I recently watched The Story of Maths on Netflix and it does a great job of explaining not only the history of mathematics but also how and why the discoveries came about - very interesting!
If you’re curious about zero, I think you’d enjoy it.
Fire existed before sentient beings ever saw it. They invented ways of creating and using it. After seeing it. But something such as the number 0, in particular. Seems an invention. The reality of it was there all along. But it was not used in methods of counting for some time. When mathematics progressed, the need for it increased. And the concept of actually reserving and using a spot for nothing, seems to me, was an invention.
Well, there’s Bleem, the integer between three and four.
There are also non-standard numbers. The set of different ones has an order of infinity larger than the set of real numbers.
Sure there are new numbers. Consider the Nemo Number System, which was just discovered today. It consists of all of the numbers included in other number systems; real numbers, complex numbers, hyperreal numbers, surreal numbers, superreal numbers, etc. But each number has an additional set of Nemo values which differentiates it into a set of distinct numbers. So 1 = 1 in the more traditional systems but both of those ones may have a different Nemo value which makes them distinct.
I’ll admit the practical applications of this system are currently limited.
Could not edit my last post.
We may discover a number that represents a relationship of some complex thing or things. But we may have invented that way of revealing it, the path, the need. All numbers exist. It is the nature of numbers. But we invent the ways to discover them. We invent their place in the universe of numbers.
I have this book. I’m not going to Google for the claim; it might have been from this book itself. In that book you will find ridiculously complicated one-point endgames.
Realize that that entire book is all about the very last point in a game of Go! If you lose to a master by 29 points, applying that book perfectly will reduce your loss to just 28 points! (The surreal numbers are more general, but mapping them to Go is non-trivial 
— however the one-point end-game moves can be mapped.)
Numbers are not invented so much as they’re second-order philosophical abstractions which have been progressively realized over tens of thousands of years. Indeed, tally sticks go back to the upper paleolithic. There’s no reason to assume we currently know all we will ever know about numbers.
NETA: No, you can’t win a one-point position in one corner, and then win a one-point position in another corner for a two-point total gain! In the one-point ending phase of Go, each player makes moves worth a point — a point for me, a point for you, a point for me, a point for you, etc… The “trick” (and indeed the whole focus of that entire book) is to get the last point; that is, to choose the one-point move which leaves, regardless of opponent’s counter-strategy, an even number of one-point moves remaining on the board.
My Master’s thesis was on the development of a Go-playing program. That was over a quarter of a century ago (when the best computer players were no better than an intelligent human novice).
The reason for Go’s long-standing resistance to computer players is largely down to the high branching factor (the number of moves available to each player at each turn). This consideration becomes less and less important as the game develops – when it comes down to an end-game with a relatively small number of possible moves, computers have long been able to out-perform human-beings.
In the ancient world, zero wasn’t a number. You couldn’t have zero of something. If you had something, you had some number of it. If you didn’t have any, then you didn’t have any number of it. So they understood the concept of not having something, obviously, but the concept of a number zero wasn’t there yet.
So joking or not, if you had something, you had at least one of it. There was no zero of something. So your statement is kinda true, actually. 
The Babylonians kinda sorta had a zero, by leaving an empty space where we would put a zero these days. But it was kinda halfway between not having anything to represent zero and representing zero as a number.
This was actually a philosophical issue in ancient Greece. There were a lot of Greeks who said that nothing couldn’t be something. On the other hand, it was the Greeks who started using a symbol as a place holder and not just leaving an empty space like the Babylonians did. The Indians (the Asian kind) took it to the next step and treated zero as a number.
The concept of having none appears to be hard-wired into our brains, as does the concept of having one of something, and two of something. I can’t remember if three is hard wired in our brains or not, but the counting capabilities hard wired into our brains ends somewhere around there. After that, our brains recognize the concept of “a few” and “many”. So even the simplest human languages typically have unique words for none, one, two, (maybe three), a few, and many.
So the OP does have a point that these sorts of things have to be “invented” (or discovered or realized or whatever word you want to use for it). Sure, having seven of something always existed, but being able to describe seven isn’t a natural concept for our brain. That’s just somewhere in the “few” range.
And treating nothing of something as a number instead of just recognizing it as a lack of something is also a concept that is not natural to our brains, so that had to be “invented” as well.
Nm
I’m not doubting you, but this seems surprising to me. In a typical master game of, say 210 moves, the branching factor decreases from 361 moves at the start of the game to 151 moves at the end, assuming no captures and ignoring the fact that a few moves may be illegal (suicide, ko, etc). While this is a fairly big decrease, the branching factor at the end is still several times higher than the branching factor of a typical chess game at any stage of the game. So I would think the go endgame would be pretty hard to handle with just a brute force tree search, even using alpha-beta pruning and all the other tricks. Furthermore, my understanding is that good evaluation functions are harder (or more computationally expensive) in go than in chess. It seems to me that there must be something else in play that makes the go endgame more tractable than just the modest decrease in branching factor.
–Mark
It’s worse than that. To ancient Greek mathematicians, ONE wasn’t a number either. One is the “unit”. “Number” implies a plurality of things. So, many Greek proofs in the original form seem very cumbersome today, since they essentially consist of two separate proofs, the first for the case of “one”, and another for the case of “more than one”. (Note this is not referring to mathematical induction, it’s just that when they say something like “take a number X”, the only valid values for X are 2 and higher.)
–Mark
You people have got to read Zero: The Biography of a Dangerous Idea.
The thing about zero is that it wasn’t “discovered” in the sense that someone found it. The change of thinking to regard zero as a number on an equal footing as the others is much deeper. It extended the mathematics possible, and arguably created a new set of “numbers”. Numbers that had new properties.
For instance zero is the additive identity. Before zero it was not possible to add a number to another and get the first number as the answer. With zero that was now possible.
Allowing negative numbers is much the same. You could argue that once you let -1 be a number, all the other negative numbers become available, and now you have a new set of numbers with even more interesting and useful properties. You can now subtract a bigger number from a smaller number and get a number. Before that wasn’t possible either. Critically these numbers are now “closed” under addition and subtraction. You can perform addition and subtraction with any of these numbers and still get one of these numbers. We call them the Integers.
And so it goes. Allow taking the square root of minus one - we invent a new number:*** i *** which begets the complex numbers. Quaterions and Octinions follow as we allow the other possible forms of square root of minus one.
Real numbers are a bit more messy - we didn’t invent a new number - we invented a way of corralling them, so that we could heard them around and generally make them do our bidding. (Although perhaps you could argue that the “infinitesimal” was the “number” that allowed this to be done.)
I never heard of that before. I like that concept. What is, is. Start counting after that. Quite Zen.
Answering that question would be self-defeating, since the post revealing that number would become a google hit.
My understanding is that we only know the first 6 perfect numbers, there are still more to be discovered. A perfect number is equal to the sum of it’s factors. The 1st is 6 (1+2+3), the 2nd is 28 (1+2+4+7+14) and so on.
There’s a few more known List of Mersenne primes and perfect numbers - Wikipedia, and possibly infinitely many (if there are infinitely many Mersenne primes).