"Mathematics is a tool invented by man, not some gift from on high. I can change the rules if I like.
But that’s irrelevant. Given the current axioms and the current rules of inference, pi is a constant, and there’s nothing that can change that fact or the value that pi has."
Ultrafilter:
I don’t think I follow you. Paragraph 1 says we can change the rules; Paragraph 2 implies that, nevertheless, nothing can change the value that Pi has. IF you are saying that we are unable to conceive of a reality in which Pi takes a different numeric value, and therefore we cannot intelligibly postulate such a reality–that’s the sort of argument I’d buy. But I’m not sure that’s what you’re saying.
A philosopher of mathematics named Paul Benacerraf has argued something like the following (if I followed it):
Such entities as numbers and constants are ultimately identified by the roles they play in the overall task of enumeration, calculation, and quantification. They are functional entities, defined by how they are used. “2” is whatever functions as 2; “Pi” is whatever functions as Pi. In other words, there are “slots” in reality that ae waiting to be occupied, and we identify a given entity by the slot it is in. BUT–important part–an entity may “fill the slot” without possessing each and every property of another entity that fills the slot equally well: not all specific slot-fillers are just the same one over again. There may be a Pi (somewhere) such that the first digit of the decimal expansion is an even number, or in which single-digit primes appear at places along the line other than they do in “our” Pi. Of course, to fulfil the function, it may well be that other parts of the mathematical universe will have to change to accomodate it–everything gets tweaked, and there’s a lot (an infinite amount?) of backing and filling.
But it can’t be written off as a logical impossibility, like a “non-triangular triangle.”
I take your point; the chemical and physical properties of matter are a completely different set of phenomena, but I still think the question is a philosophical one; Pi in Euclidian geometry is a property of the concept of ‘flatness’ (or at least that is my understanding) - it is arguable that the universe might have come into existence in such a way as that ‘flatness’ had a different set of properties.
The value of Pi has been set, just like the value of the charge on an electron, the speed of light and many other fundamental constants. It does not matter if they are physical constants or geometric.
Each reflects an aspect of how our universe is.
The constants cannot be changed. There is no evidence of a god that can change these constants.
A god didnt invent anything. Humans invented mathematics and discovered these constants, which just happen to be set at these values.
Someone else can start a discussion on the anthropic principle.
Again, I say that circles are irrelevant to working out the value of pi. Ultimately, pi can only change if we can change the concept of one-ness.
That, I think is completely inconcievable to the human mind. one is a discrete chunk of stuff, a finger, a rock, a dollar, whatever, there is a fundamental axiomatic property within each of these. I don’t see how that can be altered a whit.
BTW: Just changeing the semantics of one is not enough. You can play sophist tricks and say well, why dont we call sheeps ones then? the concept of one still exists, even if the name is different,
“What he’s basically saying is that there are other consistent theories that we could’ve picked as mathematics. But we didn’t.”
UltraFilter: Sorry, but I think you’re missing my point, and (I would argue) the real point of the OP. The question is not whether, in the universe as it is, we might have adopted some “other” mathematics with a “different Pi.” No, the question is whether an omnipotent divine being could so alter the universe (which is, more or less, to substitute a new universe for ours) that Pi in that new universe could have a different REAL value (ie, not just a trivially different value in terms of the symbols used to express it). I say yes, with some hesitation; and I interpret Prof. Benacerraf to be saying something similar.
“Analytic vs. Synthetic”
Something is analytically true if it is true by virtue of the assigned meanings of the words themselves, with no reference to any observations of the world. One might think of it as “true by definition”–as when we say, “One foot is twelve inches long.”
Something is synthetically true in the other case, wherein we base our judgment upon observations of the world. The formula for special relativity is an example.
The language comes from Kant, though the notion has a prehistory. If I recall right, the distinction has been challenged by W V O Quine.
“I thought God Is supposedly omnipotent. I also thought that an omnipotent being could do anything.”
Who:
The controversy is over what “anything” really means. The dominant view in the relevant parts of philosophy is that God can do anything that is “there to be done,” ie, any (real) “thing.” God cannot be expected to perform some task, or create some condition, if said task/condition answers to no self-consistent description. In that case, our claim “God can do X” is unintelligible, and thus not a claim at all, just noise. (Note that “self-consistency” also has differing interpretations!)
But pi isn’t just defined as “the ratio of the circumference to the diameter of a circle in our universe”, but “the ratio of the circumference to the diameter of an idealised circle on an ideal flat surface”. It can be proved that that ratio MUST be 3.1415… based only on simple axioms like a+a=2a or a+b=b+a.
And in a universe where the basic laws of logic don’t apply I could probably construct a sound, logical proof that God doesn’t exist and that I am a giant hamster, so it’s not like the question would be valid there anyway.
My opinion is that God did not create Pi, Man created Pi. God created the Universe and man created ways to explain the Universe. God can change the Universe and that will change Pi, but we wouldn’t notice the difference anyway. I think this has already been mentioned in this thread, but if it has I didn’t understand the way it was stated.
If, by “basic laws of logic”, you mean what are sometimes called the Laws of Thought (“A thing is itself, not otherwise”, “That which is true is not also false,” etc.)–then I totally and emphatically agree that we cannot justify pretending to say anything whatever about a universe in which such laws failed to apply.
I do not agree, however, that our analysis of the properties of idealized surfaces and figures is simply independent of empirical experience.
There is, admittedly, real elegance in the formulae relating Pi to (for example) the Pythagorean theorem for the hypoteneuse of a right-triangle. One feels that “another Pi” would have to manifest equal elegance. But mightn’t it be able to do that even with some change in the details? (I did say that there would have to be a LOT of tweaking all over the place.)
Ultimately, some things are just FACTS–because the universe we live in just happens to have been made (or: made itself) that way.
These axiomatic bits of reality depend on nothing else–they just ARE. But an omnipotent God MIGHT have made them otherwise.
I have been careful to state that IMHO God cannot do the unintelligible. He cannot cause something to NOT conform to its own (contemporaneous) definition. I cannot see how A+A could equal anything but 2A–because part of what we mean by “being 2” is bound up in what that equation expresses.
But it is, arguably, otherwise in the case of A+B=B+A. There are already, in fact, “addition-like” arithmetical procedures in which there is a totalling-up or incrementation process that is NOT transitive: “A+B” is one thing, “B+A” another. (Matrix addition, maybe?) I don’t know if the “lay definition-in-use” of addition requires the property of transitivity: I think it only requires that you get an accurate total. So I can imagine a reality in which even basic addition is intransitive (as is basic subtraction, y’know).
Isn’t that possibility sufficient reason to think that Pi might be other than it is?
But pi is defined as “the number you get for the curcumference over the diameter” when you’re using a transitive number system. One could probably calculate what that ratio would be in a geometric system where addition is intransitive, but per definition it that ratio wouldn’t be pi.
Good posts on this… I’ve been given much to mull over. Can am omnipotent being change the rules? No doubt. But I don’t think omnipotence is either necessary or sufficient for a being to be God.
God, the way I look at the concept, is First Cause. That can be broken down into 2 bullet points.
Aseity: God’s existence does not depend on anything else. God is only because God is. God is first, the question “Where did God come from?” is meaningless.
Created the universe. This pre-supposes that the universe is itself a conditional entity, and could possibly be non-existent.
God need not be all-powerful to create the universe. And an all-powerful being might conceivably have been created by another being. Aseity, note, can be thought a limitation of God’s power.
Then there’s those rules that we describe using mathematics. Is it necessary that those rules have an Author?
I question whether God caused pi to be pi, or whether that thing we observe as mathematics also exists independently of God.
Put badly: does math have aseity, existing independently of anything?
Those rules, while described and systematized by us, might have no author - and if not, I’d argue God can’t tweak them. The other choice is that God authored those rules.
An important note in the above is the reference to the human mind. Does a thing’s possibility of existance rely upon the ability of the human mind to understand it? I hate to think so. Of course what happens if a human is confronted with something that the human mind cannot understand? Spontaneous combustion? Or maybe it just wouldn’t be percieved at all, and the mind would simply continue in ignorance as if nothing had happened…
Depending on your definition of God in this case, the concept of one-ness may not be as firm an idea as you stated. Quite a few religions’ definitions of God involve seeming contradictions in this area. For instance, the idea that God can be everywhere at once, or the Christian concept of the Holy Trinity. How is “one” defined in those cases?
If the question assumes the existance of a being that may posess some or all of these seemingly “impossible” qualities, how can the answer be, “No”?
The Egyptians didn’t believe in what Christians or Muslims would call God. Their creator was the hawk headed Sun God Ra. We all know that this worship is truly ridiculous! In spite of this the Pyramids base equal the equivalent of Pi! Coincidence!!! I think Not!! Hail Ra!! Hail Ra!!
[sup]Why the hell would He want to do it? To impress someone? or what?[/sup][/ul]