Do infinite sets exist? Most mathematicians say yes, but to me it seems like infinite sets can only exist if we use inductive reasoning but not deductive reasoning. For example, in the set {1,2,3,4,...} we can't prove that the ... really means what we want it to. No one has shown that the universe doesn't implode before certain large enough "numbers" are ever glimpsed, so how can we say they exist as part of an "object" like a set. We can only do this by assuming the existence of the rest of the set since that seems logical base on our experience. But that seems like a rather weak argument.

We can use mathematical induction to prove that (i) infinitely many natural numbers exist from the premise that (ii) 1 is a natural number and the premise that (iii) every natural number has a successor. Although it's called mathematical "induction," it's actually deductive reasoning. I take it that (ii) is beyond dispute, and (iii) is at any rate very hard to deny! It won't do to demand proof of (ii) or (iii) before accepting this proof of (i), for if the premises in any proof must themselves have been proven, then we have an infinite regress: nothing could be proven in a finite amount of time. We've therefore proven that infinitely many natural numbers exist. The notation "{1,2,3,4,...}" is just one way of referring to the set containing all and only those infinitely many numbers. It's perhaps a fallible way of referring to that set, because it assumes that the audience knows which number comes next in the series. A more reliable way of referring to the set is "the set whose members are the...

I am looking for resources on a seemingly simple issue. I believe the seeming simplicity of this issue is quite deceptive: What is a "surface?" What allows anything to "touch?" Where does philosophy stand on this issue? Thank you for your time.

Excellent questions. I'm glad to hear you're looking into this issue. I think philosophers and scientists often throw around talk of "surfaces" much too glibly. I recommend starting your search with the SEP entry on the concept of a boundary, available here . It contains a lot of information relevant to your questions and a bibliography with several useful references.

Can we know for sure that the external world exists? I was wondering about it for a while, and yesterday I thought that it must. You see, when I drink alcohol, it is an empirically experienced factor that affects my mind. That would mean that my mind is connected to my body. And because I can observe, smell and taste alcohol, that would be a proof that my senses can be trusted, at least to a degree on which they operate. Is that a valid argument?

You asked, "Can we know for sure that the external world exists?" That will depend, of course, on what's required for such knowledge. Some philosophers have said that such knowledge requires a successful proof of the existence of the external world, but many other philosophers (especially in the last few decades) have said that no such proof is required. For those who think a proof is required, G.E. Moore famously (or infamously) offered one: see this link . If you investigate Moore's proof at that link and in other places on the web, I think you'll get a sense of how the proof you offered might be received by various philosophers.

Is knowledge based on memories?

Very interesting (if tantalizingly brief) question. There's reason to think that all human beings rely on their memories for any knowledge they possess. One might think I can know at least some facts about my present-tense experiences without relying on my memory, but what facts could those be? For example, if I'm to know that I have a headache (when I do have one), arguably I must know what counts as a headache, and isn't that something I once learned and now remember? Descartes (1596-1650) was very sensitive to the role of memory in human knowledge. He famously argued that (a) only if you're aware of the existence of a benevolent God do you have sufficient reason to trust your memory, and that (b) without sufficient reason to trust your memory you know virtually nothing. Both (a) and (b), and Descartes's arguments for them, can of course be questioned. You'll find more at this link .

I was recently at a job interview where I was informed that, if hired, I would have to sign a non-compete clause stating effectively that, if I were ever to leave my position at the company, I would be barred from taking up employment in that profession again for two years. To me, this seems extremely perverse. I invested a great deal of time and money and effort into educating myself in building a career in this particular domain, and I do not have the skills to support my family to a similar extent in any other career (the NCC is rather broad, if unambiguous, about which fields I may not enter). Is it ethical for a company to offer to end my unemployment while at the same time effectively threatening me with two years of un- or underemployment should I ever quit or be fired? This seems like an abuse of my vulnerability as a job seeker, at the very least.

I share your view that such a clause is at best very shoddy. Not being a lawyer, I wonder if the clause is even enforceable; you might ask someone who'd know. It might be unconscionable (a term I wish lawyers applied to more things than they do!) and hence unenforceable, or it might be unenforceable because too difficult in practice for a court to enforce (say, by issuing an injunction forbidding you from accepting a new job!). This may be advice of little value to a desperate job-seeker, but I'd steer clear of firms whose offers come burdened by such clauses.

"My body, my choice" is well known slogan from those who oppose laws that limit a woman's right to an abortion. Yet, the idea that a woman has a right to do what she wants to her body seems to have disturbing consequences. If a woman drinks too much alcohol or takes too many drugs then her baby will suffer the consequences. That child will then suffer many challenges in life because of his mothers supposed right to do what she wants with her body. Yet when I point this out to people they get angry and insist that I want to limit women's rights. In fact it makes me angry that anyone would disagree with the idea that a woman shouldn't be morally and legally responsible for the incalculable harm she can do to her baby by poisoning her fetus. I can grant that there are exceptions such as prescription medications but otherwise isn't the idea that women can't be held responsible for doing damage to a fetus that will then suffer after being born just a very extreme position even if its a popular belief? And I...

I'm no lawyer, but I believe that the courts in some U.S. jurisdictions allow a child to sue its mother for lasting harm she caused the child while it was in utero . (Here I'm assuming that the child is identical to something that was once in utero , an assumption not everyone will grant.) I don't know whether the plaintiff has ever prevailed in such a lawsuit. In any case, if someone can be assigned civil liability for causing such harm, then it's not a huge leap to hold her morally responsible for it as well. My sense as a non-expert is that the courts are still struggling with this legal issue, so it's an important time for philosophers to weigh in on the issue and thereby perhaps help the courts decide it wisely and justly.

Euclid in "Elements" wrote that "things which equal the same thing also equal one another." Is this true in all cases? I've read that it is only true for "absolute entities," but not to "relations," although I do not understand this exemption. Are there any examples of things that are equal to the same thing but not to one another? Are relations really exempt from Euclid's axiom, and if so, why?

If by the adjective "equal" Euclid means "identical in magnitude" (which I gather is what he does mean), then his principle follows from the combination of the symmetry of identity and the transitivity of identity . The symmetry of identity says that, for any x and y , x is identical to y if and only if y is identical to x . The transitivity of identity says that, for any x , y , and z , if x is identical to y and y is identical to z , then x is identical to z . Therefore, Euclid's principle has exceptions only if the symmetry of identity sometimes fails or the transitivity of identity sometimes fails. But I don't think either of them ever fails. Now, some relations that are similar to the identity relation aren't transitive. I might be (1) unable to tell the difference between color swatches A and B, (2) unable to tell the difference between swatches B and C, yet (3) able to tell the difference between swatches A and C. But...

How do you know that you are sure that your parent are your parents?

Your question asks about your knowledge of your own certainty : "How do you know that you are sure...?" That's a question about your knowledge of your own mind rather than about your knowledge of your parentage. So I think the best answer is "By introspection -- by looking inward -- and perhaps also by examining your own behavior for signs that you're unsure that the people you take to be your parents are in fact your parents." If, instead, you meant to ask how you can be sure that the people you take to be your parents aren't impostors, then you might have to seek DNA evidence (in the case of biological parents) or adoption records and other documents (in the case of adoptive parents). There's also a facetious answer: "Of course your parents are your parents -- that's a tautology!"

I am confused about how a conditional statement is necessarily true, and not false or unknown, when the antecedent and consequent are both false. According to the truth table, the sentence "If Bill Clinton is Cambodian, then George Bush is Angolan" is true. How can such an absurd sentence be true? It seems initially like the sentence could just as easily, or more easily, be false or unknown.

The truth-table for the material conditional says that any material conditional with a false antecedent is true. If we construe the conditional you gave as a material conditional, then (because it has a false antecedent) it comes out true. But the material conditional doesn't come out necessarily true unless it's not just false but impossible that Clinton is Cambodian (or else it's necessarily true that Bush is Angolan) . The material conditional has the advantage of being tidy, and a true material conditional will never let you infer a falsehood from a truth. Still, for the reason you gave (and for other reasons too) many philosophers say that the material conditional does a bad job of translating the conditionals we assert in everyday language. You'll find lots more information in this excellent SEP entry .