Russell as a Mathematician

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Rhubarb said:
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.
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I don’t see any problem in explaining Russell’s “paradox” using logic
 
thinkandmull said:
I don’t see any problem in explaining Russell’s “paradox” using logic
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Russell dealt with the paradox by formulating Type Set Theory. It can be done. The paradox spoke specifically to one point in Frege’s Logicism that used naive set theory. There is no ‘explanation’ of the paradox using logic. It’s cut-and-dry and showed that Frege’s system was inconsistent. You can look up Frege’s project, and see the derivations yourself.
 
Should I just search Google or do you have a more specific source? 😃
 
If I had a crockpot and it contained all the crockery that exists. It does not contain itself so then it cannot be a crockery crockpot. Is this just a labelling issue.
 
You said:
If I had a crockpot and it contained all the crockery that exists. It does not contain itself so then it cannot be a crockery crockpot. Is this just a labelling issue.
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In a way, you’re right. The paradox arises when the defined axioms of Frege’s theory lead to an inconsistency. But remember Frege and Russell were talking about sets, and their definitions. Not actual physical objects. Your crockpot that contains all crockery is a false label. We can’t really falsely define a set. I guess, unless I look at {x} and label the set {y}. But that’s not what the issue is. The issue was that Frege’s axioms allowed Russell to derive his paradox, which proved Frege’s theory was inconsistent.
 
maybe Frege shouldn’t have used sets then, if a set is defined by what it contains and should not have a container.
 
You said:
maybe Frege shouldn’t have used sets then, if a set is defined by what it contains and should not have a container.
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The point of Frege’s project was to give a basis for arithmetic in pure logic. So in the axioms he couldn’t rely on things like “number” or “counting”. His theory was supposed to explain numbers and counting, so that would be question begging. Even without number and counting, sets have meaning, so Frege was able to use that. Russell’s type theory was Russell’s solution. It aimed to distinguish sets from one another to avoid the paradox.
 
The solution seems so simple that Frege should have realized it himself, instead of complaining how “undesirable” (his word) it was to be refuted right after stating a theory
 
Aristotle misunderstood, I think, Zeno’s paradox about the grain of sand. The grain of sound doesn’t make a sound perhaps because there must be a certain threshold before there is a noise. However, I think the paradox is more along these lines: is 400 a lot of people? What about 100? Is 12? Or is it 13? We know that 400 is a lot, and that 2 is not, but we cannot pin down where the line is to be drawn. It’s a gradual slope that within which it can’t be said where one begins and another ends
 
thinkandmull said:
Aristotle misunderstood, I think, Zeno’s paradox about the grain of sand. The grain of sound doesn’t make a sound perhaps because there must be a certain threshold before there is a noise. However, I think the paradox is more along these lines: is 400 a lot of people? What about 100? Is 12? Or is it 13? We know that 400 is a lot, and that 2 is not, but we cannot pin down where the line is to be drawn. It’s a gradual slope that within which it can’t be said where one begins and another ends
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I’ve always heard that called the paradox of the heap. If you start with a heap of sand, and remove one grain, is it still a heap? What about one more? And another? etc.

It’s often presented to Intro students as removing one hair, if that makes you bald. And then another, and another, etc.
 
Rhubarb said:
I’ve always heard that called the paradox of the heap. If you start with a heap of sand, and remove one grain, is it still a heap? What about one more? And another? etc.

It’s often presented to Intro students as removing one hair, if that makes you bald. And then another, and another, etc.
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What is a heap or being bald?

When they become noticeable or recognizable? Which might mean 1 is not really 1 it is a surprising quantity that catches your attention.
 
Is a million dollars too much to spend on a car? What about a million minus one? Or two? Keep subtracting and you will reach an amount that is reasonable, but the human mind cannot find an exact amount in exact dollars. When does a one dollar difference go from “too much” to “too little”??
 
thinkandmull said:
Is a million dollars too much to spend on a car? What about a million minus one? Or two? Keep subtracting and you will reach an amount that is reasonable, but the human mind cannot find an exact amount in exact dollars. When does a one dollar difference go from “too much” to “too little”??
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The correct amount of money to spend is reasonable and possibly even surprising but never shocking. That is the limit. Numbers cannot be put on it as ultimately number have no meaning and hence do not really exist.
 
If you have only 4 million dollars, then spending 4 million would be too much. But what about 3 million, or 3 million plus 15 dollars? My point is that we have general concepts of too much and too little, but when going down or up from these there really isn’t a single point at which too little becomes too much or too much becomes too little
 
Roger Trigg, professor of philosophy at the University of Warwick, in *Beyond Matter *, writes that “Mathematics, though, could be claimed to be merely a tool created by the human mind”. Instead of believing in Idealism, I am turning more to this point of view
 
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