I think that this dialogue hides behind the questionable assumption that spacetime is quantized, which is still a subject of debate (and probably will be for a long time because we are no where near being capable of measuring anything on the planck scale.) That being said from what I know of physics, theories (not to be confused with theorems) are essentially idealized approximations of what we observe. So, while the number pi may turn out to be physically meaningless (if spacetime is indeed quantized and consequently circles are not actually circles), a circle (even an exceedingly tiny one) that is made up of arcs 10^-35 m in length is a pretty damn good approximation of a continuous circle, and therefore, the theoretical-artifact-physically-meaningless-artist-formerly-known-as pi is a damn good approximation of a non-circle's quasi-circumference to quasi-diameter ratio. Hell, it's far better than any computer in existence can do in trying to approximate the real numbers as a continuum! None of that matters anyway, because if we could know a circle's circumference and diameter accurate to the planck length, then we would know pi accurate to 10^35 digits...when was the last time you used 10^35 digits of pi in a calculation? Trick question: we don't even know pi out to 10^35 digits, we only know a tiny fraction of that (10^12 digits)! The point is that the theory would still be valid, even if spacetime is quantized, the theory would still be a good approximation to what is actually observed (assuming it was a good approximation to begin with.)
A Fantasy Dialogue
Yes, the theory would be valid, but only as a descriptive one.
Space is indeed quantized. The fact that there still may be debates about it does not mean it is not, of course.
Really? What is the evidence for spacetime being quantized? Has there been any experiment performed that has measured (either directly or indirectly) the discreteness of spacetime?
No, there is no experiment confirming that space is indivisible. But I use the same argument on your position, and more forcefully. If space is infinitely indivisible, then the uncertainty at quantum levels to measure such a small distance would produce a margin of probability and error so great by comparison as to render the small size physically meaningless. However, once we accept the Planck scale, we accept our limits. The plank scale is physically meaninful.
Next, strings may exist at the planck scale, and if string theory turns out to finally make testable predictions (which it seems like some models of it may now), then we could indirectly confirm my position.
String theory predictive? And its countless dimensionless physical constant? I must admit it is not my field, but it does sound strange to me
1: So you don't believe that circles exist physically, do you?
2: Why not?
1: There are no physical representations of a decimal expansion that goes beyond the Planck scale; it becomes physically meaningless.
2: OK....
1: So pi is ordinarily defined in terms of parts of a physically non-existent entity, right?
2: Yes, you may say so
1: Of course there are plenty of other identities with pi but I cannot deal with all of them in this conversation. I was trying to get at a point. Pi is used in so many physical theories and equations, right?
2: Yes.
1: So these theories are not really "physical" in a sense, right?
2: You could say so I guess. To me, mathematics is descriptive.
1: So we cannot really say all physical theories (or most even for that matter) are actually explanative, can we?
2: [Thinking hard] Well...mathematics is just descriptive for me...but I guess...descriptions are valid tools to make scientific explanations from for me. I think that's what I believe....
Your turn to continue the dialogue. Is it resolved here in your opinion or are there more avenues to explore?