Return to Question

Differential geometry is a rich source of examples where salvaging Leibniz's formalism could simplify many proofs. Beyond being able to say that the center of curvature is the point of intersection of two infinitely close normals, numerous arguments in Gauss and Riemann are simpler in their original infinitesimal form (including formula for curvature) than their modern reformulations; see discussion in Spivak, Differential geometry, volume 2, and related thread https://math.stackexchange.com/questions/890125/what-meaning-did-riemann-assign-to-dx

Differential geometry is a rich source of examples where salvaging Leibniz's formalism could simplify many proofs. Beyond being able to say that the center of curvature is the point of intersection of two infinitely close normals, numerous arguments in Gauss and Riemann are simpler in their original infinitesimal form (including formula for curvature) than their modern reformulations; see discussion in Spivak, Differential geometry, volume 2, and related MSE thread.

Differential geometry is a rich source of examples where salvaging Leibniz's formalism could simplify many proofs. Beyond being able to say that the center of curvature is the point of intersection of two infinitely close normals, numerous arguments in Gauss and Riemann are simpler in their original infinitesimal form (including formula for curvature) than their modern reformulations; see discussion in Spivak, Differential geometry, volume 2, and related thread http://math.stackexchange.com/questions/890125/what-meaning-did-riemann-assign-to-dx

Differential geometry is a rich source of examples where salvaging Leibniz's formalism could simplify many proofs. Beyond being able to say that the center of curvature is the point of intersection of two infinitely close normals, numerous arguments in Gauss and Riemann are simpler in their original infinitesimal form (including formula for curvature) than their modern reformulations; see discussion in Spivak, Differential geometry, volume 2, and related thread https://math.stackexchange.com/questions/890125/what-meaning-did-riemann-assign-to-dx

Berkeley claimed that Leibniz wanted to have it both ways: both dx\not=0 so as to form the differential ratio, and also dx=0 so as to get the right answer (i.e., a "standard" one).

The reason Berkeley was wrong is that Leibniz repeatedly emphasized that he is working with a generalized notion of equality. For example if y=x^2, the desired formula dy/dx = 2x does not mean that the residual dx is set equal to zero but rather that it is absorbed into the generalized relation of equality "up to" a negligible term, in an exact sense to be specified. Berkeley did not take this into account and merely misunderstood Leibniz.

In Lawvere's approach (also Kock, Bell) they replace the ratio formula f '(x)=dy/dx by the multiplication formula dy=f '(x)dx. Then they get equality on the nose by working with nilsquare infinitesimals. Thus their adequality is true equality on the nose. In this way they implement (some of) Leibniz's procedures. However, this is not entirely faithful to Leibniz because Leibniz worked with arbitrary order infinitesimals, and also divided by them freely.

In the specific case of y=x^2 the problem is the zero of the derivative where Euler's geometric equality does not work, but at any other point we are OK.

Can one redefine the relation "=" in a suitable context, so that for example one could read the chain rule as literally saying $\frac{dz}{dx} = \frac{dz}{dy}\frac{dy}{dx}$ ? This works with Euler's "geometric equality" but the problem is you can't add such equalities. Already chain rule with 2 variables requires addition.

Berkeley claimed that Leibniz wanted to have it both ways: both $dx\not=0$ so as to form the differential ratio, and also $dx=0$ so as to get the right answer (i.e., a "standard" one). Starting about 140 years ago, Berkeley's claim of inconsistency of Leibnizian calculus acquired the status of dogma to such an extent that Robinson himself felt compelled to speak of Berkeley's "brilliant critique" of the calculus, and referred to the hyperreal framework as "a small price to pay for the removal of an inconsistency"--the implied assumption being that such an "inconsistency" was real.

The reason Berkeley was wrong is that Leibniz repeatedly emphasized that he is working with a generalized notion of equality. For example if $y=x^2$, the desired formula $\frac{dy}{dx} = 2x$ does not mean that the residual $dx$ is set equal to zero but rather that it is absorbed into the generalized relation of equality "up to" a negligible term, in an exact sense to be specified. Leibniz called this principle the transcendental law of homogeneity. The principle is mentioned, for example, in the title of his 1710 paper, as reported already in 1974 by Bos. Berkeley did not take this into account and merely misunderstood Leibniz.

In Lawvere's approach (also Kock, Bell) they replace the ratio formula $f '(x)=dy/dx$ by the multiplication formula $dy=f '(x)dx$. Then they get equality on the nose by working with nilsquare infinitesimals. Thus their adequality is true equality on the nose. In this way they implement (some of) Leibniz's procedures. However, this is not entirely faithful to Leibniz because Leibniz worked with arbitrary order infinitesimals, and also divided by them freely.

In the specific case of $y=x^2$ the problem is the zero of the derivative where Euler's geometric equality does not work, but at any other point we are OK.

Can one redefine the relation "=" in a suitable context, so that for example one could read the chain rule as literally saying $\frac{dz}{dx} = \frac{dz}{dy}\frac{dy}{dx}$ ? The relation can be rewritten in a simpler form in terms of differentials: $dz = \frac{dz}{dy}dy$ but this still depends on the "cancellation" of $dy$ in the numerator and denominator. This works with Euler's "geometric equality" but the problem is you can't add such equalities. Already chain rule with 2 variables requires addition.