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This question already has an answer here:

I was wondering and came across a thought: if an object is moving at a particular speed, say 5 m/s, it is equivalent to say that the object is moving at 500 cm/s, or 5000 mm/s and so on; clearly, you can keep dividing '5m' an infinite amount of times, so there will be a particular instance where the object has to move at 5 • ∞ (infinitesimal units/s), so how does the object manage to do that; in the sense, how can it move an infinite distance of any unit in 1 second?

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marked as duplicate by sammy gerbil, Kyle Kanos, Jon Custer, stafusa, ZeroTheHero Dec 10 '17 at 22:14

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    $\begingroup$ The tortoise and the hare. Or Achilles and the tortoise. $\endgroup$ – Pieter Dec 9 '17 at 10:47
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    $\begingroup$ More on Zeno's paradox. $\endgroup$ – Qmechanic Dec 9 '17 at 12:33
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Mathematically this is simply the way continuous dimensions are, there is always a smaller subsection to consider. In this case since you are talking of infinitesimal units, it would be more helpful to consider your units to be "arbitrarily small" rather than "infinitely small". Your speed will then simply grow linearly in proportion with the decrease in unit size. Mathematically you can consider it as: $$\lim_{n\to\infty} 5 \cdot 10^{n} \cdot 10^{-n}\ \ \frac{m}{s}$$ Which of course simply reduces to 5 m/s.

It is important to remember that in physics you are developing and using mathematical models of reality, not reality itself. Therefore you are going to encounter quirks in reasoning that come from what mathematics are, rather than what reality is.

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    $\begingroup$ +1 for the last paragraph, in particular the emphasis on the "model" aspect of physics (often overlooked, in my opinion). As for the "quirks", in this case they come from a poor understanding of what the real numbers are, and not distinguishing from infinitesimals, which can be created but are not as natural as some may want to believe. $\endgroup$ – Martin Argerami Dec 9 '17 at 15:38
  • $\begingroup$ Does this expression imply that you can in fact never reach n = ∞ ? $\endgroup$ – Supernova Dec 10 '17 at 4:00
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    $\begingroup$ But how would you reach $\infty$ ? Think on the nature of this concept, infinity by its definition is not something you reach. The nature of a limit is that it approaches a value in the same way you would "reach" infinity, and in the same way an infinitesimal approaches 0. Thankfully, though, mathematics allows us to find (in certain cases) a definite answer to a limit, which is a number you reach asymptotically. Although in this equation you already start on it so its a bit of a trivial case. $\endgroup$ – Terrestrial Dec 10 '17 at 11:41
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This question was first addressed by Zeno of Elea around 400 BCE in his famous paradoxes of motion.

Aristotle, in his book Physics, pointed out that there was an adequate solution to this as per the usual solution by taking the sum of an infinite series. However he still said that this wasn't a full solution of the problem of motion in the small, and he offered a solution in terms of potentia & actuality.

The problem here is that taking the sum is merely a mathematical notion and not itself motion; it's an idealised picture. There is a difference between a physical continuum and a mathematical one.

That this problem is valid can be seen from the fact that motion in the small is given by quantum mechanics rather than classical mechanics, and quite incredibly we can view this motion in terms of potentia and actuality: the wave function - the potentia, and its measurement - the actuality. This is not to say that in the Greece of Antiquity they were doing Quantum Mechanics - far from it; what they did, by logical means, is realise that there is a problem in motion not adequately captured by our intuitive notions.

A similar problem was tackled by Al-Ghazali in a more wide-ranging way (he was addressing causality), for which he suggested occasionalism - that from moment to moment, in every place, the world in being created, decreated and recreated; and in this, he sees the hand of God.

In the early 20C the invention/discovery of QM showed to the surprise of physicists that physical motion isn't as simple as they had thought; the jury is out on its interpretation... The simplest motion is that of a single electron and here the wave function evolves in all directions, it doesn't have a classical path; this reminds me of an argument by Aristotle when he asked in which direction should a particle move in the void - his answer was in every direction, and because of this he concluded the void did not exist.

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When you say "there is a particular instance..." No, there is no "particular instance". We model our basic physics by using real numbers; and there is no "infinitely small positive number". So the reasoning is not sound, neither mathematically nor physically.

It is important to distinguish two things. First, within the physics themselves, you are using numbers to model the trajectory of your object. The point is that you are assigning numbers to (possible) measures of the object and its position. How would you measure an "infinitesimal"? Second, physics is nothing more than a mathematical model of the reality we perceive through measurement. As such, the math doesn't "decide" whether a physical object does something or not. At best, within the mathematical model some mathematical property may preclude a certain situation.

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  • $\begingroup$ In this case, though, the mathematical model is fine, and if you apply infinitesimal mathematics (i.e. calculus) to this model it gives the correct answer. $\endgroup$ – wizzwizz4 Dec 9 '17 at 16:56
  • $\begingroup$ "Infinitesimal mathematics" (limit, derivatives, integrals) is not "infinitesimals". $\endgroup$ – Martin Argerami Dec 9 '17 at 19:03
  • $\begingroup$ Actually, infinitesimal mathematics is mathematics with infinitesimals. From your link: "To give it a meaning, it usually must be compared to another infinitesimal object in the same context (as in a derivative). Infinitely many infinitesimals are summed to produce an integral." $\endgroup$ – wizzwizz4 Dec 9 '17 at 19:11
  • $\begingroup$ I guess you didn't really read the article. Infinitesimals, as in the OP and as in the the Wikipedia article, is not "calculus" (as you said, and to what I responded). It is a gimmick to do calculus without using limits. It involves creating objects that are not real numbers, and such they do not fall in the usual domain of physics. $\endgroup$ – Martin Argerami Dec 9 '17 at 19:50
  • $\begingroup$ "It is a gimmick to do calculus without using limits." Ah. I see. The confusion comes from the differences in definitions of calculus. I define calculus as a calculation involving summing infinitesimals, whether optimised or not. What's your definition? $\endgroup$ – wizzwizz4 Dec 9 '17 at 19:53

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