Knowledge is a Fractal

We sometimes imagine that our learning-journey is like a path that stretches away in front of us, and no doubt that is to some extent the case, but it is also like a fractal that reveals more and more of itself at every stage the more closely we examine it.

We might think, for example, that we learn the basic numbers 1, 2, 3, 4, ... when we are young and move on to fractions and irrationals and perhaps even complex numbers when we grow older, and that is true for many of us, but it is a mistake to imagine that we 'leave the basic numbers behind' when we 'put away childish things'. The basic numbers - the positive integers 1, 2, 3, 4, ... - continue to reveal deeper and deeper properties the more we study them.

• We might first learn to distinguish between odd and even numbers;

• We might come across prime numbers divisible only by themselves and one;

• We might come across numbers to a different base than 10 such as in binary arithmetic entirely built around 0 and 1.

• We might come across the notion of a remainder on division such as when we divide 13 by 3 and get 4 remainder 1.

• The notion of a remainder might lead us to modular arithmetic and number theory.

• We might find that there are tests for whether a number is prime, and then discover that some numbers pass those tests even though they are not prime, which leads to the notion of a 'pseudo-prime'.

• Sooner or later we will learn that integers are uniquely factorisable into prime numbers.

• We will have encountered 'The Fundamental Theorem of Arithmetic', that the factorisation of any number into prime factors is unique up to reordering (meaning that 2 x 3 and 3 x 2 are treated as the same).

• Then we might discover that these prime factors are the basis for some kinds of cryptography.

• And we learn that very large composite numbers that are made up primes may be so large and their factors so obscure that it would take the fastest computer longer than the age of the universe to find what they were.

And we are still dealing with basic numbers even though some of the things we might be interested in lie at or beyond the frontiers of human knowledge. Goldbach's Conjecture is simple to state: every even number greater than 2 is expressible as the sum of two prime numbers; but nobody can prove it, and a reward of $1,000,000 to anyone who succeeds in doing so remains unclaimed.

All this illustrates our claim: knowledge is a fractal; it endlessly enriches itself as we penetrate further and further into its depths; there is literally 'no end to our exploring' even if T.S. Eliot appeared to say otherwise.

Learning When Enough is Enough

If knowledge is a fractal, we can never exhaust it; we can never reach the end of our exploring; however much we may know, there will always be more to know.

So if complete knowledge is impossible, when should we be satisfied with what we know? When is 'Enough Enough'?

Claude 3 Opus had a nice way of putting it: sooner or later we have to abandon the search for new knowledge because knowledge is not important for its own sake; it must in the end lead to action, to some practical application or some form of enlightenment. We have to decide to give up on the search for new knowledge and apply ourselves to using the knowledge we have.

Clearly what determines whether 'Enough is Enough' is the context. A professional mathematician will want to pursue number theory further than someone only needing to count beans or their change in a shop. But everyone has to stop eventually. Once Andrew Wiles had proved Fermat's Last Theorem, that x^n+y^n=z^n has no solutions in integers if n>2, there was still the further question whether there might nevertheless be solutions to similar equations with more variables, but everyone has to stop somewhere.