Up to now, my discussions in the foundations of mathematics and logic have been more or less within a tradition of philosophy, even if it was heavily filtered through my own prejudices. Just to make sure I don't mislead anybody, I want to point out that the following is not part of any tradition I know of. I pretty much just made it up.
Here is, some, background.
The logicists tried to show that mathematics is analytical knowledge by reducing mathematics to logic. It is generally said that their program was destroyed by Godel's incompleteness theorem. But it was really in trouble before this. The problem is that some their axioms were synthetic. For example consider the law of identity. This says that if you have two objects x and y, and x=y, then for any property F, if x is F then y is F also.
It might be easier to understand with an example: George Bush is president of the United States. So we know that anything that is true of George Bush is also true of the president of the United States. If George Bush is a great war leader, then the president of the United States is a great war leader. If George is a drunken chimp then the president of the United States is a drunken chimp.
Frege and the other logicists require the law of identity or something like it to even get off the ground. But this law isn't analytic. A statement can only be judged analytically when it is no more than the elucidation of a concept. As I showed in a previous post, this is a very narrow range of statements. I can say that all bachelors are unmarried simply because the concept of being unmarried is part of the concept of being a bachelor.
When I say that George Bush is president of the United States, what I mean is that I have these two concepts in my head: "being George Bush" and "being president of the United States", and that these two concepts have the same extension --they are both satisfied by the same object. What the law of identity tells me is that when I have two concepts like this that have the same extension, then any other concept that shares the extension of one also shares the extension of the other.
But this is quite a remarkable thing to realize. It's obviously and intuitively true, no doubt about it. But the second part of this inference is not what I mean when I say the first part. Rather the second part is inferred with my mathematical intuition. It is synthetic knowledge.
In fact, analytic knowledge can only refer to concepts because it is knowledge about concepts. As soon as you bring an object into it, the extension of a concept, you are falling out of the range of analytic statements. Objects have a different sort of existence from concepts, even mathematical objects like numbers.
Concepts are the things that are directly present to our minds when we think. Concepts are about objects. When I think of the number 1, the number itself is not directly present to my mind. What is present to my mind is the concept "being the number 1". The concept is about the number. The number is not about anything.
I can no more import the number itself into my head to think about than I can import the Golden Gate Bridge to think about it. The only thing that can be "in my head" is the concept of the bridge or the number. And since I can only have concepts in my head, no objects, analytic knowledge can only concern itself with concepts, not objects.
This dichotomy between concepts and objects exactly mirrors the dichotomy between analytic knowledge and synthetic knowledge: we can have analytic knowledge of concepts, but all our knowledge of objects is synthetic. These are two different ways of knowing for two different kinds of things. Let's call these kinds "domains".
We have the domain of concepts. Concepts are directly present to the mind. They are about things, they have extensions. Concepts can entail other concepts. Concepts can exclude other concepts.
We also have the domain of objects. Objects are not directly present to the mind, but are only known mediately as the extensions of concepts. They cannot entail or be entailed, nor exclude because they are not concepts. Objects can be ideas like numbers or can be real things like bridges.
Now we can describe the project of the logicists like this: The logicists wanted to reduce mathematical objects to concepts. They wanted to show that each mathematical object was merely a stand-in for something built of concepts and that whatever you can say about mathematical objects can be reduced to something about concepts. In other words, logicism is just another species of the very common intellectual disease of reductionism.
The most pernicious species of reductionism today is physicalism. Physicalism is the extreme reductionist idea that every domain (including the domains of concepts, of mathematical objects, and of minds) is reducible to physical objects and that anything you can say about anything can be reduced to a physical fact. For example, physicalists believe that belief is a physical state of the brain (this is a self-refuting position, but try telling a physicalist that).
I'll tie the two (logicism and physicalism) together in a future post.