# "The end of Pi?", this can't be mathematically valid, right?

• Aug 23rd 2010, 11:56 AM
mfetch22
"The end of Pi?", this can't be mathematically valid, right?
So check out the second answer in this link below (it's kind of long so I just give a short description below):

How can construct pi geometrically | Answerbag

Basically, this guy is saying that physically "Pi Terminates at an exact point". Not that, when we draw a line, we can only ever get closer and closer to the exact value of pi, but we will never reach it. No, he says that since "The smallest possible measurement is the Planck Length", then Pi can theoreticall be drawn to an exact accuracy, since the Planck Length divides distance into the smallest possible length, and the accuracy would only have to lie within one planck length, or something like this. I don't buy this. Is this a valid mathematical argument?

Heres where I think its wrong: Just because there is the smallest possible physical distance, that doesn't mean that in "theory", we can't imagine (and even mathematically use and describe) a smaller distance then the Planck Length, right? And since the more "pure" mathematics is more about math theory, his "Planck Length" argument doesn't hold up against the rigor of proper mathematics, correct?

Or am I wrong? Thanks in advance for any assitance.
• Aug 23rd 2010, 07:01 PM
Ackbeet
Not a valid mathematical argument. However, this is more of a philosophical line of reasoning. The fellow you link to appears to be arguing that space is discrete, not continuous. I happen to think that the discreteness of space is a fantastic answer to Zeno's paradox (motion is then possible by quantum mechanical tunneling). I don't buy the standard calculus answer. However, that is neither here nor there.

If you assume space is discrete, then the axioms of the real numbers go out the window. If that's the case, it might be that the proof of the transcendence of $\pi$ is severely compromised. That would be the place to look: the proofs of the transcendence or irrationality of $\pi$. If all those proofs depend on the axioms of the real numbers, then this guy might be on a somewhat firm footing. If, however, there's a proof that does not depend on the axioms of the real numbers, then what he says is suspect.

That's my two bits. I could be entirely wrong!
• Aug 25th 2010, 01:56 PM
HallsofIvy
Let me also point out that physical space is NOT the space of mathematics. There is no reason to mention the "Planck length" when discussing mathematics. I will, however, say that $\pi$ is a real number- there is, in fact, a point on the number line (which is NOT a line drawn in physical space) corresponding to $\pi$. There is in fact, a precise line segment of length $\pi$. It is not a question of "getting closer and closer" because mathematics also does not involve physical time! Now, as for "theoretically draw" such a line segment, you would have to say exactly what is meant by "drawing" even theoretically. You cannot, for example, construct a segment of length $\pi$ with "straightedge and compasses", not because $\pi$ has an infinite number of decimal places or even because it irrational. 1/3 has an infinite number of decimal places and $\sqrt{2}$ is irrational but we can construct, with straightedge and compasses, line segments of both those lengths. We cannot construct a line segment of length $\pi$ because $\pi$ is not an algebraic number and the only "constructible" numbers are those that are algebraic of order a power of 2.
• Aug 25th 2010, 02:51 PM