Modern mathematics has enabled humans to develop in very sophisticated ways. It has allowed the use of technologies to cure diseases, build complex electronic infrastructures and assist in space exploration. Whilst the crux of mathematics is embedded in the ‘number’, we still do not know how to find and prove the existence of almost every number that exists.
Philosophical statements like the above arise from confusions regarding infinity, and how there are different ways to perceive the size of something that is absurdly sizeless. Many Mathematicians of the past and the present have investigated the wonderment of infinity and have created thought-provoking experiments that still perplex people today.
One such Mathematician is David Hilbert, a German Mathematician born in 1862. Starting school two years later than most German children, at the age of 8, Hilbert did not thrive in his early years at school. He didn’t enjoy the task of memorising lots of information, which is surprising for somebody who has now become a prolific Mathematician who has developed theories across many different disciplines of Mathematics.
After entering the University of Königsberg, he soon formed a great friendship with Hermann Mikowski, a fellow student studying mathematics. Mikowski helped Hilbert with his University research, and together they supported each other’s ideas and study. After graduating and becoming a working mathematician, Hilbert spent a lot of his life researching and developing new tools in Mathematics that are still used today. In 1900 Hilbert proposed 23 different Mathematical problems (“Hilbert’s Problems”) that hadn’t been solved at the time. In the century following, all but 4 of these problems have been partly solved, each signifying great Mathematical achievement. Twenty-four years later after these problems were proposed, Hilbert gave a lecture that mentioned a thought experiment, now called Hilbert’s Paradox of the Grand Hotel.
To give some insight into his lecture, we must know what infinity means in the terms of the Grand Hotel. The counting numbers, “one, two, three, …”, also called the natural numbers that repeat forever are in a class of infinity called “countably infinite”. We could count them one after another, even if it reaches into infinity (and beyond). Any set of numbers is countably infinite if we could list them in an order and uniquely draw lines between them and the natural numbers. That is, matching each number in one set to exactly one from the other. We call this a 1-1 (“one-to-one”) correspondence between two sets, and we say that they have the same “cardinality”.
Hilbert’s idea described a popular Hotel, in which there are a countably infinite number of rooms already occupied. One guest arrives late and requests to be put in a room, but there is no empty room at the “end” to put him in. The hotel requests the guests all move into their adjacent room, allowing the new guest to check into room number 1. Later on, a coach arrives carrying a countably infinite amount of guests looking to get some sleep. They all need a room, but fitting them in seems like an impossible task, until one suggests the current guests all move to the room whose number is double their own. The new guests can then be ordered and slotted in between. Whilst a very popular destination, Hilbert’s hotel doesn’t get very good reviews as guests often complain about being moved around. If you have an eternity to read all of them, that is!
What this seems to imply is that there is an equal amount of even numbers as there is even and odd! In Mathematical terms, we say that the cardinality of the set of all even numbers is equivalent to the cardinality of the set of all of the natural numbers. No matter how many more guests you bring in, as long as they are countably infinite, you will be able to fit them into this Grand Hotel that has every room occupied.
The other, scarier kind of infinity is that called “uncountably infinite” which cannot be listed out in any kind of order. It was first proven in 1874 by the German mathematician Georg Cantor that the real numbers, that is every number that exists in the number line you would recognise from school maths lessons, is an uncountable set. This idea was met with much controversy, as people didn’t believe that there could be anything “bigger” than the cardinality of the set of all counting numbers. David Hilbert strongly supported Georg Cantor’s ideas, recognised his genius mind and described his support with the quote
“No one shall drive us from the paradise which Cantor has created for us”
Many of the numbers that Mathematicians have been able to prove the existence of, fall within a countable set. That is, all counting numbers, all rational numbers (those that can be expressed as a ratio of two integers) and even more complicated numbers, such as the surds (for example, the square root of 2). If these all fall within a countable set, but the real number line continuum is uncountable, there is an uncountable set of numbers that contains other numbers not in one of these countable sets. One reason we cannot find these numbers easily is that they are infinitely precise, and as such to describe these numbers in decimal form would take an infinite amount of time and paper. One uncountable family we know of is called the “transcendental numbers”, where our favourite number celebrities, (pi) and (Euler’s constant to describe compound interest growth) can be found.
Transcendental numbers are those that are not “algebraic” i.e. cannot be used with algebra to solve polynomial equations with integer coefficients. (For example 2x+3=0 or x2+x-2=0). Whilst almost all numbers are transcendental, we know the existence of such a small amount of them that the numbers we do know take up no space at all in vast sea of real numbers. Trying to fit the Transcendental numbers into Hilbert’s Hotel would be an impossible task as we would have no way of ordering them, which is going to leave a lot of people unhappy and without a room for the night.
There are different number systems that we have created to either aid in real life application or more theoretical pursuits. The complex numbers were found to fill the hole left in algebra created with the roots of negative numbers. The quaternion numbers that are an extension of the complex numbers are used to assist in computer game graphic design. More abstract systems such as the hyperreal numbers and the surreal numbers extend what are common number analysis techniques into the weird and wonderful. In a world of numbers real and imaginary, knowing that we don’t know almost all of them and probably never will, hopefully leaves the wonder surrounding mathematics ever present for years and eternities to come.