Is the positing of an infinite regress a legitimate explanation in philosophy respectively are infinite regresses logically possible?

Are infinite regresses logically possible? Surely it's logically possible for infinitely many positive or negative integers to exist, and they represent a kind of infinite regress: for every negative integer, there's a smaller one; for every positive integer, there's a larger one. Even those who say that only potentially infinite collections (and not actually infinite collections) are possible must admit the possibility of infinite regresses of this numerical kind. Is the positing of an infinite regress a legitimate explanation in philosophy? I don't see why it couldn't be. It seems to me that the burden rests with whoever denies the acceptability of an infinite regress of explanations. Indeed, I think infinite regresses of explanations are unavoidable given some highly plausible assumptions.

We often hear people saying about how a certain artist or composer is better than another. Many people, for example, believe Bach and Verdi to be better musicians than, say, Rihanna or Justin Bieber. I share this same belief, but it is mostly based on intuition than on rational arguments. It is certainly true that Bach was able to develop a musical theme in a much more organized and logical way than Rihanna is, but does it really mean that Bach is a better musician than Rihanna? Is it true that there is such thing as a good and a bad composer or is it all just a matter of taste? Could you point out to me some arguments and readings which are relevant to this type of question?

Aesthetics isn't my area, but since no one else has responded I'll take a stab at it. To someone who thinks that aesthetic judgments can't be objectively true or false -- someone who thinks that aesthetic judgments are in that respect fundamentally subjective -- I'd pose two questions: (1) We often seem to make objective aesthetic judgments, such as the judgments concerning Bach and Rihanna that you mentioned in your question; why not take those judgments at face value? Why think we have to interpret those judgments as non-objective? (2) If there's a worry that aesthetic judgments can't be objectively true or false, does that worry extend to normative judgments in general , including the judgment that some ways of reasoning are better than others or that some ways of treating people are better than others? If it does, then it's a worry about objective normative judgments in general rather than aesthetic judgments in particular. If it doesn't, then what makes aesthetic judgments less likely to...

Are 3 and √9 the same mathematical object (in light of the fact that they have the same numerical value), or are they distinct mathematical objects? In other words, are the expressions '3' and '√9' co-referential names (both referring to the number 3), or do they refer to different objects?

If "√9" refers to the positive square root of 9 (I'm not sure what the convention is concerning the square-root symbol), then I'd say that 3 and √9 are the same object, just as Mark Twain and Samuel Clemens are the same object. (Indeed, the plural verb "are" in each case is a bit of loose talk.) Leibniz's Law (the Indiscernibility of Identicals) therefore implies that everything true of 3 is true of √9, and everything true of Twain is true of Clemens, which seems right to me.

Hi, I was hoping for some clarification from Professor Maitzen about his comments on infinite sets (on March 7). The fact that every natural number has a successor is only true for the natural numbers so far encountered (and imagined, I suppose). Granted, I can't conceive of how it could be that we couldn't just add 1 to any natural number to get another one, but that doesn't mean it's impossible. It seems quite strange, but there are some professional mathematicians who claim that the existence of a largest natural number (probably so large that we would never come close to dealing with it) is much less strange and problematic than many of the conclusions that result from the acceptance of infinities. If we want to define natural numbers such that each natural number by definition has a successor, then mathematical induction tells us there are infinitely many of them. But mathematical induction itself only proves things given certain mathematical definitions. Whether those definitions indeed...

Thanks for sending a follow-up question. Prof. Heck, who knows this territory better than I do, provided helpful corrections and amplifications in his answer to Question 5068 . I recommend taking another look there. You wrote, "The fact that every natural number has a successor is only true for the natural numbers so far encountered (and imagined, I suppose)." The claim that every natural number has a unique successor, like the claim that 1 isn't the successor of any natural number, is an axiom -- a starting point -- rather than a conclusion drawn from examining or imagining data. Your fifth sentence suggests that you know this already. You're quite right that math induction proves its results only given its starting points, but of course that's true of all proofs: all proofs (even proofs that have no premises) rely on essential assumptions. You say that some professional mathematicians would rather accept the existence of a largest natural number than accept the paradoxical features of...

I love reading the Qs and As on this site, and a recent post recommended a book "The Elements of Moral Philosophy" by Rachels. I got the book from my local library and really enjoyed it. What would be a good follow-up book on these topics? I have a hard time slogging through the original basic philosophy works, so I really value a book like this that is serious but not too technical for a layperson. Thanks.

In the category "serious but not too technical for a layperson," I'd include Russ Shafer-Landau's short paperback Whatever Happened to Good and Evil? . It concentrates on metaethics (the fundamental nature of ethics) and moral epistemology (how we might know moral truths, if there are any) rather than normative ethics (particular theories of right and wrong). It's clear, accessible, provocative in places, and enjoyable.

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 .

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