…plenty of those kinds of atheists in youtube

Plenty of all kinds of militant, weird, crazy, and too often hateful people from all backgrounds and beliefs on youtube-seriously, what is up with youtube attracting all the “venomous people” in the world?

You should be an atheist if and only if you actually believe that it’s true that no God exists. To be an atheist because you like Dawkins or Harris or Russell is just as stupid as not being an atheist because you don’t like them.

We’re getting very close to a discussion much too technical for this forum

That we are, so I’ll just answer your question and drop it.

So, what if Aristotle had rejected the law of the excluded middle and we had 2000 years of intuitionist logic rather than 2000 years of classical, bi-valued syllogistic logic?

Intuitionist logic had fairly strong support until the early 20th century. Hilbert got a lot of flack for using non-constructivist proof techniques. I think that it eventually lost broad support because we got to the point that there was just too many things that couldn’t be done with it. (So it’s possible that mathematics would have “recovered” even if Aristotle had made such a “mistake.”) Without LoEM, you basically lose case structure as well (unless you’re willing to do a lot of extra work to justify it in every instance) and things quickly become so annoying that I try to avoid bothering with them. I don’t know enough about intuitionist logic to say for sure, but I’ve always gotten the impression that it’s essentially a subset of bi-valued logic. Overall, I think the main difference is more what we know to be true than what is true. They may not think that the LoEM is justifiably true, but they usually don’t go so far as to suggest it’s false. I get the impression that olvlzl might actually be suggesting this, at least so far as it applies to metaphysics. Some Buddhist philosophy tends in this direction as well. I’ve tried my best to understand what they’re talking about, but I get lost with anything too far beyond “codependent origination.”

But my gut feeling is that the mathematical language that would exist in such a world would be radically different.

I’ll agree with this. I’m focused more on the mathematical ideas than on the language used to convey them.

I run into a bunch of “religion haters” via my blog. They send me hate mail even if I post something that is slightly supportive of religion in any way.

Mojoey, I’ve found that anything that even remotely could be taken to deny materialist fundmentalism brings them down on me. They even read things in my posts that not only aren’t there, but are refuted within the post they believe they find them in. And these are the same people who believe in the “meme”. I find it’s hard enough just to present an idea and have it read, nevermind reproducing itself like a virus.

Atheists are no less or no more likely to be rational than religious believers or agnostics. In fact I’d say that you find a higher percentage of rational thinkers among agnostics and frequently find people who are more careful about their facts among liberal religious belivers. However, any individual from any group can be any level of thinker.

About symbolic manipulation, I learned “System F” and found the experience interesting but useless and now serves mostly to show that my teacher went to Yale, though I didn’t, and that I’m a geezer.

The funny thing is, I do not know very many atheists personally, even after 30 years as an atheist. The few that I do know are not out trying to convert people. I run into a bunch of “religion haters” via my blog. They send me hate mail even if I post something that is slightly supportive of religion in any way. I’ve never understood that, there is good and bad in everything.

Great post and thread – I find these wonderful to read over a cup of coffee in the morning.

We’re getting very close to a discussion much too technical for this forum, considering it has nothing to do with the original post — and especially since we agree on the most important points about the relation between probability and propositions. (This is my way of backing down without admitting defeat.)

However, I think one key issue is this:

The affect of symbolic manipulation in our brains on mathematics is pretty small (basically, A=A, either A or not A, not both A and not A, are all we take for logic.

The history of mathematics and the history of logic are fully intertwined. So, what if Aristotle had rejected the law of the excluded middle and we had 2000 years of intuitionist logic rather than 2000 years of classical, bi-valued syllogistic logic? There’s no way of working out the truth of any such counterfactual situation. But my gut feeling is that the mathematical language that would exist in such a world would be radically different. Sure, there would be a translation manual between that weird math and our math, but that doesn’t mean that they would really be the same — just like its a mistake to say that since we can translate Navajo into English they really are, in essence, the same language.

Yes… and no. Had the history of mathematics been different, with different axioms discovered first, the kinds of mathematical constants would have been different. The exact content of mathematics isn’t set by the universe. The only real limit is internal consistency.

You’re absolutely correct that internal consistency is the key and that the axioms used can affect the kinds of mathematical constants we talk about. However, their values are still remarkable consistent. For example, in Euclidean geometry we measure pi as 3.14159265358… and under any axiomatic system categorically equivalent to it, we would find the same thing. In hyperbolic geometry, attempts to find the areas are usually unsuccessful (for one thing, similar triangles are congruent, so attempts to find area by triangulation often fail. For another, rectangles don’t exist, so attempts to find area by a Riemannian limiting process also often fail) and so the question of a circle’s area is less likely to come up in the first place. And if we do succeed, we find that the circle has area 4(pi)sinh^2(r/2), so pi is actually intimately related to this different axiomization of geometry as well. Someone who had developed hyperbolic geometry before Euclidean wouldn’t have defined pi as the ratio of a circle’s circumference to its diameter, but they would still have discovered that the number was important in and of itself.

If we embed this in the axiomization of field theory plus a bit of analysis, we can see that getting all of the real number system depends only upon having a number 1 distinct from 0 that can be added to itself as many times as one wants without ever getting the answer 0 and on a use of limits to get completion of the line. It’s hard to argue that anyone attempting to create a mathematics wouldn’t eventually come up with an axiomization (or something equivalent) that gives you this.

Mathematics is, ultimately, a language we use to describe the world. Given that (1) the world is likely unified under causal and scientific law, and (2) our language centers, abstract thought centers and symbolic manipulation centers of the brain are governed by our genes, it is likely that there is only a small range of useful mathematical systems. But I think it’s a stretch to say that there is only one.

I would say that it’s a language that can be used to describe any world. The affect of symbolic manipulation in our brains on mathematics is pretty small (basically, A=A, either A or not A, not both A and not A, are all we take for logic. Axiomization then requires only the idea that terms can be put in relation to each other). A more likely source of change would be the physical world, as some axiomizations are based on things observed in reality. E.g., if we lived in a universe where I could draw a triangle on a piece of paper and see it’s defect by sight, we would definitely have developed hyperbolic geometry before Euclidean. But it still does come down only to our choice of what to explore: the same ideas would still be out there regardless of what our universe looked like.

olvlzl, no ism, no ist seems to think that god it/him/herself is probable or improbable, but this doesn’t follow from the math at all.

It sounds to me like olvlzl is denying that the concept of probability is even valid.

The existence or not of god doesn’t depend on our beliefs in any way, just like the existence of Moscow doesn’t depend on our beliefs. So it’s absurd to say that Moscow exists with probability less than 1. Moscow exists or it doesn’t.

That’s the heart of it. Quantum mechanics aside, it’s doubtful that anything actually has a probability other than 0 and 1. However, with limited information, we can repeat an experiment over time to establish a pattern of results, so that we can understand long-term behavior without understanding short-term. The non-repeatability makes things more difficult god, but talking about a probability is still a reasonable thing to do, as long as one doesn’t try to be overly precise in quantification.

The truth of my belief depends on the universe. But the probability of my belief depends on my evidence, my patterns of inference, my trust in my sources, etc.

Exactly. Probability is sort of a measure of what odds we would need to bet on an outcome (i.e., if I get better than 6:1, I’m willing to bet that a fair die will land on any number you choose). Thus, probability is a short-hand for all of the above variables.

I didn’t invent the phrase. Ok, how about “God is unimaginable, unseeable and all mighty.” a rough translation of the beginning of Schoenberg’s great opera Moses und Aron? I could also point out that many scientists and others consider their particular means of knowing the universe as being all powerful, or at least sufficiently so in order to comprehend the entire universe. Dawkins places his theories of genetic inheritance in a position of near omnipotence on the basis of scant evidence. His “meme” leads me to believe that his opinion of his own creative impulses is way over sized.

You might find this amusing to read. I found it to be fun.

You being a mathematician I’m kind of surprised that the definition of something without a full understanding of the implications of it, even up to infinite numbers of implications, gives you a problem. It was my impression that this kind of thing happens in math and that a greater unity as yet unknown was assumed.

I’m not quite sure what you mean, but: in mathematics, terms mean nothing except what we say they mean as a precise combination of already defined terms and axiomatic relations to other terms. That doesn’t mean that there aren’t unknown results out there. It means only that talking about a term without having clearly defined what the term means is a waste of time. As it stands currently, saying god is all powerful is exactly as meaningful as saying god is bawahickijlimpi (i.e., not at all). Using the phrase “all powerful” is essentially a linguistic trick, since you’re using it in a conventional sense of implying a degree of power and then falling back on an unconventional sense of implying that we don’t understand it because it applies to god. In the end, it comes down to this: you either do or do not have a clear idea in your mind as to what you mean when you say “god is all powerful.” If you do have a clear idea in mind, then answering the stone question should be easy for you (just as it would be for any other being who’s level of power you comprehend: for example, I am not powerful enough to create a stone that I can’t lift, since I lack the power to create stones). If you do not have a clear idea in mind, then saying “god is all powerful” is no more meaningful than saying “god is bawahickijlimpi” and you ought to use this term instead, lest you confuse people into thinking that you’re claiming that you do know what the words you are using mean.

I’d love it if religion would disappear. But my goal is not to destroy religion. My goal is to make people think critically and to think for themselves.

Agreed. Indeed, the problem I have with the so-called militant atheists (or whatever you want to call “those” atheists) is that they aren’t that good at showing critical thinking by example. This is a side effect of crossing the line from brisk critique of religion, which can be done rationally, to demonization of theists, which, like demonization of any other group, requires distortion of facts or logic. The Rational Response Squad is ironically often an example of this. Dawkins goes back and forth across that line in TGD. He’ll point out valid problems in the cosmological argument and the argument from personal experience on the one hand, but on the other hand, selectively quote the Catholic Encylopedia to give the false impression that it dismisses atheism without much discussion. I doubt that few people recognize in detail the errors that these atheists commit, but they sense that something isn’t quite right, even if they can put their finger on it.