according to Russell, Weierstrass's solution was to throw out the very notion of an "infinitesimal" all together. — Gregory

Correct. The modern epsilon-delta formalism finesses infinitesimals. It makes infinitesimals go away. We replace the phrase "infinitely small" with "arbitrarily small," and that turns out to make all the difference. I'd be glad to expand on this but I don't know if you want a discourse on basic calculus here, probably not. Suffice to say that (1) There are no infinitesimals in the real numbers; and (2) We can base calculus on the notion of arbitrary closeness between quantities.

That is: To say that the "limit" if 1/x is 0 as x gets large without bound, is to play a game. You give me a tiny tiny tiny positive real number; and I'll give you an x such that 1/x is strictly between 0 and your number. If I can always win that game by finding a suitable, x, no matter what tiny positive real number you give me, then we say 1/x has the limit 0. No infinitesimals are used nor do they exist. They've been banished from mathematics.

I know that non-standard analysis in the 60's brought "points" back — Gregory

NSA is based on the hyperreal numbers, which are a nonstandard model of the first-order axioms of the real numbers. The hyperreals contain true infinitesimals. One calculus book, by Keisler in 1976, has been written based on NSA. If the idea had pedagogical value we'd have seen more since then; but in fact studies show that students come out of an NSA-based calculus class just as confused as they do from a course in standard calculus.

It's important to mention that the hyperreals require, for their mathematical existence, a weak form of the Axiom of Choice. So NSA requires stronger logical assumptions to get off the ground than standard math does.

, but was calculus taught before this without infinitesimals? (How is that possible?) — Gregory

It's done every day. We don't use infinitesimals in math, they've been banished. [To be fair, NSA does have its uses, and Fields medalist Terence Tao and others them in their work. But in terms of mainstream math, NSA is a niche idea with hardly any mindshare in the curriculum].

Pedagogically you don't need to know any of these subtleties to teach basic calculus, which is mostly a grab bag of tricks and techniques that ignores these logical subtleties. It's not till a course in real analysis, offered mostly to math majors, that the logical underpinnings of calculus are clarified. So there are a lot of educated physicists and engineers around who really don't care about any of this; and in fact physicists freely use infinitesimals in their reasoning.

As I say I would be glad to expand and expound at length on these ideas but it's best to keep it short an respond to questions, if any.

Thanks. The phrase "The devil is in the details" was used by the scholastic school mean in the Middle Ages. You weren't allowed to talk about infinitesimals back then. It's actually understandable because questioning how many parts a geometric object has leads to a lot of trouble I fear. It's sounds as if math professors kind of agree that it's best not to tilt at that particular windmill

It's sounds as if math professors kind of agree that it's best not to tilt at that particular windmill — Gregory

If I gave that impression I failed in my post utterly. And I see I did.

The epsilon-delta formulation of limits is one of the crowning intellectual achievements of humanity. From Newton and Leibniz's calculus of the late 17th century, it was another 200 years till the late 19th century before we had a fully rigorous account of limits.

There's no windmill tilting being avoided at all. Rather, it's extremely difficult to come up with a logically rigorous theory of infinitesimals. It wasn't done till 1948, by Hewitt, and then Robinson came up with NSA in the 1960's. But to this day, the epsilon-delta idea has all the mindshare, and as I say you need extra strong logical assumptions to get infinitesimals off the ground. And NSA has not turned out to be any improvement in pedagogy. So there are good reasons why they're not used in the calculus curriculum.

There are schools here and there that teach calculus using infinitesimals, but by far most use the epsilon-delta approach.

https://matheducators.stackexchange.com/questions/5989/which-universities-teach-true-infinitesimal-calculus

If the idea of infinitesimals is a black hole in a garden, I see the point about this. Heidegger started saying "what IS being", and in the 60's and beyond everyone is asking "what IS consciousness" . Ideas too big to grapple with one step at a time will spring up and sometimes catch followers. Maybe there is a dialectic behind the whole thing

If the idea of infinitesimals is a black hole in a garden — Gregory

An infinitesimal is an element of an ordered field that is greater than zero, but less than 1/n for every natural number n. If your original remarks were about mathematical infinitesimals, this is a strange comment; and if not, why the quotes from Russell and the mention of NSA?

I see the point about this. Heidegger started saying "what IS being", and in the 60's and beyond everyone is asking "what IS consciousness" . — Gregory

Surely this has nothing at all to do with infinitesimals.

Ideas too big to grapple with — Gregory

That's funny. Since an infinitesimal is something that's infinitely small, it's ironic to call it an idea too big.

Maybe there is a dialectic behind the whole thing — Gregory

Maybe you could say why you totally changed the subject from your original post. Did I totally waste my time discussing mathematical infinitesimals? Were you never interested in them at all, but rather about something else?

I'm very very interested in infinitesimals. Berkeley called them ghosts of dead space as if space dies as it approaches infinity. My question is why does it approach infinity when we get smaller and smaller but not when going in the opposite direction. With the former you get nowhere and in the later we get limited finitude. How can some thing be infinite and finite in regard to its spatial component? If matter is pure extension as Descartes said there results confusion. Yet Hegel said space was "outside itself" and I try to understand this as curved space. If we have a globe, you can do non-Euclidean geometry on the surface but inside it you can still do Euclidean stuff. However if curvature is prior to other aspects of extension than the whole globe is permeated with a curve. It's from this angle that I am trying to understand infinitesimals and how they loop back into finitude. So you can see I do take this subject seriously.

I guess I'll add to the discussion about spatial reality the observation that with numbers the finite make up the infinite while with space the infinite makes up the finite

Maybe my concept of "finite" doesn't correspond to anything but it seems to me it merely says "The finite means that which has beginning and end". That mathematicians don't scratch their heads at the idea of a geometric object having infinite parts but also having a beginning and end, well that makes me scratch my head lol

I'm very very interested in infinitesimals. Berkeley called them ghosts of dead space as if space dies as it approaches infinity. My question is why does it approach infinity when we get smaller and smaller but not when going in the opposite direction — Gregory

Have another glass of wine, my friend. :roll:

Berkeley called them ghosts of dead space as if space dies as it approaches infinity. — Gregory

A misquote and out of context as well, unless you are referring to something else. If so please provide a reference so that I can learn something. I assume you are referring to "the ghost of departed quantities."

Berkeley called Newton's fluxion (what we now call the derivative) the ghost of departed quantities. He was calling attention to the logical problems in the definition of the derivative; problems that as I indicated earlier were not fully resolved for another 200 years. Berkeley had a point; and, like the intellectuals of the day, could snark with the best. If these guys came back today they'd be right at home online.

The question at issue was the meaning of the derivative. I'll use modern notation rather than Newton's original notation and terminology. If we have some function $y = f(x)$, we can form the difference quotient

$\displaystyle \frac{f(x + h) - f(x)}{h}$

where $h$ is some tiny increment. As $h$ gets very close to 0, the difference quotient approaches what Newton called $\dot x$ and what we today call $f'(x)$ or $\frac{dy}{dx}$. In fact Newton's dot notation is still sometimes used in physics and engineering.

As you can see, as $h$ gets very close to 0, $f(x + h)$ gets very close to $f(x)$, so that the numerator of the difference quotient gets very close to 0; and likewise, the denominator gets very close to 0 as well.

The idea, by the way, is that $f(x)$ gives the position of some object at time $x$; and the derivative turns out to be its velocity. The derivative of your car's position function is exactly what's shown on your speedometer at any moment.

Berkeley rightly pointed out that if $h$ is nonzero, the difference quotient is not the derivative; but if $h$ IS zero, then the difference quotient is the entirely meaningless $\frac{0}{0}$. Newton had no logical explanation and Berkeley was correct to point this out in the witty manner he did.

Yet, the procedure "gave the right answer" and allowed Newton to calculate the orbits of the planets and derive the universal law of gravitation. As so often happens in the history of science, the physicists had a clever procedure that proved incredibly useful and gave the right answer; but that was not mathematically legitimate. It was left to the mathematicians to put derivatives on a sound logical footing, and this is what took 200 years.

Apologies for the freshman calculus lesson (augmented by history, which sadly they DON'T teach in calculus class); but this is what the quote was about.

They were NOT talking about space. They were talking about a mathematical formalism that seemed to be the key to understanding the universe, yet which could not be defended logically with the mathematics of the day.

My question is why does it approach infinity when we get smaller and smaller but not when going in the opposite direction. — Gregory

Not clear what "it" is in this context, but nothing's approaching infinity. Rather, as $h$ gets close to zero, the difference quotient gets close to the true velocity of a moving particle whose position at time $x$ is given by $f(x)$.

Using this idea, Newton was able to work out the inverse square law of gravitational attraction and describe the workings of the universe, and show that the fall of an apple from a tree was exactly the same phenomenon, described by the same law of gravity, as the motions of the planets in the heavens. That was a profound scientific breakthrough. As Berkeley pointed out, we did not understand mathematically what the derivative was; only that it seemed to work.

With the former you get nowhere and in the later we get limited finitude. How can some thing be infinite and finite in regard to its spatial component? — Gregory

I wonder if you can be more specific here. I can't correlate this with what I just described as the mathematical context of the "ghosts of departed quantities" remark as applied to Newton's early concept of the derivative.

If matter is pure extension as Descartes said there results confusion. — Gregory

Nothing to do with the nature of matter; only with the logical nature of a mathematical formalism that allowed Newton to discover his law of universal gravitation; yet did not seem to have a proper logical basis.

Yet Hegel said space was "outside itself" and I try to understand this as curved space. — Gregory

Berkeley's quote has nothing to do with any of this. The context was Newton's definition of the derivative (or as he called it, the fluxion) of a function. The method worked but the logical foundation wasn't clear, nor would it be clear for another 200 years. That's the subject. The mathematical formalism of the derivative as the limit of the difference quotient.

If we have a globe, you can do non-Euclidean geometry on the surface but inside it you can still do Euclidean stuff. However if curvature is prior to other aspects of extension than the whole globe is permeated with a curve. It's from this angle that I am trying to understand infinitesimals and how they loop back into finitude. — Gregory

Yes but this has nothing to do with the development of the logical foundation of the definition of the derivative, which was the actual context of Berkeley's brilliantly snarky remark.Yet even so, Berkeley was the moon to Newton's sun. Berkeley was picking at the logical problems, correctly; but Newton was revolutionizing our understanding of the world.

So you can see I do take this subject seriously. — Gregory

You threw me off a bit going off into the social upheavals of the 1960's. But also with regard to your comments on the nature of space. That was never the subject of Berkeley's remark.

Leibniz wrote about infinitesimals as monads, that's something I don't know much about. Perhaps that would be of interest to you. And of course there is an ongoing question of the ultimate nature of space. Is it made up of little infinitesimal thingies, or what?

So perhaps I'm being too literal, which is a constant fault of mine. Someone asks a question, I answer it correctly with precision, yet totally miss the point. A bit like Berkeley's logical carping next to Newton's revolutionary discoveries. So be it.

But if the question is what did Berkeley mean by the ghosts of departed quantities, he was referring specifically to the numerator and denominator of the difference quotient as $h$ gets close to 0, and as the difference quotient gets close to the derivative.

Newton himself referred to this as the "ultimate ratio" of the difference quotient and struggled over the course of his life to make logical sense of the derivative. He never succeeded. It took the geniuses of the 19th century to finally nail it down.

The tl;dr:

* Berkeley's quote is about the technical definition of the derivative; a definition that actually makes no sense till you have a rigorous theory of limits, which didn't show up till the 19th century; and

* None of this has anything to do with the actual nature of space, which is a different subject entirely. After all, both the infinitesimals of NSA and the standard epsilon-delta approach are mathematical formalisms that we use to try to model space. They tell us nothing at all about space itself.

I may have used Berkeley's statement too broadly. Sometimes I generalize too quickly. Thanks for your explanation however. The world truly is not pure geometry, and this actually may be why we can't understand the world completely. If we have Cartesian extension on one side and absolute nothingness on the other, gravitational potential energy would be at the center. The way numbers move in the mind is analogous to how objects move in space, as Kant implied. In the dispute between Berkeley and Newton, numbers are not being added up one at a time though. And there is still an infinity in the equation which I took to be Berkeley's real problem with Newton's calculus. Berkeley was an idealist and people who denied the reality of the world had traditional used Zeno's paradoxes to "prove" their point. The question of whether "many is always many" or whether a multitude can be subsumed into something else is of interest I think. As an equation reaches zero, do we in the next step have infinity as a negative zero? I think that might be a Liebnizian monad, but idnk, I have to read an actual work by Leibniz sometime (it will be my first). Anyway, if someone mentally scales his mind down to a point, perhaps he will here the words of Sartre: " … the total disappearance of being would not be the advent of the reign of non-being, but on the contrary the concomitant disappearance of nothingness."