# Does geometry have any bearing on the fine tuning argument.

## Main Question or Discussion Point

Now I think there are arguments for the Fine Tuning of the universe. I like Martin Rees' book but I know there are others who disagree with what he said in his book (Just Six Numbers)
On another forum I've got involved in a discussion on this topic and I've seen an argument that I think has no proper foundation but I'd like to try it here:

Me:
"If anyone were to come up with the reasons why the magic constants have to be constrained (or not of course!)within very narrow limits they'd be up for the Nobel prize for physics. Until then all this thread is speculation v counter speculation. However we do know, through simulations, that the constants have to lie in narrow ranges, and that (the fact that they have to lie in narrow ranges rather than the extreme narrowness of the ranges) is remarkable."

Strange (to my mind) argument
It is no more remarkable than the fact that the range for pi constant is rather narrow. Or for the sum of the angles in the triangle. Or for the sum of 2 + 2. Yet no fine tuner claims that the range for pi values on an Euclidean plane could be wider and it would take luck to have it as it is now.

I think this is misapplying geometry to a physics problem and not appropriate. Any thoughts?

Yet no fine tuner claims that the range for pi values on an Euclidean plane could be wider and it would take luck to have it as it is now.
Well, of course. That's how we define pi (usually.) So the fact that pi is, well, pi comes directly from that. Just about anything else that, seemingly coincidentally, seems to relate to pi we've proven has the given relationship. So we define it as the ratio between the diameter and the circumference of a circle on a Euclidian plane and any other relations to pi come as consequences of how we define everything. Pi's also uniquely determined from which terminating decimals it's larger than and smaller than, so if we can show that those are uniquely determined (shouldn't be too hard,) we can easily show that pi's unique in Euclidian geometry.

Same goes for 2+2, though from what I understand, from the Completeness theorem, any proof that 2+2 can't be anything other than 4 is absurdly difficult (if one exists) and would imply that the current axiom and definition set we base math on (ZFC) is fundamentally flawed. Thus we're just assuming that 2+2 is only equal to 4.

The distinction here is that the only reason we think the constants of the Universe are within a habitable range is that we exist, and so the Universe must be habitable. There's, as far as we know, nothing we can use to prove that they must be within this small range; that is, the best current theories we have describing the Universe (notably the Standard Model) give no reason for the Universe to be habitable. I suppose one could use the Strong Anthropic Principle to explain it, though from what I understand the Strong Anthropic Principle isn't universally accepted.

I've also come across this:
"I'm not talking about a circle, I'm talking about a simulation. It is perfectly possible to simulate a reality where pi=5. Simulated circles in that reality would still be understood to have points equidistant to their centre, they would simply have a different ratio between their diameter and their circumference.

... I know it isn't a circle, it is a simulation of a circle in a world where pi is different. What you have described isn't a universe (or set of universes) it is a simulation of a universe where the cosmological constants are different."

and I'm trying to explain that pi depends on "circle" which is defined for the Euclidean plane so there aren't any other worlds with different pi values.

and I'm trying to explain that pi depends on "circle" which is defined for the Euclidean plane so there aren't any other worlds with different pi values.
Basically. It's defined in terms of Euclidian circles, and the properties of Euclidian geometry is independent of what world one lives in. :)

I've also come across this:
"I'm not talking about a circle, I'm talking about a simulation. It is perfectly possible to simulate a reality where pi=5. Simulated circles in that reality would still be understood to have points equidistant to their centre, they would simply have a different ratio between their diameter and their circumference.

... I know it isn't a circle, it is a simulation of a circle in a world where pi is different. What you have described isn't a universe (or set of universes) it is a simulation of a universe where the cosmological constants are different."
Well ... sorta. What we usually do is we consider the set of "Universes" with different constants, and while it isn't the Universe, we can study their properties. We then let our Universe be the specific one with the constants we measure. There's no reason we can't let them be Universes other than, of course, the fact that, as far as we can tell, we don't live in them. And those studying them rarely care about that.

So they somewhat are simulations, given that we can't perform experiments in them, though they're simulations in the same sense a Euclidian plane is a simulation of a Euclidian two-dimensional Universe. What you're asking about is what properties a Universe would have if the ratio of the circumference to the diameter of a circle were, say, 5 instead of pi. While I'm not sure if there are any Universes imaginable in which such a ratio is constant and not pi (all the ones I can imagine have a ratio which is dependent on the radius of the circle,) the argument should still hold.

It just occurred to me that exactly your question might be a big area in modern cosmology. While gravity leads to local variations in the Universe's geometry, no one's quite sure what the net curvature of the Universe is; the "measured value" of pi, if you will, is highly dependent on what the overall geometry of the Universe is. The Universe is, as far as we can measure, Euclidian overall (specifically, if I remember the experiment correctly, the angles of a huge triangle were measured to add up to 180°,) though the best cosmological models we have end up predicting a hyperbolic geometry for the Universe (the angles of a triangle add up to less than 180° and the ratio of the circumference of a circle to its diameter is >pi.) Most of the current speculation I can think of is either than it's immeasurably close to Euclidian or that something's wrong with our current formulation of inflation.

pbuk
Gold Member
Let's just clear up a couple of things...

Pi's also uniquely determined from which terminating decimals it's larger than and smaller than, so if we can show that those are uniquely determined (shouldn't be too hard,) we can easily show that pi's unique in Euclidian geometry.
These numbers do not exist: given any x,y ∈ ℚ, x < ∏ < y it is simple to construct x' and y' such that x ≤ x' < ∏ < y' ≤ y and either x < x' or y' < y. Come to that your statement is not even true if you replace ∏ with 1/3.

Same goes for 2+2, though from what I understand, from the Completeness theorem, any proof that 2+2 can't be anything other than 4 is absurdly difficult (if one exists) and would imply that the current axiom and definition set we base math on (ZFC) is fundamentally flawed. Thus we're just assuming that 2+2 is only equal to 4.
Informally, from definitions and the theorem a + (b + c) = (a + b) + c: 2 + 2 = (1 + 1) + (1 + 1) = ((1 + 1) + 1) + 1 = 4. The formal version is a bit longer.

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These numbers do not exist: given any x,y ∈ ℚ, x < ∏ < y it is simple to construct x' and y' such that x ≤ x' < ∏ < y' ≤ y and either x < x' or y' < y. Come to that your statement is not even true if you replace ∏ with 1/3.
Ah. I didn't mean that, given terminating decimals x,y, that π is uniquely determined by the ordered pair (x<π,y<π); I meant that, if we let D be the set of all terminating decimals, a real y is uniquely determined by the function f:D->{0,1} given by f(x)=(x<y) (though such a real doesn't necessarily exist for all f.) While I'm not sure how to prove this and I might only be giving something *close* to something true, I think this is true and any counterexamples are either really bizarre or stupidly obvious.

Informally, from definitions and the theorem a + (b + c) = (a + b) + c: 2 + 2 = (1 + 1) + (1 + 1) = ((1 + 1) + 1) + 1 = 4. The formal version is a bit longer.
Ah. I'm certainly recognizing that it's provable that 2+2=4; but, if I remember correctly, any proof that addition is well-defined, and I imagine a proof that 2+2 is well-defined is enough to construct one, would be a proof that arithmetic is self-consistent, which violates the Completeness Theorem. Note that I'm a bit more tentative about this and if I'm wrong I'd appreciate it if someone could clarify.

speed of light in another universe

This thought came to me. Given another universe with different physical constants but with 3 dimensions (4 with time) the speed of light could be different but its invariance and other features of special relativity would still hold?

Chronos
Gold Member
This all assumes time is a 'fundamental' property of the universe - a position not supported by GR.

Sorry Chronos what do you mean by "this all" - the whole thread, the previous post or something else?

Chalnoth
Ah. I'm certainly recognizing that it's provable that 2+2=4; but, if I remember correctly, any proof that addition is well-defined, and I imagine a proof that 2+2 is well-defined is enough to construct one, would be a proof that arithmetic is self-consistent, which violates the Completeness Theorem. Note that I'm a bit more tentative about this and if I'm wrong I'd appreciate it if someone could clarify.
I don't see why this would be a proof that arithmetic is self-consistent, especially as arithmetic is more than just addition.

Chronos
Gold Member
Apologies for any confusion. Only speaking with reference to post 7.

pbuk
Gold Member
Ah. I'm certainly recognizing that it's provable that 2+2=4; but, if I remember correctly, any proof that addition is well-defined, and I imagine a proof that 2+2 is well-defined is enough to construct one, would be a proof that arithmetic is self-consistent, which violates the Completeness Theorem. Note that I'm a bit more tentative about this and if I'm wrong I'd appreciate it if someone could clarify.
I wasn't going to pick up on this, but as someone else has and as I suppose others might be interested I guess I will. There is not room here for even outline proofs of any of this, but they can easily be found by googling.

Addition over the integers is well defined, consistent and complete (google Presburger arithmetic), and integer arithmetic (which includes multiplication) is both well-defined and consistent but it is not complete (google Peano arithmetic). Gödel's completeness theorem is a statement about a property of complete systems and is not really relevent here, I think Whovian meant to refer to Gödel's incompleteness theorem, which would indeed be violated by a proof that real arithmetic is complete.

Ah. I didn't mean that, given terminating decimals x,y, that π is uniquely determined by the ordered pair (x<π,y<π); I meant that, if we let D be the set of all terminating decimals, a real y is uniquely determined by the function f:D->{0,1} given by f(x)=(x<y) (though such a real doesn't necessarily exist for all f.) While I'm not sure how to prove this and I might only be giving something *close* to something true, I think this is true and any counterexamples are either really bizarre or stupidly obvious.
Yes this is close to some things which are true and can be inferred from the construction of the reals from the rationals using Dedekind cuts or Cauchy sequences.

However the construction of ∏ using its Dedekind cut does not determine the value of ∏; quite the opposite as that (one-sided) cut is defined as the set of all rationals less than ∏. That's the problem with transcendentals - they transcend attempts to define them algebraically.

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pbuk