The answer is 'proof'. In every other field of study, a problem is considered solved when a theory has enough evidence to back it up. For example, people wondered how human beings came into existence. Darwin studied organisms, saw evidence of genetic lineage and natural selection (eg. Galapagos Islands) and so came up with the theory of Evolution. Problem Solved. 

However, this approach doesn't work for mathematics. The Riemann Hypothesis (a very, very important and famous problem in mathematics) has been checked for billions of numbers but this is not even close to what constitutes proof in mathematics. This is because there is an infinite number of numbers. So even if we check Graham's number (This is the largest non-trivial number currently known. It is so big, there is not enough atoms in the universe to write it out as a tower of powers, let alone in full!) of numbers, there's still infinite numbers to go!

So how can you prove something is true in maths? You have to generalize, and all students have done this countless times. Every time you use x in some equation, you are proving something true for all numbers, because if it works for x, it'll work for any number. What's more, once you prove it, your theorem (like Pythagoras') can never be disproved because there are no opinions, new evidence or anything else, only logical proof which can never change. Today mathematicians use methods of proof which are considered 'rigorous' (something which I will devote an article to soon):

Pythagoras' Theorem is a lovely reminder that mathematical theorems will last forever. So, for those of you who crave immortality, prove something new in mathematics (much, much easier said than done :p )

Reflection of a point through a given curve (original)

Hello reader,

This will be my first 'original work' post (NOTE: Chances are, this is not undiscovered knowledge. Somebody, somewhere must have already discovered this. However, I am not aware of any work already done on the problem, and all work has been done independently ie. I worked this out, I didn't copy it from somewhere else). Hope you guys like it :)

I was on a plane back from South Africa (which is an AMAZING place to visit if you ever get the chance) and anxiously awaiting GCSE results. To take my mind of the impending judgement which, on the basis of a poorly designed exams, would give me a grade to decide my aptitude for the given subject for the rest of my life... (-_- ugh...GCSEs)

I started thinking about Maths (as you do) and the problem that had been bugging me for a while. If you draw a curve:

y= f(x)

and a point 

(x₁ , y₁)

you should be able to reflect the point in the curve such that the line connecting the point and the reflected point  is a normal (perpendicular) to the curve (what I like to call a 'proper' reflection) like so:

I got thinking about it because it's easy to estimate the 'proper' reflection point on paper just by drawing (like above), so the maths should be pretty simple too...right?

Well, this is the equation I came up with:

                                           f(x)  + [(x₁ + x)/f '(x)] – y₁ = 0

The solutions (values of x which satisfy the above) to the first order differential equation above can be input into the following formula to give the set of 'proper' reflection points:

                                               (2x – x₁  ,  2 f(x) – y₁)

So, there we are. Sorry if this was a bit confusing, blogger is a bit rubbish for equations and diagrams. Please comment with any difficulties in understanding or general questions and I'll do my utmost to answer. If you have some free time, why not try to see how I proved this solution. The proof of this is unfortunately far too large to contain in this margin (hehehe, sorry, couldn't resist ;) ). Not difficult really, but interesting and with enough pitfalls to dodge to keep you on your toes. 

If you're wondering whether to comment or not... please comment :p

Thanks for reading :)
Anand

Millennium Prize Problems

Just a quick development on the P=NP article for anyone unaware of the Millennium Prize Problems.

In the year 2000, the Clay Mathematics Institute offered a prize of $1,000,000 for a proof for any one of the following 7 open (ie. unsolved) problems:

1.       P versus NP problem – A major unsolved problem in computer science essentially asking whether if a solution to a problem can be verified in polynomial time, can the problem itself be solved in polynomial time i.e. does P=NP?

                                      

2.       Hodge conjecture – A major unsolved problem in algebraic geometry on the subject of the algebraic topology of non singular complex solutions to polynomials.

3.       Poincaré conjecture (solved) – Solved by Grigori Perelman using the principles of Ricci Flow, the problem was on the subject of the topological characterization of spheres.

4.      Riemann hypothesis – A major unsolved problem on the subject of the distribution of the zeros of the Riemann Zeta Function.

                                                                

5.       Yang–Mills existence and mass gap – A major unsolved problem in quantum field theory underlying the Standard Model in physics. It asks for a proof of the existence of a non trivial Yang-Mills.

6.       Navier–Stokes existence and smoothness – A major unsolved problem in engineering and theoretical physics on the subject of the proof of the Navier-Stokes equations which are one of the pillars of fluid dynamics.

7.       Birch and Swinnerton-Dyer conjecture – A major unsolved problem in number theory on the subject of the elliptic curves over given number fields.

Each of these is an extremely difficult (to the extent that many brilliant mathematicians have devoted many years of their lives to solving the problem and still failed to find a complete solution) and very interesting problem and as yet only one has been solved. Furthermore, each problem’s potential solution would have extremely far reaching consequences for their field. 

P = NP (And Some Basic Computer Science)

The computer science and maths problem P = NP is one that many people have heard of, mostly because a $1,000,000 prize has been offered for a solution (that's right, though this is probably the hardest way imaginable to become a millionaire!). The Clay Mathematics institute named it as one of their seven 'Millennium Prize Problems', and as such it has gained a certain notoriety. Another, less provable but I think equally true reason it is famous is because its name ( ie. P=NP) is not difficult to remember, at least compared with other such problems (eg. Navier - Stokes existence and smoothness). But anyway, onto the maths:

P=NP is a problem to do with the complexity of certain questions that can be solved by computers. Say you wanted to write a program that would give you all the even numbers from a set of numbers that you give it. For example, say you give it x numbers. For each number, it would divide by 2 and then if there is no remainder, the program tells you the number is even. Therefore, for each number, the program takes one step to 'solve' it, and then moves onto the next number. The running time for this program is therefore x steps. However, if you wanted all the numbers which are not divisible by 2 or 3, then the same program would take 2 steps per number, giving a running time of 2x.

Now this is a simplistic introduction to the concept of 'running time', and by extension 'complexity'. However, running time is not the same as complexity. If you gave the 'even number program' described above 10¹⁰⁰ numbers, it would have a very long running time (10¹⁰⁰ steps). However, complexity is about the equation which gives us the running time. In the case of our even number program, the running time = X steps, with X being the number of input numbers. This is considered an 'easy' problem.

Now some problems have a complexity equation that is a polynomial (ie. the sum of powers of x). These problems are considered 'easy' as the running time increases relatively slowly as x increases. Other problems do not, and rather have a complexity equation that is an exponential function (eg. 2^x). These problems are considered 'hard' as the running time increases very quickly as x increases (see how big 2¹⁰ is, and then 2¹⁰⁰ if you don't believe me).

So where does P=NP fit into all this computational complexity theory? Well P-problems (or Polynomial problems / easy problems) and NP-problems (Not Polynomial problems / exponential problems / hard problems) both exist, and P problems are a subset of NP problems (ie. all 'easy' problems have ways of solving them which are 'hard', but not all 'hard' problems have ways of solving them which are 'easy'). 

Once the problem is solved however, another problem to verify the solution arises, and this second problem is also either 'easy' or 'hard'.

P=NP states that all problems that can be verified in polynomial time (ie. the second problem is 'easy') can also be solved in polynomial time (ie. the first problem has an 'easy' way of solving it, even if people don't know how yet). For example, if you make a program that, given a set of integers, finds if some nonempty subset of them sum to 0? For instance, does a subset of the set {−2, −3, 15, 14, 7, −10} add up to 0? The answer "yes, because {−2, −3, −10, 15} add up to zero" can be quickly verified with three additions (three steps). However, there is no known algorithm to find such a subset in polynomial time (there is one, however, in exponential time, which consists of 2ⁿ-1 steps) (Source of Example: Wikipedia)

P=NP would state that there must exist an algorithm to find the subset in polynomial time, as all problems that can be verified 'easily' can also be solved 'easily'.

There is great debate as to whether P=NP is true, with many mathematicians and computer scientists working on the project. The current majority lies in the P does not equal NP camp, who believe there exist problems that can be difficult to solve but easy to check (like the one above), but more than enough believe that P does indeed equal NP. A proof either way would have huge practical and theoretical implications in many fields including, of course, mathematics which is why this is hailed as one of the most important problems of the modern age. Any takers?

Hello World

Hello World,

Welcome to My Maths Blog (not the most inventive name I'm aware but it's late and it's the best I've got). My name is Anand (recently found out this is Sanskrit for 'joyful' :) ) and I am 16 years old. A quick bit about myself: I live in Hertfordshire, UK and attend a local school called Haberdashers' Aske's Boys' School. My favourite subject is mathematics (no, really), and has been for the last 2 years. A few non-academic interests of mine include sports (namely hockey, cricket and football), music, theatre and film, novels, economics and philosophy. I support Saracens RC, Spurs FC and Southgate Hockey Club (who I also play for in the junior section)....right, that's enough about me for now.

I've created this blog because I want to share both some of the 'original' (someone else has no doubt come up with them already, but by original I mean I haven't read/learnt it somewhere else) ideas I think of from time to time but seldom write down on paper, and the interesting mathematical tidbits I come across as I try to learn as much as I can about this awesome subject. Hopefully, this should be fun :)

How I foresee this will work is every so often I will post an article on some point of interesting maths. Comments either with questions (which I will do my utmost to answer), points of interest, developments or even just thanks would be greatly appreciated. So here goes...