More in infinitesimals

I criticized Sylvia Wenmackers, and she posted a rebuttal in the comments.

She explained what she meant by the hyperreals being incomplete. She is right that the hyperreals do not have the least upper bound property if you include non-internal sets. That is, the infinitesimals are bounded but do not have a least upper bound. But the bounded internal sets have least upper bounds.

The distinction is a little subtle. Arguments involving hyperreals mostly use internal sets, because then the properties of the reals can be used.

On another point, she refers me to this Philip Ehrlich article in the history of non-Archimedean fields. He says:

In his paper Recent Work On The Principles of Mathematics, which appeared in 1901, Bertrand Russell reported that the three central problems of traditional mathematical philosophy – the nature of the infinite, the nature of the infinitesimal, and the nature of the continuum – had all been “completely solved” [1901, p. 89]. Indeed, as Russell went on to add: “The solutions, for those acquainted with mathematics, are so clear as to leave no longer the slightest doubt or difficulty” [1901, p. 89]. According to Russell, the structure of the infinite and the continuum were completely revealed by Cantor and Dedekind, and the concept of an infinitesimal had been found to be incoherent and was “banish[ed] from mathematics” through the work of Weierstrass and others [1901, pp. 88, 90].

I think that it is correct that the continuum ( = number line = real numbers) was figured out in the late 19th century by Weierstrauss, Dedikind, and others, and widely understood in the early XXc, as I explained here. See Construction of the real numbers for details on the leading methods. The standard real numbers do not include infinitesimals.

But "banished" is not the right word. It is more accurate to say that infinitesimal arguments were made rigorous with limits.

The Stanford Encyclopedia of Philosophy entry on Continuity and Infinitesimals starts:

The usual meaning of the word continuous is “unbroken” or “uninterrupted”: thus a continuous entity — a continuum — has no “gaps.” We commonly suppose that space and time are continuous, and certain philosophers have maintained that all natural processes occur continuously: witness, for example, Leibniz's famous apothegm natura non facit saltus — “nature makes no jump.” In mathematics the word is used in the same general sense, but has had to be furnished with increasingly precise definitions. So, for instance, in the later 18th century continuity of a function was taken to mean that infinitesimal changes in the value of the argument induced infinitesimal changes in the value of the function. With the abandonment of infinitesimals in the 19th century this definition came to be replaced by one employing the more precise concept of limit.

Traditionally, an infinitesimal quantity is one which, while not necessarily coinciding with zero, is in some sense smaller than any finite quantity. For engineers, an infinitesimal is a quantity so small that its square and all higher powers can be neglected. In the theory of limits the term “infinitesimal” is sometimes applied to any sequence whose limit is zero. An infinitesimal magnitude may be regarded as what remains after a continuum has been subjected to an exhaustive analysis, in other words, as a continuum “viewed in the small.” It is in this sense that continuous curves have sometimes been held to be “composed” of infinitesimal straight lines.

Infinitesimals have a long and colourful history. They make an early appearance in the mathematics of the Greek atomist philosopher Democritus (c. 450 B.C.E.), only to be banished by the mathematician Eudoxus (c. 350 B.C.E.) in what was to become official “Euclidean” mathematics. Taking the somewhat obscure form of “indivisibles,” they reappear in the mathematics of the late middle ages and later played an important role in the development of the calculus. Their doubtful logical status led in the nineteenth century to their abandonment and replacement by the limit concept. In recent years, however, the concept of infinitesimal has been refounded on a rigorous basis.

This mentions and explains what I have been calling my motto or slogan, only it calls it a "apothegm", whatever that is. My dictionary says "A short pithy instructive saying". The "g" is silent, and it is pronounced APP-u-thum. Okay, I'll accept that, and may even adopt the word.

Consider the above statement that a continuous curve is composed of infinitesimal straight lines. Taken literally, it seems like nonsense. It took mathematicians 3 centuries to make it rigorous, and you can find the result in mathematical analysis textbooks.

The common textbook explanations use infinitesimal methods and limits, but not hyperreals.

My problem with Wenmackers is that she treats infinitesimals as sloppy reasoning until hyperreals came along, and the conventional epsilon-delta arguments as something that might only merit a footnote as it might distract casual readers.

This is just wrong. The mainstream methods for rigorous infinitesimal methods use epsilons, deltas, limits, tangents, and derivatives. The hyperreals have their place as a fringe alternative view, but they are not central or necessary for rigor.

In quantum mechanics, the momentum operator is an infinitesimal translation symmetry. This does not mean that either sloppy reasoning or hyperreals are used. It means that infinitesimal methods were used to linearize the symmetry group at a point. This is essential to how quantum mechanics have been understood for almost 90 years.

I also mentioned that special relativity is the infinitesimal version of general relativity. So yes, infinitesimal analysis is essential to XXc physics. How else do you understand relativity and quantum mechanics?

She writes:

What I mean by "loose talk involving infinitesimals" "frowned upon by mathematicians" is that physicists often talk about infinitesimals in a way that is close in spirit to Leibniz's work (and hence to non-standard analysis), which is not compatible with the definition of the classical limit as mathematicians use it in standard analysis.

No, I disagree with this. Limits and standard analysis were invented to make work by Leibniz and others rigorous, and they are still accepted as the best way.

She is essentially saying that the hyperreals are a better way to make Leibniz's work rigorous. If that is so, then why do all the textbooks do it a different way?

There are a few hyperreal enthusiasts among mathematicians who believe the hyperreals are superior, but they are a very small minority. I doubt that there are any colleges that teach analysis that way.

I think many of your other remarks also boil down to the same point: I use the term 'infinitesimal' in a more restricted sense than the way you seem to interpret it.

This is like saying:

When I refer to atoms, I am not talking about the atoms that are commonly described in college chemistry textbooks. I mean the hyper-atoms that were recently conceived as being closer in spirit to the way that the ancient Greek Democritus talked about atoms.

Russell did not just try to banish infinitesimals. He also tried to banish causality from physics, and convinced modern philosophers that there is no such thing.

This is a problem with modern philosophers. They can say the most ridiculous things just because they are accepted by other philosophers and historians. If they want to know how Leibniz's work was made rigorous, they could knock on a mathematician's door and ask, "Did anyone make Leibniz's work rigorous?" He would say, "Sure, just take our Calculus I and Analysis I courses." If she said, "what about the hyperreals?", he would say, "Yes, you can do it that way also, but we do not teach a class on it."

If I am wrong, please explain in the comments.