In Search of Continuity: Thoughts of an Epistemic Empiricist by Ian Durham

Hi Lev,

I'm not sure I understand the second part of your comment - unless, of course, you agree with my conclusion (continuity is merely a mathematical "ruse").

But I take "discrete" to be the opposite of continuous. In my essay I hint at (and would have expanded on, given more space) the fact that there are different notions of these things - some mathematical, some physical.

Ian

Hmmm. Why does his equation negate a particle model for gravity? Coulomb's law is similar and yet we have a very successful particle model for electrostatics.

Thanks Mike. It was quite a shock and we're still grappling with it (especially my kids who were very close to their "opa").

Anyway, as you well know I'm not a huge fan of the multiverse concept so I'm not sure Deutsch's argument resonates with me, but I believe I said something similar on my blog.

Nevertheless, it's good to have healthy debate on the issue!

Thanks Roger.

I guess I don't have a problem with "bad experimental technique" in mathematics because, to me, mathematics is not an experimental science. I think our mistake is in assuming that mathematics represents some kind of universal "truth." I have no trouble accepting the fact that we can have infinitely countable and uncountable sets and other oddities in mathematics. Mathematics is either right or wrong, in a sense. It's more black and white than science which is fine. We just have to remember that science is about modeling and sometimes our models contradict one another. Mathematics, unlike science, is entirely self-consistent.

Ian

Juan,

Thanks for your comments. I have a few replies.

First, regarding Robinson, I certainly am no fan of his and I did not cite him in such a way as to say I supported his conclusion. I simply cited him in order to point out that someone had attempted to "vindicate" Newton and Leibniz in recent decades.

I agree that time is a complicated and funky problem (and I'm looking forward to the FQXi conference in August when we'll be discussing the nature of time in greater detail).

Here is what I mean about the uncertainty relations when applied to a radar gun. A radar gun relies on a measurable change in the wavelength of the light it emits and then reabsorbs. Since the wavelength is directly related to the energy, as Delta t goes to zero, Delta E must go to infinity by the uncertainty relation. But Delta E *can't* be infinite if the radar gun is to work (it *must* be finite). Therefore, Delta t must have a non-zero lower bound.

One of my points is that if we tried to truly measure quantities exactly and in a truly continuous manner, we need to have more and more accurate measurements. But as we make more and more accurate measurements, we eventually leave the classical realm and end up in the quantum realm and the quantum realm is constrained by the uncertainty relations. This limits our knowledge to discrete "chunks." So our knowledge of the universe is limited to discrete "chunks."

As for the old chemists, they certainly may have believed in the discrete nature of matter, but did they necessarily think that the universe itself was necessarily discrete? I seriously doubt that. The atomic hypothesis applies to matter. I am unaware of it having been applied to the universe as a whole by anyone between the Ancient Greeks and the twentieth (maybe late nineteenth?) century.

Regarding the non-geometric interpretation of gravity, while there certainly are non-geometric interpretations of gravity, they are by no means mainstream. The geometric interpretation of gravity has been the paradigm since Einstein. Nevertheless, I certainly was not defending that interpretation. In fact, that was my point. I don't like that interpretation because it is difficult to reconcile with quantization (despite what the field theorists think).

I am familiar with how quantum field theory handles quantization and localization. In fact there is a rich history of foundational discussions surrounding this but, alas, I was limited to 25000 characters and I know how Dirac viewed it. But it is still a fact that QFT is built on top of QM and thus includes its postulates and thus its limitations, i.e., as different as they may be, they do *not* contradict one another.

You say, "I think that this is a reflect of the traditional epistemological approach to physical reality, where science is perceived as a sequence of approximations to one supposed fundamental true." I don't know if I agree that what I'm saying is necessarily a reflection of this. In fact, assuming that classical physics is just an approximation and that the world is really quantum only seems to allow for *multiple* truths, in my opinion. After all, the quantum world is a strange place.

The reason I say classical physics is a "myth" is because as soon as you start to make classical measurements more and more accurate (to more and more decimal places) you will eventually bump into the quantum realm. In other words, you can't have perfectly accurate measuring apparatuses without getting into quantum mechanics. Just think about how we "define" the meter now. It used to be a rod in Paris, but then copying it introduced small defects. The only way anyone could see to it that the meter was the same everywhere was to redefine it in such a way that it can only be measured in ways that involve quantum mechanics! (The current definition is based on the speed of light, which is classical, but all laboratory measurements of the speed of light run up against quantum uncertainties and quantum photon counting statistics.)

Ian

Thanks Rick.

Maybe. I mean, I think I see where you're going, but it's quite a novel idea, if I'm understanding you correctly.

Personally, I don't think discreteness is a "mathematical ruse." I only think that about continuity. Or, rather, I think our knowledge about the continuity of the universe itself is a mathematical ruse. It is entirely possible that the universe *is* continuous, but we simply can't determine if it is or not because we're limited by a discrete "lens," as it were.

Juan,

I beg to disagree with you on a number of points.

First, I completely disagree with your claim that simply because c is a constant, Dx/Dt must trivially go to dx/dt. This is true *mathematically.* My point is that it makes no sense *physically.* I think I made it fairly clear *why* this makes no sense in my essay.

Now, before addressing your next points, let me first address your comments concerning quantum mechanics and quantum field theory. You are taking the difference between the two as being a contradiction. But two things can be different and not contradict. More to the point, quantum mechanics treats time as absolute (i.e. it ignores time) much as Newtonian mechanics (NM) does and thus both are generally interested in obtaining information about positions as functions of time. In the corresponding relativistic extensions of both QM and NM, space and time are now considered together and thus we are interested in functions of position and time (together). This does not, however, mean they contradict each other. In fact, as an example, it is a rather simple affair to derive NM from GR (see for example Shutz or Misner, Thorne, & Wheeler). They can't contradict if one can be derived from the other. The uncertainty relations still hold in QFT (in fact they are sometimes invoked in order to "explain" the spontaneous pair creation).

Now, regarding the points you made regarding discreteness (atomism, or whatever), let me start by quoting from Griffiths: "In principle, the force of impact between a bat and a baseball is nothing but the combined interaction of the quarks and leptons in one with the quarks and leptons in the other." So, ultimately, our classical interactions like that between a baseball and a bat, are really the sum of a bunch of quantum interactions. As you yourself just said, "there exist limits where that discreteness is indistinguishable from a continuum." Precisely my point! That limit is the macroscopic realm of classical physics! Classical physics works because we don't look closely enough or don't care for an increase in precision! But as soon as we do, we run into discreteness. Imagine you're an engineer making a speedometer for a car. Your boss asks you to make this speedometer more precise - say to 2 decimal places. Then he/she comes back and asks you to make it accurate to 4 decimal places. Then he/she wants 6 decimal places, etc. Eventually, though you're measuring a classical value, you're going to run into a *physical* - perhaps engineering is a better term - problem of *how* to get that information from the universe! The most accurate machines are discrete! In fact, the most accurate physical theory ever developed, i.e. in which theory comes closest to experiment, is QED *which is ultimately a discrete theory!*

As for the atomic chemists, I disagree, but then I note that the difference between your view and mine is simply a matter of interpretation. I have done a lot of work on the history of science and have reached a different conclusion. But then again, I know a lot of people who disagree with Thomas Kuhn's take on the history of science (including myself) and yet others who staunchly defend him. It's hard to be "right" when talking about the history of science in such a way.

Juan,

Let me add a few other points that just occurred to me.

I want to make it clear that it is entirely possible the universe is continuous. I'm arguing that we can't get truly continuous *knowledge* about it.

Now, my view is perhaps colored a bit by my experiences. Oddly enough, I have degrees in mathematics, physics, and engineering and was a practicing engineer for awhile before entering academia. I think it this strange combination of all three that has given me the opinion that I have. While continuous measurements sound plausible, the engineer in me wants to know how in the heck we can make something that can truly measure something continuously.

It goes back to that speedometer example again. Ask yourself this: whenever someone needs a truly accurate measure of speed, do they use an analog or a digital speedometer? How about a bathroom scale? If your response to the latter is to say we could use a traditional scale balance, I ask you what we used to accurately measure the weights we use on the balance? Another balance? And how accurate are those weights when compared with the *current* international standards for mass and local measurements of g? And I could go on asking questions like these and eventually you'd have to cite something that was quantum mechanical! Do you see my point?

Actually, the most successful theory of all time, as measured by how closely it matches experiment, is QED.

"It is my understanding that this essay contest deals with the ontology of spacetime, not our epistemic states. These states are modified as science progresses and new experiments are performed."

Actually, this essay contest deals with reality. Reality is more than merely spacetime. Regardless, the point of my essay is that the epistemic states through which we access the ontology of spacetime (reality, whatever) necessarily limit the amount of knowledge we can obtain about the ontology of reality.

One final point to add: mathematics (as with logic, linguistics, and computer science) is a formal science, i.e. one that follows from stated axioms. Formal sciences, by their very nature, are entirely self-consistent. The natural (or empirical) sciences (e.g. biology, physics, chemistry, etc.) are not necessarily self-consistent, e.g. quantum mechanics and general relativity don't quite mesh.