Wouldn’t a Planck grid be better thought of as a mass of closely packed and incredibly tiny spheres rather than a grid?
Even so, unless you are constrained to movement along the gridlines, then a roundabout trip would leave you at some fractional distance from your origin.
And if we are constrained to movement along the gridlines, that breaks conservation of momentum.
Would it necessarily break conservation of momentum at any scale that we can observe, though?
My only familiarity with ultrafinitism comes from the wikipedia article linked to in the OP, and so everything I write below is based only on that and my own reasoning and intuition, but I don’t necessarily think that the issue you raise would actually be a problem. Apparently you are allowed to use indefinately large numbers.
Shaughan Lavine has developed a form of set-theoretical ultrafinitism that is consistent with classical mathematics.[4] Lavine has shown that the basic principles of arithmetic such as “there is no largest natural number” can be upheld, as Lavine allows for the inclusion of “indefinitely large” numbers.[4]
The point seems to be to avoid using number that are so large that they have no constructive relevance. I think you could similarly argue that numbers that numbers close enough to zero are constructively indistinguishable from zero.
I was thinking that useful application of ultirafinitism might be in computer science where algorithms need to store the numbers that they are dealing which limits certain applications.
In general I don’t think that finitism hurts limits, with the one exception being, limits at infinity. Often Mathematicians like to using an augmented real or complex number line that has an additional point at infinity (or maybe two points one at infinity and the other at minus infinity) that sequences can converge to. This is called the compactification of the real line, and means that every sequence of real numbers has a sub-sequence that converges to a point, and allows for all sorts of nice theorems.
We never should have put the mathematicians in charge of math.
(I’m only about 90% joking there. I do actually think it’s pedagogically damaging that we don’t use infinitesimals, just because some mathematicians once falsely thought limits were more rigorous.)
Hasn’t been true for a couple of decades. Everything is floating point now. Except (ironically?) some AI math, which is now back to being so resource constrained that you really need to use as few bits as you can, and that means fixed-point math is making a comeback.
A good reason why floating point is so popular. At the expense of a few bits for the exponent, you get sufficient range to handle any scaling needs. No need to keep track of your scale factors or adjust them when they get too large/small.
Unless the spheres are packed so closely that there is nothing between them, we need to worry about what is in the spaces between spheres. If you do squish things down, it becomes some form of grid - with the shape depending upon the packing.
However, I don’t think anyone thinks space at the Planck scale looks like a grid of any kind. Some see space as a random seething foam with an average size of the Plank length.
The Planck length is, at best, a minimum discernible length. It isn’t something that divides space up. More like having blurry partly overlapping markers on your ruler than anything physical - and some interesting consequences if you try to make the markers too fine and close together.
I remain unconvinced by many Planck scale arguments. The Planck length is in part justified by a combining of GR and QM at a scale where we are pretty sure we don’t understand what is going on.
If the universe is basically running on integer math at the smallest scale, we can posit all kinds of experiments where an odd number must be halved and a remainder is lost.
Someone is always going to complain about something. Suppose you did teach “Mathematics for Engineers” using non-standard analysis. Are people going to say it’s “too hard”? “Too abstract”? That they don’t want to encounter bridges designed to take infinite loads…
It is not true that they thought infinitesimals were less rigorous. Infinitesimals, until Robinson came along, had no mathematical rigor at all. Robinson gave them a rigorous basis, but only by using non-standard ultrapowers of the reals. Which requires the axiom of choice to show existence. Cauchy type limits require nothing of the sort, just ordinary logic.
But it is a shame that calculus with infinitesimals has not caught on as a teaching tool. The argument that you need fancy constructions to justify infinitesimals fails since elementary calc books make no real attempt to explain the logic behind limits either.
Here is another approach to infinitesimals. It has its own problems, which I won’t get into.
Say a “number” is a sequence a_1,a_2,\ldots of ordinary real numbers that either tends to infinity or to a finite limit in the usual sense. Two numbers are “equal” if they are except for a finite truncation. So the sequence 1,1/2,1/3,\ldots is equal to 1/2,1/3,\ldots. An ordinary number appears as a constant sequence. Say 1,1,1,\ldots and anything equal to it represents the ordinary number 1. Now an infinitesimal is any number that in the usual sense has limit 0. It is now clear that any finite number is the sum of an ordinary number (the limit of the sequence) and an infinitesimal. An infinite number is the inverse of an infinitesimal. 0 still has no inverse.
The problem is that you cannot extend discontinuous functions to these “numbers”. And there are differentiable functions whose derivative is not continuous. But these will not be studied in elementary calculus.
If the position on the pre-existing grid varies randomly, the remainder would be lost according to some probability distribution.
And, for that matter, an even number being halved would also sometimes have the wrong answer.
But the fact is, we have no idea whether space is quantized at all, and if it is, we have no idea the characteristic length scale on which it’s quantized, nor the manner in which it’s quantized. For the former question, the Planck length is as good a guess as any other, but it’s really just a guess, barely even worth calling a hypothesis.
What is beyond the edge? Where is the edge? When you look out from the edge, what do you see? At the end of something, there is something else which defines that end. How far does that extend?
We already have the physics in quantized spacetime worked out in some specialized circumstances. For example, see atomic orbitals and electronic band structures. Without going into the math, some states and some transitions are disallowed. If a transition from one allowed state to another allowed state violates conservation of energy or momentum, then it won’t happen.
Just because a space has finite extent does not imply it has a boundary. The 2-dimensional surface of a sphere or of a torus is finite but has no boundary. There are similar 3-dimensional spaces, also having no boundary. As well as in any dimension.
These seem analogous to walking in a circle on a flat plane. Also. If you are limiting to the 2 dimensional surface of a sphere then there is a boundary. The third dimension. Look up. Or dig down. There it is.
I think you’re missing the picture - a 2D space that can be mapped to the surface of a sphere (or torus) NOT the surface of a sphere itself. There is no third dimension one can look up to or dig into in such a space.
ETA for bad languager-y: There are such n-dimensional structures for any value of n, i.e. finite extent without a boundary.
This was depicted in a 2D setting in the video game Asteroids which wiki characterizes as, “a two-dimensional view that wraps around both screen axes.” Similar geometry was portrayed in The Flintstones.
If you arbitrarily pick a construction and or rules that fit your argument for finite or infinite, then you can prove the argument. But it seems the original question was more open ended in relation to the universe. I disagree with the idea that the surface of the sphere is infinite. This supposes that any point on the sphere is the same as any other point. That would be one dimensional? If one walks a circle round the sphere, you come back to the same place. You have traced out at least one boundary.
The OP wasn’t about the universe to begin with, so I’m not sure what the objection is.
Also, to repeat myself: it’s not the surface of a sphere but a 2D analog that can be mapped to such. Or you can refer to MfM’s excellent example of Asteroids, which exhibits such behavior.
And, yes, you can envision a 1 dimensional space that has no boundary. You can think of it as mappable or analogous to the surface of a circle but not a circle itself.