All you can do is attach the sum of numbers in the system to your address.
I can only attach it to my machine, but not to others'. I can change it to 18,000,000 BTC to my machine, but this won't make it true, because I won't have the necessary digital signature to prove it so. I need other people to sign that I'm the new owner of their coins. I can't just make those up.
As for semantics, that you keep saying, you're the only one who's playing with semantics all the time... "numbers", "math symbols", "Satoshi's system"... Get real, man.
If you have 0.0000001 attached to your address, and a guy has 18,000,000 attached you both can show equal number of keys and math symbols.
Ugh.. Same shit... All over again... You don't (want to) get it...
Again, you don't have coins. You have fake number attached to your address that provides false information about the existance of a thing named "coin", while in reality you own nothing. Anyone can write down a number a name of a non-existent thing and than clam to owe it. I can write down to own 20,000 Ferraris. Satoshi, via his software, can write down that the address owns 20,000 bitcoins. Someone else can counterfeit $20,000 dollars and claim do own bank debt. All that would be false information. If I would keep insisting I own 20,000 Ferries, while only pointing to what I wrote down, that would be called a delusion. If I would do that publicly, that would be spreading misinformation.