Joe N says: >Take a look at the Stanford article on "Principia". The main problems with the logicist program were perceived to be the "axiom of infinity", and that of "reducibility".
OK, let me try again, I was just trying to get you to reiterate in your own words what your "propositions" meant to you, while touching on the fact that there's a big long history here.
Here's the deal as I understand it, very very roughly, in my own words. Don't be offended by my lecturing style, or hold it against me if you already know all this stuff, I'm just laying it out in case anyone wanted to challenge these understandings or delve deeper into something more specific or somewhat different than your propositions (about "One IS INLCUDED IN the other.")
Logic is that which seeks to understand the "laws of thought", independent of subject matter. So if we agree that the following is logical:
1. All men are mortal, and 2. Socrates is a man, then we must agree with Woody Allen that 3. Therefore, all men are Socrates.
If I replace "All men are Mortal" with 1. All toves are slithy...
The logic of the syllogism is the same. Obviously also, the above is not generally considered math. So its not generally considered reasonable to say that "Math includes all of logic" or "All logic is just math."
But the reverse proposition, that "all math is just logic" is less obviously false, and was seriously considered at the end of the 19th century.
Now, I'll cut to the chase, and be somewhat purposely vague. It seems to me, the big stumbling block, is the subject matter of mathematics. The axiom of infinity, for example, claims "there is an infinite set". This notion of "actual" or "completed" infinities has along history going back to the days of Aristotle. Gauss seems to have railed against the concept, saying actual infinities are *never* treated in mathematics, yet a hundred years later most everbody is accepting them. Strange! (You mention the intuitionists -- they disagreed with the changing fashion and are still a minority.)
If you dig into the "reducibility" axiom, its a lot harder to understand, but I think it comes down to a similar thing - just what is it the mathematicians are talking about, really? What is a "set"? When do we know we have something really mathematizable, versus something plagued with paradox? What are the rules to avoid paradox?
Anyway, for whatever its worth, and this is off-hand, I could easily be shown wrong, but I have a lot of sympathy and respect for the logicist, intuitionist, formalist, and computationlist movements. On the other hand, I have a picture of Kurt Gödel hanging by my desk, and he was a famous Platonist. Still he was a logician, and I think the *best* part of mathematics is *logic*, actual infinities or no actual infinities, sets or no sets.