TITLE: I need mathematical proof that the distance from zero to 1 is the equal to the distance from 1 to 2
QUESTION [62 upvotes]: I didn't know how to phrase the question properly so I am going to explain how this came about.
I know Math is a very rigorous subject and there are proofs for everything we know and use.  In fact, I am sure that if there was anything that we couldn't prove Mathematically, then we wouldn't use it any where in Math or Science.
Anyway, here is the story:
Not to long ago I was having a conversation with someone who insisted in telling me that any human knowledge based on Math is a sham, this of course includes Science.  He then proceed to tell me that he thought this was the case because according to him there is no proof that zero is a number and second he insisted even if we were to say that zero is a number,  that there is no proof that the distance from 0 to 1 is the same as the distance from 1 to 2.
According to this guy, there is no way to measure the distance from something that doesn't exist to something that does.  If I understood him correctly he said that is because he just can't see how we can measure from non-existence to something that does exist.  In this case, he said that there is now way to know for sure that the length or distance from zero or non-existent number to one would be the same as the distance or length from 1 to 2.
No matter what I said to him, he just disregarded offhandedly.  I am so upset that I want to punch him in the face, but I would rather get the proofs and show him that he is completely wrong.  To me this would be the equivalent of slapping him on the face with a gauntlet and I would have the satisfaction of knowing that Math and Science are in solid ground and that Mathematicians and/or Scientist haven't pulled a fast one on us.
So please post links to where I can read the Mathematical Proofs for these.  Or links that show that to be a fallacy.
Thanks.

The question was, is there a place where I can get proofs for both of those things.  First, for Zero being a number whether an integer, complex or any other kind.  And second, for a proof that would show that the distance from 0 to 1 is equal to the distance from 1 to 2.  
But I guess what I should really have asked is whether or not these proof exist and where I can find them.

REPLY [46 votes]: Balthus: "Zero is a fictional concept!"
Robespierre: "You are right, friend! Mathematicians are fools! By the way, when are you going to pay me the money you owe me?"
Balthus: "I don't owe you any money!"
Robespierre: "Why, certainly; but that's as much as to say, you owe me zero francs; and we just determined that there's no such thing as zero."
Balthus: "Um..."
Robespierre: "Therefore, since you cannot owe me zero francs, you must owe me some other amount. I demand that you pay me!"

In short, rather than arguing from the essential truths (which your friend will ignore anyway), argue from utility. Mathematicians choose axioms based on utility for their current purpose, after all.
More than likely, this person just wanted an all-purpose counterargument for science he doesn't want to believe. If you get rid of that motivation to disbelieve, the person would doubtless abandon their arguments.

REPLY [4 votes]: Here's a quick thought... The problem is two fold - the proof, and the guy.
Ask the guy how one might define 'stupid'.
"Would someone who didn't know that 1 plus 1 is 2 be stupid?"
Then ask if the two ones in the this sum are the same size. Get him choose which is the one from 0 to 1 and which is the one from 1 to 2...
Actually, if you want a really good proof, from a philospical point of view, try Alfred Whitehead and Bertrand "History of Western Civilization" Russell's "Principia Mathematica" which takes a good 200 odd pages If memory serves to prove that indeed 1+1 probably is 2... (Just don't mention Godel to the guy ;-) )