Recent Responses

Is Russell's Paradox a problem for our confidence that 2+2=4 is true? I've never understood how big a problem it represents in math. Does it throw everything into doubt, or just some things? The <i>Stanford Encyclopedia</i> entry is a bit technical.

Russell's Paradox is a problem for set theory--or at least it was when Russell discovered it. The most popular modern approach to set theory is based on the axioms developed by Zermelo and Frankel, and the Zermelo-Frankel (ZF) axioms are formulated to avoid the paradox. So Russell's Paradox is not a problem for modern set theory.

The reason paradoxes in set theory are considered to be such a serious matter is that most mathematicians regard set theory as the foundation of all of mathematics. Virtually all mathematical statements can be formulated in the language of set theory, and all mathematical theorems--including your example 2+2=4--can be proven from the ZF axioms.

But you ask about our "confidence" that 2+2=4 is true. I don't think anyone's confidence in 2+2=4 is based on the fact that it is provable in ZF set theory, even though ZF is regarded as the foundation of mathematics. It's hard to imagine anyone having serious doubts about whether or not 2+2=4, and having those doubts relieved by seeing the proof in ZF. So if a new paradox were discovered that showed that the ZF axioms were flawed, I don't think anyone's confidence in 2+2=4 would be shaken. We'd just try to fix the flaw in the axioms in a way that would allow us to still develop mathematics.

How do we resolve the fact that our finite brains can conceive of mental spaces far more vast than the known physical universe and more numerous than all of the atoms? For example, the total possible state-space of a game of chess is well defined, finite, but much larger than the number of atoms in the universe ( Obviously, all of these states "exist" in some nebulous sense insofar as the rules of chess describe the boundaries of the possible space, and any particular instance within that space we conceive of is instantly manifest as soon as we think of it. But what is the nature of this existence, since it is equally obvious that the entire state-space can never actually be manifest simultaneously in our universe, as even the idea of a board position requires more than one atom to manifest that mental event? Yet through abstraction, we can casually refer to many such hyper-huge spaces. We can talk of infinite number ranges like the integers, and "bigger" infinite ranges like rational numbers, and yet still "bigger" infinite ranges like the irrationals. In what sense can we say all of these potential huge spaces exist (and they must, since we can so easily instantiate well-formed members of them, at will) yet we don't have even the slightest fraction of sufficent space for them in our known universe?

To add a word or two to Dan's great response: there is no questionthatmathematics deals with infinite collections, but what those are, whatwe mean when we make claims about them, which claims are correct —these have been hotly disputed issues for thousands of years. (Inthe history of mathematics, concern for these foundational questionshas waxed and waned. There have been times, for instance in theearly part of the twentieth century, when disputes over these issues,were very heated and split the mathematical community. There have beenother times, for instance now, when mathematicians have been lessinterested in these issues — although of course there are alwaysexceptions, like Dan.) The basic question — what does it mean to call aset "infinite"? — is so fundamental that it's simply astounding that wedon't know how to answer it.

Onone way of looking at the matter, what Dan called "platonism", to saythat a set is infinite is simply to have given a measure of its size.To say that a set is infinite is much like saying that it's got 17elements in it: if you counted up the elements in the second set you'dfind there were 17 of them, and if you counted up the elements of aninfinite set you'd find there were infinitely many of them.

But on another of way looking at the matter, this is insane. How can one finishcounting up the elements in an infinite set? Isn't that what "infinite"means, that the process of counting never stops? On this way of lookingat things, to call a set "infinite" is not to describe the size of someactual collection, but rather to mark it off from all finitecollections: finite collections are ones for which the process ofcounting their members eventually stops, while infinite ones arecollections whose elements we can keep on generating without end.

The first conception accepts the existence of the actual infinite:a collection that actually contains infinitely many objects. The secondconception rejects this as unintelligible and talks instead of the potential infinite:to say that a set is infinite is not to make a claim about the size ofan actually existing object but rather to say that each of its elementscan potentially be brought into existence. (The two conceptions will beconfused if you think that an entity that can potentially be broughtinto existence really exists after all — and has the property ofpotential existence attached to it. See here for some comments on a comparable error.)

Should education be a means to an end?

I don't see anything wrong with using education as a means to an end, as when I suffer through a dreary course on car mechanics so that I can learn how to fix my own engine. Having said this, I don't think education is always merely a means to an end: not only can it be fulfilling to learn certain things even if this knowledge is put to no practical use, but the very process of educating oneself can be fulfilling independently of any value practical or otherwise in the things learned.

If you were sent back 100 years in time and met a fellow philosopher, what advances in the field since his or her time would you tell him or her of? Would you be able to convince him or her of what you said?

To my mind, the formulation, discussion, appreciation, and absorption of the work of Gottlob Frege, Ludwig Wittgenstein, W.V. Quine, Donald Davidson, and Saul Kripke have allowed for far deeper, sharper, and more sophisticated discussions in the philosophy of language than have ever been possible before. Could I convince someone of this after traveling back in time, you ask. Could I convince someone of that now?

A discussion with a philosopher friend got me all bewildered. He claimed that we cannot say that animals feel pain, because a mind is necessary to feel pain, and animals don't have a mind. My argument was twofold: 1. How do we know that animals don't have minds? 2. Pain is a result of stimulus to certain parts of the brain. If we assume that animals don't have minds, we can still see that their brains respond to pain stimuli the same way as ours. Even if they are unable to cognitively translate an external factor into a thought train like "I stuck my hand on a hot plate, it hurt, so I removed my hand from the hot plate", surely we can watch them pull back from things that we would experience as painful. I was wondering what your thoughts are on this subject. Thanks.

I know of no good argument for the conclusion that animals cannot feel pain, and given the behavioral and physiological similarities between us and some animals the evidence seems very strong that some do. A biologist friend of mine told me about an experiement with, yes, rats. These rats had severe arthritis, a condition very painful in humans. They were given a choice between plain water and water laced with a tasteless drug (tylenol, perhaps) that does nothing to improve the arthritis, but in humans reduces pain. The rats quickly came to prefer the water with the pain-killer. This is no proof that rats feel pain, but it is a telling argument. And remember that you have no proof, in the strong sense of that term, that people other than yourself feel pain either.

Presuming that it is impossible to write unbiased history, does that make the discipline invalid in that it can never be what it would ideally (at least for many) be: a completely truthful presentation of the past?

I'm not sure what you're presuming until you say what "biased" means. Do you believe that contemporary physics is biased? If not, then what is it about historical research that makes it impossible for historians to attain the same degree of rigor and truth that physicists do? And if so, then what would inquiry have to look like in order for it to be "unbiased"?