One of the most fundamental pieces of logic in mathematics, is that if the number n
exists, then its successor n plus 1 must also exists this type of existence is called mathematical Platonism
It seems so obviously true that we don't give it much thought
But it is such a key part of mathematical logic that if this is wrong, then almost all of mathematics is wrong
So, let's give it a bit more thought but instead of natural numbers, let's take n to mean 'written numeral'
As far as we know, only a finite amount of numerals have ever been written
And so there must be at least one written numeral whose successor has not been written yet. In other words
the statement is false if n means written numeral
But we assume it is true if n means 'natural number'
Because we've already assumed that all natural numbers must exist all infinitely many of them
This begs the question what exactly is a natural number and in what sense does it exist?
Questions like these were fiercely contested by the ancient Greek philosophers seeking the source of all knowledge
On one side of the fence, there was one of the founders of Atomism; Democritus
He believed that knowledge comes from the study of physical things.
On the other side of the fence we have the founder of Platonism; Plato himself
He believed that knowledge comes from the study of metaphysical things which are things that don't exist in the physical world
Consider the number 3 in the context of 3 apples on a table
If everything can be explained in terms of physics then so come the number 3
and so can the brain, which might be nothing more than just a biological machine
So perhaps the number 3 can be explained in terms of the functioning of the brain?
Perhaps one part of the brain counts how many apples there, are and the result could be stored somewhere else
And we might keep an note somewhere about what the data refers to
Then when our thinking process (our consciousness if you like) uses this data item it will realise what this three refers to
On the other hand, perhaps the brain allows us to conceive of things in a realm beyond all physics
Maybe the concept of three-ness exists outside brains
Perhaps every group of three things has the property of three-ness
Regardless of whether or not any brains exist to perceive this three-ness
So now let's consider a world where all brains or thinking machines have suddenly stopped working. At first
we'll consider a written numeral that represents three. With no thinking entities to perceive and understand the written symbol
it would just be a meaningless physical collection of atoms
It requires a thinking process to understand the symbol and to perceive the number it represents
But what about the three-ness of the apples on the table?
Again, let's assume there are no thinking entities. Arguably, it would just be a meaningless physical collection of atoms
Because it requires a thinking process to identify the different objects and to describe them as three objects on a table
On the other hand, perhaps the brain is not just a physical machine
limited to physical activities
I mean, what about consciousness and self-awareness?
Could this be evidence that the brain is somehow working with non-physical entities?
And surely if we say we can imagine something that is non-physical then surely it must be true that we can!
Or maybe both of these are mistakes
Perhaps we're just fooling ourselves into believing in non-physical things, after all people believe in all sorts of strange things
If we imagine a creature that doesn't exist in the real world
Then arguably we've just taken bits of knowledge acquired from the physical world and used them to create this new imagined creature
The image exists in the machinery of the brain and everything can be explained in terms of real-world physics
Maybe we shouldn't worry about physical or non-physical things. Maybe all that matters is consistency?
Maybe mathematical logic doesn't have to be valid in the real world. Maybe if it's consistent then it's valid
Yes, if mathematics was all about describing the physical world then it would be a slave to empirical evidence
But if mathematics is allowed to be metaphysical then knowledge will mysteriously reveal itself to us just like magic
So who won the battle for the source of all knowledge? Was it physics or was it metaphysics?
Well in those days everyone had to accept that the gods were infinite had perfect forms and were beyond all physics
And so there was only ever going to be one winner.
So in modern times should we accept this statement as being valid?
Well, almost all of us do and we don't give a second thought to its metaphysical nature
We are so comfortable with metaphysical existence that it underpins all formal logic
If we dared to contemplate that only physical things exist
Then we might conclude that almost all of mathematics is rotten to the core
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