Here’s my thought process. Numbers are infinite, and decimals are infinite. Therefore units of measure are therefore infinite. A ruler may not be 1 foot long, it may be 1 + ( 1 × 10^-63836275783837472827263728) feet long.
To me, for a measurement to be exact, it has to be to the point where as you get to smaller and smaller decimals of the measurement, you get to the point where the additional decimals are infinitely 0. But if numbers are infinite, than the ability to continuously shrink down and get those decimals is infinite too. Doesn’t that mean it’s impossible to get these measurements?
And if these measurements are physically impossible to find, since the requirements for them are physically impossible to occur, doesn’t that mean that no object in the world has a physical set size? And moreso, that the space in between any two objects, as well as the size of those objects, are both by definition infinite?
In: Physics
> Numbers are infinite, and decimals are infinite.
Don’t confuse the map for the territory. Numbers are abstract concepts that we use as an idealization of things in the real world. Numbers aren’t themselves physical objects and they don’t have to correspond precisely to physical reality. We can only measure anything in reality approximately. And for most purposes we only need to know things to within a certain accuracy. For example if you want to build a table to fit in a certain space, it’s good enough to know that the space is approximately 1.35m wide – knowing that it’s actually closer to 1.352383746m wide doesn’t make it any easier to build the table at all.
> doesn’t that mean that no object in the world has a physical set size?
Once you get down to small enough scales everything is an indeterminate size anyway. Most objects grow or shrink very slightly depending on factors such as temperature, and quantum-scale objects such as atoms and electrons don’t really have a well-defined size.
> And moreso, that the space in between any two objects, as well as the size of those objects, are both by definition infinite?
You mean because there are infinitely many points between them?
In terms of physical reality, we can’t really say that there are infinitely many points there. How could we demonstrate that there are? We can’t exactly do an experiment in which we move an object to infinitely many different positions.
In terms of the mathematical idealization of space, it’s perfectly possible to define a mathematical space in which there are infinitely many points between 0 and 1 and yet the distance between 0 and 1 is still just 1. The concept of distance has nothing to do with the number of intervening points.
Something that is a foot long will always be more than 11 inches long and less than 13 inches long the exact measurement depends on circumstances and measuring tools but it has limits.
In engineering there is the concept of “tolerances”
Though our definition of how long something is gets defined as how far a beam of light goes in a set time, exactly none of our measurements is ever 100% accurate.
So engineers usually say something like “this is 1 meter long +/- 5 mm”, we don’t know exactly, if you paid more money to use more precise measurements , you could go down to 1mm, or 1 micron, or lower, but do you really need that?
To add to this, lots of things can cause an object to change size and it would be stupid not to take this into account. Bridges have bits built into them specifically to handle changes in size as the bridge expands/contracts because it’s getting hotter/colder.
More precise machines, like stuff that places stuff withing 2-3 microns, use electronics and cameras to continually adjust themselves to keep that accurate.
The fact that digits are infinite has no bearing on physical reality. A ruler only has X atoms in it, no more, no less. So it does have a fixed, set size, because it has a fixed, set mass. Whether or not we can accurately describe that size using our numerical system has no bearing on the fact that the ruler, physically, has a fixed, set size.
If you want to look at it another way, pi is infinite, right? We will never find the last digit of pi. But we can confidently say that pi is less than 4, no matter how many digits you drill down. So a number with infinite decimals is not the same as an undefined or “infinite” number.
>And moreso, that the space in between any two objects, as well as the size of those objects, are both by definition infinite?
This isn’t the case, for an easy example consider the numbers 0 and 1, the difference between them is just 1. There are an infinite number of real numbers between them, but there is still a finite ‘distance’.
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