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Can pi be reduced to a rational number?


Dennisg
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Willa, your argument is based on the idea that no perfect circles exist in nature. But that is fine as a circle is a mathematical construction and not a physical one.

 

In physics one uses mathematical models to describe nature. The question here is how does this model relate to nature?

 

So to the bubble question. The mathematical description of a bubble (the 2-d projection of) as a circle is a good model as it describes the shape up to the molecular scale. That is, up to this scale any deviations from the circle are small and within experimental accuracy. (Lets forget any electron-microscope like experiments where we can see the molecular structure).

 

But please note, any mathematical model of nature will have some feature like this. It is not really possible to ask if the model is correct or not, but just how accurately it describes the phenomena you are interested in within the range of parameters /scales you set.

 

The model is good if it describes phenomena within the accuracy you require. Otherwise, it is a bad model.

 

So, asking if a physical bubble is a perfect circle is a very loaded question. It depends on what you mean. Ultimately, if it looks like a circle up to the accuracy you require then it is a circle.

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So to the bubble question. The mathematical description of a bubble (the 2-d projection of) as a circle is a good model as it describes the shape up to the molecular scale. That is, up to this scale any deviations from the circle are small and within experimental accuracy. (Lets forget any electron-microscope like experiments where we can see the molecular structure).

Just to be pedantic:

A circle is not such a good model if you want to explain all those pretty iridescent colors one sees on the bubble. A better model is needed to explain soap bubble iridescence -- and that better model is a spherical shell, or in cross section, an annulus. Circles are infinitely thin. An annulus is not.

 

It depends on what you mean. Ultimately, if it looks like a circle up to the accuracy you require then it is a circle.

I disagree with you here. Scientists should always be cautious in reading too much into their math models. Math models are approximations of reality. While some models do a bang-up job of representing reality, they are not reality. In the words of Alfred Korzybski, The map is not the territory. This thread is chock-full of statements that equate the map with the territory.

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Just to be pedantic:

A circle is not such a good model if you want to explain all those pretty iridescent colors one sees on the bubble. A better model is needed to explain soap bubble iridescence -- and that better model is a spherical shell, or in cross section, an annulus. Circles are infinitely thin. An annulus is not.

 

It is up to whoever is using the model to define if it is good or not, depending on what they want to describe. I agree that a spherical shell is better. But none of this distracts the point I was making about modelling nature.

 

 

I disagree with you here. Scientists should always be cautious in reading too much into their math models. Math models are approximations of reality. While some models do a bang-up job of representing reality, they are not reality. In the words of Alfred Korzybski, The map is not the territory. This thread is chock-full of statements that equate the map with the territory.

 

You have misunderstood what I was trying to say. In order to describe nature all we really have are mathematical models. The only real things are what we can measure. Even then we usually attach a numerical value to this, which means a model is needed.

 

Models are not really an approximation. They are exactly that, a mathematical model. You can of course make approximations within a given model or relax some of the starting assumptions/axioms.

 

Models are not nature, but they are used to describe nature. If my theory says something is a circle and then I go away and measure it to be (within whatever accuracy I decide) then it is fine to say that within the context of my theory it is a circle. In any discussion about describing nature one should be fairly specific about the theory being used, even if it is incomplete.

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Maybe its time to have a rethink on the defination of what a point is. The old one is based on a world view that no longer exists.

 

I have to agree with Gilded... you need to expound on this quite a lot.

 

The mathematical notion of a point has served mathematics quite, quite well for a very ling time. And there is no compelling reason at all to change a fundamental definition of mathematics.

 

In fact, though the physical world and the mathematical world don't have to necessarily coincide as has been pointed out in this thread many time, there are many physical situations where using a mathematical point -- a zero dimensional object -- yields exceptionally good results. Treating an ideal gas as a collection of points can be quite accurate for the correct gas (like Helium) and for high temperatures. Modeling the trajectory of a probe on its way to Mars, you can treat the gravitational influence of the distant planets like Uranus and Pluto like there are just point masses. There are others -- the main thesis here being that the physical and mathematical notions of a point can and often do coincide with exceptionally very good results.

 

The biggest thing is that the mathematical notion of a point is still an exceptionally useful learning tool. Sure, a ball flying through the air isn't really a point. But, the full problem taking into account the drag, the lift, the spin of the ball, the turbulence of the air, etc. is a very tough problem. Treating the ball as a point and neglecting the air resistance generates a problem that can be solved. And, in learning to solve the problem, the students get a feel for the skills necessary for problem solving and thinking logically and working it out for themselves. That way they know the basic problem before tackling the more complicated ones with drag, etc.

 

So, you're going to have to present so very, very compelling reasons why the "world view" is no longer compatible with the mathematical notion of a point.

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Maybe its time to have a rethink on the defination of what a point is. The old one is based on a world view that no longer exists.

 

Q) What is the definition of a point?

 

 

A) An element of the underlying set of a topological space.

 

This is the generally accepted notion. If the space can be charted, such as a manifold then we can define a point to be a zero-dimensional object. That is we can describe a point by a single coordinate (in a given chart).

 

As I have said, we can deal with pointless geometry, i.e. spaces that are not topological spaces, that is non-set theoretical objects.

 

In noncommutative geometry, you can (sometimes) use the word point very informally to mean a specified coordinate. (Assuming we have local coordinates) This is sometimes called a running point. If we say "an object O at a point", we mean O(x) if x is the coordinate. This can be a useful idea (locally) but one must realise that the coordinates here do not describe classical points.

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Ok, there is another notion of a point that comes from algebraic geometry. For topological spaces, this notion gives the same notion as a "set theory point". This notion is probably closer to what you want.

 

I am a little reluctant to give you the definition as it requires some knowledge of algebra and algebraic geometry. But basically, in algebraic geometry one describes a space by the complex (or real) valued functions on it. This algebra can be made into a c*-algebra and we have a theorem that states that the categories of topological spaces is equivalent to the category of c*-algebras. (subcategories really) (This is the Gelfand-Naimark therem).

 

So we can state that algebraic geometry is the following diagram;

 

Differential geometry [math]\rightarrow[/math] algebraic geometry

coordinates [math]\rightarrow [/math] coordinate functions

 

What is the notion of a point? A point is a maximal ideal of the algebra of functions. It represents the "smallest piece" in some sense.

 

The idea here is that we can now do everything we can on a topological space in terms of the algebra of functions on that space. Including defining a point from the algebra and not the other way round. Geometric objects can also be defined in terms of this algebra.

 

Now we can define a noncommutative space to be the "space" that has a noncommutative algebra of functions on it. Notice that this definition makes no use of the notion of a point. All geometric objects are defined in terms of this algebra.

 

So (roughly),

 

commutative c*-algebras = topological spaces

noncommutative c*-algebras = noncommutative spaces.

 

We can now define a point to be a maximal ideal of this c*-algebra.

 

This is the closest well defined notion I know of that sounds like what you are trying to describe.

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You can definitely think of space-time being divided into "Planck-cells" of radius comparable to the Planck length, but I would think of this as an "artists impression" because geometric shapes like the circle become lost in NCG.

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You can definitely think of space-time being divided into "Planck-cells" of radius comparable to the Planck length, but I would think of this as an "artists impression" because geometric shapes like the circle become lost in NCG.

 

 

Yes I agree. Shape would have to be deifined by something smaller than a plank length.

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I agree that shapes such as squares and circles do not naturally occur. What interests me is why pi is an irrational number. This means that the exact area of a circle can never be known. Why is this? We can know the exact area of a square etc but not a circle. Maybe an irrational number is nature’s way of telling us that we have got something basic wrong - in this case the definition of what a "point" is. I think for a point to occupy a location it must have a dimension.

 

Wrong, all the way from the first sentence to the last

All this signals your lack of understanding basic mathematics concepts

 

We can know the exact area of a square etc but not a circle.

Basic geometry would tell you the area of a square and a circle. Just because a circle involves more calculation doesn't mean it's more complex or mysterious.

 

Maybe an irrational number is nature’s way of telling us that we have got something basic wrong - in this case the definition of what a "point" is. I think for a point to occupy a location it must have a dimension.

 

I don't know how to explain but your way of thinking is reversed somehow.

A point in mathematics doesn't occupy anything. A point is 1-dimension, a plane is 2-dimension. A plane doesn't occupy anything either.

 

And irrational numbers are not as special as you think at all. They are as plain as rational numbers.

 

If I were you I would worry about complex and imaginary numbers instead. Those are really mysterious to me.:rolleyes:

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Wrong, all the way from the first sentence to the last

All this signals your lack of understanding basic mathematics concepts

We've already worked out that he doesn't have much in the way of an education in mathematics, it's nothing to shout about.

 

And irrational numbers are not as special as you think at all. They are as plain as rational numbers.
Actually, rational numbers could easily be argued to be more special, since there are infinitely fewer of them.

 

If I were you I would worry about complex and imaginary numbers instead. Those are really mysterious to me.
You'll come to terms with them.
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Wrong, all the way from the first sentence to the last

All this signals your lack of understanding basic mathematics concepts

 

A bit harsh I think. Its like saying that someone who is poor at spelling can't write a meaningful sentence. Relax a little we're just exchanging ideas here.

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The proof of [math]\pi[/math] being an irrational number is a beauty. There is a few proof, such as seen on wikipedia:

 

http://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational

 

I had a look at these proofs. It seems to me that they assume that pi has only one value and then conclude that pi is irrational on that basis. But what if as noted above pi varies with the radius? They then prove that pi is irrational by not finding a rational value - so what if there are an infinite number of pi values would one get the same result with this logic?

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Pi by itself though is not just a number, it’s a product of mathematics. I mean you can do an equation and graph it that does not provide some min and max that is finite, you can get infinite also. So why pi may be in whatever is the best current approximation, again its not as if its just a number by itself. So if math has to derive a circle or a sphere in some equation, it might be the nature of math itself that is giving the result. Such as I think this may be why irrationals numbers exist and don’t only take the form of pi.

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I had a look at these proofs. It seems to me that they assume that pi has only one value and then conclude that pi is irrational on that basis. But what if as noted above pi varies with the radius? They then prove that pi is irrational by not finding a rational value - so what if there are an infinite number of pi values would one get the same result with this logic?

 

perhaps because pi is one number. it is the ratio between the circumference and the diameter of a circle in the euclidean plane. that is, a hypothetical construct where it is impossible for it to change with radius. anything else is not pi.

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perhaps because pi is one number. it is the ratio between the circumference and the diameter of a circle in the euclidean plane. that is, a hypothetical construct where it is impossible for it to change with radius. anything else is not pi.

 

This assumes that shape of the outside of the circle is perfectly round. While pi applies to space, seems to be independent to the number of dimensions, that is the pi of two dimensions is the same as the pi of three dimensions. This seems logical enough but it does some something about pi itself. What about pi in four dimensions – would the features of four dimensional space alter pi?

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This assumes that shape of the outside of the circle is perfectly round.
Perfectly round is the outside of a circle.
While pi applies to space, seems to be independent to the number of dimensions, that is the pi of two dimensions is the same as the pi of three dimensions.
Pi is, as we really should have established, a real number, this means that in continuous space of at least one degree, you can get pi. The same applies to every other real number ever.
This seems logical enough but it does some something about pi itself. What about pi in four dimensions – would the features of four dimensional space alter pi?
Why on earth would it?
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perhaps because pi is one number. it is the ratio between the circumference and the diameter of a circle in the euclidean plane. that is, a hypothetical construct where it is impossible for it to change with radius. anything else is not pi.

 

I wouldn't say that pi is "just one number" because at the plank lenght scale pi is close to 3.0 Having an absolute limit on how small things can get does change things - don't you think so?

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