I have wanted to start writing a series on mathematical concepts in an alphabetical order. Finally, I will start Mathematics A – Z. On our Facebook page, I asked what concepts or ideas you want to see that start with A. A couple of people were asking for Axioms… so here it is: Axioms for A.
General definition: An axiom is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from a Greek word (axíōma), which means ‘that which is thought worthy or fit’ or ‘that which commends itself as evident.’
Thinking about AXIOMS, you realize how important they are for everything in the mathematical world. All the topics in mathematics need this wonderful, interesting and still not fully understood things. Sometimes, it feels like we [mathematicians] cannot live without them.
Historically speaking, the first person that came up with this genial idea was Euclid. Around 350 B.C, he wrote the set of axioms for the plane geometry. Using these axioms he proved every theorem for plane geometry. All of this is presented in his book- collection “Elements” – one of the most successful textbooks in the history of mathematics. It is well known that he was not the first mathematician to have proven all of those statements, but his organization and clear style made the work so significant. After so many years, we are now depended on those axioms for our day to day life: construction, architecture and much more. Axioms are to mathematics more important then a mobile phone (or a smartphone) to our society, because mathematics without axioms is nothing, null.
Creating New Universes
The beauty of these axioms is the fact that they are the building blocks of any topic. So what happens when you change the color of one brick? Or what happens when you remove a brick from your tower? In the second case, you might think that the tower is completely destroyed, but it’s not always the case. It will not crumble just from one small brick. What if you put another more stable brick in its place? Well, you get a more stable and different tower. This is exactly what mathematicians have done.
Think again at the axioms for plane geometry: one of the most important is related to parallel lines. In modern vocabulary, the parallel axiom sound like this: “In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.” Mathematicians have played with this axiom so much that they came up with multiple different ideas, thus creating new geometries. For example: no parallel axiom created the hyperbolic geometry.
“[…] this same effect applies to any system of mathematics: changing the underlying axioms opens up a new mathematical realm where different rules come into play.” (David Darling and Agnijo Banerjee – “Weird Maths at the End of Infinity and Beyond“)
Breaking the Tower
It is strange to think that mathematicians are so dependent on somethings for all their work. Can you imagine how devastating it can be when their lovely tower is just blown out by someone that claims that THE Axioms are not good enough? Even more… it becomes more badly when he can actually prove it. This is the story of Bertrand Russel and the inconsistency of set axioms. For anyone that knows about it: the Russell Paradox is wonderful and devastating at the same time. Russell’s Paradox is about sets in Set Theory; it is sometimes presented as the Barber Paradox:
“[…] the village barber, who shaves everyone who does not shave themselves. Who shaves the barber? If he does shave himself, then by definition he is shaved by the village barber – himself! If he does not shave himself, then he is shaved by the barber – which again is himself.” (Ian Stewart – “Taming the Infinite“)
“Of all the things that can be wrong with an axiom system, inconsistency is absolutely the worst. It is possible to work with axioms that are hard to understand. It is possible to work with axioms that are counter-intuitive. And all might not be lost if your axioms don’t accurately describe the structure you intended to capture – maybe they will find some other application, as has happened on more than one occasion. But an inconsistent set of axioms is completely useless.” (Keith Devlin, “The Language of Mathematics – Making the invisible visible”)
How do they know it is true?
Besides the Russell Paradox, that shocked me first time I heard about it, there was something else that bothered me a lot. Every single time a teacher wrote an axiom on the board or I was reading an axiom from a textbook, there was one question that always came through my mind: “How do they know it is true?”. Even to this day, I ask myself the same question every single time.
After all this trouble I was giving myself, I read the following:
“a young Austrian mathematician named Kurt Godel proved a result that was to change our view of mathematics forever. Godel’s theorem says that if you write down any consistent axiom system for some reasonably large part of mathematics, then that axiom system must be incomplete: there will always be some questions that cannot be answered on the basis of the axioms.” (Keith Devlin, “The Language of Mathematics – Making the Invisible Visible”)
At first, the shock was huge!! I was like WHAT???!!!! Are you joking?!!! I screamed inside and I had to read the whole paragraph all over again. This really needs to be a trick or something, because it just does not seem natural, it does not seem correct. On the other hand, I thought again about Russel’s Paradox and maybe, MAYBE, I can give this idea a chance. Maybe, just maybe, it fits in this hole “axiomatization game”.
“The incompleteness theorems are analogous in some ways to the uncertainty principle in physics in that they expose fundamental limits to what can be known. And like the uncertainty principle, they’re including purely intellectual reality – behaves in ways that prevent us from being omniscient about everything that we try to penetrate with our minds. To put it bluntly, truth is a more powerful concept than proof, which, for a mathematician especially, is anathema.” (David Darling and Agnijo Banerjee – “Weird Maths at the End of Infinity and Beyond“)
Hope you liked this post and you find it useful. Have a great day. If you have ideas for future blog posts, let us know. Don’t forget to check our last post: Weird Maths | Book Review. You can find us on Facebook, Tumblr, Google+, Twitter and Instagram. We will try to post there as often as possible. Enjoy the day!
Lots of love and don’t forget that maths is everywhere! Enjoy!
Here are some short reviews for some of the books mentioned above: