Inspirational math

Posted: May 20, 2023 | Author: Einar | Filed under: Uncategorized |Leave a comment

It’s pretty cute that self-help people have discovered exponential functions. Just improve 1% every day, and you will have improved almost 38-fold by the end of the year! Because you see, 1.01^365 is 37.78. Wow! So inspirational!

But what does it mean? What does it mean for me to improve 1%? What is it that I improve? My creativity? My output? My writing? My talks? My interpersonal skills? Something else? Is it everything I do? Because that sounds like a lot, to get 1% better in total, across everything I do. I’d spread myself thin, wouldn’t I? So I guess I must choose, but what will happen to my other skills then? Because skills in general need to be maintained, right? Will they somehow stay unchanged? Or will they drop 1% per day? But that would be terrible – an accumulated skill loss of more than 97% by the end of the year!

Where did the 1% number come from? How much does 1% improvement entail? Is it realistic? I guess it’s supposed to sound like a minuscule improvement, something easily achievable, but is it really? How can we tell? In fact, if 1% is so easily within our reach, why stop there? Why not put in some extra effort and go for 2% improvement? That would yield a truly mind-blowing 1377-fold improvement!

But that’s obviously ridiculous. We can’t just arbitrarily decide to improve by 2% every day! That’s absurd! But how is 1% any different? In fact, how can we be sure that the improvement – whatever it is, and whatever it even means – is not something like 0.1% instead? It makes for less impressive math of course – 1.001^365 is approximately 1.44. It is still a significant improvement though! To get 44% better at something in a year (even if I’m not entirely sure what that means unless I have a very clear metric, like time spent on something, or weights lifted or whatever) is truly remarkable in many cases. But of course it could be even bleaker. Maybe daily improvement is unrealistic, and we have to settle for weekly? We’re down to 5%. I’m not sure about you, but I’m getting less inspired. But which calculation is most realistic? How can we tell?

But it gets worse. Is it likely that we’ll be able to improve any skill or process consistently by any fixed percentage over time? If you consider an athlete, do you think they can run 1% faster every day? Or even 0.1%? Most things don’t improve that way at all! Improvement tends to be easiest in the beginning. As you get better, it gets harder and harder to improve. Even with deliberate practice, it is very hard to improve by a fixed amount every day – not to mention an exponentially growing amount! After all, 1% at the end of the year is a much larger improvement than at the beginning – that’s the whole point of exponential functions! That growth of daily increment is where the impressive 38-fold improvement comes from! It makes you wonder if exponential functions really are a good fit for most optimization problems.

Of course, continuously trying to make things a little better, or to learn something new every day, remains good advice. But we don’t need to dress it up in arbitrary, ill-fitting math to make that point.