Question

Is the derivative an approximation, or, is it non-existent?

Answer 1

if the function is not continuous at a certain point (like it skips an x-value) then it has no derivative at that point additionally if there is like a spike in the graph (not like a curvy turning point but a sharp spike) there is no derivative....all linear functions have derivatives

Answer 2

From the lecture, "x is fixed, delta x moves". Since the derivative always involves delta x, and delta x always moves, then there is no "certain point" that is the derivative. This seems to imply that the derivative does not exist. That is, the derivative is a verb, not a noun, a process, not a thing. Of course, it can be reified by calling it an approximation. Yet, as an approximation, it is not the real thing, not the derivative.

Answer 3

The derivative as precise as any number, so there is no sense in which it is an approximation. Likewise, it exists in the same sense that numbers and functions exist. Perhaps the key point to understand is that the derivative is a _limit_. In conceptualizing the limit, we may think of delta x as moving, but the derivative at a given point is a limit, a precise number that does not move.

Answer 4

From the lecture: "The left hand side, lim f(x)/x->xzero = F(xzero), is completely independent, is evaluated by a procedure that does not involve the right hand side…They are separate things".

The left hand side has an identity that is "evaluated by a procedure that does not involve the right hand side".

"From the lecture, "you always avoid the limit point, its like a paradox…the tricky part of the definition of a limit." The left hand side, its identity, only exists within a dimension of reality different from the right hand side. The derivative is not the limit. It does not have an identity in the same way a number does.

Pi, sqrt2 are also irrational. They do not exist in the same way a rational number does. That's why the length of the diagonal of a triangle with sides of length 1 was such a problem for the ancient Greeks. The length of the diagonal did not exist in the same sense as the length of the sides. The problem was ignored, glossed over by the lecture comments like, "its a paradox…its tricky".

I agree that the limit, the right hand side exists, but not in the same way as the left hand side. There are different (unacknowledged) dimensions of reality.

Answer 5

The issue of what it means for a number to be real has a rich history in the realm of philosophy of mathematics. I believe it's more accurate to say, though, that the reason Pythagoreans had problems with the square root of 2 is that it DOES exist. They wouldn't have been troubled by a nonexistent number, but the existence of a number that couldn't be represented as a ratio of integers challenged the exalted position integers held in their philosophy.

More pertinent to this course, in the early days of calculus the methods had no secure grounding, leading some to challenge their validity despite their obvious effectiveness. It was only much later, with the development of a clear conception of the limit, that we could do calculus without a bit of magical hand-waving.

Answer 6

The existence of irrational numbers implies there are still holes in the number line that cannot be described as a ratio of two integers. Natura non facit saltus (nature does not make jumps) Leibniz (New Essays, IV, 16). Of course you can argue that just because they cannot be represented by integer ratios, that does not mean they don't exist, that they do in fact have identities. But then, you would necessarily have to change the mode of identity. Following Leibniz's axiom, the square root of 2 does not exist in nature, does not have an identity in the same way that "2" has. He waved his magical hand and created the discrete out of the continuous. His infinitesimal calculus was the practical solution to this problem. So, if sqrt2 does exist, it exists elsewhere. It is in a different (unacknowledged) dimension of existence.

The issue here, with respect to the Philosophy of Mathematics, can be found in Logic, and the The Liar Paradox. There seem to be 2 levels of existence. There is the objective (the limit-ed limit), and the subjective (the un-limited derivative.) The Liar can be said to equivocate between the objective perspective, what it says, and the subjective perspective, what you think it says. Unfortunately, like many other philosophers and scientists, you have glossed over this difference. You have therefore equivocated between different dimensions of existence, the objective and the subjective.

This also suggests, to me, why science is having such difficulty coming to terms with such issues as "Consciousness" and "Quantum Mechanics". I think it would help if you go back and find out what Leibniz had to say about it in his "Monadology". Monads exist, but are not physical. God exists, but is not physical, etc.