One is an imaginary concept about some series going on without end <snip>.
Do you think you could chop something real up infinitely?
Suppose you chop up the movement of some object infinitely and the object makes the smallest possible move forward.
How far has it moved?
Ah, here is something interesting. Several posters have already commented on that but I think there's a bit more to say.
UM's suggestion to "chop up the movement of some object infinitely" shows he's not interested in the actual concept of infinity and how it is effectively applied in scientific theories to the physical movement of material objects. But it's nonetheless interesting to look into the details of this question.
I think the best and most admirable example of how the concept of infinity works is in the work of the guy who invented it, Isaac Newton. So, it goes back more than three hundred years ago already. Analytical geometry had just been invented by Descartes and UM wasn't even born.
So Newton invented the concept of infinitesimals that is crucial to what came to be called "calculus", or "infinitesimal calculus". It is also interesting to note that Leibnitz, more of a logician than a physicist like Newton, is also regarded as having independently discovered calculus, at the same time, which does provide some respectability to such small thing as infinitesimals might be.
Anyway, my dictionary says for 'infinitesimal", "infinitely or immeasurably small".
Let's note straight away that "today, calculus has widespread uses in science, engineering, and economics." (Wiki, on Calculus). And that "Calculus is used in every branch of the physical sciences, actuarial science, computer science, statistics, engineering, economics, business, medicine, demography, and in other fields wherever a problem can be mathematically modeled" (Wiki).
So, the concept of infinitesimal works rather very, very well when applied to the physical world. And so anybody who would think of dismissing this concept in just a few words would be a "triple buse" ("triple buse", in French, was the preferred insult of our chemistry teacher for her students back in the sixties. A "buse" in French is a buzzard but also a total idiot).
So, what are infinitesimals? Well, mathematicians themselves seem to have had to wrestle with this notion for a very long time. Prior to Newton and Leibnitz, notions of calculus can be found in all traditions (Ancient Egypt, Ancient China, Ancient Greece, medieval Europe and India) and we all know the Zeno paradox. And European mathematicians still struggled to find a rigorous definition of infinitesimals. I'm proud to say that the French mathematician Augustin-Louis Cauchy got there first, but it was already 1821 and it came after much debates among mathematicians. Let's note Cauchy was also a physicist.
So, infinitesimal is clearly a difficult concept but then again it received the full attention of brilliant people for the time necessary to elucidate the question. So, again, someone who thinks he can dismiss such a concept in just a few words must have some unresolved understanding issues.
Now, what about the specifics? Does calculus assumes it is physically possible to do something like "chop up the movement of some object infinitely"? Well, there's a prior question to that, which is what does it mean to do something "infinitely". 'Infinite' just means "not finite", i.e. "without" limit". So, to do something infinitely just means that you start doing it and you never stop. So, rather obviously, there won't be a time when it's done. And this is really how infinitesimals are understood (except by UM of course). This is also why we talk of the "logical possibility" of infinity rather than the physical possibility.
This is a rather uncontroversial issue. No sane person would interpret the concept of infinitesimal as a suggestion that it is physically possible to "chop up" something, a length, a material object, energy etc. infinitely. It can't be done physically if "done physically" means that you have to have finished the job at some point in time, UM's claim notwithstanding.
Perhaps something should be said about the metaphysical assumptions behind the concept of infinitesimal. Basically, the idea is that time is taken to have no end and that it is therefore at least conceivable that a particular activity could go on and on without end. So, we can conceive of starting to chop up a length in two bits and then repeat this operation on one of the two halves so obtained so that we would get smaller and smaller lengths as time would go on and on. Again, no idea of ever actually finishing. We can physically start such a process but we won't ever have done it to the end precisely because there's no end to it. So, it's all notional, conceptual, virtual. What matters is that for practical applications, we can start to chop some thing up and repeat the operation as long as we need to until we're satisfied that the physical result complies with our theoretical expectations. The second metaphysical assumption is that space is continuous. We obviously don't know that it is but we can assume that it is without contradiction until such a time that someone finds a particular length that couldn't be divided into two smaller lengths. So, it would be an empirical discovery, like the 1899 discovery that energy was quantified.
Can these two assumptions be true? Well, what do you mean by "can"? Either they are true or they are not. We just don't know. We know of only of two ways to find that something is not true: empirically or logically. It is conceivable that we will find empirically that time and/or space are not continuous and therefore couldn't be chopped up infinitely. But since Zeno two millennia ago, philosophers have tried to find logical arguments showing that the concept of infinity was wrong, without success.
UM at least is still trying. As if he was never going to stop. As if to prove time is really infinite.
There's some beauty in that at least.
EB
