I'm not sure about the relationship to Plato's theory of ideas, but there are many connections between programming and philosophy. I'll mention just a few. Some of the earliest investigations into natural language semantics appealed to ideas connected to the notion of compilation. Roughly, the thought was that understanding an uttered sentence might be something like compiling a program, i.e., translating it into the "machine language" of the brain. My own view, which is probably the majority view, is that this is seriously confused, but it has been attractive to many people. The idea that "the mind is the software of the brain" has also been very attractive, since it was first articulated (though not quite in those terms) by the great British logician Alan Turing. There are many ways to implement this idea, perhaps the most familiar of them being the various forms of functionalism. Finally, philosophers are often interested in formal languages, and software languages are certainly a variety...

What Prof Pogge has said represents one perspective on this issue, but it involves assumptions that can be rejected. The central issue is whether you are prepared to speak of "how many sets there are". If so, then let Fred be how many sets there are, that is, the number of things that are sets. It is sufficiently clear that Fred is the largest number. In standard set theory, by which I mean Zermelo-Frankel set theory (ZF) and its extensions, there is no such thing as the number of things that are sets. There just isn't such a number. But there are other set theories in which there is such a number, and one can, in fact, consistently add to ZF an axiom known as HP which allows us to speak of (cardinal) numbers in a way different from how ZF by itself allows us to speak of them. And then there is a number of all the sets there are, and it is again the biggest number. How can the question how many sets there are simply fail to have an answer? The idea to which Prof Pogge is giving expression is...

This depends in part upon what you mean by "mathematicians". Ordinary mathematicians, by which I mean mathematicians who aren't particularly or specially interested in logic, have generally, as a group, been utterly uninterested in Gödel's theorems. Reactions vary from case to case, and some are based on ignorance. But we do know, generally, that a huge proportion of "ordinary mathematics" can be done in what are, by the standards of set theorists, very weak theories. So the incompleteness of these theories tends not to be an issue. We've hardly exploited the strength they have. Another way to put this point is that, by and large, we know of very few "interesting" mathematical claims---claims that would be interesting to an "ordinary mathematician"--- that can be shown to be independent of these same, quite weak theories, let alone independent of Zermelo-Frankel set theory plus the axiom of choice, which is what most logicians would regard as sufficient to formalize the principles used in ordinary...

One of the things usually taken to be distinctive of mathematical and logical truth is that such truths are in some very strong sense necessary , i.e., they could not have been false. Assuming that it is in fact true that 2 + 2 = 4, how could that have failed to be true? (Or, to take a logical example: How could it fail to be true that, if Goldbach's conjecture is true and the twin prime conjuecture is also true, then Goldbach's conjecture is true?) Presumably, the answer to this question depends upon what, precisely, one thinks "2 + 2 = 4" means, but it is hard to see how one could accept the statement that 2 + 2 = 4 as both meaningful and true and think that it might not have been true. It's important to be clear that this statement does not say anything about how actual objects behave, e.g., that if you put two oranges on a table with two apples and no other pieces of fruit, then you'll have four pieces of fruit. Weird things might happen in some worlds, but that would not make it false in...

I would put the question slightly differently, if I understand it right: The question is whether the cardinal number 3, used to say how many of something there are, is the same or different from the real number 3, which is used to report the results of measurement. There is of course a different between the cardinal number 3 and a length of three meters, but the question is whether, when one says, "There are three apples" and "This board is three meters long", we refer to the same number three both times. Mathematicians and people who work on foundations of mathematics tend to have different views about this, at least in practice. The way one defines the cardinal numbers in set theory, for example, is very different from how one defines the reals. But working mathematicians will often speak of "identifying" the cardinal with the real and often seem impatient with such niceties as whether they are really the same. A more difficult question, I think, concerns cardinals and ordinals....

I think it's important to distinguish the two sources of "failure", not so much as regards Principia but as regards logicism quite generally. I'll stick, as Prof Smith did, to arithmetic. Here's a way to put Gödel's (first) incompleteness theorem: the set of truths expressible in the language of first-order arithmetic cannot be listed by any algorithmic method, i.e., it is not (as we say) "recursively enumerable". Now why is that supposed to show that logicism fails? Because the set of theorems of any first-order formal theory is recursively enumerable. This is a consequence of Gödel's first great theorem, the completeness theorem for first-order logic (and also of what we mean by a "formal" theory). So the truths aren't r.e. and the theorems are—you can list the theorems but not the truths—so the theorems can't exhaust the truths. Now why is this a problem for logicism? Obviously, as the argument has been stated, it depends critically upon the assumption that the proposed logical...

To add a bit more, there are some interesting applications of urelements in set theory. Perhaps the most famous example is Quine's theory New Foundations. NF, as it is known, which does not permit urelements, remains something of a puzzle: It is not known if it is consistent. But NFU, which is just NF plus urelements, is known to be consistent if Peano arithmetic is. See the wikipedia entry on NF for more. I seem to recall some interesting results due to Vann McGee about ZFU, as well, but they do not come immediately to mind.

Thank you for this wonderful question. I don't myself know whether to say that mathematics is something divine, but the idea that it is has a long history, going back at least to the early modern Rationalists. Many of them suggest, directly or indirectly, that, in uncovering the (as they saw it) fundamentally mathematical principles that describe the operation of the universe—this is the very birth of mathematical physics—one is thereby limning the very thoughts of the Creator, which gave birth to, and continually sustain, the universe. Even today, one often reads remarks by physicists (and other scientists) that have a similar bent: that there is an astonishing beauty to the fundamental laws that, in some way, seems to provide a glimpse of the divine. Note that this essentially aesthetic response needs to be sharply distinguished from any form of the argument from design. It is not that people think, "Well, this is all so incredible, someone had to design it". It's more, "Wow", followed by an...

You aren't the only one who finds mathematical fictionalism puzzling. But the nature of the analogy between mathematics and fiction needs to be spelled out carefully and, once it has been, I think a sensible fictionalist will have the resources to deny that there is an equally good fiction in which 2+2=5. Simple equations like this one are in fact a good case for fictionalism, because there is a clear sense in which their content can be reduced to pure logic. Fictionalists ask us to think about the application of such statements. So how is "2+2=4" applied? Well, if you have two apples, and you have two oranges, and no other fruit, then you have four pieces of fruit. More generally, if you have two Fs, and you have two Gs, and none of the Fs are G, then you have four things that are either F or G. (That can be written out in logical notation fairly easily.) The thought, then, is that our talk of numbers as objects is a fiction that we build on top of these sorts of simple equational facts, one that...

There are a few kinds of support. One is that one can prove certain special cases of the conjecture that seem inherently unrelated, so one thinks that these special cases must really be true because a certain generalization of them is true---and that's what one conjectures. But conjectures are often based upon a dim and hard to express appreciation for "what is going on", so that it just sort of seems as if the thing ought to be true. One can sometimes give reasons to think things ought to work out that way, but they wouldn't be the kinds of reasons that would count as a proof.