we will prevail where frege failed...

[Jon]

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]

[Victor]

Originally Posted by Jon

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...

Frege and Russell may have failed but George Spencer-Brown succeeded in his book, "The Laws of Form".

In which all of mathematics begins with an injunction -

"Make a distinction".

[ygolo]

Originally Posted by Victor

Frege and Russell may have failed but George Spencer-Brown succeeded in his book, "The Laws of Form".

In which all of mathematics begins with an injunction -

"Make a distinction".

Many people on the net rave about it, but I hadn't heard of it. How is it different from a simple explanation of boolean algebra?

[Jon]

Originally Posted by Victor

Frege and Russell may have failed but George Spencer-Brown succeeded in his book, "The Laws of Form".

In which all of mathematics begins with an injunction -

"Make a distinction".

I've never read it. But you're saying that it purports to save logicism? Or that it provides another basic account of number?

Because I think the last real attempt was Wright's neo-logicism. Meh, I'm a formalist about math, lol.

[BlueWing]

Originally Posted by Jon

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).

[Jon]

Ok, good. But I have a completely different way of conceptualizing: (1) russell's paradox, (2) the responses, and (3) the fate of Frege's project.

Very briefly (because I'm not sure how much you care about my version of the story):

(1) Russell's paradox is generated by the combination of (a) the Naive Comprehension Axiom (every concept has an extension), and (b) the Rule of Substitution (every open sentence which defines a condition on objects corresponds to a concept).

(2) Responses to the Paradox: The two solutions are (a) The Theory of Types (which tries to find a non ad hoc way to restrict what counts as a genuine property), and (b) Zermelo-Frankel Set Theory (denies that there is a set for every property).

(3) The Fate of Frege: Two respects in which the elements which generate the paradox are deeply entrenched in Frege's entire programme. (a) getting big enough classes (he needs there to be an infinite number of objects in order to secure the result that every number has a successor), and (b) Hume's Principle (the fact that Frege derives Hume's Principle from Basic Law 5)

I'll elaborate if asked.

lol: I posted this as a response to my thoughts regarding online threads. Most threads try to square an issue posed by a member. The benefit of the community then is to bring a group of minds together to deal with, and resolve, the issue. I found it funny that this purpose is meaningless in the face of certain issues (ie: salvaging Logicism) even though thousands could work on it. Is this medium therefore only useful for trivial issues? Depends on what you consider trivial. But read my first post with this in mind and you'll see what I mean.

[BlueWing]

Originally Posted by Jon

(1) Russell's paradox is generated by the combination of (a) the Naive Comprehension Axiom (every concept has an extension), and (b) the Rule of Substitution (every open sentence which defines a condition on objects corresponds to a concept). .

This does not define Russell's paradox and I fail to see the difference between your conception of the notion and mine. Clarify.

Originally Posted by Jon

((2) Responses to the Paradox: The two solutions are (a) The Theory of Types (which tries to find a non ad hoc way to restrict what counts as a genuine property), and (b) Zermelo-Frankel Set Theory (denies that there is a set for every property)..

I am familiar with the theory of types, yet you should explain what Zermelo-Frankel Set theory is.

Originally Posted by Jon

(((3) The Fate of Frege: Two respects in which the elements which generate the paradox are deeply entrenched in Frege's entire programme. (a) getting big enough classes (he needs there to be an infinite number of objects in order to secure the result that every number has a successor), and (b) Hume's Principle (the fact that Frege derives Hume's Principle from Basic Law 5))..

Why must there be an infinite number of objects in order for every number to have a successor? Numbers are not objects, but merely abstract figures of objects. Here we encounter the distinction between existing mathematical entities, finite sets and mathematical possibilities. Numbers inhere within mathematical possibilities and there is no reason why every number could not have a successor owing to the fact that a set of natural numbers is infinite.

Originally Posted by Jon

(((I'll elaborate if asked.))..

Yes you should? Which of Hume's principles are you talking about? His epistemology is very extensive.