alt Hacker News> To Zeilberger, believing in infinity is like believing in God. It’s an alluring idea that flatters our intuitions and helps us make sense of all sorts of phenomena. But the problem is that we cannot truly observe infinity, and so we cannot truly say what it is.
When the author says we cannot truly observe infinity, what does that mean? Infinity is a mathematical symbol we can observe. We can't observe infinitely many objects, but even if we could, it wouldn't be the same as observing infinity. You can't observe the number one by observing one stone.
I think there is some confusion in this article between symbols and what they can stand for, and I can't help but wonder if that same confusion is at the root of ideas like ultrafinitism.
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You cannot observe infinity operationally. Take 0 and add 1 repeatedly. For what n does n+1 become infinite? Never. Since you can't construct infinity you can only believe in it like God.
Hence the jargon "completed infinity". Semantically --- but not in the symbols themselves 0+1+1..." one can pass from finite to infinite by arguing since every n has a successor define Z to be the set of all successors "completing into infinity".
Not having infinity is the real reason a+b=b+a can't be proved in ultra finitism. Induction which depends on the idea of completed infinity is what is otherwise is used.
saying infinity is a mathematical symbol we can observe is simplifying it way too much, all mathematical symbols are abstractions.
i can observe two apples. i cannot observe infinity apples.
I think Douglas Adams had one of the best quotes regarding observing infinity:
"Infinity itself looks flat and uninteresting. Looking up into the night sky is looking into infinity – distance is incomprehensible and therefore meaningless."
It seems to me that you're the one confused?
The mathematical symbol is just a representation of a concept, it's not infinity itself, you've got it backwards.
The symbol is not the thing. The map is not the territory. Ceci n'est pas une pipe.
> Infinity is a mathematical symbol we can observe.
This is like confusing the map for the territory.
Symbols live in syntax (like the syntax of programming languages), while mathematical concepts live in semantics. Infinity is not a symbol, it's not ∞. ∞ is the symbol we use to represent infinity.