I want to know if a number exists when there is no one around to count it. Furthermore does thinking about a number, conceiving of it, imply that it exists? Or is a number that is not associated to a set of actual objects just a chimera, where we concatenate distinct and separate impressions. Like piecing together a mystical animal out of the body parts of animals that actually exist. So if we say that we believe that a number exists when not associated with an actual object, our statement is equivalent to stating that we believe a unicorn is real simply because you can conceive of it?

If that is the case, well then I suppose that would imply that numbers are more than just linguistic, symbolic structures. It would also mean that we live in a world inhabited by unicorns and fairies.

I’m no mathematician, but I look around me and something tells me that the implications of our previous assertions don’t appear to have much bearing on reality.

To be honest, I have never understood mathematics. I can use a trivial amount of it, but I hardly understand it. In truth I think that some of the fundamental axioms that underpin mathematics smack of the meta-physical and the mystical. But I assume there is just an awful lot I don’t understand. So humour me while I try and figure out the basics on my own here:

If we enumerate 1 and then 1 then we get 2, correct? But what is that 2 supposed to be representing, two real world objects or two numbers? Are the symbols referring to two chickens or are the numbers referring to themselves? The later proposition doesn’t mean anything as far as I can tell, because 1 apple explains something. We have placed on that apple the attribute of being 1 of something generally described as an apple. But 1 number 1, explains nothing; how could a number explain anything when it is just referring to itself.

So unless a number is an extension of an object then it is basically a self-referential statement, which has no value beyond it being a linguistic construction. Meaning the purely abstract number 1 is essentially an empty set ({}) with the potential to hold and be attributed to 1 object, 111 to one hundred and eleven objects and so on.

I feel like I am probably mixing up different fields of mathematics, but I don’t see a problem as of yet. Sets are groups or numbers which represent objects, so that {Apple1,Apple2,Apple3} represents the number 3 or the statement, ‘I count three apples’. That seems to make sense, but what if we start filling sets with purely abstract numbers. Numbers we have already decided represent empty sets when not attributed to and thus holding instances of objects, so that: The number 3 can be represented by { {},{},{} }, which is a set of three potential objects. This number 3 could then be simplified by just indicating the absence of objects in a set by saying {}. But we already said that the number 1 or any other purely abstract number can be represented by {}.

What does that all mean? Well I don’t know honestly, as I said I am just trying to figure it out as I go along. I am sure there are children out there with a better grasp on this stuff than me, but let’s just keep going and see what conclusions are drawn.

I suppose that the most obvious implication of this logic would be to say that, any number that is not an attribute of an object or set of objects, is essentially representative of the absence of value and meaning. Represented by {}, which is essentially a placeholder for absence as true absence is beyond our comprehension.

That would mean that unless there are an infinite number of objects in the universe, then there cannot be a meaningful use for numeric infinities. As once the last object has been accounted for in the universal set, once you add an empty set of 1 to that, you lose all meaning and you end up with a rather large number representative of nothing, which can again be represented as {}, or if we want to be more accurate we just wouldn’t refer to it with a symbol at all.

Essentially, you can’t just add one to the highest number you can think of, because unless that number is representative of distinct and unique objects or collections of objects, then it is meaningless and thus represents absence.

Another result of this logic would be that all distinction of objects as unique and definable as atomic instances of a certain type, namely an ‘apple’, is completely arbitrary. As ‘Apple1’ and ‘Apple2’ could be different sizes, density and more than likely have more or less atoms in one or another of the two apples. Yet we put them in the same set of objects, {Apple1, Apple2}, and represent that with the number 2. Not a particularly accurate measurement and furthermore, we have just shown that counting is merely a mental construct, wherein we generalise and group objects together. My point being that if there is no entity around to do the counting, then numbers don’t exist independent of thought. It seems that numerical infinities are contingent on the existence of an entity capable of counting for an infinite length of time. So even if an infinite number of objects do exist then numerical infinities are still in doubt.

I agree that something doesn’t sound right here. But common sense is a biological constraint and it would be foolish to image that the universe should be consistent with what seems ‘right’ to the human race.