But still, didn’t we already know that if you take a line, two distinct points A and B on it, there are an infinite number of points between A and B, and yet an infinite number of points outside [AB] ? Didn’t we know that since the ancient greeks ?
That’s a good point. Spinoza himself was a mathematician of no mean talent, so we should assume that he was aware of it as well. So the question is, does his argument avoid the mistake of taking ‘infinite’ to mean ‘all encompassing?’ without any argument to that effect? There are certainly questions to be raised about his argument, but I don’t think this is one of his mistakes. If you don’t want to take my word for it, here’s the opening argument of the Ethics. Good luck, it’s quite a slog.
The idea seems to be that the one substance has to be infinite and singular, because substances can’t share attributes (see his definitions), and things which have nothing in common can’t interact. Therefore substances can’t cause each other to exist, and therefore if any exists, it must exist necessarily. If that’s true, then existence is an attribute of a substance, and so no other substance could exist.
At any rate, the argument concerns an ‘infinity’ of attributes, and I think these are reasonably taken as countably infinite. Spinoza also defines infinite as ‘not being limited by anything of the same kind’, so by that definition he would say that with reference to the ‘kind’ ‘number’, the even numbers are finite, though they’re infinite with reference to the ‘kind’ ‘even number’.
That’s a good point. Spinoza himself was a mathematician of no mean talent, so we should assume that he was aware of it as well. So the question is, does his argument avoid the mistake of taking ‘infinite’ to mean ‘all encompassing?’ without any argument to that effect? There are certainly questions to be raised about his argument, but I don’t think this is one of his mistakes. If you don’t want to take my word for it, here’s the opening argument of the Ethics. Good luck, it’s quite a slog.
The idea seems to be that the one substance has to be infinite and singular, because substances can’t share attributes (see his definitions), and things which have nothing in common can’t interact. Therefore substances can’t cause each other to exist, and therefore if any exists, it must exist necessarily. If that’s true, then existence is an attribute of a substance, and so no other substance could exist.
At any rate, the argument concerns an ‘infinity’ of attributes, and I think these are reasonably taken as countably infinite. Spinoza also defines infinite as ‘not being limited by anything of the same kind’, so by that definition he would say that with reference to the ‘kind’ ‘number’, the even numbers are finite, though they’re infinite with reference to the ‘kind’ ‘even number’.