thinking about issues of scale

I recently went deep back into the archives and read an old post on the number liberation front. The attempt at humor aside, there was a serious idea tucked away therein. Namely, that how we view a concept (in this case, a number) depends on how that concept is presented to us. In this case, the difference in presentation of a number as a word or as in digits, and the view that presenting large numbers as words (million, billion, trillion) acts as a shield to their true size and scale, given the similarity of the words used.

I can’t claim this is an idea that’s original to me and I suspect (though my reading here is still wildly incomplete) that this is something commonly known in psychology circles. Perhaps it’s related to issues of cognitive load and what our brains can properly handle, and I’m sure there’s a good idea for a story about first contact with an alien species, where we as humans don’t have the cognitive capacity to understand the aliens and what we then have to do to mitigate this.

But this is one facet of the larger issue of how we handle discussions and contemplation of scale, particularly when the scale gets very large. We develop tools to help us handle these, and one example of this that I find very interesting is how mathematicians handle the infinite. To some extent, the infinite is the ultimate scale problem, particularly when we touch on issues like the different sizes of infinity, the infinitude of infinities.

For this, we have developed a structure of notation and conceptual tools that allow us to manipulate and explore infinity, but there is a small part of me that wonders what we’re missing. Are there aspects of the infinite that we haven’t yet encountered, perhaps that are shielded from us by the very conceptual framework we’re using to explore the infinite. And this contemplation I find exciting and interesting, because there is always something more to do.

But this is only one small example. When we consider the world in all its glorious expanse, I find it hard to wrap my head around the whole of it, and to understand what direction to take moving forward, out of all the many possible directions. The scale issue here is that the space of all possible futures is a wildly massively high dimensional space, and we are navigating a path through this space.

There are lots of difficulties with this process of navigation, only a few of which I’m sure I have sight of at present. Do we for instance want our path to be a geodesic path, one that best reflects the changing geometry of this space of possible futures. But this requires that we’re able to get a handle on this changing geometry, and that then runs back into this issue of scale and being able to capture and reasonably manipulate the amount of quantity of information needed to understand this space.

And this is one of the things I most love about being a mathematician. We have the opportunity to explore such spaces and to develop the tools to understand such spaces, and there’s always another horizon over which to journey.

~ by Jim Anderson on 22 March 2022.

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