I have no facts to back this, but I would say America's problems are due to human failings (including failings of the earthly church), whereas many of Saudi Arabia's issues are encouraged by their text.

America's problems are caused by religious texts too, but people have been brainwashed into not realising it's a religion.

I also disagree that Mathematics is not about the "real, physical world we live in". Even something as detached and abstract as a Turing machine embeds serious physical assumptions, such as a tape cell only containing a single symbol or blank at any given time (this assumption is broken in quantum computing models). Solomonoff, Kolmogorov, and Chaitin have produced an extensive body of work demonstrating that Mathematics itself is an empirical science, requiring the same inferential tools we apply to the "real, physical world" (see Algorithmic Information Theory).

I disagree, because I prefer to distinguish between mathematics as a body of knowledge and mathematics as a science.

By mathematics as a science, I mean the process by which we (humans) discover the body of knowledge that is mathematics. This process is frequently motivated by questions in the real world, that is without doubt true. You might even call parts of it empirical, e.g. when we write computer programs to explore a conjecture before we set out to find a proof or a counter-example. An example of this kind of "empirical" work would be the many computational partial verifications of the Riemann hypothesis.

By mathematics as a body of knowledge, I mean the collection of definitions, proofs, theorems. Those can be understood to exist entirely independently of our physical world, and contain no physical assumptions.

Turing machines are a good example to explore this. The definition of a Turing machine, as a mathematical definition, contains no physical assumptions at all. It is simply a definition. The entire body of rigorous theorems (as opposed to "theorems") in computability and complexity theory contains exactly zero physical assumptions. However, there is a boundary where physical assumptions come in: the (physical) Church-Turing thesis, and its complexity theoretic variants. If you believe in the physical Church-Turing thesis, then you are making physical assumptions (you could say the same about the original Church-Turing thesis as well, depending on your philosophy).

Admittedly, many people are often sloppy in drawing such boundaries. People make statements like "SAT is NP-complete, and therefore cannot be solved efficiently", even though that statement by itself is very sloppy. However, these kinds of imprecisions are usually unproblematic, because there is usually a shared understanding of the underlying assumptions, and taking shortcuts is then more convenient and efficient for communication. However, the fact that communication about such mathematical statements is often somewhat imprecise does not imply the same for the actual underlying mathematics (the body of knowledge). When you hear that example statement on a TCS conference, for example, you can usually assume that both speaker and listener are aware that "SAT is NP-complete, and therefore cannot be solved in polynomial time on a Turing machine, assuming P != NP" is a mathematical statement that is true (and I suppose you could go even further down the rabbit hole and talk about axiom systems), whereas "SAT is NP-complete, and therefore cannot be solved in polynomial time on any existing or potential physical computational device" is not a mathematical statement, but rather contains a bunch of underlying physical assumptions.

I would even say that this kind of clear separation between mathematics and the physical world is one of the greatest achievements of recent mathematics, because it allows us greater confidence in evaluating the correctness of proofs and thus the truth of statements. It is true that this separation does not necessarily play a big role in everyday mathematics as a science (unless you happen to be a set theorist or something), but a good mathematician always has it ready as a tool to reason about what is really true, and where she might be mislead by physical thinking.

By mathematics as a body of knowledge, I mean the collection of definitions, proofs, theorems. Those can be understood to exist entirely independently of our physical world, and contain no physical assumptions.

People often say this, but what does "no physical assumptions" really mean? Equivalence, grouping, and ordering may seem very abstract, but they require assumptions about preservation of state through time and repeatability of measurement. Kinds of mathematics can be defined that do not require these assumptions, though they do not appear to be terribly useful so far.

This is related to the thread in general, because it is precisely the problem I have with people going around saying they have an understanding of God (loosely defined as the created of the universe). The leap that one must take to claim any understanding, however minute, of God from the infinitesimal dot of their own existence is simply beyond me. This is usually where faith would come in, but all I see is an infinite space of potential gods with no principled way to choose (unless I have faith in some choice mechanism, ad infinitum).

I would even say that this kind of clear separation between mathematics and the physical world is one of the greatest achievements of recent mathematics, because it allows us greater confidence in evaluating the correctness of proofs and thus the truth of statements. It is true that this separation does not necessarily play a big role in everyday mathematics as a science (unless you happen to be a set theorist or something), but a good mathematician always has it ready as a tool to reason about what is really true, and where she might be mislead by physical thinking.

This is not about a sloppy distinction between theoretical and practical Mathematics. It may be the case that some useful mathematical "truths" cannot be proven under the set of axioms we have chosen, and must instead be adopted as axioms themselves. Gödel showed us that such truths must exist, but it's hard to tell if they're just a bunch of silly, self-referential statements or something much more useful. Chaitin conjectures that the Riemann Hypothesis is one such case, though I don't think he has any grounds for it.

[quote name='Prefect' timestamp='1296464252' post='4767392']
By mathematics as a body of knowledge, I mean the collection of definitions, proofs, theorems. Those can be understood to exist entirely independently of our physical world, and contain no physical assumptions.

People often say this, but what does "no physical assumptions" really mean? Equivalence, grouping, and ordering may seem very abstract, but they require assumptions about preservation of state through time and repeatability of measurement. Kinds of mathematics can be defined that do not require these assumptions, though they do not appear to be terribly useful so far.
[/quote]
Equivalence and ordering do not require assumptions about preservation of state or repeatability of measurement. Have you ever read a text on formal languages and logic? All these things can be boiled down to totally mechanic operations on strings of symbols. Perhaps you take issue with that, but then I'd like to understand in a more concrete way what those issues are.

For your reference, I would say that I've been heavily influenced in my thinking by Gödel, Escher. Bach. That book was my starting point in understanding such abstract systems, though it's been a very long time that I've read it.

as a close up to this thread I'd like to state that everyone who feels in the urge of proving that God doesn't exist is a freaking dumbass.

That said. Grow up.

I logged back in to GDnet after several weeks of hiatus, to find that the staff have dramatically updated the look, added polls and voting system, a C vs (insert other languages) thread, and a religious thread that is 15-page long.

But I have yet to see kittens.

I logged back in to GDnet after several weeks of hiatus, to find that the staff have dramatically updated the look, added polls and voting system, a C vs (insert other languages) thread, and a religious thread that is 15-page long.

But I have yet to see kittens.

[quote name='alnite' timestamp='1296558214' post='4767900']
I logged back in to GDnet after several weeks of hiatus, to find that the staff have dramatically updated the look, added polls and voting system, a C vs (insert other languages) thread, and a religious thread that is 15-page long.

But I have yet to see kittens.

http://www.youtube.c...h?v=vdQj2ohqCBk
[/quote]

Datss soo cuuute!!

Have you ever read a text on formal languages and logic? All these things can be boiled down to totally mechanic operations on strings of symbols.

I certainly hope so, since I'm working on a PhD in Computer Science and currently doing programming languages research! Yes, these things can be boiled down to mechanistic string operations, but that mechanism assumes that (1) the string will remain stable if it is not operated upon and (2) that operations are regular, in the sense that the same operation on the same string will always give the same result. Proofs do not float in some abstract space, they are relative to both a set of axioms and a method for checking (or executing) the proof. In practice, I think the methods we have are pretty solid, but they are certainly not the only game in town.

For your reference, I would say that I've been heavily influenced in my thinking by Gödel, Escher. Bach. That book was my starting point in understanding such abstract systems, though it's been a very long time that I've read it.

Yes, I've read GEB and even attended lectures by Dr. Hofstadter. I obviously can't speak for him, but my intuition is that he would not be happy with your notion of Mathematics as a science being purely a notion of abstract symbol manipulation. He talked briefly at one point about losing faith in Mathematics as a source of objective truth, and I've been meaning to get more details about it.

Since this thread is now being taken over by cats, how about we call it even?

People often say this, but what does "no physical assumptions" really mean? Equivalence, grouping, and ordering may seem very abstract, but they require assumptions about preservation of state through time and repeatability of measurement. Kinds of mathematics can be defined that do not require these assumptions, though they do not appear to be terribly useful so far.

Before you both yell at each other more, the problem is that things like preservation of state through time(we'll say time is the more abstract definition of a changing global state) and repeatability of measurement aren't physical assumptions.