We sometimes imagine that our learning-journey is like a path that stretches away in front of us and leads towards a fixed destination, but what if there is no path and even less a fixed destination? What if knowledge is like a fractal that reveals more and more of itself and - far more significantly - creates more of itself at every stage?
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.
When we learn to count as tiny children we hardly have any sense of the rich world of possibility that hangs on that simple, basic skill. Fraction and deminals and percentages and primes, algebra, equations, geometry and shapes: these all lie beyond our horizon. And at every stage, as we grow and explore these mysteries, we find that there are limitless riches. We never ‘leave simple numbers behind’ because they are endlessly fascinating, rich and perplexing.
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.
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