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Is there a philosophical argument supporting the hypothesis that an infinitesimal instant of time has zero duration?

The reference to infinitesimal includes the modern presentation of it in non-standard analysis by Abraham Robinson in 1960 and H. Jerome Keisler's Elementary Calculus:

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The infinitesimal instants are non-zero-it is infinitely small.

Infinitesimals have a colorful history.

In the later part of the 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.

A major development in the re-founding of the concept of infinitesimal took place in the nineteen seventies with the emergence of synthetic differential geometry, also known as smooth infinitesimal analysis (SIA)[50].

Based on the ideas of the American mathematician F. W. Lawvere, and employing the methods of category theory, smooth infinitesimal analysis provides an image of the world in which the continuous is an autonomous notion, not explicable in terms of the discrete.

It provides a rigorous framework for mathematical analysis in which every function between spaces is smooth (i.e., differentiable arbitrarily many times, and so in particular continuous) and in which the use of limits in defining the basic notions of the calculus is replaced by nilpotent infinitesimals, that is, of quantities so small (but not actually zero) that some power—most usefully, the square—vanishes.

Since in SIA all functions are continuous, it embodies in a striking way Leibniz's principle of continuity Natura non facit saltus.

Ref.-

https://plato.stanford.edu/entries/continuity/#9

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An infinitesimal is arbitrarily small, but greater than zero. Here is how Wikipedia describes it:

In mathematics, infinitesimals are things so small that there is no way to measure them. The insight with exploiting infinitesimals was that entities could still retain certain specific properties, such as angle or slope, even though these entities were quantitatively small.

Note that if they were of zero length or duration they would no longer have the property of a "slope". Hence there is value in making sure they are not actually zero.


As Conifold informed me in the comments below there is a modern development of the concept of infinitesimal:

There is a movement in math education to reintroduce infinitesimals into the curriculum, see e.g. Keisler's text based on non-standard analysis, one of their selling points is affinity to intuition and its uses in physics, etc. So we do not necessarily need limits.

Keisler's discussion of infinitesimals contain these initial axioms (page 1):

AXIOM A
   R is a complete ordered field.
AXIOM B
   R∗ is an ordered field extension of R.
AXIOM C 
   R∗ has a positive infinitesimal, that is, an element ε 
such that 0 < ε and ε < r for every positive r ∈ R.

For the purposes of this question, note that Axiom C assumes that the infinitesimal is strictly greater than zero.


Reference

Keisler, H. J. (1976). Foundations of infinitesimal calculus (Vol. 20). Boston: Prindle, Weber & Schmidt.

Wikipedia, "Infinitesimal" https://en.wikipedia.org/wiki/Infinitesimal

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    That particular Wiki article is wretched. "So small there's no way to measure them?" What the heck does that mean? Measure is a technical term in math that doesn't apply here at all. The entire article is one misunderstanding after another. – user4894 Sep 27 '18 at 18:53
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    @user4894 I do think you have a point. I was surprised by the "measurement" idea as well when I read it. But I think it might have value. We want the infinitesimal so small the margin of error is irrelevant even when integrating. The idea that without some distance or duration the infinitesimal would not have a slope (which requires two instants) is a useful way to think of it. – Frank Hubeny Sep 27 '18 at 18:58
  • Infinitesimals are supposed to be additional numbers on the number line, so what does "infinitesimal having a slope" mean? I also do not understand "infinitesimal so small the margin of error is irrelevant even when integrating"? What does it have to do with integrating??? And I'd delete the first sentence from the Wikipedia quote, "in mathematics... there is no way" just makes it comical. In mathematics there is no problem with measuring infinitesimals, infinitesimal intervals have infinitesimal lengths. – Conifold Sep 28 '18 at 3:35
  • @Conifold The 19th century replacement of infinitesimals with limits put calculus (analysis) on a firmer foundation. Guiseppe Veronese may have considered infinitesimals as additional numbers, but one can simply look at them metaphorically and ignore adding them or infinity to the number line. The point of this answer is that metaphorically one should not think of them ever being of zero duration and length. – Frank Hubeny Sep 28 '18 at 10:27
  • Why Veronese? His was just one of many end of 19th century theories on infinitesimals. There is now an equally firm foundation that includes them, non-standard analysis, that explicitly adds more numbers to the number line (hyperreals). Infinitesimal intervals are assigned non-zero hyperreals, although each hyperreal itself is of course just a point, same as 1 is a single number representing non-zero length. But a slope? Also, metaphorical use is inadequate for doing mathematics (e.g. integrating), which requires definitional precision. – Conifold Sep 30 '18 at 20:52
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If I have a clock stopped at 12:00, there are two times a day when it's accurate. It's only accurate for an instant, less than any nameable quantity of time, hence infinitesimal at most. However, it happens. If the amount of time the clock was correct was actually zero, it would never be correct.

  • That's very interesting, thank you. But it's not logically necessary that a clock is correct: it's just an human tool. I agree that if an instant were 0 time would not exists, but we could still imagine a minimum unit of time (a discrete time). – Francesco D'Isa Sep 27 '18 at 17:07
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    The last inference seems invalid to me, "never" and "for zero amount of time" are different things. Rational numbers occupy zero length on the real line, but they do not never occur. There is no logical connection between existing and lasting a positive time, only our habit to associate the two. – Conifold Sep 28 '18 at 0:09

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