The problem of differential calculus

A discussion of geometry and its lack of basis in phenomenal reality reminded me of this problem, mentioned in passing by an authority on the matter, to which I elucidated a little.

The problem of differentiation from first principles: It appears at first that division by 0 is a problem (cue dunning-kruger scoffing), though mathematicians (or those who use maths) tend to brush aside this and tell you to go and learn about “limits” and there is no problem. This says (presumably hard mathematicians write this stuff, Penguin dictionary of mathematics):

“Differential calculus is concerned with the rates of change of functions with respect to changes in the independent variable. It came out of problems of finding tangents to curves”

A change in the independent variable necessitates a non-zero difference between 2 points on a curve, hence we have a chord rather than a tangent. A tangent to the curve seems satisfactory to provide an exact value for the slope of the curve at one point, whereby there is a change of ‘direction’ of points entering and leaving the point in question. However, this is exposed as fallacious: we require an infinitesimal approach of neighbouring points, there can be no variation in the direction of such points.

They go on to say:

“In the 1820s, Cauchy put the differential and integral calculus on a more secure footing by using the concept of a limit. Differentiation he defined by the limit of a ratio”

Limit is defined as:

“A value that can be approached arbitrarily closely by the dependent variable when some restriction is placed on the independent variable of a function”

The example is given of ‘the limit of 1/x as x tends to infinity is 0’.

As is clear in this example, the limit will not be reached by the dependent variable. “Arbitrary” is used to describe the lack of a determined value of separation between the DV and the limit, it’s just really really close. So we do seem to merge, alas fudge, the necessity of a change in the IV while also reducing the delta to 0 in the algebra. Not sure about a ‘sure footing’. Perhaps this is why the idea of ‘linear approximation’ is used. The arbitrary limit is not actually reducing the delta to 0 as the algebra would suggest.