in this thread we shall become logicists and take up their ambition to reduce mathematical truths to logical truths...
we'll get out of russell's paradox and establish the true foundation of mathematics!
it took Frege and Russell their whole life to fail at this...but today...on this thread (and before bedtime)...MBTI Central will win...
[hahaha, i would consider this spam...so lame....hahaha]
For general information on the subject, I recommend this video. Very informal discussion between Bryan Magee (popularizer of philosophy) and A.J Ayer (a philosopher who is well acquainted with Russell personally and his work). This video is intended to be understood by a layman, does not presuppose much knowledge of philosophy of mathematics.
YouTube - flame0430's Channel
Russell's paradox has already been resolved by Russell himself later in Principles of Mathematics. Principles of Mathematics, P.527 (the very end) the section "Contradiction arising from the question whether there are more classes of propositions than propositions".
Russell's paradox is as follows. Suppose we have a team of baskebtall players, all players are part of the class. Yet the team itself is not a basketball player, therefore it is not part of the class, contrary to what Frege has maintained.
Russell's Paradox [Internet Encyclopedia of Philosophy]
This could be avoided by carefully defining our terms where our logical operations will not require for the group of entities we are dealing with to have the same membership status as the members that inhere within it. " The only method of evading this difficulty is to deny that propositional concepts are individuals; and this seems to be the course which we are driven". Principles of Mathematics P.526 More is stated on this in the 'Theory of Types' which is the second section in the article below and the section in PPs 525-527 in Principles of Mathematics. With the way types are defined, the question does not even arise whether or not a type is a property of itself.
Russell's Paradox [Internet Encyclopedia of Philosophy]
As for Frege's attempt to reduce Mathematics to logic, this cannot be done as Frege has envisaged, as translation of some logical ideas to mathematical requires the use of some non-logical symbols. In the most formal sense of the terms, the two do not share an identity, as they operate on different symbols. However, the essence inherent in both of them is the same. Namely, they are both concerned with the proper laws of our reasoning. Logic represents elementary patterns of proper reasoning, mathematics is the sophistication thereof. As Russell himself wrote in the Introduction to Mathematical Philosophy,
Amazon.com: Introduction to Mathematical Philosophy,
in paraphrase, logic is analogous to mathematics as boy to man, if it was not so, where in Principia Mathematica logic ends and mathematics begins?
As for Frege's treatment of the subject, I recommend this article,
Frege's Logic, Theorem, and Foundations for Arithmetic (Stanford Encyclopedia of Philosophy).