Mr. Wheeler explained, last night in class, that an integral is just the limit of a Riemann sum. He said that we’ll learn clever ways to integrate (e.g. by parts, the substitution rule, completing the squares, etc), but that in each case we’re really just finding the limit of the corresponding Riemann sum. This is all good because it turns out that its pretty hard to find the limit of the Riemann sum and relatively easy to integrate using these other techniques (once you’ve drilled them into your head).

To illustrate graphically: lim n->∞ ∑[i=1,n] f(x_{i})Δx = ∫f(x)dx

That thing on the left is, by definition, equal to the thing on the right of the equal sign. This is just messing with symbols. Like saying 1+1+1 is equal to 1*3 or like the little stick man on the doors of bathrooms. There is something to that symbol-play that sets our feeble little minds at ease, though. For some reason, the simplicity in this new set of symbols on the right of the equation make it easier to fiddle with integrals. To me, this is amazing. We haven’t introduced anything but a metaphor and it makes the job that much easier. Why does this magic happen?!

Is it that our brains can only process so many little symbolic units at a time? Anything that can compress the number of units can simplify the processing the brain has to do. With this understanding, the way to understand the integral metaphor is by counting the symbols… the metaphor reduces the number of characters in the Riemann sum (20) down to 7 for the integral on the right hand side. So introducing this new notation reduced the complexity that the mind has to deal with by a factor of almost 3.

The genius of inventing these metaphoric mind tools, the genius of Riemann, is that they make concepts like integrals easier for us mere mortals to grasp, not that they tell us something profound about the truth of integrals. If we were gods, omnipotent, we wouldn’t need these metaphors, the complexity of Riemann’s Sums would be transparent, and there would be no need for integral notation. This fact stands in strange contrast to our notions of the divine beauty of metaphors, the idea that they get us closer to the truth, don’t you think?

This contradiction begs another question: is it possible that metaphors get us further from the truth, rather than closer?

The thought that I have to keep distant, for now, is that if these metaphors are just tools, there is no truth in them besides the object of the metaphor, and they don’t help to uncover more deeply hidden truths, why do I need to learn them? I have to keep this question distant because I’m in the middle of taking a 10 week course where I’m learning those worthless metaphors!

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