3 ms·

Got to the place where he says "As you can see, this is identical to the d/dx() operation except that the result is not divided by dx." What does it mean with

Got to the place where he says "As you can see, this is identical to the d/dx() operation except that the result is not divided by dx." What does it mean with his d() operator to "divide by dx"? All of a sudden it seems like he has changed dy/dx from unfortunate notation that sort of looks like a division into something that actually is dividing two meaningful things, dy and dx? And so what the hell are dy and dx?

mkl 1mo ago
They are differentials. https://en.wikipedia.org/wiki/Differential_(mathematics) https://en.wikipedia.org/wiki/Differential_(mathematics) has some info but is not great as a beginning introduction. dx is an infinitesimal bit of x, and dy is an infinitesimal bit of y. dx here is the same dx as in an integral, which you can think of as the width of one of the infinite infinitesimally thin rectangles whose areas are being added up to find the area under the curve: https://en.wikipedia.org/wiki/Riemann_integral https://en.wikipedia.org/wiki/Riemann_integral
simonreiff 1mo ago
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.
- archermarks 1mo ago
  This is effectively how dual numbers work! https://en.wikipedia.org/wiki/Dual_number https://en.wikipedia.org/wiki/Dual_number
  - ogogmad 1mo ago
    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!
    - srean 1mo ago
      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 https://www-cs-faculty.stanford.edu/~knuth/calc From the Arxiv paper: > 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.
- u1hcw9nx 1mo ago
  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.
  - simonreiff 1mo ago
    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.
jjmarr 1mo ago
> And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities? -- George Berkeley, namesake of UC Berkeley, in 1734, critiquing infitesimal approaches to calculus. Math uses limits because "dx" as a concept is hard to define and relies on faith that such an object can exist. It behaves as zero when convenient yet is non-zero when that breaks math. Limits have a more rigorous footing.
- augustusseizure 1mo ago
  Infinitesimals don't rely on faith any more than any other mathematical idea. The "normal" calculus sequence being based on limits is solely due to the fact that limits were the first discovered method for rigorously formalizing the subject. Infinitesimals do a better job representing how most people intuitively think about calculus though, as evidenced by being the vehicle through which the entire field was discovered in the first place, and so now that they too rest on a formally rigorous foundation we should probably consider rebuilding calculus education around them.
- gloveone 1mo ago
  Your comment was correct 100 years ago. But today it is highly inaccurate. Initially Calculus was developed using the ideas of infinitesimals throughout, although this was not yet fully formalized. The first to ground with mathematical rigor was O believe Cauchy with the epsilon-delta definition of limits. For historical reasons this caught on and is the standard way we introduce students to the subject till today. But since then we have already discovered fully rigorous and zero faith ways to define and work with limits: Robinson's non standard analysis and using nilpotent infinitesimals a la synthetic differential geometry. These provide completely rigorous way to view all the classical intuitions that initially develop the subject and are much easier to work with than the current standard epsilon-delta gymnastics. Unfortunately, mathematicians are very conservative and we tend to stick with the conventional way of doing things way more than we should. In fact a huge part of mathematical community have not fully engaged with the beautiful way of defining and using infinitesimals for calculus, even though it would greatly aid the students learning and intuition and solve the disconnect when working with physics using infinitesimals.
anthk 1mo ago
dx and dy are the smallest change ever in a function. Think about the speed of something (that's the reason of derivatives, you can't trace the speed of a stopped car in a track, you can't divide by zero, of course, there's no motion).

mkl 1mo ago

They are differentials. https://en.wikipedia.org/wiki/Differential_(mathematics) https://en.wikipedia.org/wiki/Differential_(mathematics) has some info but is not great as a beginning introduction. dx is an infinitesimal bit of x, and dy is an infinitesimal bit of y. dx here is the same dx as in an integral, which you can think of as the width of one of the infinite infinitesimally thin rectangles whose areas are being added up to find the area under the curve: https://en.wikipedia.org/wiki/Riemann_integral https://en.wikipedia.org/wiki/Riemann_integral

simonreiff 1mo ago

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.

archermarks 1mo ago
This is effectively how dual numbers work! https://en.wikipedia.org/wiki/Dual_number https://en.wikipedia.org/wiki/Dual_number
- ogogmad 1mo ago
  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!
  - srean 1mo ago
    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 https://www-cs-faculty.stanford.edu/~knuth/calc From the Arxiv paper: > 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.
u1hcw9nx 1mo ago
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.
- simonreiff 1mo ago
  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.

jjmarr 1mo ago

> And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities? -- George Berkeley, namesake of UC Berkeley, in 1734, critiquing infitesimal approaches to calculus. Math uses limits because "dx" as a concept is hard to define and relies on faith that such an object can exist. It behaves as zero when convenient yet is non-zero when that breaks math. Limits have a more rigorous footing.

augustusseizure 1mo ago
Infinitesimals don't rely on faith any more than any other mathematical idea. The "normal" calculus sequence being based on limits is solely due to the fact that limits were the first discovered method for rigorously formalizing the subject. Infinitesimals do a better job representing how most people intuitively think about calculus though, as evidenced by being the vehicle through which the entire field was discovered in the first place, and so now that they too rest on a formally rigorous foundation we should probably consider rebuilding calculus education around them.
gloveone 1mo ago
Your comment was correct 100 years ago. But today it is highly inaccurate. Initially Calculus was developed using the ideas of infinitesimals throughout, although this was not yet fully formalized. The first to ground with mathematical rigor was O believe Cauchy with the epsilon-delta definition of limits. For historical reasons this caught on and is the standard way we introduce students to the subject till today. But since then we have already discovered fully rigorous and zero faith ways to define and work with limits: Robinson's non standard analysis and using nilpotent infinitesimals a la synthetic differential geometry. These provide completely rigorous way to view all the classical intuitions that initially develop the subject and are much easier to work with than the current standard epsilon-delta gymnastics. Unfortunately, mathematicians are very conservative and we tend to stick with the conventional way of doing things way more than we should. In fact a huge part of mathematical community have not fully engaged with the beautiful way of defining and using infinitesimals for calculus, even though it would greatly aid the students learning and intuition and solve the disconnect when working with physics using infinitesimals.

anthk 1mo ago

dx and dy are the smallest change ever in a function. Think about the speed of something (that's the reason of derivatives, you can't trace the speed of a stopped car in a track, you can't divide by zero, of course, there's no motion).