Is Planck's constant a paradox?

[ribbie]

The following is a quote from Wikipedia on Irrational Numbers (the bold is mine):

The first proof of the existence of irrational numbers is usually attributed to a Pythagorean (possibly Hippasus of Metapontum), who probably discovered them while identifying sides of the pentagram. The then-current Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the other. However, Hippasus, in the 5th century BC, was able to deduce that there was in fact no common unit of measure, and that the assertion of such an existence was in fact a contradiction. He did this by demonstrating that if the hypotenuse of an isosceles right triangle was indeed commensurable with an arm, then that unit of measure must be both odd and even, which is impossible. His reasoning is as follows:

* The ratio of the hypotenuse to an arm of an isosceles right triangle is c:b expressed in the smallest units possible.
* By the Pythagorean theorem: c² = a²+b² = 2b². (Since the triangle is isosceles, a = b.)
* Since c² is even, c must be even.
* Since c:b is in its lowest terms, b must be odd.
* Since c is even, let c = 2y.
* Then c² = 4y² = 2b²
* b² = 2y² so b² must be even, therefore b is even.
* However we asserted b must be odd. Here is the contradiction.

If this is the case and there is no "small indivisible unit that could fit evenly into one of these lengths as well as the other", then how can Planck's Constant be true, unless it is a paradoxical number that is both odd and even at the same time, or neither odd nor even?

 

[Pengwuino]

You're confusing the mathematical possibilities with the physical possibilities. For example, mathematical there is nothing wrong with something traveling faster than the speed of light. The physical implications are just strange. Mathematically, there's nothing wrong with a length shorter than the Planck length, but physically we don't think it's meaningful to our laws of physics.

 

[ribbie]

So what you're saying is that even though Planck's length as "a small indivisible unit" exists, it has no significance in mathematics because mathematics is essentially infinitessimal and has no bearing on the laws of the physical world, whereas physics measures the physical world in which everything is finite and rational?
In other words, the world is not run by mathematics, rather mathematics is a tool used by physicists to describe the world.


Philosophically speaking, you could say that math is kinda Divine, while physics is "human" (until you get to quantum mechanics, that is).

 

[Pengwuino]

In other words, the world is not run by mathematics, rather mathematics is a tool used by physicists to describe the world.

Yes

Philosophically speaking, you could say that math is kinda Divine, while physics is "human" (until you get to quantum mechanics, that is).

As much as a hammer can be considered "divine" to a carpenter

 

[ribbie]

As much as a hammer can be considered "divine" to a carpenter

perhaps more like a word is divine to the speaker - not that humans need to pay homage to it, but that it is relatively more "spiritual" and ethereal.

 

[Demystifier]

In other words, the world is not run by mathematics, rather mathematics is a tool used by physicists to describe the world.

"As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality."
Albert Einstein

 

[Meir Achuz]

The following is a quote from Wikipedia on Irrational Numbers (the bold is mine):

If this is the case and there is no "small indivisible unit that could fit evenly into one of these lengths as well as the other", then how can Planck's Constant be true, unless it is a paradoxical number that is both odd and even at the same time, or neither odd nor even?

The numerical value of Planck's constant has no mathematical or physical meaning.
The numerical value depends on the system of units used. It is a different number for
eV-sec, than for Joule-sec.

 

[maverick_starstrider]

The following is a quote from Wikipedia on Irrational Numbers (the bold is mine):

If this is the case and there is no "small indivisible unit that could fit evenly into one of these lengths as well as the other", then how can Planck's Constant be true, unless it is a paradoxical number that is both odd and even at the same time, or neither odd nor even?

I think you're thinking about this on far too philosophical a level. In our first attempt at mathematically describing the universe we assumed that energy and something called action could have any value at all (these quantities were given by real numbers (rational + irrational)). We now call this classical mechanics. Later we realized that this was not the case, it is not the way reality is, reality is quantized (i'm avoiding all subtlety as to exactly WHAT is quantized here). The "unit" of this quantization is on the order of Planck's constant. We call this quantum mechanics. Why the number Planck's constant and not some other number? Who knows, because it is, (well actually the exact number is really just an artifact of the fact that long ago we chose to measure energy in these things called Joules).

Anyways, imagine a computer screen, at first flush one might think that you could render any image, no matter how small, on a computer screen and thus position could take on any value (within the dimensions of the screen). But then you look closer and you see that a computer screen is really made up of 1024 by 768 (depending on your resolution) little pixels. It cannot render any image smaller than 1/(1024*768)th of a pixel.

That's essentially the idea here. Our universe is quantized, assuming it wasn't was wrong. (NOTE: My example is really describing an example of SPACE quantization which isn't what is really getting quantized in quantum mechanics, but it illustrates the idea).

 

[maverick_starstrider]

The following is a quote from Wikipedia on Irrational Numbers (the bold is mine):

If this is the case and there is no "small indivisible unit that could fit evenly into one of these lengths as well as the other", then how can Planck's Constant be true, unless it is a paradoxical number that is both odd and even at the same time, or neither odd nor even?

On an additional note the actual NUMBER of Planck's Constant is arbitrary. It's only 6.63*10^-34 or whatever because in SI units we use Joules as our unit of energy. One could just as easily work in what are called "fundamental units" (a bit of a misleading name) in which case its value is simply 1. To re-iterate, the crazy value it has is entirely due to the fact that we are choosing to measure energy in terms of kilograms times meters squared divided by seconds squared. Of course, kilograms, meters and seconds are entirely arbitrary units, we could have just as well measured it in (average mass of an american apple) * (length of my pinky finger)^2 divided by (the time it takes me to blink)^2 and we'd get an entirely different number. And, we could just as easily choose to work in units where Planck's constant is simply 1.