If numbers are infinite, and they can get infinitely smaller, then how is it possible that we can touch anything? Wouldn’t the distance between my foot and the floor, for example, just keep getting infinitely smaller?
In: Mathematics
When two things “touch” we usually take this to mean that the distance between them is zero. In a mathematical sense, things can touch simply by having a portion of them occupy the same space. Moving things that start apart can approach each other and eventually touch. Issues you arise are solved through things like limits and calculus.
In the real world, things can’t touch in the mathematical sense because everything is made up of volume-less point-particles and physics says that two particles cannot occupy the same space (barring black holes).
In practice, we say that two things are touching in the real world when there is no noticeable space between them. Generally this becomes apparent when repelling forces between the electrons in the atoms in each object becomes noticeable (in the sense that you cannot bring the objects any closer and the friction between the two objects is noticeable).
In a mathematical sense…
What you are describing is when things get asymptomatically closer. Like, you cut the distance between them by half every second, they never actually touch.
When you put your foot down, they get closer in a linear way. Like, they get a centimeter closer every second. Eventually, they crash together and touch.
In a chemistry sense…
The electrons in your atoms repulse the electrons in the thing you are touching, so you really never completely touch something unless it chemically combines together.
I think a physicist would have to get in here, but you are correct… theoretically. There is a small distance between anything you touch and you. But this is happening at the atomic level, I believe.
Number can get infinitely smaller; distances cannot. There is a finite unit of length, called the planck length, that is theorized to be the smallest possible unit of distance. In any case, no widely accepted model of physics can make meaningful statements about any distance smaller than that. That being said, objects don’t “touch” in the sense that there is truly zero distance between their atoms. Even atoms themselves are mostly empty space.
Technically nothing ever touches anything the repulsive force between subatomic particles just equal the forces trying to push them together.
But I don’t think that’s the question here so in a more practical sense yes there can be infinitely many set steps to reach a goal but if the time for each step gets smaller you can do those infinite steps in a finite time.
https://www.reddit.com/r/explainlikeimfive/search?q=Zeno%27s+Paradox&restrict_sr=on&include_over_18=on&sort=relevance&t=all
What you’re describing is the Achilles and the Tortoise paradox. You can keep making infinitely smaller and smaller segments to pass. However, the time it takes to travel those segments also becomes infinitely smaller and smaller. In these paradoxes, those infinities effectively cancel each other out.
This is a classic paradox (Zeno’s paradox). The thing is, while the number of divisions approaches infinity, the length of each division approaches zero. So mathematically, the problem of traversal ends up being a summation of zero times infinity. The two end up cancelling each other out, leaving you with a finite value.
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