Hacker News Reader

Comment by getnormality | Hacker News Reader

> 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.

View on HN

> 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.

There is a way to look at mathematics as just a bunch of rewrite rules for things on paper. It might not be particularly inspiring, but it's a valid way to look at things.

Indeed, there's a way to get a semantics for free, based on the syntax alone. For example, in the first order logic this is the Herbrand interpretation

https://en.wikipedia.org/wiki/Herbrand_interpretation

The point of mathematical semantics is that for any given theory, we can have other interpretations that don't just interpret symbols as themselves.

So we could conceivably imagine an interpretation where ∞ doesn't just mean literally ∞ and nothing more.

The number 42 is also a mathematical symbol we can observe. (Or two symbols, depending on how you want to define symbol).

You can observe the symbol. You can observe 42 of some object, 42 sheep for example.

You can observe a pie chart, or an actual pie, with 42% of it missing.

You can observe a plank of wood that is 42 inches or centimeters long.

But you can't observe 42 itself.

It is not like a hill on a map, where there is a symbol, and there's an actual hill.

It is an adjective and not a noun. It's not real unless it is describing something else.

My point being that regular finite numbers are not real either. So what's wrong with infinity? Or the square root of 2, or pi?

Well importantly like scrubs points out in a sibling comment

42+1 = 43, 42 + 1 ≠ 42, ∞ + 1 = ∞

Infinity plays by very different rules than numbers.

I actually agree, there's nothing wrong with infinity. I think finitists are silly and ultrafinitists are ultra silly.

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 problem to me seems to be that we are trying to map everyday language onto the mathematics. Even though we have a symbol for infinity, infinity is not necessarily a "thing" that the symbol points to.

In analysis, when we write "the limit as x goes to infinity" this translates into a logical statement like "for all x, there exists some y > x such that ..." I don't really see anything conceptually difficult or contradictory here.

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."

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.

Mathematical concepts don't have to have an obviously physical analogue. I mean, you'd find it difficult to observe minus two apples and certainly tricky to observe i apples.

To my mind, maths is like a "what if?" puzzle and whether or not infinity makes sense in the physical world, there's still fun to be had by considering the consequences of it.

That also means that it can be interesting to consider limited number systems which don't have any concept of infinity.

Can you observe 2.34 x 10^456789 apples?

No. I believe that is more apples than there are atoms in the universe, so not only it is impossible to observe, it is a fundamental contradiction with our universal reality. No one and nothing will ever be able to observe or interact that many apples, and so a reference to that many apples is only an abstract mathematical convenience that has no direct bearing to reality.

Like infinity.

I'm not sure I actually believe that, I'm just thinking out loud. But it leads me to think the question "Does infinity exist?" should be answered with the question "An infinity of what?"

You say that as if we knew the number of atoms in the universe, or its size, age, and "duration".

But none of this can be observed either, which in my book makes your argument a bit weak.

Your "universal reality" is a construction relying in big parts on the mathematics relying on infinity as a concept.

Some might say that 2 is as made up as infinity. Let me elaborate a little - your brain together with society made an abstraction "apple", and only by not distinguishing between these "sets" of atoms you can have numbers.

> some might say

Well do you say it or are you just playing devils advocate? The post you are responding to seems very straightforward.

If you wanna go all philosophical, “real” might just be anything that is useful. In that way infinity is real because you can use it to do calculus. On the other hand, there are ways of doing calculus that do not involve thinking about infinity. But if you’re gonna count to three apples you pretty much have to go through “two” no matter what.

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.

The symbol is not the thing. The map is not the territory. Ceci n'est pas une pipe.

Reply on news.ycombinator.com