Friday, September 14, 2012

what does "continuous" mean in real life?

A very interesting and important point was raised (more than once) during the discussion in one of my calculus classes today. Essentially it asked, “Why can we assume that certain functions, like height or weight or cost, vary continuously? After all, isn’t money measured in cents, and mass in atoms? Aren’t these indivisible? But we know that numbers can always be considered on a smaller scale—remember Zeno's paradox—and so how can anything that changes in fixed amounts, no matter how small, be changing continuously?”

There’s a lot in that set of questions, regarding the philosophy of what numbers really are, how well they match the world we inhabit, and whether we can reasonably conclude anything about the world based on fanciful mathematical flights. I’m going to try to give two answers to the questions above. You may find one more convincing than the other. You may not like either, and end up finding your own. But the question cannot be disregarded lightly.

First, we use continuous models for physical systems because doing so works. We study the world, try to abstract from it, and pick the easiest mathematical model that fits the data (until it doesn’t). And somehow this produces results. We make simplifying assumptions about complicated questions, and as long as the mathematical model keeps matching what we observe, we continue to use it, along with its simplifying assumptions (like true mathematical continuity). We may need to revisit those assumptions later, but the evidence points to the effectiveness of these models, and so it’s worth hanging on to them, even if philosophically they don’t seem to be taking every factor into account.

Second, one could argue that any mathematical model only exists in our heads at the level of detail we can currently imagine, not at arbitrarily small scales—all those extra numbers filling up the space don’t really exist in the pure mathematical realm any more than they do in the physical world. In this sense, “continuity” is nothing more than a convenient fiction. What we look at can never be actually continuous, mere functionally so. We could claim that the height of a person must always increase by at least an atom at a time, but generally our methods of measuring are too coarse to detect this. When we say that a particular phenomenon is continuous, what we really mean is “continuous within a margin of error”. Those tiny little changes are practically (and I really mean “in a practical sense”) nothing in comparison to the large scale we're considering. From this perspective, practical continuity, or continuity neglecting tiny details, is enough to make the math and science work together.

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