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There are many ways to conceive of a universe where physical constants are different to our own; however, what would the universe be like if mathematics itself were different?
For instance, what would a universe where every number was either a square or a prime look like?
You imply that mathematics is some fundamental root of the entire universe. While there are many who agree, this is not a fully agreed upon assumption. Many would say what you describe just any other world, only with different math, that looks feels and tastes just like this one, because mathematics is nothing more than a human construct. (as SJuan76 says in the comments)
For those who do believe the fundamental root of the universe is mathematics, this question would have more meaning.
First off, you can declare a mathematical system to do anything you want. It's not hard. However, most mathematicians value consistency, which is a precise term meaning that I can never prove something to be both true and false at the same time. For example, I can declare "6 is a prime number" and "6/2=3" and still call it math, but it's not consistent because the former proves that the latter should be false. It's a murky world if you ditch consistency, so let's presume we want to stay consistent.
What's really interesting about mathematics is that they keep digging deeper and deeper. Primality and squareness are built off of the laws of arithmetic, so you're suggesting that the laws of arithmetic change. However, the laws of arithmetic are, themselves, built on a deeper set of axioms called the Peano axioms. Have you ever joked about having to prove that "1+1=2?" If you have to prove it, the Peano axioms are the things you assume:
And then we define a function S(n), which is the "successor" function. This is basically adding one, but we can't use that terminology because addition isn't defined yet:
And finally we add an inductive axiom, which is hard to write out in technically correct notation, but it's the basic mathematical induction we had in school: if you can prove f(0) is true, and prove f(n+1) is true given f(n), then f is true for all natural numbers.
That's at the bottom of the concepts of arithmetic. When you talk about changing things, it has to be done at a layer where we don't even know 1=1 until we prove it. We actually use these axioms to define addition in another layer:
a + 0 = a
a + S(b) = S(a + b)
It goes on and on, but this is a good example of how far down the rabbit hole you have to go before discussing the concept of changing mathematics even starts to be meaningful. Any change done at a higher layer will either be inconsistent, or simply defining a new concept meaninglessly.
There are mathematicians who have explored numeric systems that don't follow these axioms. Dan Willard is my favorite. Instead of building a world up using addition and subtraction, he starts with subtraction and division. This sounds like its a minor detail, but its actually enough to completely change the nature of his systems. For example, it is possible to construct a countable infinity in one of his systems, and the create a sub-system within it which can prove that countable infinity is actually uncountable (which would be paradoxical and inconsistent if it were not for the quirks that were set up by starting with subtraction and division).
So, in summary, go for it. Go make a world with different math. However, you'll find the rabbit hole for math goes quite deep. You will learn something along the way though. I know I have, and I continue to learn something new each time I travel down it.
In a way, mathematics is the science of turning axioms into theorems using logic (which it itself based on axioms).
A famous example is Euclidean Geometry. You start with four axioms and add the parallel axiom as a fifth, and you get one kind of mathematics. Without the parallel axiom, you get a non-Euclidean geometry.
The same could be done with the Peano axioms of arithmetic. You change something, you add something, and you get a different kind of mathematics.
Imagine an arithmetic that is much like ours except that 13 = 1. That means I'm adding numbers like the dial of a clock, after 12 o'clock comes 1 o'clock again. You can do plenty of useful theorems in that kind of arithmetic, but you can't balance your checkbook.
So I guess what you are really asking is something like this:
Could there be an universe where other mathematical axioms are a better fit to reality?
mathematics is the science of turning axioms into theorems and "A mathematician is a device for turning coffee into theorems"
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This is something I have pondered since I was very young (possibly 8 or so)... and my main conclusion is that it all depends on the number system(s) used and the level of intellectual evolution attained by the race(s) involved.
As humans, we have 10 digits... or put in more correct terms our number system is decimal (base 10, having 10 distinct numeric value representations 0-9).
Why? Well, we have 10 fingers. it makes sense that all throughout our planet, every culture has some sort of base 10 number system because all(in general) humans by default have 10 fingers.
Imagine a race with only say 2 fingers... well, they'd likely have a binary number system or some derivative thereof. Same as our computers and electronics. This also changes the way one might think about maths, the rules and mechanisms for performing operations differ. some things are much easier to do in binary than in decimal - like for instance multiplying by 2. to multiply by 2 in binary, you simply add a zero to the end of the line (effectively shifting all the numbers left by one digit). likewise to divide by two you simply shift the numbers to the right. ...and if you have any power n of 2 then we simply shift the digits n times. ie: to get the number multiplied by 8, simply shift the number left 3 times (2 ^ 3 = 8)
in binary this looks as follows:
each column represents a power of 2 with the lowest power (0) on the right
1: 00000001 (1 x 2^0)
2: 00000010 (1 x 2^1 + 0 x 2^0) drop the second term to simplify to (1x2^1)
3: 00000011 (1 x 2^1 + 1 x 2^0)
4: 00000100 (1 x 2^2)
5: 00000101 (1 x 2^2 + 1 x 2^0)
...
8: 00010000 (1 x 2^3)
However, in a decimal system the same holds true for powers of 10... to multiply by 10^n, simply shift the numbers left by n digits and add n trailing 0's.
For example:
123 * 10^3 = 123 * 1000 = 123000
Where:
10^0 = 1
10^1 = 10
10^2 = 100
10^3 = 1000
...
There are an infinite number of number systems... In computing we use: base 16 (Hexadecimal), base 8 (Octal), base 10 (decimal), base 2 (binary)...
It could also be further complicated By including the number of appendages and the division of digits among those too. You could have numerical systems based on some odd, nonsensical (to us) sequence of values. for instance, beings may have devised some number system that uses/combines properties from the octal, binary and hexadecimal number systems...
Imagine an octopus with two "fingers" at the end of each tentacle... it could count in binary on each tentacle (base 2 - 2 fingers), it could count in octal if counting each limb (base 8 - 8 tentacles)... or if combining both tentacles and fingers, it could count in some hexadecimal fashion (base 16 - 8x2 digits = 16).
Multiple, symbiotic, sentient species living together... well, it's likely that they will find a common number system with which to interact. for instance, beings with 8 digits interacting with beings having 4 digits will likely work in a number system somehow common to both (in this case, most likely base 8 - Octal).
Prime-base number system intersections...
what about beings with say 7 digits living with beings having 13 digits...
The first common number system that intersects across both is probably going to be base (7x13) = base 91. Unlikely, but who are we to say.
you will also see that in nature, there are many references to the fibonacci sequence (simply google "sequences in nature" to see what I mean). This may be true on earth, but may be completely different on some other planet.
We also have an abundance of plants and creatures that adhere to specific numbers and structures which are purely mathematical... this goes as far as crystalline structures of molecules and elements.
All these things play a role in what one might use to create a base number system.
for instance, if everything around us had six branches, or six sides or six leaves or six legs... then it's quite likely that a hex number system would evolve despite our having 10 digits. OR... rather there is a likely chance that the number 6 or the hex number system would play an important role despite our having a decimal number system.
Programmers and scientists work in different number systems. Cryptographers and information specialists will think in other, higher base number systems - the reasons for which are because they are more condensed, so you can represent more data in a single digit than you can in a decimal or binary system.
How to see densification of numeric representation... simply open a calculator app and put it into programmer mode, enter some number and switch between the different base representations. The lower the base, the longer the number representation has to be... the higher the base, the shorter the number representation (but the more complex the number system and number of symbols).
Quantum computing This is another area where higher order number systems COULD used, but it gets far more complex because the cubits could in fact represent multiple values simultaneously. There is no reason why a race couldn't have evolved to such a degree that their number system could be represented in a similar manner - We have. We might only be at the initial stage of such an evolution, but we are at the point where the number systems we use are no longer constrained by the prehistoric notions of counting in factors of 10.
Advancements in math if you really want to see what we have evolved our math to... simply look at the the number (-1/12) ... it's derived from a mathematical summation of all the natural numbers Sum of all natural numbers
1 + 2 + 3 + 4 + 5... = -1/12 (negative one-twelfth)
What about imaginary numbers and other such formulated concepts which make impossible formulae very possible and elegant to solve.
Using math to describe everything
I believe that to some degree, we can. the limitations are in how much detail you want to portray and across how much variation and scope etc. DNA is essentially a really well constructed mathematical function that permeates through some pseudo-random modifications in each iteration (hence evolution)..
You could for instance use a mathematical formula to describe any set of data to some degree of accuracy - a fourier transform could be used to describe a drawing of homer simpson for instance Homer Simpson described by Fourier Transform function
Conclusion My advice is think about this from all angles... it might even be influenced by factors such as how many eyes does the creature have, how their brains work and think, sleep cycles, moons around the planet... It's simply a question of how the perceive the world they live in (or LIVED in) and how they represent that and everything else.
What is "different mathematics"? You can consider different axiomatic systems, but that's not different mathematics; indeed, some mathematicians are doing exactly this. You may even consider modifying the rules of logical reasoning and seeing what happens then — again, there are mathematicians who are doing that. In short: Whatever you think of changing, you will find that as long as it is consistent, is will still be mathematics as we know it, in the most extreme case a new branch of mathematics, but still not a different mathematics.
Why is that so? Well, that's the case because mathematics is basically the science that explores the logical connections between abstract ideas. Therefore any logically consistent set of abstract ideas is included in mathematics.
So, how is this related to the real world? Strictly speaking, not at all. There are tons of mathematical constructs which are completely unrelated to our world. You'll be hard pressed to find the first uncountable ordinal somewhere in our universe (except as abstract concept in mathematics books or articles). However, if we can perceive any part of the world, we can abstract it, because fundamentally the act of perception is an act of abstraction (you don't really see the world, you see a reconstruction of the world based on the input of your sensory organs and a set of assumptions built into your brain). If we can perceive something, we can speak about it. And if we can speak about it, we can apply logic to it. And thus we can do mathematics on it.
There are certain mathematical models which fairly well match out observed world; the science that develops mathematical models that fit our world is known as physics. Now the specific mathematical models tell us something about our world, but the fact that we can create mathematical models of the world does not. Whatever the world is, as long as we can perceive (or imagine) it and talk about it, we can create mathematical models to describe it. We may need to invent new mathematical constructs to do so. But we don't need a new mathematics to do so.
I have to disagree with those who say that math is a "human construct" and so any set of math rules is possible. It is true that math is pretty much built by taking starting axioms -- assumptions -- and then investigating what you can prove by applying rigorous logic to those assumptions. So if you started with different assumptions, you'd get widely different conclusions. But most of the math that non-mathematicians are familiar with is based on axioms that are rooted in real-world observations.
If you consider very simple math, like arithmetic, a universe where it is different is very hard to even imagine. Like in a universe where 2+2=5, that means that if I put two rocks in an empty box, and then add two more rocks, then I look in the box, there are now five rocks. How many are left if I take one out? What if I put them in one at a time? It's just very difficult to conceive of how this could hang together.
As the math gets more complex, the problems may become less obvious, but they don't go away. Like suppose you say that in your hypothetical universe the surface area of a sphere is 4 * π * r^3 instead of 4 * π * r^2. At first glance that may sound plausible. But somebody didn't just make up that formula one day because it sounded cool. Look at methods for deriving that formula, and it's not easy to see how you could change underlying assumptions to come up with a different formula, and still have those assumptions be mutually consistent and make sense.
Mathematicians regularly find that math all connects together. Prime numbers turn out to be related to powers of 2. The square root of -1 turns out to be related to sines and cosines. Etc. If in an alternate universe 4 is a prime number, all sorts of things would have to be different for that to make sense, and all those differences would have to connect together.
Yes, there are aspects of mathematics that are purely abstract, and perhaps you could imagine those being different. But the "routine" stuff, like arithmetic and trigonometry and Cartesian geometry ... imagining an alternative system that is internally consistent is very difficult.
The universe would look exactly like a giant frosted donut, but it would also be square, and the passage of time would be experienced as the simultaneous existence and non-existence of birds. Other than that, everything would be exactly the same as it is now.
Even the reasoning we would use to answer a question like "How would change X affect the universe?" is subject to change in this scenario, because it is based on fundamental principles of logic and cause-and-effect, which are closely tied our current mathematical system and therefore can no longer be assumed to apply.
I think simulationism can help us make sense of the possibility of a different mathematics. Remember the Pentium floating point division bug? A processor simulating a universe could have a similar problem. But really it wouldn't exactly be that math was different. The inhabitants of the universe might construct a mathematics exactly like our own, it's just that the universe wouldn't perfectly conform to mathematical patterns. At a certain level of sophistication the inhabitants of the universe might begin discovering small anomalies, measured values different very slightly from expected values derived from otherwise very well-confirmed theories. Maybe there could be even more severe manifestations of the mathematical errors, like an inability to ever collect a particular number of particles in a single structure. (an atom with X protons & electrons and Y neutrons might be predicted to be stable by the best atomic theories, but discovered to actually decay after an immeasurably small interval in experiments) The bug might also be observable via occasional, repeatable conservation law violations.
(The thought of this possibility actually delights me personally, because it would be such an incredible nightmare for reductionism.)
My opinion is that there can't be another mathematics. Maybe somewhere in the universe or in another universe you will find another kind of hydrogen, helium, f. ex metallic carbon, gaseous iron, in low temperature, etc. but you will never find another kind of mathematics. Even god can't make 2+2=5
Mathematics is God of the Universe, there is no god above mathematics, because nobody can rule mathematics but mathematics rule everybody.
