"I perceive that I now exist and recall that I have previously existed for some time. And I have various thoughts and know how many of them there are. It is in doing these things that I acquire the ideas of duration and number, which I can apply to other things." (Meditation III, 45)
This is one of the more exciting moments in philosophy, and I just realized that people may not know just how huge this sentence is. Arguably Descartes may not even have noticed; he may have been concentrating on something else that would result from his saying this, or ever more, he may just have taken this for granted that everyone understands this - proof of how genius he was.
But have you ever been in a class and had the question asked to you: "what are numbers?" I have, and at that point I was equally clueless as to the answer, and it just struck me that almost everyone I've ever met probably hasn't had a clue either.
Well, Descartes did. Kant finished, but this is really the first place I've seen this concept so concretely asserted.
When most student think about this, it is a logical (or just good enough) conclusion to come to that there are many things that exist and when presented with a finite quantity of them we are able to distinguish each and every one separately in some sort of cognition that enables us to give the correct meaning to the symbols "1, 2, 3, 4, 5, etc." Of course, I am being generous; the most common problem people have when defining numbers, when asked "are numbers just out there?" is that they try to fit the symbol "3" into a basket filled with 3 apples. Once you get over the symbols, then the real fun begins.
As I mentioned Descartes has the answer, and though he took giant steps in philosophy, he couldn't make the jump from where intellectual history was to the Doctrine of Transcendental Idealism. But Kant had the history, the education...well, the mind at least, with which to accomplish that task:
So what are they saying? From whence come numbers? From a Succession of Thoughts. Kant's answer is that you have two a priori sense, and inner and an outer. The outer is that of spaciality, the inner of time. These two senses, in my analysis, must cooperate in order to for a coherent grounds for "the possibility of experience."
What? I'm not going to explain the Transcendental Aesthetic right now. We'll keep it to numbers:
So the inner sense of time is a priori, we have it 'hard wired' into us and we need it to experience anything (along with the outer sense). So in our mind we have these a priori cognitions prior to any experience, our minds must for a template, shall we say, on which experience becomes a possibility.
That being said, time is a sense which precedes experience, this is important enough that I say it so many times because it will get the objectivists to shut up. So before we can have experience, sayeth Kant, we develop a sense of time. And what comes of this sense of time? Exactly what Descartes says.
The concept of number in your mind is due to your undergoing a succession of conscious thoughts, which you experience in a rational order.
So you derive numbers, to which you later ascribe symbols. You experience infinitesimally small moments - the smallest increment of time in which you can process the smallest increment of a thought - you experience these moments successively, one after the next, and you grow an idea of number because you can recall that are processing each thought and the past thoughts.
It took us a while to think up making symbols and our base ten system, etc, so avoid the fallacy that one person in one life could think up the whole system of symbolized numbers, this cognition I've attempted to describe simply gives you the basis, along with the outer sense, to form the judgment that there are many objects, many different things, however many are in your field of sensibility (sight, hearing, feeling, etc), that are being represented to you. There are seven pads of paper on the table, I know this because I have experienced an a priori succession of thoughts which enables me to judge that there are these differing objects being represented to me in space and time.
Consider this theoretical. A person who has both the senses of inner and outer, spaciality and temporality, but has of yet not experienced any sensible stimulus. If this person were to somehow be shown a picture in their abyss of consciousness, then they would see a single object presented in time. They would never have had the empirical experience to distinguish that this object is meant to represent other things within it. The person would not see the table and chairs and guitar, they would just see a colorful rectangle, some strange form of stimulus which she has never been privy to before, and which represents only one thing to her mind.
I doubt I have explained this well enough. Note that I have used experience in terms of experiencing a sucession of thoughts which I also dubbed a priori; experience, as in the a priori a posteriori distinction, is empirical experience, sensible experience: we must experience these a priori elements of exstence to make empirical experience possible. The word experience doesn't have any adequate synonyms that I can think of.