Something from Nothing
I've always been interested in the nature of the universe, and the ultimate question of "Why are we here?".
Invoking a god doesn't seem compelling, as it only pushes the question back to "Why do god(s) exist?". If a god can exist without a creator, then why can't the universe exist without a creator? Note, that doesn't imply that gods don't exist, it just doesn't answer the question of how or why there is something rather than nothing. The "something" is a god for theists, and the laws of physics for non-theists.
I really enjoy discussing these issues with clever people (especially theists), and was fortunate to meet a christian while on jury service who was very engaging. I struggled to put my point across, but it did give me a chance to firm up some of my ideas.
Were Circles Invented?
It sounds like a stupid question, but I don't mean a shape drawn on a piece of paper, I mean the mathematical idea of a circle. A circle drawn on paper is only an approximation of a circle. After all, a drawing is made from carbon atoms, so even if you could place all the carbon atoms in exactly to right places, you won't have a perfect circle (where are the points between two adjacent atoms?).
If we write :
x² + y² = 1
this defines a perfect circle, but I could have written down a different equation, and created a different shape, for example
2x² + y² = 1
defines an ellipse.
I didn't get out more than I put in. I wrote down a simple equation, and got out a simple shape (a circle). By writing down a slightly more complex equation, I got out a more complex shape. So at this point, I can't say that I discovered something that pre-existed.
Were numbers invented or discovered?
Millennia ago, the only numbers that humans had where whole positive integers (1, 2, 3, ...). They are great for counting cattle and other physical objects. Then came fractions (a half, two thirds etc). Great for cutting cakes and diving land. Zero is cropped up some time (I'm not sure of the chronology). Negative numbers are useful for recording debts, and at this point we have a rich set of numbers for most practical uses, but there are other numbers too (e.g. irrational numbers such as Pi).
If we go backwards, and start off with just the number 1, and addition, then we can generate all the positive integers (2 = 1+1, 3 = 1 + 1 + 1 etc). Now take the mental leap to ask "What is the opposite of addition?". Take 1 + 1 = 2. I'll now put a question mark in place of the 2. Giving :
1 + 1 = ?.
Finding the "opposite" of addition, is like knowing the answer 2, but not knowing one of the other numbers. i.e. :
1 + ? = 2
What do we have to add to 1 in order to get 2. This may seem slow and obvious, but using this simple concept, I'll soon introduce you to "new" numbers, that were beyond the greatest mathematicians until the 16th century).
We started with only 1 and addition, giving rise to all positive integers, but with subtraction, we now have zero : 1 - 1 = 0 We have already created/discovered a new number that eluded the great minds of ancient civilisations. We also create/discover negative numbers (e.g. 1 - 3 = -2)
Multiplication is just repeated addition, so I argue that once we have addition, multiplication is inevitable. It isn't something truly new.
2 x 3 = 6
can be replaced with a series of additions :
2 + 2 + 2 = 6
What's the opposite of multiplication? e.g. 2 x ? = 6 Obviously we call this division, and this gives rise to another set of numbers, fractions. e.g.
2 ÷ 4 = ½
Using multiplication, we can define squaring.
3 x 3 = 3² = 9
We then do ask the same question : What is its opposite i.e. ?² = 9. We call this the square root and is written √9 = 3.
The square root is strange, because it has two solutions. √9 = 3 (because 3² = 3 x 3 = 9) and also √9 = -3 (because -3² = -3 x -3 = 9). What about √-9. The answer isn't 3 or -3, and for millennia this was considered illegal. i.e. there is no answer.
When we only had positive integers, and invented/discovered subtraction, we could have said that 2 - 3 was illegal, because there is no positive whole number that fits the bill. We didn't though, and instead we invented/discovered negative numbers.
We can do the same for square roots, instead of insisting the equation is illegal, we say : Hmm, that's weird, none of our existing numbers fit the bill, so there must be another type of number we hadn't thought of before.
So let's consider √-1
We don't have a name for this new number, so let's call it "bob" (after all, names aren't that important, most of the world doesn't use the names "one", "two" or "three" etc. The French call them un, deux trois.
Does that mean we need a new name for √-9 too? Actually no. It turns out that
√-9 = √(-1 x 9) = √-1 x √9 = bob x 3
or just "three bob". We only need one new name.
As you've probably guessed, mathematicians don't call it "bob". They call it "i" which is short for "imaginary". IMHO, this is a horrible name, because "i" is no more imaginary than -1. You can't have -1 sheep in your field, you can't have -1 acres of land etc.
What about 1 + i? Can this be simplified? No, but we do have a special name for such a number, its called a complex number. It is called complex, because it is compose of a real number (one) and an imaginary number (bob or i).
While it is common place to use negative numbers to describe debt (owning a negative amount of money), debt is imaginary, you cannot hold -1 dollars in your hand, you cannot see it, you have to imagine it! It is only due to quality eduction that we find negative numbers useful and common place. If we were all well versed in degree level maths, then imaginary numbers would be as "real" (by which I mean useful, and common place) as negative numbers.
We are used to thinking of the real numbers existing on a line stretching out to infinity with zero in the middle and positive numbers on the right and negative numbers on the left. So where is bob? It turns out that bob is the same distance from zero as 1 and -1. But instead of left and right, bob is one unit upwards. and -bob is one unit downwards.
Numbers are two dimensional. If you limit yourself to thinking of numbers on a 1 dimensional line, then you will fail as soon as you start using square roots.
BTW, Using square roots, we can also discover/invent irrational numbers. For example √2 has an answer which is NOT a fraction. No matter how hard you try, to write it down, you'll need an infinite string of digits.
When will this all end?
Can we perform similar tricks and keep getting ever more types of numbers? AFAIK, the answer is no. We can apply any of our operations (addition, subtraction etc.) to complex numbers, and we get another complex number, and we cannot (AFAIK) generate a new operation from the existing ones which forces us to embrace new types of numbers.
So, there is no need to go to higher dimensions. Yes, you could choose to create a number with three dimensions, but there isn't any mathematical operation which forces you to do so. One dimension is insufficient, but three is superfluous.
So are numbers invented or discovered?
I started off with just the number one and the addition operation, and ended up with whole numbers, zero, negative numbers, fractions, irrational numbers and complex numbers. For example, I didn't invent zero, it naturally occurred when I asked 1 - 1 = ?.
So I feel confident saying that these numbers were discovered, they weren't invented. An intelligent alien race will discover the same numbers (but obviously they will use different names for them, and different ways of writing them down, but the concepts will be the same).
I feel equally confident that if/when the human race dies out, that these numbers will still exist, ready for a new race to discover them. Also, these numbers existed before humans existed.
Numbers in Alternative Universes
Imagine a universe with different rules of physics, with different kinds of particles and different kinds of forces. Will these numbers exist in those (make believe) universes? Given that my arguments for the existence of numbers didn't depend on physics, only maths, I suggest that these numbers are there to be discovered in all possible universes.
I venture one step further : Numbers exist without a universe, they existed before the big bang. BTW, I hate the phrase "before the big bang", because time itself was created as part of the big bang. But that's another story!
Conclusion
If I'm right, then numbers existed before humans, they have always existed, waiting to be discovered by us. They existed before the earth, before the sun, before the milky way. There was no point in time that they sprang into being. In fact, they exist without reference to physical matter.
Were circles invented or discovered? If numbers were discovered, then circles are too, because a circle of radius 1 is defined by all of the complex numbers whose magnitude is 1. Magnitude is another word for distance. So 1, -1 bob and -bob all have a magnitude of 1.
This definition of a circle didn't require a seemingly arbitrary equation (x² + y² = 1). It only required the "magnitude" operation. So, IMHO, circles were discovered, they weren't invented. They don't require reference to physical matter, and don't require an arbitrary equation to define them. I expect any sufficiently intelligent alien life to discover the same shape; it isn't a human invention.
As an aside, the properties of a circle are also discoverable, and not invented. The ratio of its diameter to its circumference is Pi.
We can imagine difference universes with different laws of physics, and yet the circle's existence, and its properties will remain the same. Our universe has many spherical objects (stars and planets), but I could contrive a universe whose planets were different shapes (due to different laws of gravity), but mathematical circles would still exist in cube-shaped planet universes!
My next next question is What Else Exists Beyond the Universe. This question gets me really exited, because my answer is surprisingly "quite a lot". A heck of a lot. Much more than you might imagine!