We are accustomed to thinking that physical objects have “positions” which we can come along and measure. We think of the position of a thing as being a property of the object and as existing to an infinite degree of accuracy, even though we never measure positions to an infinite degree of accuracy.
But is this view of position really correct? Is it possible that measurements of position are not one-shot events that try to measure something infinitely precise “as well as possible”, but rather iterative processes that in themselves determine the positions of objects?
Measuring the position of an object might involve first pinning the object down to some particular space; we must make some measurement that convinces us that the object to be measured exists to some degree of probability within some particular volume. We then make further measurements, determining that the object lies within the scope of our measurement, that it lies between certain rough borders, and then determining ever more precisely which two other objects in each dimension it lies between.
In quantum physics our view of the nature of position and measurement has caused us to create infinite-dimensional mathematical spaces in order to deal with the phenomenon of position. We cannot say exactly what causes us to even think that an object has a position, and we cannot currently break the measurement of position down as an iterative process, into a series of steps. Perhaps one day we might manage to do this, just as Einstein broke down the measurement of length into certain very rough initial steps involving time. When that day comes, we may finally be able to rid quantum mechanics of infinite mathematical spaces.