Dear Robert, You are right. The key to understanding the paradox is that although Achilles must complete an infinite number of tasks in order to catch up to the Tortoise, he can do so in a finite amount of time, since each successive task takes much less time than its predecessor (as you noted). Of course, today we understand how to add an infinite sequence of terms that converge to a finite quantity. But this wasn't well understood until millenia after Zeno -- and the logical foundations for doing so required Cauchy and Weierstrass in the nineteenth century. So we shouldn't be too hard on old Zeno. By the way, you might find it amusing to consider some more recent Zeno-like puzzles, such as the "New Zeno" discussed by Stephen Yablo in the journal ANALYSIS, vol 60 (April 2000).

I do not think that specialized philosophical training is needed to dissect the speeches of candidates for elective office. Only the rudiments of critical thinking, a willingness to follow good arguments wherever they lead, and some knowledge of the relevant issues is needed. In universities and colleges, philosophers teach classes that aim to cultivate the skills useful for analyzing arguments and recognizing their strength (or weakness). We do our part in that way -- by helping to produce an electorate that can think critically, reason carefully, and express itself clearly. But the rest is up to you!

Well, I cannot speak for all philosophers. But it seems to me that Hawking has not kept up with the developments in philosophy. Of course, he need not do so ... unless he plans to say something about them, as he apparently did. There is a tremendous amount of very scientifically informed philosophy of science. People in philosophy departments and people in physics departments both work on the conceptual, logical, and metaphysical foundations of physics (and analogous points could be made about evolutionary biology or economics, for instance). Even a cursory glance at the literature would bear this out. I apologize if this sounds somewhat defensive. I guess it is. But physicists do tend to deprecate philosophy of science without having taken the trouble to familiarize themselves with it. See, for instance, Steven Weinberg's book "Dreams of a Final Theory" (1992), to which Wesley Salmon replies in "Dreams of a Famous Physicist", an article reprinted in his book "Causality and Explanation".

What is the difference between a living thing and a non-living thing? What is "vitality"? This is a difficult question. Once upon a time, it was widely believed that living things are distinguished by possessing a certain substance (an "elan vital") or perhaps by a certain force being present in them alone. This was a legitimate, testable scientific theory ("vitalism") that now appears to be false, since living processes can take place outside of living things (as when digestive enzymes can break down food in the test tube). Another notable family of views on this question is that living things are alive in virtue of the fact that they carry out certain "life functions" such as growth, self-motion, metabolism, reproduction, and so forth. This view would account for the intermediate cases between life and non-life (such as viruses and whatever entities existed in the early stages of the origins of life on Earth). The intermediate cases could presumably carry out some but not all of the life functions. ...

I don't know what the precise definition of a "paradox" is, but roughly speaking, it is an argument that begins from premises that are too obvious to deny and ends by deriving from them a conclusion that is too ridiculous to accept. (Did Bertrand Russell say that somewhere?) By this standard, a paradox need not involve self-reference or lead to an infinite regress. Carl Hempel's famous "Paradox of Confirmation" (a.k.a. "Paradox of the Ravens") fits the above rough definition but involves neither self-reference nor infinite regress. Now the Fermi Paradox, as you say, begins from several considerations that aim to show that it is highly likely that extraterrestrial civilizations exist. Perhaps none of these considerations is really too obvious to deny, but all of them are intended to be well-grounded (e.g., the number of potentially life-supporting planets in the universe). With a few further premises about interstellar communication, the conclusion of the argument is supposed to be that (it is...

Granted, 10 years in comparison to infinity is as short as 10 seconds is in comparison to infinity. But it does not follow that 10 years and 10 seconds are equally long (or short). In comparison to any finite span of time, 10 years is longer than 10 seconds. The same applies to lengths and heights. I see no reason to say that there is no common reference for duration. The amount of time it takes for the earth to go once around the sun (or to spin once on its axis) is commonly used as a unit of duration.

My view (which I defended in my recent book, "Laws and Lawmakers" from Oxford University Press) is that symmetry principles in physics are widely regarded as meta-laws. For instance, the principle that all first-order laws must be invariant under arbitrary displacement in time or space explains why all first-order laws have this feature (and, in a Hamiltonian framework, ultimately explains why various physical quantities are conserved). The symmetry principles function as constraints upon what first-order laws there could have been. Had there been an additional force, for instance, then the laws governing its operation would have obeyed these symmetry principles, since these symmetry principles are meta-laws. Eugene Wigner and others have suggested that the relation of symmetry principles to the first-order laws they govern is like the relation of those first-order laws to the particular events they govern. I see no reason why symmetry principles would differ from first-order laws by holding in...

The paradox concerns the logic by which we are justified in forming our expectations about the future. Suppose we observe a bunch of emeralds and at the time that we observe them, each is green. This evidence is generally thought to support our expectation that on the first occasion on which we observe an emerald after the year 2100, let's say, we will find it to be green. The pattern of reasoning seems to be "If every F [emerald] we have examined has been found to be G at the time we observed it, then we should expect any given F to be G at any future time at which we might observe it." However, this pattern of reasoning (Goodman shows) cannot in fact support our prediction. For suppose that instead of making G = green, we make G = grue, where an object at a given moment is "grue" at that moment if and only if that object is green at that moment and the moment is (let's say) before the year 2100, or the object is blue at that moment and the moment is during or after the year 2100. So every emerald we...

Thank you for your question. The standard answer is that an instant lasts for no time at all. That is to say, the start of an instant and the end of an instance occur at exactly the same time. An instant is indivisible; it has no separate beginning, middle, or end. You might think of time as like a number line, with (for instance) zero as the time when you started reading this sentence and 1 as the time when you arrived at the end of it. Then each number between zero and 1 corresponds to an instant of time. None of those instants is any length of time at all. Of course, that an instant of time lasts for no time at all might lead you to wonder how a span of time lasting, say, for an hour could possibly consist of a bunch of instants each lasting for no time at all. This is closely related to some of the paradoxes first proposed by the Greek philosopher Zeno thousands of years ago. Bear in mind as well that between any two instants of time, there is another instant of time -- and, indeed,...

There are several considerations that count strongly against Popper's "falsifiability" criterion, but I'll mention just two. Remember that Popper's criterion is intended to distinguish science from non-science (or pseudoscience) on the grounds that a theory is scientific if and only if it is 'falsifiable', i.e., there are possible observations that would logically contradict it. Now for two quick arguments against this view: 1) Consider a statistical hypothesis, such as "This coin has a 50% chance of landing heads and a 50% chance of landing tails on any toss." Statistical hypotheses play a very important part in many important scientific theories (such quantum mechanics, evolutionary biology, statistical mechanics). But no possible sequence of coin toss outcomes logically contradicts this '50% chance' hypothesis. Some outcomes support the hypothesis; some disconfirm it. None falsifies it. 2) Many significant scientific hypothesis make no observable predictions all by themselves, but can...