Mr. Tea
Let's Talk About Ceps
OK, well I had a good deal of difficulty sleeping last night for some reason, and I had all these thoughts turning over in my head, some of which I might be able to remember.
Firstly, as to whether things like numbers and shapes 'exist': I said I earlier that I was leaning towards the idea that they don't really exist in any meaningful way, which presumably means they're just signs or something. But then it struck me that within the context of maths itself numbers are undoubtedly objects, in a way that they're not in tangible actuality. For instance, in the context of physical objects an empty box is practically and formally identical to box containing zero things, so here it makes sense to identify the number 0 with the concept of nothing. But in set theory, a set containing only the number 0 is not empty (symbolically, {} =/= {0}). So in this context, a set containing nothing is different from a set containing nothing! This is because although zero signifies a vanishing quantity of something, the concept of zero has an existence of its own, the same way dragons don't exist but the concept of 'dragon' - the word itself and the associations it has in people's minds - certainly does exist. Though it is of course a human construct, so if there were no humans it would not exist; by contrast the concept of 'rabbit' would also cease to exist if there were no humans around to have concepts, but of course rabbits would carry on existing regardless.
This led me to surmise that numbers exist as objects within the context of mathematics at a similar ontological level to the way words exist as objects in the context of a language. I then came to something like the following conclusions:
Firstly, as to whether things like numbers and shapes 'exist': I said I earlier that I was leaning towards the idea that they don't really exist in any meaningful way, which presumably means they're just signs or something. But then it struck me that within the context of maths itself numbers are undoubtedly objects, in a way that they're not in tangible actuality. For instance, in the context of physical objects an empty box is practically and formally identical to box containing zero things, so here it makes sense to identify the number 0 with the concept of nothing. But in set theory, a set containing only the number 0 is not empty (symbolically, {} =/= {0}). So in this context, a set containing nothing is different from a set containing nothing! This is because although zero signifies a vanishing quantity of something, the concept of zero has an existence of its own, the same way dragons don't exist but the concept of 'dragon' - the word itself and the associations it has in people's minds - certainly does exist. Though it is of course a human construct, so if there were no humans it would not exist; by contrast the concept of 'rabbit' would also cease to exist if there were no humans around to have concepts, but of course rabbits would carry on existing regardless.
This led me to surmise that numbers exist as objects within the context of mathematics at a similar ontological level to the way words exist as objects in the context of a language. I then came to something like the following conclusions:
- numbers, shapes, equations and so on exist as objects within the context of mathematics; it's not that they cease to exist outside this context, they just cease to be meaningful (in much the same way that if I'm speaking English to someone who doesn't understand it, it's not that my voice is inaudible to him, it's just that my words aren't recognisable as meaningful semantic objects);
- mathematics, like a verbal language, a belief system, a set of laws or the UNIX operating system, is a human construct, essentially a set of conventions and in some sense a highly formal and symbolic language in its own right;
- we could maybe define mathematics as "that area of human discourse where statements are true if and only if they are formally tautological".
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