My post on sets and numbers was adapted from the of the work of Gottlob Frege (1848-1925), a brilliant German logician, mathematician and philosopher. The article I linked to with his name credits him with founding modern mathematical logic and modern analytical philosophy. I thinks that's a bit extreme, but he was certainly influential.
Frege was a logicist --he believed that arithmetic can be reduced entirely to logic. Much of his work (including the part I adapted in that post) was intended to prove this thesis. Although the logicist conjecture is interesting in its own right, the real significance of the logicist project was that it was intended to disprove an argument of Immanuel Kant.
I'll get to Kant in a moment, but first a comment on the logicist project. It is generally accepted that the logicist project foundered on the incompleteness theorem of Kurt Godel. I've never quite understood how this works. Godel's incompleteness theorem basically proves that any calculus that is powerful enough to do arithmetic is incomplete --there are theorems of arithmetic that are true but cannot be proven in the calculus.
Clearly the incompleteness theorem is devastating to the formalists --people who believed that all of mathematics is just a calculus-- but I don't see how it effects the logicists. The logicists claim that arithmetic can be reduced to logic, not that it can be reduced to a calculus.
Here I'm going to give a brief synopsis of part of Kant's metaphysics of knowledge and then present an adaptation largely inspired by the famous anti-Kantian: Gottlob Frege:
Kant argued that there are two kinds of knowledge. Analytical knowledge is what we know just by understanding. The most common example is that we know all bachelors are unmarried just because we understand how being a bachelor is related to being married. There is very little knowledge of this sort and in a sense it isn't even real knowledge, just understanding.
By contrast, synthetic knowledge requires more than just understanding, it requires sensing or grasping something outside of the thought itself. For example I know that all dogs live on the earth. I can't just know this by understanding how being a dog is related to living on the earth. I have to know something about actual dogs and something about the earth. I have to sense or grasp something outside of my own understanding. Kant called this ability to sense or grasp something "intuition".
So far so good. What Kant did that really upset a lot of people, including Frege, is to argue that arithmetic and geometry are synthetic knowledge. He argued that we have to grasp numbers with our intuition. There is nothing in the thought of 5 and 7 that tells you they must sum to 12 --it's a form of outside knowledge. Keep in mind here that Kant did not claim that you can't do arithmetic purely in the mind, quite the contrary: he claimed that we can know arithmetic with no outside knowledge at all. He argued that although arithmetic is purely done in the mind it still requires something more than pure understanding. It requires a special intuition about numbers that tells you things like "if x is a smaller number than y and y is a smaller number than z, then x is a smaller number than z."
The logicists wanted to prove that no special intuition is needed. They argued that logic is analytic and that arithmetic is just logic. Therefore arithmetic is analytic. And Therefore arithmetic only requires the faculty of understanding --mysterious faculties of intuition need not apply.
Now, on to the Fregean/Kantian/Gudemanian theory of concepts: In the post on sets and numbers I explained that properties (as I define them) are intensional objects as opposed to extensional objects. Intensional objects are distinguished by the concepts we have of them. Being an odd number between 2 and 8 is a different concept from being a prime number between 2 and 8. These two concepts describe the same numbers but they are distinct concepts. We say that the extensions of the concepts are the same, but the intensions --what we understand of the concepts-- are different.
(In the previous post I used "property", but the text flows better with "concept" so I'm switching to that. If I use the word "property" again, consider it a common term, not the technical term I made it in the previous post. In this post I use "concept" as the technical term.)
Given any concept C and any object o, we can ask whether o falls under C. This means that o satisfies all that we think of in C or that o is one of the C's. For example 3 is one of the prime numbers between 2 and 8 so 3 falls under the concept "prime number between 2 and 8". Of course 3 also falls under the concept "odd number between 2 and 8". In general, every object falls under many concepts.
Given an concept, there may be one, many, or no objects that fall under it. The concept "person writing this post" only has one object that satisfies it. The concept "prime number between 13 and 17" has no objects that fall under it.
In order to know whether any given object falls under a given concept, you have to understand the concept and have some knowledge that tells you the object satisfies the concept. Where does this knowledge come from? It must come from some grasp of the object, some intuition. Objects are not concepts and are not even capable of being understood in the sense that concepts are. So whenever you judge that some o is a C, you are using synthetic knowledge.
One concept can entail another. If C1 and C2 are concepts, we write C1 |- C2 and say "C1 entails C2" if and only if being C2 is part of what we mean by being C1. For example being a bachelor entails being unmarried. Don't confuse this with the relation "every C2 is a C1" as in "every dog is an earth-dweller". The "every-is" relation is extensional: it only depends on the extensions of C2 and C1. By contrast the relation of entailment is intensional: it expresses our understanding of C2 and C1. And since it only relies on our understanding of C1 and C2, entailment is analytic knowledge.
Entailment implies the every-is relation. Since being a bachelor entails being unmarried, we can conclude that every bachelor is unmarried. But every-is does not imply entails. Just because every dog is an earth-dweller, that does not imply that being a dog entails being an earth-dweller. To know where dogs live, we have to use synthetic knowledge.
One concept can disentail another. We say that C1 _|_ C2 and say "C1 disentails C2" if and only if C1 entails that not C2. For example being a bachelor disentails being married. Disentailment is analytic. To judge whether one concept disentails another, we need only understand the two concepts.
Under this theory of concepts --which I believe is consistent with both Frege and Kant-- Frege has a serious effort ahead of him. He has to define all of arithmetic using only concepts, entailment, disentailment, and similar intensional relations between concepts. As soon as he appeals to an object and asks for us to grasp some property of the object, he has slid onto the synthetic side of the field.
Frege would be in trouble even for asking us to believe that some object exists. Our mere understanding of a concept can never let us know whether there is anything that falls under that concept. In order to know that, we need some intuition of the objects that fall under the concept.