I guess you are confusing two things - think of a mark on a real line at x somewhere between 17 mm and 18 mm from the origin. If you measure its location with a ruler, you might only be able to say that it is somewhere between 17 mm and 18 mm from the origin, but this uncertainty in your measurement in no way constraints x to only be located at integer millimeter positions or something like that.

There is not even a real limit imposed by the uncertainty principle, you can measure positions as precisely as you want, you just have to pay in momentum uncertainty. Where we really seem to run into a wall is that if you keep making the measurement more and more precise, you need higher and higher energies to achieve ever shorter wavelengths and dumping a lot of energy into a small volume to measure the position really precisely will eventually result in black holes.

> There is not even a real limit imposed by the uncertainty principle, you can measure positions as precisely as you want, you just have to pay in momentum uncertainty.

Yes, that's the current continuous model of how this works. That isn't necessarily reflective of reality though.

> Where we really seem to run into a wall is that if you keep making the measurement more and more precise, you need higher and higher energies to achieve ever shorter wavelengths and dumping a lot of energy into a small volume to measure the position really precisely will eventually result in black holes.

Exactly. In other words, all measurements necessarily have finite precision due to various uncertainties or other physical limits.

I didn't pull this limit out of the aether, I came across it from one of Shor's posts [1] where he states that physical constants can't be defined to greater precision than what I specified. If the physical constants can't have more than 60 digits of precision, then neither can any calculations or measurements based on them.

The fact is, we're bounded on all sides to finite precision.

[1] https://physics.stackexchange.com/questions/16889/non-comput...

My point is just that limited measurement precision does not imply anything about the underlying structure, could be continuous or not, I am not arguing for one side or the other.

Well then you seem to be agreeing with my original statement:

> So yes, current theories use continuous real numbers, but I wouldn't generalize that to say that's confirmed because we're nowhere near being able to test that level of precision.

Yes and no, depending on what exactly you intended to say. We agree, I guess, that we are currently using real numbers but that does not mean that the universe is actually continuous. Where I am not so sure that we agree is about the experimental side. We could run into some barrier when probing shorter and shorter distances, but this would not necessarily imply that space is not continuous. On the other hand we could also observe effects that clearly indicate a non-continuous structure of space without running into some measurement limit.

> Where I am not so sure that we agree is about the experimental side. We could run into some barrier when probing shorter and shorter distances, but this would not necessarily imply that space is not continuous

Agreed, though I personally find continuous quantities implausible, despite preferring them at one point. As long as we had a discrete theory with equal predictive and explanatory power and it was equally parsimonious to a continuous theory, I would likely prefer it. I think the next revolution in physics will see an expansion of discretization or other forms of finitism.

> On the other hand we could also observe effects that clearly indicate a non-continuous structure of space without running into some measurement limit.

Agreed as well.

Because if position and momentum are not quantized like this, you cannot get discrete bits of information (aka negative entropy) and the whole (so-called) "2nd law of thermodynamics" becomes impossible to derive, which is kind of a problem ?

http://www.av8n.com/physics/thermo/entropy-more.html#sec-pha...

Note that this underlying structure is subjective, relative to the observer, not something objective... (but we already know that there's no such thing as an "objective underlying structure" from elsewhere from quantum mechanics and also from relativity)

> (but we already know that there's no such thing as an "objective underlying structure" from elsewhere from quantum mechanics and also from relativity)

No we don't. Don't confuse the map with the territory.

In the sense that physics is about "map-making", not the "true nature of reality", and has been for a while now (since it split from philosophy ? since the beginning of postmodern physics in 1905 ?).

Even philosophy has pretty much given up that claim : with Gödel/Church/Turing having blown up to smithereens the positivist project of a "theory of everything" for mathematics, and Wittgenstein/Kuhn/Derrida/Foucault/Chomsky having redirected the rest of philosophy towards "the naming of things".

And that project had been ironically doomed from before its start anyway : Descartes both laid the groundwork for it by elevating epistemology to "fist philosophy" and for solipsism - which, while a dead-end, cannot be ruled out !

(Also honorable mention for Max Tegmark's Mathematical Universe I guess, which, in a way, indirectly achieves the claim by positing a mathematical multiverse so vast that "our" reality can only be contained inside it.)

So the only discipline left that still lays claim on Truth and the True Nature of Reality is theology. (Note that this is how Descartes "solved" the problem of solipsism.)

Physics is map making, but from that your claims - subjective nature of reality, no objective structure - do not follow, at best you could claim that physics has nothing to say about the nature of reality. And I have doubts about that, at least to some extend. What kind of answer would you expect for a question like what is the true nature of an electron? You can describe the properties of an electron, what more do you want? What is a thing above and beyond the sum of its properties?

The tricky issue here is that "the electron" itself is a specific model that only makes sense under some paradigms...

There's also a point in how one paradigm might be ontologically radically different compared to the previous one... but what science cares about more is the new paradigm being a "tighter fit" between its new models and the results of the new experiments.

Also, beyound a single "thing", it's when we start considering collections of things that the situation can get very tricky very fast, like chaotic behavior from something as simple as 3 masses under the Newtonian paradigm ! (See also : "emergence")

Or the concept of temperature : it doesn't go "down" to the "true nature of reality", but is a statistical one that is not even always defined, yet is still quite useful.

But yet again I would like to emphasize how in several subfields of physics we now are in a situation where we had to give up an objective viewpoint of the situation for a subjective one, and where the information itself that we have about a system (aka negative entropy) is another variable in a super system that includes us (and our instruments) and the system being studied, and we are forced to consider that supersystem instead, or at least also, in order to go "deeper".

(It's also impressive how in some cases we now study "turtles all the way down" situations, with models having an infinite number of correction steps that we can cheat around thanks to the use of advanced mathematics. But maybe in the future these will be seen as trivial as we see the Zeno paradox today ?)