At the very foundation of modern mathematics you will find a belief in a mysterious realm that lies outside this physical world. Inside this realm all numbers from all number systems inherently exist. This abstract world has always existed and always will. And without this imagined place it is claimed we would not even be able to prove that numbers exist.
But how could this strange state of affairs have occurred? Maybe it was the desire to prove that numbers ‘exist’ rather than can be constructed that led down this extraordinary path. Should we really accept that saying numbers exist in an abstraction is somehow a valid reason to bypass the need for any physical existence?
An idea has a physical presence in our mind (via the chemistry of the brain) or where it is documented (e.g. via the ink it is written in). In this respect, abstract ideas need a physical manifestation in order to exist. Ideas might include numbers, the rules of how to create more numbers, or even the rules of a how objects behave in an abstract system.
If I imagine some cats controlling a spacecraft and landing on Mars then this idea can be said to exist in my mind. But I cannot conclude that space cats must exist on Mars simply because I can imagine it. Instead of Mars I could use an imaginary planet and call it an abstract thought, but this still does not prove that space cats or the imaginary planet actually exist themselves. Similarly, being able to think about numbers does not prove that numbers themselves exist in an abstract world. To say something can exist without any physical presence at all is a belief in the supernatural, and this is exactly what mainstream mathematics is based upon!
If I write down a big number that has never been written before, I have simply constructed it for my own use. I have not brought it into existence for everyone else to use, and I have not fetched it out of a mysterious realm containing all numbers. But mainstream mathematicians disagree. Rather than talk in terms of construction, they prefer to talk about existence as they insist all numbers already exist.
This belief (called mathematical platonism) is that with absolutely no physical manifestation at all, an infinite number of numbers (or possibly many infinities) just ‘exist’ in an abstract realm that has existed since the start of time.
Now let’s see how this concept is used. Consider this simple fundamental proof…
Given any two different real numbers a0 and b we can construct another real number halfway between them. If we call this new number a1 then we can show how a new number (a2) can be created halfway between a0 and a1.
This is claimed to inductively show there are an infinite number of reals between a0 and b.
To me, this is just two statements of fact that prove we can construct more numbers endlessly. But most mathematicians see this as two statements of fact that apply to all the numbers between a0 and b. And since all the numbers already ‘exist’ (in an eternal abstraction), then it must apply to an infinite number of numbers.
Importantly, endless and infinite are not the same thing. Infinite implies you can have an infinite number of something whereas endless implies an open-ended process where you can never reach an end point, and so you can never have an infinite number of anything.
The word ‘abstract’ has been used to supposedly justify numbers can inherently exist in a realm outside of reality. And once this strange idea has been accepted, the notion that infinity must also exist is a very small step indeed.
Mathematics would be a more staid and boring subject without all the weird and wondrous things that infinity brings with it. But do we really want mystic beliefs at the heart of a discipline that is traditionally associated with logic and rigour?