Am I wrong? Is this correct thinking? Am I weird? Am I crazy?

[pergradus]

I certainly don't think Pi can be thought of as a physical unit.

Consider the way we measure the length of string of length 10 cm:

You can start by placing your ruler down, and starting at 0 cm, count by 1 cm until you get to the end of the string, and at at 10 cm - which is equivalent to adding 1 + 1 + 1... + 1 until you get 10.

Suppose it was now 10 and 1/3 cm:

You add 10 cm together, and then add an extra 1/3 cm. It doesn't matter if you can't write down what 1/3cm is in terms of mm or any other decimal, physically you can divide 1 cm into exactly three pieces - there's nothing stopping you from doing that.

Now try to get to 10 + pi cm - you start with adding 10 cm up, then you can add three more, then you have to add 1 mm, and then a little more, and a little more, and just a little bit more... and so on forever. You have to add an infinite number of smaller and smaller measurements to ever achieve that extra pi cm. You can never actually measure out an irrational physical quantity - and hence I don't think it's possible for physical quantities to obtain irrational values.

The fact that the billiard ball doesn't have an irrational surface area is because it's not a perfect sphere, and never will be. Just like you can't ever draw a perfect triangle with a hypotenuse of Sqrt(2).

 

[1MileCrash]

I certainly don't think Pi can be thought of as a physical unit.

Consider the way we measure the length of string of length 10 cm:

You can start by placing your ruler down, and starting at 0 cm, count by 1 cm until you get to the end of the string, and at at 10 cm - which is equivalent to adding 1 + 1 + 1... + 1 until you get 10.

Suppose it was now 10 and 1/3 cm:

You add 10 cm together, and then add an extra 1/3 cm. It doesn't matter if you can't write down what 1/3cm is in terms of mm or any other decimal, physically you can divide 1 cm into exactly three pieces - there's nothing stopping you from doing that.

Now try to get to 10 + pi cm - you start with adding 10 cm up, then you can add three more, then you have to add 1 mm, and then a little more, and a little more, and just a little bit more... and so on forever. You have to add an infinite number of smaller and smaller measurements to ever achieve that extra pi cm. You can never actually measure out an irrational physical quantity - and hence I don't think it's possible for physical quantities to obtain irrational values.

The fact that the billiard ball doesn't have an irrational surface area is because it's not a perfect sphere, and never will be. Just like you can't ever draw a perfect triangle with a hypotenuse of Sqrt(2).

Again, counting exactly 1 cm and counting exactly pi cm are equally daunting tasks. You can measure 1.00000000000 cm perhaps, but that's no easier than measuring 3.141592653589 cm. Both are approximations to 1 and pi respectively when we are talking about measurement. 1.0 cm is not equal to exactly 1 cm, it means we stopped measuring after millimeters. You can't just ignore that in the rational number's case.

 

[Number Nine]

The issue here that computations with pi (and irrational numbers) can only work with rational approximations. This is a technological limitation, it has nothing to do with "not knowing pi exactly"; we do know pi exactly, we just can't work with it exactly. It's just a number.

 

[Deveno]

assuming we know how to define it, any real number is "knowable" in the sense that we can compute it to any desired accuracy.

i mean what is a mathematical relationship?

when we say a = f(b), for some formula involving b, there is a tacit appeal to a and b being "the same kind of something". if our formula is:

A = \pir², than A is not a rational number. so we have to have some kind of notion that allows us to say what equality of two real numbers IS.

analytically, we say that A converges to \pi. some people have trouble resolving this notion of convergence to our everyday notion of "is-ness". probably because it relies on the notion of a "completed infinity" (\pi is not finitely expressible as a rational, it is an infinite sequence of such rationals).

my point being, saying a = b, when a and b are two real numbers, sweeps a lot of complicated machinery "under the rug", the truth of such a statement (and how complicated it actually is) is hidden behind a formalism which makes it appear as if it is just as simple as a statement like 2+2 = 4, or 1/3 + 2/3 = 1.

to go a bit further, how does one actually prove that the formula for an area of a circle actually holds? if you can show me a derivation that does not, in fact, rely on some limiting process (and thus a "completed infinity"), i will withdraw my statements.

 

[pwsnafu]

to go a bit further, how does one actually prove that the formula for an area of a circle actually holds?

Before one can do that you need to define the problem of finding "area of a circle" without analysis.

 

[Deveno]

Before one can do that you need to define the problem of finding "area of a circle" without analysis.

if by this question, you mean: "what is 'area' "? then i agree.

the most delicate and complicated mechanisms underlie our spatial notions of "n-dimensional content" (length/area/volume/etc.). it is not immediately clear that a region has such a number associated with it (i can think of some very bizzarre regions in the plane), nor is it clear how to find a coordinates-free method of assigning such a number.

the answer (if i recall correctly) has something to do with pull-backs, determinants and n-chains. and even this appeals (at some level) to assigning "1" to some basic figure (a unit interval, disk/square, ball/cube, etc.).

frankly, i am quite impressed that people even decided (long before calculus came around) that the circumference of a circle was rectifiable.