Actually Leibniz invented the modern dy and dx notation and did view the differentials as genuinely nonzero, which is generally speaking a safe assumption. In other words, dy/dx really was a quotient, albeit of really tiny values (at least we assume dx can become arbitrarily small while remaining nonzero). The calculation Leibniz would do looked something like this. First he would consider an equation y = x^2. Then he would consider a nonzero difference so something like y + dy = (x + dx)^2 = x^2 + 2x dx + (dx)^2. At this point he would use his starting equation to subtract y from the LHS and x^2 from the RHS, leaving: dy = 2x dx + (dx)^2. Then he would divide by dx leaving dy/dx = 2x + dx and since dx is infintisimal, he would just lop it off. Suffice it to say, just ignoring the nonzero dx on the RHS, or casting it to 0 while conveniently ignoring the division by 0 on the LHS, was rather disturbing to many critics. A lot of work had to be done by Riemann, Cauchy, and Weierstrass over the following century after Newton and Leibniz invented calculus to answer the question you are asking.
I think the best way to understand this is that division by dx is always allowed. It is genuinely a nonzero quantity. Later, we think more in a more abstract way in a tiny neighborhood around (x,y), considering what happens to dy as dx becomes arbitrarily small, but it never vanishes entirely. That explains why we can say dy = 2x dx or dy/dx = 2x and both are completely true and reasonable. I think the author's argument is that d() is a bit easier to understand because we aren't dividing by dx but it makes no sense (to me) that way. If you cannot divide by dx, a nonzero number, then why not? And if you can, why doesn't dy/dx involve zero division, which is clearly not well-defined? I think answering those questions makes calculus a lot easier to understand and that they are in a sense the hardest questions. The notation this author uses doesn't really illuminate those points and the fact that the author realizes that he is basically teaching the students to accept zero division for most of the year suggests he is basically saying we should go back to a Leibniz-era approach to calculus. I would rather make rigorous what is meant by dx/dy and what exactly dx and dy are.
Personally I dislike hyperreals for the reason that we lose the all-powerful Archimidean property (for all real numbers x, there exists a natural number N such that N > x or N = x). But yes, hyperreals are completely consistent. Robinson showed, I want to say in the 1960s, that any statement about the logical consistency of the hyperreals is true if and only if the same is true for the reals. Then hyperreals basically save Leibniz by introducing the standard part function st(), which is rigorously defined.
Again, for me, this seems a little costly. If cookbook calculus is a struggle, the student probably won't take real analysis. And once you get to analysis, the fact that the reals are Archimidean grounds the whole endeavor on a very intuitive basis. By contrast, the hyperreals are not Archimedian so we can find omega bigger than every natural number. I don't personally feel like that helps intuition and indeed requires development of non-standard analysis but I admit it is pretty cool that Leibniz-style computations can be salvaged this way.
I think the cost is just a question of tradition what you find harder and the point author makes that it's easier to learn trough non-traditional way.
I see losing Archimedean property as more of a win than loss. It's a one simple thing that turns calculus operations into single algebraic evaluation, where you can avoid dynamic limits. Things become simpler and easier to prove.
Dual numbers are basically big/little O notation. They combine beautifully with Robinson's NSA to give the most 18th century-like approach to deriving integral/derivative formulas that I know. And it's fully rigorous!
Students have to memorize a diversity of processes for essentially performing the same task.
Is that true for differentiation ? I don't recall having to memorize many things, just how differentiation composes over +,-,×,÷, function composition and the differential of a few standard forms.
Symbolic integration, on the other hand, is a whole can of worms.
Comments
Actually Leibniz invented the modern dy and dx notation and did view the differentials as genuinely nonzero, which is generally speaking a safe assumption. In other words, dy/dx really was a quotient, albeit of really tiny values (at least we assume dx can become arbitrarily small while remaining nonzero). The calculation Leibniz would do looked something like this. First he would consider an equation y = x^2. Then he would consider a nonzero difference so something like y + dy = (x + dx)^2 = x^2 + 2x dx + (dx)^2. At this point he would use his starting equation to subtract y from the LHS and x^2 from the RHS, leaving: dy = 2x dx + (dx)^2. Then he would divide by dx leaving dy/dx = 2x + dx and since dx is infintisimal, he would just lop it off. Suffice it to say, just ignoring the nonzero dx on the RHS, or casting it to 0 while conveniently ignoring the division by 0 on the LHS, was rather disturbing to many critics. A lot of work had to be done by Riemann, Cauchy, and Weierstrass over the following century after Newton and Leibniz invented calculus to answer the question you are asking.
I think the best way to understand this is that division by dx is always allowed. It is genuinely a nonzero quantity. Later, we think more in a more abstract way in a tiny neighborhood around (x,y), considering what happens to dy as dx becomes arbitrarily small, but it never vanishes entirely. That explains why we can say dy = 2x dx or dy/dx = 2x and both are completely true and reasonable. I think the author's argument is that d() is a bit easier to understand because we aren't dividing by dx but it makes no sense (to me) that way. If you cannot divide by dx, a nonzero number, then why not? And if you can, why doesn't dy/dx involve zero division, which is clearly not well-defined? I think answering those questions makes calculus a lot easier to understand and that they are in a sense the hardest questions. The notation this author uses doesn't really illuminate those points and the fact that the author realizes that he is basically teaching the students to accept zero division for most of the year suggests he is basically saying we should go back to a Leibniz-era approach to calculus. I would rather make rigorous what is meant by dx/dy and what exactly dx and dy are.
Hyperreal numbers make Leibniz notation literal and algebraically consistent and rigorous, rather than a convenient shorthand for limits.
Author suggests using them. Resistance to Hyperreals seems to come from era before there were rigorous definitions for them.
Personally I dislike hyperreals for the reason that we lose the all-powerful Archimidean property (for all real numbers x, there exists a natural number N such that N > x or N = x). But yes, hyperreals are completely consistent. Robinson showed, I want to say in the 1960s, that any statement about the logical consistency of the hyperreals is true if and only if the same is true for the reals. Then hyperreals basically save Leibniz by introducing the standard part function st(), which is rigorously defined.
Again, for me, this seems a little costly. If cookbook calculus is a struggle, the student probably won't take real analysis. And once you get to analysis, the fact that the reals are Archimidean grounds the whole endeavor on a very intuitive basis. By contrast, the hyperreals are not Archimedian so we can find omega bigger than every natural number. I don't personally feel like that helps intuition and indeed requires development of non-standard analysis but I admit it is pretty cool that Leibniz-style computations can be salvaged this way.
I think the cost is just a question of tradition what you find harder and the point author makes that it's easier to learn trough non-traditional way.
I see losing Archimedean property as more of a win than loss. It's a one simple thing that turns calculus operations into single algebraic evaluation, where you can avoid dynamic limits. Things become simpler and easier to prove.
This is effectively how dual numbers work! https://en.wikipedia.org/wiki/Dual_number
Dual numbers are basically big/little O notation. They combine beautifully with Robinson's NSA to give the most 18th century-like approach to deriving integral/derivative formulas that I know. And it's fully rigorous!
True. Knuth had a paper where he suggests using (a slightly modified) O notation for teaching calculus.
https://www-cs-faculty.stanford.edu/~knuth/calc
From the Arxiv paper:
Is that true for differentiation ? I don't recall having to memorize many things, just how differentiation composes over +,-,×,÷, function composition and the differential of a few standard forms.
Symbolic integration, on the other hand, is a whole can of worms.