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 )