Principle of Contradiction: The possibility of something being thought rests on noticing the impossibility of something being.
Hume: Either an abstract idea is a fuzzy picture or a faint idea, or it is something that is at once both large and not-large, equilateral and not equilateral, evergreen and not-evergreen, etc. But the second option is a straightforward contradiction.
Response: In fact, it must be both: it’s the first so far as the first principle of thought is the nervous system, and the second so far as the first principle is the principle of contradiction. All ideas are understood along with their contradictions.
Hume 2: There are no abstract beings in reality (just look around), and so also none in thought. Forming an idea means forming something real. Ideas represent multitudes like a coupon for a free ice cream represents 31 flavors – it’s a conventional or agreed-upon token that you can redeem for any one of a multitude, not some monster that is all flavors and none of them.
Response: We don’t see abstract things when looking around because we see both what mind transcends and what transcends mind, and the only way to account for this is to recognize the abstract character of human cognition.
In looking around, I see continuous progressions in motion, growth, etc, not the discrete jumps of logical inference; I see time measured by clocks that can always be corrected, not the absolute time that allows me to know that they all can be corrected (ditto for the measurement of space). What’s true of space and time is also true of causality: the world of experience has causes known mostly by repetition and custom and not necessity while the world of logic has conclusions caused by necessity. That said, the world of experience always contains more variety and information than the average and statistical abstraction that I use to understand it. The world of thought and the world we think about have to be understood in opposition to each other – we can’t just assume a uniform “reality” that captures them both. But this only shows that the argument is a non sequitur – we can go further than this and show that it’s precisely the abstract character of the world that explains its peculiar relation to the world of experience.
Looking around shows you nature, or mobile reality as mobile. This world both is transcended by thought and transcends it in different ways. On the one hand, Hume is right that nothing can be without being some definite amount or measure, but perfectly definite measurement exists only as an idealized and unattainable limit, as is explained in the first chapter of any science textbook. Mind alone can form a precise account of quality or quantity, which cannot exist in reality. In this sense the precision of measurement is abstract because it is more definite and precise than nature can be. In this sense nature always falls short of the sort of existence it has in thought. On the other hand, thought learns about nature only by abstracting from its variety and by overlooking all properties that are not studied, and in this sense nature always transcends the thought of one who knows by way of abstraction from it. In this sense, the abstraction is less definite than the world, but this definiteness of the world is also as unattainable to it as a perfect measurement. So it is not that an idea must be as definite as the world, but rather that it is impossible for the two to be definite in the same way. One and the same mental reality must be at once more and less definite in different respects, and this intrinsic opposition between the definiteness of the mind and world counts as an abstract idea by Hume’s criteria.
And so the sense we get from looking around and not seeing the abstract arises from both the way in which thought transcends nature and the way it is infinitely beneath it. Both nature and mind are unattainable limits to each other precisely because we know by abstraction.