The simpler and more fundamental our assumptions become, the more intricate is our mathematical tool of reasoning; the way from theory to observation becomes longer, more subtle, more complicated. Although it sounds paradoxical, we could say: Modern physics is simpler than the old physics and seems, therefore, more difficult and intricate. The simpler our picture of the external world and the more facts it embraces, the stronger it reflects in our minds the harmony of the universe.
These wise words were written by Einstein & Leonard in 1967. And since then mathematicians still work to boil down our understanding to some mysterious essence.
Let us suppose, not only that irrational numbers (which Pythagoras feared), but also that complex numbers exist. If we accept this, we not only open a pandorean box into infinite problems of a "known" size, but a door of perception beyond our "normal" minds.
Ask yourself this very simple question:
If we add all positive numbers, what will be the sum?
The most intuitive answer is that the sum will be an infinitely large number, "∞" (lemniscate)?
However, reality is a bit more complex than our (normal) minds can (easily) comprehend. In fact, the answer to our question is -1/12 (!).
Leonhard Euler and Srinivasa Ramanujan showed us that: 1 + 2 + 3 + 4 + 5 + ... = -1/12.* So, adding all positive numbers gives us minus one twelfth.
Why is this so? Nobody (except Ramanujan?) really knows, but apparently -1/12 is what is left when you throw away all the "infinite garbage", and so in some mysterious way, this number is what the sum of all positive numbers boils down to.
If the doors of perception were cleansed every thing would appear to man as it is, Infinite. – William Blake
Ok, so far so good, but this was not where we were headed for. Let us now take a look at the very nature of this exploration into numbers itself. If we study Gödel's theorem, we find that his logical proof indicates that there are countless true statements that can never be proven, and vice versa. This perhaps indicates that the spider that produces the web of thoughts can never free itself from the same web. If so, we have a limited ability to understand the source of our thoughts, because our attempts of understanding them are themselves thoughts. It is hard for us to invent a thought that thinks itself – we do not call spelling the word "spell" second-order spelling.
What would be a practical implication of this? Well, for instance, Mandelbrot showed mathematically that a coastline's measured length changes with the length of the measuring stick used: So, there are no definite length, but knowledge only through objects of measure.
This is perhaps disappointing, but at the same time it opens up an universe of endless discovery. Stephen Hawking wrote it in 2002: "Some people will be very disappointed if there is not an ultimate theory that can be formulated as a finite number of principles. I used to belong to that camp, but I have changed my mind. I'm now glad that our search for understanding will never come to an end, and that we will always have the challenge of new discovery. Without it, we would stagnate. Gödel’s theorem ensured there would always be a job for mathematicians."
So, why should we care? Well, because as Roger Penrose wrote: “When we convince ourselves of the validity of Gödel’s theorem we not only see it, but by so doing we reveal the very non algorithmic nature of the seeing process itself”. It is a way to clean the doors of perception.
* From this rather unexpected equation we derive the 26 "dimensions" of reality in M-theory, and this "-1/12" is also used many places in physics. Which demonstrated that it does indeed "exist" somewhere. Further, if you add all positive numbers cubed (^3), Euler showed that the answer is 1/120. But, sorry to say, if you summarize all positive numbers squared, the answer is "nothing" – e^ipi+1 (i.e., 0).