Russell as a Mathematician

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thinkandmull

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Hey there,

Does anyone here have an opinion of Bertrand Russell’s (and I guess Whitehead two, since they wrote a book together) math?

P.S. A barber shaves everyone who does not save themselves, so who shaves the barber? The barber… Where’s the paradox even in that?
 
If the barber decides to stop shaving and grow a beard, then he is one of the people in the group of those who do not shave themselves. But all those who do not shave themselves get shaved by the barber. Therefore the barber shaves him.

On the other hand, if he shaves regularly instead of growing a beard, he is one of the group of people who shave themselves. But the barber doesn’t shave the people who shave themselves. Therefore the barber doesn’t shave him.
 
At the turn of last century, there was a push by some philosophers to base arithmetic solely on logic. Previous attempts to give a solid foundation of it was, and still is, difficult. Frege thought he could use logic and sets to give that basis to explain how arithmetic works. This camp in philosophy of math is called ‘logicism’. What Russell did was show that the rules that Frege was using is contradictory, by bringing up Russell’s Paradox, which you mention.
The paradox is shown when you consider impredicative sets - sets that are defined in terms of itself. Consider the set of all sets that do not contain itself. This set contains itself if and only if it does not contain itself. This is a contradiction,(which in logic means the proposition is false) but, it’s also true by definition. So there’s paradox because we have a situation where something is true when it’s false, and false when it’s true. This was a major blow to Frege, who abandoned his logicist project and fell back to Kant’s conception of math. Like, it was so devastating to Frege he just gave up on trying to deal with math. Which is startling because Russell was, I think, in his 20’s at the time. And Frege is an absolute giant in philosophy and logic. Though, so was Russell.

As for Russell’s own math, I’m not very versed in set theories and such so I can’t talk to that. I know Russell tried to ‘fix’ logicism by introducing ‘rammified set theory’, where there are sets of different orders. Other than that, I can’t say.
 
So in a multiverse there should be false universes, which would make an infinite multiverse hypothesis false?
 
“the barber doesn’t shave the people who shave themselves”? That’s not part of the premise. It just starts by saying that the barber shaves everyone who doesn’t shave themselves. If you add that as a second premise, its just a linguistic game. What kind of idiot thinks this is actually a mathematical problem? That’s why I wonder about Russell
 
Has anyone read the Principles of Mathematics by Russell and Whitehead?
 
thinkandmull said:
“the barber doesn’t shave the people who shave themselves”? That’s not part of the premise. It just starts by saying that the barber shaves everyone who doesn’t shave themselves. If you add that as a second premise, its just a linguistic game. What kind of idiot thinks this is actually a mathematical problem? That’s why I wonder about Russell
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Then move on to “a set of sets that are not elements of themselves”. Is it its own element? And then we get the same problem as in case of the barber. And in this case it is obviously a mathematical problem. And it shows that defining a set as a group of elements for which something (some predicate) is true (and that’s the definition that was used previously) leads to self-contradictions. Thus another definition is necessary etc. en.wikipedia.org/wiki/Russell’s_paradox lists still more details.
 
^

And yeah, remember, Russell’s Paradox underminded Frege’s Logicism, which was an attempt to provide the basis for arithmetic.
 
I tend to think that Logicism could be garbage and those mathematicians had minds that were too rigid. My math teacher in college told me that Russell said that he couldn’t think as well about other topics after he wrote his book on math. I think a person should gain strength in other topics if what he is thinking about is true. The barber shaves everyone who doesn’t shave themselves, and what of the others? The barber doesn’t shave the people who shave themselves. What of the barber? He is either an exception or not. You’d have to shave the barber. I see no mathematical problem here. Russell did say that Zeno’s paradox was profound, and I admit it is a very problematic paradox. Russell’s is just not in the same league though.
 
 
They are actually optical illusions, but I don’t know how they are made
 
You’re stuck on the barber. The barber is just an analogy. The problem is impredicative definitions of sets. The problem is Frege’s attempt to give a basis for arithmetic in pure logic.
 
Even if I put numbers instead of barber in the “paradox”, I don’t see the problem. It takes two premises that are incomplete together and need further explanation.

Has anyone heard the radio debate between Russell and a priest on God? I heard a part of it once, it was on a website, awhile back
 
With Zeno’s paradox, you think of a light going on upon the line at every halfway point as B approaches A. The mind jumps between finite distance and a point not existing. The mind doesn’t jump, and know our minds are lacking something, with Russell’s paradox.
 
thinkandmull said:
Even if I put numbers instead of barber in the “paradox”, I don’t see the problem. It takes two premises that are incomplete together and need further explanation.

Has anyone heard the radio debate between Russell and a priest on God? I heard a part of it once, it was on a website, awhile back
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It’s not putting numbers in for barbers. Russell’s Paradox is prior to numbers. Frege’s project was to provide the basis or arithmetic, recall. To explain where numbers come from, and how they work. Or, maybe to say better, to provide a model that explains number. It’s a very controversial topic, even today. Many philosophers and mathematicians have tried to do this. From Plato and Aristotle, to JS. Mill who tried to make the basis purely empirical and physical, to Frege, Russell, Zermelo, Hilbert, Dummett, Carnap, Brouwer, Kant, etc. bringing up so many theories. Platonism, Empiricism, Formalism, Intuitionism, Constructivism, Fictionalism, Logicism, etc.

Russell’s Paradox speaks to Frege’s attempt to base arithmetic in logic, making it a purely analytic and a priori matter. This would help solve the problem of ontology and epistemology of math. In order to do that, Frege had to provide axioms - logic requires that. In an argument it’s called ‘assumptions’, in these theories they’re called axioms. They’re the foundation of the rest of the theory that is derived. Like how Euclidian geometry can be derived from Euclid’s axioms. What Russell’s Paradox did was show that Frege’s Logicism using Naive Set Theory was inconstant. This is a fatal flaw in a logical system. Russell and other mathematicians and logicians HAVE tried to continue the Logicism, altering the axioms to resolve the paradox.

All this happens before we even get to numbers. Frege was giving us a way to define numbers.
 
thinkandmull said:
Has anyone heard the radio debate between Russell and a priest on God? I heard a part of it once, it was on a website, awhile back
Click to expand...
I take it you are referring to his debate with Father Copleston? I have never heard it, but I read some of the transcript of it years ago, in an old philosophy textbook (Paul Edward’s and Arthur Pap’s A Modern Introduction to Philosophy, 2nd edition).

I don’t remember much of that particular debate, but that was the book that converted eleven-year-old me to atheism. Every time my mother tried to make me to go to Mass, I’d quote Bertrand Russell and Clarence Darrow and a few less notable names at her.

(Not an atheist any more, have not been for many years now - I guess I kind of grew out of it.)

Simon
 
You got his name wrong. You mean “Russel Olson.” Don’t lie; it won’t do you or your soul any good. Prepare for eternity mingebag.
 
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