Can there be a universe where mathematics is different?

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So it actually sounds like you're agreeing with me. But your post only touches on part of what I covered. Before you can say 4 = s(s(s(s(0)))), you still have to have the concept of a counting object. Otherwise how would you know that '4' is four (i.e. the concept of this many * * * * of something). What I'm getting at is spelled out in more detail here:

www2 dot latech.edu/~schroder/slides/fund_slides/nat_arith.pdf (sorry I'm not officially allowed to post URLs yet. you would think an exception would be allowed for .edu's)

But the conclusion I think is if you define a field of counting objects, it's going to behave the same as Z0+ with addition, or in plain English 2 + 2 is always 4. If you use a different field whose elements do not behave like counting objects (it's possible and they exist), 2 + 2 truly mean something else entirely.



You mean mathematical platonism? It's an interesting idea, but I don't really have a grasp for what it really means to say that math "exists".

But, can 2 + 2 = 5?

Math does not define the set (or class) which is a group of discrete items to count. This is taken as a given. Similarly to the Euclidian axioms of geometry, it might be possible to define the set in such a way that your mathematics is different. But, given that you're grouping things so you can count them, this wouldn't make any sense.

Similarly, the concept of counting is defined by Peano's postulates and it would be difficult to redefine these, since they basically postulate the unit (one discrete item), the successor, and induction. I do not belive that there could be any conceptual way to count differently.

So the answer to the original question seems to be no. Math could not be defined differently and still make any sense. Any redefinition of math where 2+2=5 would be nonsense. No actual reality could exist where nonsense made sense. We could notionally think of such a reality, where kids built tesseracts out of toothpics and Donald Duck could do math magic, but it could not exist.

Consider an operator @ such that a@b = a*b + 1.

Then 2@2 = 5. So if one regards the + sign to mean what we defined prior substituting + for @ will give us
2 + 2 = 5. So it all depends on what you mean by +.

ruveyn

Hey, those are fighting words. I ain't no Platonist.

All I'm saying is that the axiomatisation of something as natural as nonnegative integer arithmetic is mostly just a way for obsessive people to be able to sleep at night. It's a way of tying the balloon to the ground, but it doesn't create the balloon.

In all of those centuries when mathematicians didn't have an axiomatisation for nonnegative integer arithmetic, were they just spouting nonsense when they claimed "2 + 2 = 4"?

That's interesting. Are we talking about a universe where the concept of math is different, or one where the symbols of math are different? Redefining + to not be addition doesn't change addition. If I have three marbles in a bag, and put two more marbles in a bag, I'll have five marbles in the bag regardless of how 3, 2, +, and set are defined. Are we talking about a universe where we use different symbols for the same underlying notions, or are we talking about a universe where the underlying notions of math are different? Are we talking about use (of the underlying concepts) or mention (of the symbols)? I don't think the original question was detailed enough to know.

If you are talking about combining finite disjoint sets of separate objects then sure enough 2 + 2 = 4. That is just a fact that goes with the disjointness of the sets of pairs and the fact that the objects can be distinguished from each other.

ruveyn

2 = 1+1
4 = 1+1+1+1
Thus, 4 = 2+2

It's hard to imagine a context in which "1" is meaningless, as the context will have at least one singular concept, that being itself. Identity as a concept is thus somewhat inescapable.

However, it's conceivable that a context might exist which is all-encompassing. Completely infinite and uniform. In that case, addition might be nonsensical, as nothing can be added.

So while that doesn't mean 2+2 wouldn't equal 4, it does suggest that 2+2 might never arise even as a concept.

However, for that context to exist, our own could not, since it would either be outside of that context, and thus it's finite, or within it, and thus it's not uniform.

Hmm.

That's a damned good question, OP. This should be crossposted to the philosophy subforum.

Change a few parameters and it would be very unlike for life to be able to form at all in a universe. As such, there would be nobody in the universe to even do mathematics at all.

That said, I think that in a universe in which it is impossible for life to have formed, 1+1 would still equal 2 even though there would be nobody there to count.

Can there be a universe where mathematics is different? We don't know for sure. In our own universe U it is a brute fact that 2 + 3 = 5. If there somehow could exist a universe U* where it is a brute fact that 2 + 3 = 4, its inhabitants would find it as difficult to answer the question "Can there be a universe where mathematics is different?"

I'm inclined to think that U and U* cannot coexist no matter how they are separated. As soon as there is U or U* there cannot be the other. Things have a different ring when we go back to when nothing yet existed. Could it be that from the zero point on all there is woud be a U*? This is what we don't know.

In theory, it's possible that there are different universes where the laws of physics are completely different. It is also possible, though, that most of these universes are just empty spaces with no matter whatsoever.

With that being said, there's no proof whatsoever of the multiverse theory, and most of the adherrants of this theory are simply using an advanced version of the God of the Gaps fallacy.

Two and two ALWAYS equals four no matter how many times God recycles the universe.

And he has already tried thousands of times to get it right.

But certain aspects of math do change from universe to universe.

You can be sure that the next time around- the diameter of a circle will go into the circumphrence of the same circle EXACTLY three times!

This Pi nonsense just doesnt cut it!

Thats the conclusion I came to after three bong hits of Hawiian many years ago.

Laws of physics may differ in different universes, but two plus two always equals four, no matter what universe one is in.

Depends what you mean by +. On a four hour clock 2 + 2 = 0

ruveyn

Last edited by ruveyn on 03 Jan 2013, 12:22 am, edited 1 time in total.

In an empty universe or in a universe containing only a single "particle" there would be no "two".

It doesn't matter what universe you're in. All of mathematics depends only on the axioms of set theory and logic. These would be the same in any universe and do not depend on physics.

Does every universe have the Axiom of Choice?

How about Banach-Tarski?