Can there be a universe where mathematics is different?

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One where 2 + 2 doesn't equal 4, for instance?

In a world in which distinct objects either do not occur or cannot be perceived arithmetic would be different.

2 + 2 = 4 only if you have individual entities to count separately.

ruveyn

The question needs some more amplification. Do you mean the mathematical models of the physical universe (ie all those equations like F=ma and so on) would be different? The axiomatic foundations of math itself?

With physics, no, the universe with different equations couldn't exist. Except notionally. We can imagine a 1-dimensional universe where everyone lives on a line, but it can't really exist.

In axiomatic math, numbers are labels for successor operations (based on Peano's postulates). When you say 2, you're saying 1' (the successor to the unit), and 4 is 1''', 5 is 1'''''

So numerals are in essence abbreviations for a series of successors. You could redefine the symbols, but no one would know what you are talking about (ie redefine 5 = 1'''). This has actually happened, btw, since some early computer programming languages like the very early FORTRAN compilers allowed you to do this. You could literally redefine numerals to have new values. This was considered a bad thing and eliminated.

(My = should be the three-line version of material equivalence, but this forum doesn't do LaTeX.)

So yeah I guess it is true, like I said in another thread. While you can imagine a universe where the laws of physics are different, you can't imagine one where mathematics is different. Math is always the same, no matter what. It can't be different. The way it's done with the symbols used and the base 10 that is used could be different, but math itself is never different. 2 + 2 always equals 4.

On a deep level, one could possibly have some changes. Perhaps if some basic axioms were to be changed or eliminated.

On a very shallow level, it would be quite easy to change arithmetic. I think it was Stephen Jay Gould who suggested that if we had evolved from some certain once common dinosaur that could walk upright and had four fingers on each hand/paw, that we would naturally do our arithmetic in octal rather than decimal.

2+2=4 may be hard to understand in fact, and it may be that in a way this may not be true; it depends on the understanding of each element in the sentence. For instance may 2+2 represent two distinct quantities where the element of 4 represents a quality which can not be divided into two elements without losing its qualitative feature. Said in another way, the continuity of 4 would be broke if reduced to the sum of discontinuities. In short this means that 4 can not be reduced to 2+2, but 2+2 may become 4, and to equal them make up for a problem.

No, I think that mathematics doesn't depend on the nature of the universe you're in. Of course, the bits of mathematics which interest you would probably depend on your universe, since you want mathematics which can be used to describe the physical laws of your universe.

In this universe, the number "367" is a sensible practical concept, whereas "i" is a strange silly concept that only specialists care about. However, in a different universe, where there are only 300 objects and where complex numbers are required to understand macroscopic physics, it would be the other way around.

Most people seem to think that math is somehow the distilled essence of the universe. It was invented to model physics and other sciences, but it doesn't strictly have anything to do with the universe.

2 + 2 = 4 because of the axioms that define counting, addition, and equality. You could create another mathematical system that uses completely different axioms. Most people don't see this since they aren't exposed to higher math, plus there's a lot of overlap between fields since re-inventing the wheel on basic things like real numbers or set theory would be a waste of time. If you study say linear algebra you'll see linear/vector objects axiomatically defined with properties similar to numbers, but they are a new creation independent of previously existing math.

The fact that when you model a physics problem using our math only to have the result match up with reality seems to indicate that our math somehow matches the underlying nature of the universe, but I think that's impossible to prove.

That's a pretty extreme position. Certainly you could find a set of axioms such that "the consequences of these axioms" seems to match up exactly with "facts about nonnegative integers", but is it really true that when we are talking about nonnegative integers, we are actually talking about the consequences of a certain set of axioms? After all, there are lots of different obsessive ways of boiling down topics to axioms, and they all seem just as "real" as each other.

I met a guy once who had just learned about a set-theoretical foundation for numbers, and he said that he now actually believed that 2 is an element of 3. (Because it is, when you define what 2 and 3 "really" are.) I think that's a bit silly.

Such a universe would be impossible for us to concieve.

I'm not sure what's extreme about it. Without getting into the arbitrary nature of how the actual numbers were chosen or how they're written, you have a predefined set of counting objects S. By counting object I basically mean Z0+ (non-negative integer).

S = {0, 1, 2, 3, 4, ...}

For these objects addition can be defined as follows:

x,y,z element of S:

z + 0 = z
z + 1 = the element that comes after z
z - 1 = the element that comes before z.
x + y + z = (x+y) + z

(I think that's all you need)

Thus 2 + 2
= 2 + 1 +1 (or 2 + 1*2)
=3 + 1
=4 + 0
=4

Or written in plain English, 2 + 2 means take the counting object "2" and move forward "2" counts in the sequence. The real trick is that the counting object represents a position in the counting sequence as well as the concept of of distance between the counting objects.

This touches on the issue I mentioned previously. The axioms were chosen such that the math works for a certain kind of problem. The average person probably doesn't appreciate the axioms, but the mathematicians do, and the simplest theorems about numbers definitely rely on axioms, since what other property do they have to work with. You could argue that any counting object scheme would basically look the same as what we have, but I've heard just enough about field theory and abstract algebra to think that creating a totally different scheme could be possible, but I can't give an example.

Not for me; I actually find the experienced universe easier to conceive.

Imagine that there are two people, X and Y.

When X says "2 + 2 = 4", what he means is "the string of symbols '2 + 2 = 4' is a consequence of the axioms laid out by stands2reason". If X is asked to prove that "2 + 2 = 4", he will do this:


When Y says, "2 + 2 = 4", she means something different. She thinks that '2' and '4' are actually just code for 's(s(0))' and 's(s(s(s(0))))', and she has a set of axioms (called the axioms of Peano Arithmetic) which talk about a single object 0 and a single function s (the "successor function"). For Y, the symbol '+' is defined inductively. If Y is asked to prove that "2 + 2 = 4", she will do this:


So they are talking about entirely different things. Or are they? They never seem to actually disagree about the facts about nonnegative integers. Both of the axiom sets get them to the same place. So I would prefer to say that both axiom sets are models of the facts about nonnegative integers. In other words, the facts about nonnegative integers are something "out there" which axiom sets can either capture or fail to capture.

I'd say that a universe where 2 + 2 = 5 is not possible. Math is always the same. It can't be different.

In math there is a two element field called Z[sub]2[/sub] is a two element field where 1+1=0 but I only had this beifly disscussed in an upper division linear algebra course. I am not sure if this makes sense to people but it is not true for real numbers.

That's just an example of symbols having multiple uses. The symbol '0' can be used for any type of additive identity, and the symbol '1' can be used for any type of multiplicative identity. So it's a bit misleading to say "1 + 1 = 0". Really you should say "1* +* 1* = 0*, if you see what I mean.