[FredPret]

It’s more complicated than that.

For a number like 2 to exist, the implication is that there are two things in the universe that can be exactly equal.

Even if there are two or more of something (not a settled question in physics), the idea that you can add two such groups of two and have it be equivalent to some other group of four of that thing is only an abstraction in our heads.

In reality, if you have two apples in two pockets and I have four in mine, all we can say is that there are eight regions of the universe we like to call “apples”. 2+2=4 isn’t true in the same way that Sherlock Holmes’s address is because the former depends on a shared illusion/set of abstractions.

[tremon]

For a number like 2 to exist, the implication is that there are two things in the universe that can be exactly equal.

I'm not convinced. You are defining the number 2 as a material cardinal, and then coupling the existence of the number 2 to the existence of two completely identical material objects. But why would the existence of those objects give rise to the existence of the number 2?

Can't I similarly define that the number 2 is a materialized ordinal, and simply count the revolutions of a moving object, and state that the number 2 is instantiated by observing the periodicity of a single material object?

[Koshkin]

> regions in the universe

This has nothing to do with natural numbers. What does have to do with them is simple practical considerations like that you need two apples to treat two horses. This is why (small) natural numbers do, in fact, exist - in a very clear practical sense. And if you only have one apple and get bitten by the second horse who is now upset, you can only blame yourself for ignoring such an obvious fact.

[FredPret]

Totally agree with you that it’s practical to think of apples and horses as more or less alike. We evolved this mental tendency because it works.

But it is a style of thinking that is adopted because it is useful to human lives on human timescales, not because it is true.

The truth is there are no apples, no people, no objects even. There is just an immense swarm of subatomic particles interacting woth one another across the universe.

If you could observe the particles in that “apple” over a billion years instead of a more human timescale, it won’t make any sense to think of it as an object anymore. The particles started out in stars far and wide, are briefly frozen together in a fruit on earth, and will soon be spread apart widely again.

Some of the particles in the universal soup have combined into a chemical reaction that thinks it is you, and has evolved useful ways of perceiving patterns in other particles (for instance, object permanence). Because this phenomenon (you) is able to perpetuate itself by thinking in this way (perhaps by, among other things, finding and eating apples?), these thought patterns are sticky. Or in your words, practical. But not the whole picture.

[Koshkin]

This looks like a classical case of someone missing the forest for the trees.

[jonnybgood]

At what point does one apple and another one apple become two apples? Is a matter of distance? Likeness? Practicality doesn’t say there must be two apples, just one apple and another one apple. Two (apples) is just an abstraction in my mind.

[templeosenjoyer]

>At what point does one apple and another one apple become two apples? Is a matter of distance? Likeness?

Perhaps I am misinterpreting, but it seems to me like this would occur whenever you formed a coherent question. "How many apples are there?" would be insoluble, whilst "how many apples are there in this room?" or "how many red apples are there resting on that table?" would result in an answer belonging to the set of naturals.

[Koshkin]

You are right in that one apple and another apple do not make two apples. You need a matching pair of happy horses for the number two to emerge.

[talaketu]

It's a matter of language and logic. "another" is a contraction of "one other", and "other" means "distinct".

So, given you have "one (element) and one different (element)", at that point, you have, ipso facto, 2 elements.

[jonnybgood]

It's not ipso facto 2 elements. You need to define what it means for there to be 2 elements, or 2 apples. Which goes back to question: "At what point does one apple and another one apple become two apples?"

I can simply say there's only one's of apples. The existence of the number two is optional in this case. Its existence is completely up to my own mind (Distance? Likeness? Etc.). Prove to me in this case that the number two exists outside our own minds.

[Koshkin]

Natural numbers are objective, observable relations between collections of things (and also between a collection and its sub-collections). They can only be understood by observing differences and similarities between collections. (Similar to how the color red would not exist if everything was red.)

[GoblinSlayer]

Split one apple into two halves and feed two horses with that.

[adrianN]

No, I don't think there is any such implication. Mathematics works by defining some axioms and some rules on those axioms and then people think hard to see what theorems are true in this system. Being able to define something doesn't seem to imply that such a something must exist.

[talaketu]

You are unsure whether "2 + 2 = 4" is real, but certain that "2 + 2 + 4 = 8" is.