The topic of infinity is a favorite. I know a lot of people think that all that can be said on the subject has been said - but they're wrong. There is always more.
Infinity is one of those topics people don't like me to talk about because I can go on, and on, and on, and on about it. (Another topic people don't like to hear me talk about is transcendence - because I bring transcendence to a whole new level).
In philosopphical discussions, I often use the phrase, “that beyond us which conditions us.” This is my way of speaking of infinity.
This notion of that beyond the self that conditions the self has import in philosophical discussions involving mathematics. What is mathematics?, What accounts for our ability to do mathematics? In what way does math refer to reality? I believe that answering these questions involve a notion of infinity - of that beyond the mind and conditions the mind. Humans do mathematics with an intuitive connection with the beyond - knowing, for instance, that one more always can be added. The "always can" in "always can be added" is telling - it implies knowledge of a condition which presupposes a beyond.
I think that mathematics is really real. However, the reality of mathematics depends on the mind. That is not to say that mathematics is imaginary (it is not created, exactly). Mathematics is mediated through perspectives of finite beings; through the mind. Mathematical reality is somehow intrinsic to our minds. However, our minds are finite - that is, conditioned by that beyond the mind, such as time, chemistry, the laws of physics, etc. So, of course, mathematics is real - but a reality that involves the consciousness that discovers it. Because time and chemestry are real, and time and chemestry condition the mind, then mathematics is real. So the reality of mathematics, even though it involves the imagination, is grounded in those conditions, constraints, and boundaries that are preconscious, but are also parrt of of those things through which consciousness emerges.
I have been told that my position as to the whether or not mathematics is 'created or discovered' is closer to Kantian idealism than traditional realism. If finitude is the main thing differentiating the finite phenomenon and the infinite noumenon, then, yes, I can see how my view is a Kantian idealism. On the other hand, to be finite is to be conditioned by the infinite. It is that 'being condiitoned" by that "beyond the creative self" that ensures that mathematics is discovered and a real connection with reality. Then again, perhaps my understanding of realism and idealism are naive. Perhaps both need to be refined or done away with.
The notion of infinity as that which is beyond the self and conditions the self extends to other areas of philosophy. In epistemology, for instance, I would argue that a good critical realism relies on a theory of knowledge where it is recognized that one cannot rely on what one knows as an excuse to not learn something new; that is, that what one knows can always be challenged. This is because the validity and coherence of any epistemic foundation is dependent on something beyond the horizon of the knower (even when the knower is not aware of it).
Derrida's deconstruction, as well, can be expressed with this notion of infinity. Deconstruction relates to the negation of the consummation of meaning; how a word or a notion (such as 'gift' or 'hospitality') depends on something that is beyond it's immediate use and yet conditions it's meaning.
Below are three books I recommend on the topic of infinity.
