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.

I don't think anybody's yelling, just a (hopefully) friendly debate. We most likely just have different definitions of what constitutes a physical assumption. Care to give your definition?

[quote name='way2lazy2care' timestamp='1296584985' post='4768048']
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.

I don't think anybody's yelling, just a (hopefully) friendly debate. We most likely just have different definitions of what constitutes a physical assumption. Care to give your definition?
[/quote]

I'd say something whose definition exists only when physical observations or properties are included. Something like gravity is defined entirely because we have observed it to be that way. 1=1 will always be true regardless of how we observe it.

I am of course ignoring that things like numbers and equivalence have been defined by man and man is physical therefor they could be considered physical. The concepts of equality or numbers do not require a physical world to hold definition.

I'd say something whose definition exists only when physical observations or properties are included. Something like gravity is defined entirely because we have observed it to be that way. 1=1 will always be true regardless of how we observe it.

I am of course ignoring that things like numbers and equivalence have been defined by man and man is physical therefor they could be considered physical. The concepts of equality or numbers do not require a physical world to hold definition.

I think this is a decent definition, but I feel that even something like 1=1 has the hidden assumption that two things can share the same identity in some sense. I couldn't tell you what a reality would be like where that wasn't true, but I can imagine defining a mathematical system where two things can only ever be approximately equal, and thus 1=1 is never true under that system. Admittedly, I'm seriously pushing the limits of what "physical" means.

What I really want to get at is the idea that even something as simple as 1=1 relies on an axiom (usually S(x) = S(y) => x = y under Peano) and that this axiom appears to be inspired by the reality we live in, where objects can be reliably compared and classified. One of my professors told a great story about a visiting IBM researcher who talked about how the design of one of their original cryptographic co-processors had been formally verified by a proof checker. It was hacked within days of its release because people figured out how to mess with the power input to the chips and get them into undefined states. The moral of the story was: logic depends crucially on external factors; the process of abstraction does not remove those factors, it merely assumes they are within tolerable ranges.

[quote name='Prefect' timestamp='1296547760' post='4767862']
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!
[/quote]
Good. I was asking because what you said before can certainly also be said by somebody lacking that understanding, and in that case any further discussion would probably have resulted in the two of us talking about entirely different things

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.
[/quote]
Okay. The underlying difference in thinking is that I would say that proofs do float in an abstract space. They are relative to a set of axioms and a method of operating with them, but that does not mean that all those things do not float in some abstract space. At least that's my point of view, and clearly yours is different, which is where the differences come from.

If I understand you correctly, you are essentially taking our physical world as the foundation of how you think about these things. That certainly makes sense, given that we personally operate on and talk about proofs in that physical world, and we can't know what goes on outside that physical world, or if there even is an outside. We can certainly speculate on it, but of course our speculation is limited by the constraints of the physical world.

I do not take our physical world as the basis, because the question of why things exist and why our physical world is the way it is is simply too appealing to me. Once you start down that line, having the physical world as the basis of everything is rather unsatisfying because it seems so arbitrary, and having a very abstract "space" (the term can be misleading) as the basis, within which our physical world is simply one object besides many others, becomes much more plausible. It is this abstract "space" that I consider proofs and the rest of mathematics (as the body of knowledge) to exist in. Naturally, this means that they are free of physical assumptions.

Both points of view have their merits and it doesn't make much sense to talk about which one is The Right One.

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?
[/quote]
Careful: I did not say that mathematics as a science is purely a notion of abstract symbol manipulation. That would be very weird considering my experience as a mathematician

What I'm saying is that I consider there to be a distinction between mathematics-the-thing and mathematics-the-science, which is the science of studying mathematics-the-thing. I consider those to exist in very different settings: mathematics-the-thing "floats in an abstract space", as you called it, whereas mathematics-the-science ultimately studies mathematics-the-thing through whatever means there are available to us in the physical world. I believe that we have a pretty accurate representation of the real thing, although we can never be entirely certain of that. I also believe - as I've indicated before - that all this is actually pretty irrelevant to the daily life of a working mathematician, much like talking about what science is or is not is pretty irrelevant to the daily life of a physicist.

As for Hofstadter, who knows. Frankly I don't care. His book certainly influenced me, but it's not unheard of that books influence their readers in ways that the author originally did not anticipate or intend. I mean, I've read Atlas Shrugged, but its influence on me was most likely rather different from what Ayn Rand intended.

I will cede the floor to the cats now.

[quote name='way2lazy2care' timestamp='1296596097' post='4768141']
I'd say something whose definition exists only when physical observations or properties are included. Something like gravity is defined entirely because we have observed it to be that way. 1=1 will always be true regardless of how we observe it.

I am of course ignoring that things like numbers and equivalence have been defined by man and man is physical therefor they could be considered physical. The concepts of equality or numbers do not require a physical world to hold definition.

I think this is a decent definition, but I feel that even something like 1=1 has the hidden assumption that two things can share the same identity in some sense. I couldn't tell you what a reality would be like where that wasn't true, but I can imagine defining a mathematical system where two things can only ever be approximately equal, and thus 1=1 is never true under that system. Admittedly, I'm seriously pushing the limits of what "physical" means.

What I really want to get at is the idea that even something as simple as 1=1 relies on an axiom (usually S(x) = S(y) => x = y under Peano) and that this axiom appears to be inspired by the reality we live in, where objects can be reliably compared and classified. One of my professors told a great story about a visiting IBM researcher who talked about how the design of one of their original cryptographic co-processors had been formally verified by a proof checker. It was hacked within days of its release because people figured out how to mess with the power input to the chips and get them into undefined states. The moral of the story was: logic depends crucially on external factors; the process of abstraction does not remove those factors, it merely assumes they are within tolerable ranges.
[/quote]
I like such stories. To me, they reinforce the notion that mathematics has nothing to do with the real world. The mistake that was made is not that there was an incorrect assumption in the mathematics, but that a mistake was made in translating a mathematical statement into a statement about the real world. In that sense, I think this is not a good example to illustrate your point about axioms. (*) It is a very good example though to stress the fact that too great a belief in mathematics, and only mathematics, can be a problem.

(*) I do see your point about axioms, but I don't think it's possible to really explain it well, and give examples, without becoming very technical.

All I was trying to say is that religion is a world view that is not, necessarily, illogical or irrational or a "fantasy land".[/quote]
It is, by definition. Religion without the supernatural aspects isn't religion anymore.

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.[/quote]
I assume you say this because it is impossible to prove the non-existence of something, and thus the burden of proof falls the other way on to those making the extraordinary claims? And not because you are just being mean to people who you don't agree with?

All I was trying to say is that religion is a world view that is not, necessarily, illogical or irrational or a "fantasy land".

It is, by definition. Religion without the supernatural aspects isn't religion anymore.[/quote]
Explain how you jump from point A to point banana.

Both points of view have their merits and it doesn't make much sense to talk about which one is The Right One.

Agreed. When we start talking about things this "meta", correctness gets a lot fuzzier. I see your points and I can say that I'm less confident in my views than before. I need to make friends with some Mathematicians.

As for Hofstadter, who knows. Frankly I don't care. His book certainly influenced me, but it's not unheard of that books influence their readers in ways that the author originally did not anticipate or intend. I mean, I've read Atlas Shrugged, but its influence on me was most likely rather different from what Ayn Rand intended.

Reading Atlas Shrugged helped propel me into learning more about philosophy in general, which ended up eroding all of the certainty I had in Objectivism during my early teens. Probably not Rand's intention either, though she may have been delighted to hear me parrot John Galt to my family's pastor. I must've been an annoying little sh*t

I will cede the floor to the cats now.

Meow.

Religion * humans = insanity


Joshua: Shall we play a game?


1.) A man died on a cross which somehow saves the entire human race (only a TINY fraction of 1% of life that has been on the planet) from living in eternal damnation.
2.) Somehow we are still all born into sin and guilt and should spend our lives repenting and asking for forgiveness.
3.) If you don't do this, you will spend an eternity in a burning bad place - if you do this you will spend eternity in absolute happiness.

My question isn't which God is at the top... it's.. what could possibly make someone read a relic written thousands of years ago that says these things, and nod their head.. "yea... yea.. this makes sense, I'm going with this one."

If you frame the debate in such derogatory language then of course it doesn't make sense that rational people would believe it. But how about you rephrased it like this:

1.) We were granted the gift of life by an omnipotent ruler who had no obligation to do so
2.) The whole reason we were given life was to live in relationship with that generous ruler, and yet we ignore him and live as though he wasn't there
3.) That ruler went out of his way to call us back to him by stepping into his own creation, taking on human form and suffering the consequences of our rejection in our place, and yet STILL we ignore him
4.) That ruler eventually gives people what they ask for and puts them out of his presence forever, but graciously allows those who accept his rescue to be in his presence forever

Of course, that's also a pretty biased way of presenting it, but hopefully you will see that it's simplistic at best to say "I can phrase this in a way that sounds ridiculous and that's sufficient reason to dismiss all those who believe it as loons"