It's more that all the interesting things that come from prime numbers don't really work if you include 1 as a prime. So every time you talked about primes you would have to say "something something prime number, other than 1, yadda yadda".

For example: The Fundamental Theorem of Arithmetic states that every natural number can be expressed as a finite product of primes, and that prime factorization is unique.

So

12 = 2^2 x 3

is the only way to express 12 as a product of primes. But if we include 1 as a prime, then

12 = 1 x 2^2 x 3
    = 1^2 x 2^2 x 3
    = 1^3 x 2^2 x 3

etc. There are infinite prime factorization of every natural number, and the most interesting part of the Fundamental Theorem of Arithmetic no longer holds.

It is actually more useful for 1 to not be considered prime in terms of application of the concept, as the other comments mentioned. Of course it's artificial, all of math is. But the artificial definitions in science only stay in place if they're actually useful.

Imagine a distance of any length. How long does it take to cross an infinite number of that length? It takes an infinite amount of time.

Divide the length between A and B an infinite number of times. We now have an infinite number of lengths, which means it will take an infinite amount of time to cross them.

Which means nothing ever actually moves and movement itself is an illusion.

Well, this Zeno guy was an ancient philosopher who existed long long ago.

A "series" is the summation of all the terms in a sequence. In the modern day, we now know that an infinite geometric series, where there is a common ratio r, can be finite if the absolute value of r is less than 1! There's a neat bit of maths behind that proof too.*

So in the case of Zeno's paradox, it's an infinite geometric series with r = 1/2. So if you had a distance of 1 metre, the next ones are 0.5m, 0.25m, etc. Each term is half of the previous. And since the absolute value of r is less than zero, the sum of the infinite series is finite, in this case it's simply 2.

Infinity is a little weird.

*See replies below if you want to see my attempt at conveying it within the confines of an internet comment

That's the paradox and that's why I love this thought experiment.

There is infinity in everything and everywhere ... yet we are told that we can't cross infinity ... yet we do it all the time.

What is the chance anteaters would evolve on a planet with the perfect atmosphere and biome for them to survive? Do you think the perfect food just randomly evolved by happenstance? Obviously this universe was created for anteaters.

Ah, so that's why looking it up first got me "retarded matter".