Can there be a universe where mathematics is different?

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Simply, yes it's possible. This is a rather simple question. With physics we can conclude that there are an infinite amount of universes. So Its obviously possible for math to operate differently in one of those universes.

If we see this universe as just one of infinitely many possible universes out there (with each possible universe having its own laws and constants brought forth into being via random singularities like the Big Bang) I don't see why not.

The question is whether mathematics could be different or not. The answer to that is a definite yes, such as my insane concept of a stochastic number line where every number on the line is different every time you look at it. Such a universe would have insanely different mathematics than our own, and would not be able to exist because it would be internally inconsistent. (Imagine, for example, a Turing machine in a stochastic universe where the tape is different every time you look at it.) In a stochastic universe, sure, 2+2 = 5. Or 10,012. Or 94. Just depends on when you look at the answer.

The specific example of 2+2=5 is what's causing the debate. Are we talking about math as a notional thought exercise, or a description of physical reality (the marbles in the bag)? As a thought exercise, sure, 2+2 could be 5. (Imagine a laniappe universe where each operation had a bonus unit thrown in with every operation performed!)

The deeper question is whether a universe could have a different method of counting discrete units (such as marbles) without using the basic concepts of unit, successor, and induction. I can't see a different method of counting existing.

If there's one thing I've learned for quite a while now, it's that this whole existence (whatever you want to call it) does not ultimately conform to our human logic and/or intuition.

Quite so. Nature is in no way obliged to make sense to us.

I forget who said this: The Universe in not only stranger than we imagine, it is stranger than we -can- imagine.

ruveyn

What I'm saying is that mathematical laws are independent of the universe since as I pointed out you only need basic logic and the axioms of set theory. While mathematics can be used to analyze and better understand the universe, this doesn't mean that math and physics are the same thing. Abstractly speaking, mathematics is just the set of all true statements about sets since any mathematical statement can be expressed in this way.

I'm not sure why you feel the axiom of choice and the Banach-Tarski paradox would demonstrate a flaw in my argument. The axiom of choice is an abstract assumption and does not correspond to any physical law. Also, the Banach-Tarski Paradox is a mathematical statement that tells us nothing about physical reality.

First of all, we can't conclude anything yet.

Secondly, even if there are an infinite number of universes (multiverse), their random laws would likely have finite boundaries that would rule out any violation of how basic arithmetic operates. Meaning every single universe would be describable using our math; Instead of imagining the set of rules for the multiverse as an infinite cloud of possibilities, think of it as an x-dimensional cone of possible rules, with edges defined in basic set theory and where the set of rules governing a given universe is an (x - 1)-dimensional cross section of that cone.

One such cone-edge could represent the lower boundary of the speed of light; Speed can never be negative, so that edge represent 0 speed.

Yeah, I'm no expert when it comes to cosmology and physics and such, but doesn't everything math-related break down at the initial point of the Big Bang anyway? If so, why do some members insist that mathematical laws have to be the same in all possible universes?

There are many mathematical systems in our universe (some of which are pairwise contrary to each other [i.e. both cannnot be true at the same time]). The question is this: is the set of all possible mathematical theories (i.e. those which are consistent) universal for all possible universes? I have not the foggiest idea how to find out if that is true or false.

ruveyn

The real point is that maths are "just" symbolic logic systems. Most of them happen to be useful for modeling the universe, but there is "pure math" which describes math with no known real world application. Yet it still works like other math in that it's a bunch of theorems proved based on axioms and existing theorems.

Is there anything about the Fundamental Theorem of Calculus that makes it only true in some universes? Let's say that for some reason that another universe is so different that its physics wouldn't use calculus. That doesn't say anything about the correctness of the theorem. A math system is its own self-contained symbolic logic.

The real philosophical questions in my mind are: Do math concepts exist independent of something understanding them? Also, while math aims to be technical and unambiguous, it is actually a human language. Can something that is not human understand a human math concept by observing the symbols?

When I started seeing and working on math I was unpleasantly surprised how arbitrary it seems compared to say a programming language where a language's evaluation is completely mechanical and well-defined. In math you can literally say whatever you want as well as you're using well-defined symbols. It's about an expressing an idea to others, who will decide if there are any errors in your proof.

This is easily the best answer yet.

I think that if the answer to 2+2 depended on when you "looked" at it then you are not referring to addition.

What you would have is the mapping $f: T \times \mathbb{C} \times \mathbb{C} \to \mathbb{C}$ where $T$ denotes time modeled as $\mathbb{R}^+$.

There may exist a $t \in T$ such that if we consider the derived mapping $f_t: \mathbb{C} \times \mathbb{C} \to \mathbb{C}$ then we have that this is in agreement with the usual definition of addition on $\mathbb{C}$. But I do not think you can refer to $f$ as addition in any normal sense of the word.

There do exist alternative axiomatizations for the natural numbers but we only agree that they refer to the natural numbers if they are logically equivalent to the schema that the mathematical community have so far agreed refers to the concept they all intuitively sense to exist.

For example Deskins in his 1964 book Abstract Algebra provides an alternative schema for the natural numbers using addition and multiplication as primitive binary operations. He goes on to prove that it is logically equivalent to the Peano-Dedekind axioms and so we agree that he is talking about the same thing. If his didn't then we would just say that he was describing something that was only similar.

It is important then that we dig deeper, turning our attention away from arithmetic and start looking at logic.

Have you ever wondered why mathematics, which is purely abstract, works so well in describing particular physical processes and properties in the real world?

You might want to look up an essay by Eugene Wigner on "The Unreasonable Effectiveness of Mathematics in the Natural Sciences".

See this site: http://www.dartmouth.edu/~matc/MathDram ... igner.html

ruveyn

I wondered the same thing when the movie Contact came out. They talked about how math is the only universal language. is it also inter dimensional though?

Mathematics itself would be. The laws of physics in any given dimension described by the math might not be. But then that dimension might not be able to exist (due to instability); it's more of a gedanken experiment.

I have trouble imagining any reality where the two main building blocks of math (grouping stuff together and then counting what you grouped) don't hold true. Well, I can imagine a "stochastic universe" where every math operation results in a random number, but it's hard to posit such a place actually existing.

People were doing mathematics (for instance, the ancient Greeks) long before set theory and logic.

ruveyn