How small can we measure space? [closed]

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.

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 depends on what you mean by "measure"

The numbers I give in this answer are inexact. The purpose of this answer is to help you clarify what you mean to ask, so that you can ask a more targeted question of experts who might be able to answer it (like on physics stackexchange).

Unaided Direct Observation

The coarsest scale we can consider is a unit of distance that can be observed by direct observation, where we can count and enumerate things that have differed by that value. This is probably around the 0.1-millimeter level (10^-4). Below this level, we probably could tell that an object of size X and an object of size 2X are different, but we wouldn't be able to quantify how much different they were.  

Unaided Indirect Observation

The next scale up would be the scale of what we can prove to exist using our five senses, even if we can't directly observe individual changes on this scale.

For example, with our sense of touch, we can feel differences in texture caused by even very small differences in size (for example, threadcounts in high-quality bedsheets versus medium-quality ones). We can't measure how much smaller the threads are, but we can tell that the texture is different on a scale we cannot observe. Thus, we can prove the existence of a scale smaller than our observations.

The limit here is probably not much smaller than the size of a human cell, or tens of micrometers (10^-5 meters).

Tool-assisted Direct Observation

The best tools I'm aware of right now are electron microscopes. These can get us down to 0.1 nanometers (10^-10 meters).

Tool-assisted Indirect Observation

By taking many direct measurements and constructing probability models, we can measure the sizes of different kinds of atoms. This is an indirect observation, because even with our best tools, we cannot observe or count single atoms, but we can tell that (for example), a sample containing mostly-pure uranium is made-up primarily of structures with nuclei of 12 femtometers (1.2m * 10^-14). Our precision with these sorts of indirect measurements goes to about the tenth of a femtometer (10^-16).

Implications of our Models

Physical models (which predicted the existence of nuclear power) assume that these nuclei are made of subatomic particles (neutrons, protons, and electrons), and can make some predictions about how many of each there must be in the nucleus of any given element. Since we can already estimate the size of the nuclei, we can then make some estimates about how large these subatomic particles are. We can prove that electrons are smaller than an attometer (10^-18), but we don't know how much smaller, yet.

I consider this measurement to be the practical limit of our technology. With better technology, we might be able to measure the radius of an electron - or at least reduce the maximum size an electron could be.

Limits of Our Models

Based on our models, we can conceptualize what the smallest observable change might be - that is to say if our theories are correct, then there is no possible technology that will allow us to observe changes smaller than 10^-35m. Even if changes smaller than this are possible, then any change smaller than this amount would be literally inconsequential - there would be no effect on the rest of the system.

Of course, it's possible that our models are incomplete, and there is yet another layer of physics which can account for consequences on a smaller scale than this.

Answer 3

pi, being a transcendental number doesn't actually exist.

some transcendental numbers, like pi, happen to be computable, so one might say that pi 'exists' on the grounds that algorithms for approximating it 'exist'

I don't think we have measured any thing or any length accurately in the absolute sense.

yeah, there's little sense in speaking of 'absolute' measurements

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.

even if we use the real (or even rational) line as a convenient tool to operate with numbers after measurements provide these numbers, it's not clear why such a 'function' from measurements to the number line would be surjective, and it's in fact easier to suppose that it cannot

Does the mental image of numbers map properly into reality or are numbers and our view and conception of reality not 'reality'.

it's more likely the other way round, as in the previous point

Are numbers just tools for us to make sense of the world for ourselves

from a pragmatic point of view, yes, but you can also have fun with them, of course