How small can we measure space?

Question

I got this question after looking into transcendental numbers and I noticed how there are some distinctions that should be made from numbers and reality especially in measurement of length for example there are no perfect circles in reality and only exist in the mind and $\pi$, being a transcendental number doesn't actually exist. I don't know if it is the fact my classical education on numbers was taught by a number line but the way I think of it is if we measure any length say an object we can never be absolutely accurate because there is an infinite amount of numbers between every number and therefore I don't think we have measured any thing or any length accurately in the absolute sense.

This brings me to my question, "Is there a limit to how small a unit of Space as in the spacetime could be?" because I am thinking about space as in the spacetime continuum and assuming we are using the number line and therefore we could infinitely even go smaller. Does the mental image of numbers map properly into reality or are numbers and our view and conception of reality not 'reality'.

This also brings me to another question, "Is there anything 2-dimensional in our 3-dimensional universe or even better 1-dimensional?" yet we discuss them and attribute these properties to things that are actually not. Are we so disconnected from 'objective reality' and we usually just exist in our minds.

P.S. Are numbers just tools for us to make sense of the world for ourselves and how did we develop it? Was it just inherent or did it follow from language and other things to begin with, do animals have a concept of number?

Answer 1

We can measure down to somewhere between 10^-15 meter and 10^-18 meter, these are exceedingly small sizes.

There is no experimental evidence for a theoretical "bottom limit" on size measurement all the way down to those sizes.

There is a fundamental issue which arises when you wish to measure something far, far tinier than that- on what is called the Planck scale which you run into at distances of order ~10^-35 meter. Below this limit, the physical models we have stop yielding meaningful results and some physicists think that sizes smaller than that do not have meaning.

Recently, models have been developed in which 3-dimensional space itself comes in discrete "chunks"; nothing smaller than a single chunk can then exist. But the length scale at which space becomes discretized is of order ~the Planck length or smaller which cannot be probed even with the most powerful particle accelerators we could ever possibly build. So although we can put together mathematical models of this on paper, we can't test them.

The impossibility of probing really tiny distances in the laboratory arises because energy and distance are related: tiny distances require enormous amounts of energy to explore, and probing the Planck scale would require a particle accelerator that was ~light years in length.

For comparison, to explore between 10^-15 and 10^-18 meter requires an accelerator two miles long.

Answer 2

It's generally expected that the Planck length is the smallest length that can be probed before quantum gravity effects kick in. Its also generally expected by quantum gravity researchers that spacetime at that scale will have a granular texture. The general consensus also seems to be that the Planck length is the minimal length, but of course we do not know - we can't probe at that miniscule scale. Moreover, this is likely to remain the case for the forseeable future.

Ontologically speaking, the case that space does not decohere into a set of points was made by Aristotle. He argued that space was not infinitely divisible. By this he means being infinitely divisible into bare points. Mathematicians have more or less acknowledged this with their invention of topology and smoothness structures. The same is just as true of physics, except here we have to think about quantum effects.