If math "governs" the universe, as mathematical physicists sometimes claim, why is there so much more math than universe?
It seems wasteful.
The earliest evidence we have of counting things--which is the core of mathematics--is from the Sumerians and was used to count goods: sheep, grain, ingots of low-quality copper, and so on.
"Counting tokens" were simple, small, clay shapes--round, tetrahedral, square, all about the size of marbles--that had a one-to-one correspondence with the things being counted, and the shape indicated the kind of good involved.
Collections of tokens were often kept in "envelopes" made of a folded sheet of clay, and the tokens inside it were impressed into the clay surface before it was sealed, so the contents could be verified without breaking the envelope. This was the very first instance of writing, from which so much else follows. Later on, fancier tokens were marked with dots or lines to indicate number while the shape indicated what was being counted.
The nineteenth century German mathematician Leopold Kronecker said, "God made the integers, all else is the work of man", but as nearly as we can tell, humans made the integers too. They were useful.
And in fairness to Kronecker, Sumerian archaeology didn't get under the ground for a good half-century after he spoke.
Once we had the integers, we found it useful to define operations on them that reflected the things we did in the real world: addition, subtraction, multiplication, and division.
This created problems.
Addition is easy: if you've got ten sheep and I've got five sheep and we run them together in the sheepfold of Uruk then we have fifteen sheep, and if we put our counting tokens together we have fifteen of them.
Multiplication starts off easy: if we repeat the process above, adding the same number of sheep each time then we end up with the number of sheep times the number of times we added them. Multiplication is repeated addition.
Subtraction is where it gets weird. The operations of addition and subtraction are commutative, which means the order of the numbers doesn't matter: if you throw ten tokens into a basket and I then throw five tokens in, there are the same number of tokens in the basket as there would be if I went first and then you did: 5+10=10+5.
Likewise, adding ten sheep three times gives the same number as adding three sheep ten times: 3*10 = 10*3
Multiplication is also distributive over addition: if you add ten sheep three times and then five sheep three times you end up with 45 sheep (3*10+3*5), which is the same number you get when you add fifteen sheep three times (3*(10+5)). "Distributing" the operation of multiplication-by-three across the sum doesn't change the result. This is just a reflection of the way things like sheep actually behave in the world.
Subtraction messes this up, and what's worse, it gives us numbers that don't seem to correspond to anything.
Consider: if I have fifteen sheep and take five away, I get ten sheep: 15-10. But what about the reverse process: 10-15? If I have 10 sheep and take fifteen away... does this even mean anything?
Sumerian mathematicians could have stopped there and called it a day. Maybe they did. For all we know there could have been some centuries-long prohibition on "unnatural subtraction" until some Babylonian Galileo insisted that operations like 10-15 could have useful and meaningful results... probably to do with money, or possibly distance.
Because while "negative five sheep" doesn't make a lot of sense, the idea of direction does, and so does the idea of "Ea-nasir owes me five ingots of fine-quality copper" which has a different meaning than "I owe Ea-nasir five ingots of fine-quality copper." The quantity is the same. The sign encodes the direction of the relationship.
So as soon as one moves away from counting things--which have an absolute zero-point--to counting measures that are relative to a nominal zero-point, like the balance of an account or the distance along a road from a particular spot, then negative numbers become meaningful again.
Which is great, until we want to multiply them.
A negative number multiplied by a positive number is pretty clearly meaningful: if negative three means you owe me three sheep, then three times that means you owe me nine sheep. "Multiplication is repeated addition" implies that "subtraction is adding a negative number", so -9 = (-3) + (-3) + (-3) = 3*(-3)
But what could it mean to "negatively add" a negative number? That is, what does -3*-3 equal?
The simplest way of seeing the answer is to consider the distributive property of multiplication. -3*1 = -3 according to our rule for multiplying a positive number by a negative one. But we can re-write 1 to be the difference between two numbers, like 2-1=2+(-1), say. In that case: -3*1 = -3*(2 +(-1)) = -3*2 + -3*(-1) = -6 + -3*(-1) = -3, which means -3*(-1) must be +3.
Two negative numbers multiplied by each other results in a positive number because that's the only way multiplication can remain distributive over addition, and that's the way the things in the world we want to describe with multiplication work.
Could we define it differently? Probably, but we'd get math that didn't describe the world we live in, which might be entertaining but wouldn't have much use beyond that.
Division also creates a problem. If multiplication is repeated addition, then division, its opposite, is repeated subtraction, but sometimes it leaves things over. If we have fifteen sheep and want to divide them into three groups, we can after a little experimentation figure out that if we take away five, and then another five, we end up with three groups of five.
But what if we want them divided into four groups? There's no way to do that without part of a sheep left over, which is a problem, especially if you're a sheep.
For a few thousand years mathematicians dealt with these cases using fractional numbers, which were just the bit of the division problem that got left over: 15/4 = 3 3/4. These fractions were called the "rational" numbers because they were ratios of integers. Rational numbers were an accounting mechanism that allowed people to keep track of fractional sheep without actually cutting them up.
Then the Greeks came to realize--and were able to prove--that there were numbers, like the square root of two and the ratio of a circle's circumference to its diameter (tau/2), that couldn't be written as a ratio. These "irrational" numbers apparently had no connection to the integers at all, even though they came up in everyday geometric contexts.
Today we call these the "real numbers" and use them to measure everything, confident in the knowledge that whatever value something has, from the mass of the electron to the distance to the Andromeda galaxy, there will be a real number that corresponds to it.
At this point, it sounds like we have just as much math as we need... but wait, there's more!
Negative, rational, and real numbers were invented as a result of operations on integers or simple geometric shape that produced results which couldn't be expressed within the number system that produced them.
Complex numbers--which involve the square root of negative one--were invented to deal with equations between real numbers that required something that represented the square root of negative one, which is not a real number.
Complex numbers are two-dimensional, being made of both an ordinary real number and another real number times the "imaginary" square root of negative one. They turned out to be so useful in physics that 19th century British physicist William Hamilton invented quaternions as four-dimensional generalization of them, and around the same time mathematicians Arthur Cayley and Leonard Dickson came up with a scheme that would take any number system and build a new one that had twice the dimension of the previous one.
The reals are one-dimensional, complex numbers are two dimensional, quaternions are four-dimensional, and above that there are octonions (eight-dimensional) and sedenions (16-dimensional) and on to the nameless heights of 32 dimensions and beyond.
All of these are "numbers" in that they have quantity and are subject to operations much like addition and subtraction and multiplication and division, although the specific behaviours of those things get weirder and weirder as you go up the tree. Complex numbers don't have an ordering like the reals. Quaternions multiply non-commutatively, so A*B =/= B*A. Octonians are non-associative, so A(BC) =/= (AB)C. Sedenion arithmetic is so strange I don't even have the vocabulary to describe it.
None of these number systems beyond the quanterions--which are useful representations of real objects in three dimensions--seem to describe anything that exists outside of our heads. If they do, it is a reality that is sufficiently strange it seems more likely to come from the imagination of HP Lovecraft than Albert Einstein.
And there are a literally infinite number of numbering systems that can be constructed, without even going into even more peculiar things like the surreal numbers which cover the reals but also include numbers that are smaller than the smallest real number and larger than the largest real number, which are hard ideas to get your head around because there is no smallest or largest real number, at least not in the sense that "smallest" and "largest" are usually meant.
So much math, so little reality.
As I said at the beginning, mathematical physicists sometimes claim that math "governs" the universe, or "god speaks in mathematical language" or similar ideas. If that's really the case, one has to wonder what god is saying with an infinite series of number systems that correspond to nothing in reality as we know it.
Perhaps god sees more than we do.
Or maybe all of this really is the work of humans, and just as we took a system of vocal signalling that was useful for coordinating large troops of social primates and turned it into language, which can express any idea at all, including how mimsey the borogoves are, mathematics too has its nonsense verse, capable of bringing pleasure and delight, and needing no more justification than that.
Do I understand this correctly, that you have disconnected science from the necessity of mathematics? That there could be things in our universe that are not well described by the mathematics that we use and need another system of representation? That some part of this system that our mathematics doesn't work well for might be included in what could broadly be termed 'magic'?
After reading this I can just about believe that mathematics is “capable of bringing pleasure and delight “! Thanks!