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Is it possible to divide something into an infinite amount of parts?

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It depends on what one means by the 'thing' in question. According to the principles of geometry, a line is infinitely divisible, although of course it has basic components, points. But what of ordinary, middle-sized objects, such as tables and chairs? Certain early modern philosophers, such as Leibniz, believed that material things were not only infinitely divisible, but that they were also actually divided. (Hence, Leibniz concluded, material things weren't real things, because they lacked what he called a 'true principle of unity', and the only 'real' things were souls, which, being immaterial, could not be infinitely divided.) In a recent article, "Van Inwagen and the Possibility of Gunk," the metaphysician Ted Sider argues that it is possible that objects in the world are infinitely divisible and even lack basic constituents, like the points of a line. According to Sider, it is logically possible that the world consists only of 'atomless gunk', that is, "that it divides further into smaller and smaller parts," and, unlike a line, "does not even have atomic parts..._all_ parts of such an object have proper parts. Now one might think that current physical theory is incompatible with the possibility of what Sider calls 'a gunk world'. Sider anticipates this objection: "Scientists discovered that hydrogen ‘atoms’ have proper parts. Then they discovered that protons have proper parts. At one point, at least, it was a legitimate scientific hypothesis that this process could go on forever, that there is no end to the world’s complexity. A metaphysical theory should not have the consequence that a legitimate scientific hypothesis is metaphysically impossible. So we ought to accept the possibility of material objects made of gunk." I agree with Sider that "a metaphysical theory should not have the consequence that a legitimate scientific hypothesis is metaphysically impossible"; however, it's not clear to me that it is indeed a live scientific hypothesis at this point that everything is subdivisible. Since I, at least am inclined to take current physical theory as the best guide to understanding the nature of reality--although, of course, this is a controversial assumption--if current scientific theory does not allow for the actual divisibility of things into an infinite number of parts, then I think that the answer to your question is 'no'. (Of course, to be sure, even if current scientific theory does not allow for a 'gunk world', future discoveries might reveal that it is indeed possible. In that case, I would of course revise my answer to your question.)

I've nothing against Sean Greenberg's answer, but I figured I'd just add a word or two on a further relevant distinction here. Infinite divisibility is not the same as the possibility of dividing something into infinitely many parts. At least, it doesn't need to be understood in that way. There's a distinction that goes back at least as far as Aristotle, between the actual infinite and potential infinite, and the notion of infinite divisibility can be interpreted in either way. If we interpret infinite divisibility in the sense of the potential infinite (which, for what it's worth, is how Aristotle himself understood -- and endorsed -- the concept), this will mean that, no matter how small something might be, it can still be divided into still smaller parts. You can cut something into two halves, divide each of those to yield four quarters, divide each of these to yield eight eighths, and carry on going without ever needing to stop dividing. Mathematically, there is no greatest power of two: so, no matter how many pieces this process of subdivision has already yielded, there is always the potential for that number to be doubled by further subdivision. But the thing to appreciate is that the number of pieces will always remain finite. Mathematically, there are infinitely many positive integers, but each one of them individually is finite. The whole point about an infinite process is that it can never be completed. The very word suggests this: the prefix 'in' is a negation while the term 'finite' (from the same etymological root as 'finished') indicates a terminus. As Aristotle put it: 'The infinite turns out to be the contrary of what it is said to be. It is not what has nothing outside it that is infinite, but what always has something outside it.' (Physics, 206b34-207a1).

So much for the potential infinite. One might alternatively maintain that it should in principle be possible for an infinite process -- of division, or whatever else it might be -- to be completed. For instance, if one believes that God is actually omnipotent, then He at least ought to be able to divide something infinitely many times. After all, what good is infinite power if it can't be exercised infinitely? Perhaps He makes the second cut half a minute after the first one, makes the third cut a quarter of a minute after that, makes the fourth cut an eighth of a minute later, and so on. Then, after the whole minute has elapsed, His work will be complete. And how many parts will He then have produced? Why, infinitely many of them.

Historically, I think it's fair to say that the notion of infinite divisibility has more frequently been embraced in the potential sense, but there have been a few who've pressed for actual infinite divisibility too. But note that there's a problem arising here, one that doesn't arise under the potential interpretation. I don't say an insurmountable problem, necessarily, but it is one that will certainly need to be addressed. Suppose we do allow a process of infinite division to be actually completed, so as to yield infinitely many parts. Are these parts extended or aren't they? If they have no extension, no size whatsoever, then where did the extension of the original object go? It seems that the bulk of the original object has not merely been dispersed but has actually been annihilated. And that doesn't seem right, that we should be able to annihilate the bulk of something just by moving its parts around. On the other hand, if each of these parts has some extension individually, then it seems that infinitely many of them together must have infinitely much. So, simply by moving the parts of our original object around, we'll have generated an infinite bulk out of a finite one. And that doesn't seem right either. Consequently, many of those who favoured the 'actual' interpretation of the notion of infinite divisibility rejected that notion in favour of a theory of necessarily indivisible atoms. That, I think, is the reason why infinite divisibility has found more support when understood in the 'potential' sense, for there the same problem doesn't arise. Even if there's no limit to the number of parts you can get through division, that number will still be some finite number n, and the size of each one can unproblematically be an nth of the size of the original object.

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