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calculus and how we use the notion of the possible infinite to get an essential insight; the concept of truth

The concept of the integral is introduced by asking the student to imagine breaking the area under a curve into smaller and smaller parts (each of which is a column whose width is delta x or change in x) and then figure out how, as delta x (the change in the value of x) approaches zero and then number of columns (each of whose width is delta x) approaches infinity, ... how one better forms an approximation of the area under the curve.  This series of approximations is then superseded by an insight into another a formula that can more simply calculate for the area of that curve.

It might be helpful to see this process as illustrative of how we use our imagination to stretch our experience and thereby arrive at new insights that go beyond possible experience.  Also, it illustrates how the notion of the infinite is at work in the formation of new concepts.

Let's try to apply it to our concept of truth: we think that claim which is obvious to ourselves and those we know as true.  We might stretch that to include folks who have come before and those who have come afterwards, and believe or know that they would agree with us the truth of the claim we have in mind.  We might stretch it further to any sort of rational being, including those we lack the imagination to conceive of.  We might stretch it further to any possible rational being would think about that same claim.  To say that this claim is true in the strongest sense of that word is to relate it to all this and say, yes, they would recognize what I recognize when I say that it is true.

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