# Russell's and Frege's Definition of Number

1. ### Barbie

38
As you may know, Russell wrote "Definition of Number" inspired by Frege's earlier definition.

I have heard that he did this for a specific reason, but hours of research have proven to be hopeless. What was the confusion about definition of number that Russell needed to do this?

Also, Im not so sure I truelly understand it...

"The number of a class is the class of all those classes that are similar to it." thus "A number is anything which is the number of some class." -Bertrand Russell

(He is only talking about whole cardinal numbers here of course)

I did have a spark somewhere in my brain ( ), but I wanna hear opinions of others as well. What do you think of this definition? How do you understand it?

2. ### mathwonk

9,690
I think it is utter trivial BS. I agree with Hausdorff, a real mathematician: What is of interest to us is not what numbers "are", but how they behave.

Russell's definition is like saying that a property is defined simply by collecting together all those things that have that property.

E.g. he would define "green" as the set of all green things. so he defines the cardinal number "5" as the class of all those sets which admit a bijection with the set {1,2,3,4,5}.

Last edited: Mar 8, 2005
3. ### Barbie

38
I see where you are coming from. Russell and Frege are clearly essentialists.

4. ### mathwonk

9,690
i suspect they are people who do not have to earn a living.

and i have verified this in the case of russell who was born a lord of some kind.

i have enormous respect for russell's courageous advocacy of peace, especially during wartime, but little regard at all for his mathematical work, which does not impress me.

Last edited: Mar 8, 2005
5. ### Barbie

38
Yeah, Russell did not work to earn money.

I personally dont understand why he felt the need to publish this when it was already published by Frege??

6. ### Hurkyl

16,090
Staff Emeritus
Well, that is how sets model logic. :tongue2: Each unary relation of the language is modelled as the set of all objects satisfying the relation.

It sounds like an attempt to define a number by:
(1) Defining equivalence classes of sets. (i.e. sets are equivalent if they are the same size)
(2) Selecting a single representative from each class
(3) Defining a number to be that representative.

Effectively, that's what it means to be a cardinal number today... but a more interesting construction is required: steps (2) and (3) in general can't be done in modern set theory.

7. ### guitry

3
Yes, the class of classless classes...all this leads to Goedel, doesn't it?

8. ### inquire4more

47
Actually, it seems to me that what you are referring to is Russell's paradox. Russell's paradox makes reference to the set of all sets who are not members of themselves. It was with this paradox, as I recall, and I may be entirely wrong, that he dispelled Frege's notion that all sets may, in some sense, be defined implicitly. After this Russell and Whitehead worked on the theory of types for use in the Principia Mathematica. Later came Gödel's answer to the Principia Mathematica and the assertion that number theory, or set theory perhaps, could be complete AND consistent. Someone please check me on this, its been a while since I sat in my mathematical logic and set theory courses.

9. ### inquire4more

47
Mathwonk, Russell may not have had to work for a living, but this in no way makes his contributions to set theory and the foundations of mathematics trivial. He may have been wrong, but his was still an important step. Also, some of us are interested as much in what an object is, as how it behaves (further, some of us believe how it behaves is very much related to what it is). To many his efforts may have seemd useless. But I am comforted somewhat by the efforts of some people to provide some rigorous foundation for the rest of the maths. Even if Gödel later showed that not all things could be proved in such a system. Frege and Russell were both brilliant men with great contributions under their belts.

10. ### mathwonk

9,690
you could be right. But in my opinion the substance of this definition of number is really due to Georg Cantor, before 1883, and hence before either Frege's 1884 paper, published when Russell was about 8 years old.

I.e. I prefer Cantors Contributions to the Founding of the Theory of Transfinite Numbers to Russells work, thats all.

Last edited: Apr 15, 2005
11. ### inquire4more

47
No, I agree, I am every bit as interested in Cantor's work with the continuum and the origin of transfinite numbers as I am with Russel's work. Cantor's work was part of what motivated me to persue pure, and not so much applied, mathematics in uni. I still consider his proof of the equivalence of the cardinality of the rationals with that of the naturals, as well as his proof of the uncountability of the continuum, as among the most intuitive and beautiful I have yet encountered. I was simply stating that Russell's work was important. Though we use the naturals, the rationals, etc. all the time, I believe some firm definition was still warranted, even though we all know pretty much what number is and how to operate on it. I liken it to seismic retrofitting of structures. Yes, the building stands as it is now, but it's nice to know it can withstand a little more severe an attack due to our work on it. I think his work towards some rigorous foundation for the naturals helped in this manner. It certainly helped Gödel arrive at his result.

12. ### guigus

9
You mean a pragmatic mathematician -- which is not necessarily a good one. A philosopher, on the other hand, will be always interested in what numbers are.

This is the tautological interpretation of Russell's definition, which prevents us from truly understanding it -- a true definition does not presuppose whatever it defines. Russell's definition, without resorting to the concept of a number, is that "a number is a class of similar classes." And what are similar classes? Two similar classes have, of course, the same number of elements, but such a definition is again tautological regarding the concept of a number. You must find a definition of the similarity between any classes that does not depend on the concept of a number at all. When you do so, then you will understand Russell's definition.

242
14. ### AlephZero

7,300
Well, you don't need to find it yourself - it's all there in "Principia mathematica".

As the simplest case, he defines the notion of "a set with one element" starting from axioms of logic, without making any reference to the idea of "the number one".

The interesting philosophical problem is why these man-made things called numbers have any relation to the way the universe seems to behave. Personally, I have a deep suspicion that the correct answer to that is "well, actually, they don't, because the foundations of math as understood in 2011 are in no better state than the foundations of physics were in 1811".

15. ### guigus

9
Are you saying that the "one" in "a set with one element" is not the number one? What is it, then?

If you don't know what a number is, then you don't know if it is "man-made." First you must find out what a number is, then you will be able to tell who made it, if someone. Otherwise, you will never have more than a "suspicion," no matter how deep.

9
17. ### disregardthat

1,811
These bastardizations of the concept of numbers doesn't appeal to me at all. Formally, he might define what he means by "number" with respect to his axiomatic setting as he like; incorporating it in some theory of his, but to consider this as the fundamental aspect of the notion of numbers, a logical definition, is wrong and absurd. In language, definitions are descriptions of common use, and can't be more than that.

Last edited: Feb 2, 2011
18. ### Hurkyl

16,090
Staff Emeritus
In the definition AlephZero was referring to, "set with one element" cannot be parsed into smaller pieces -- the sequence of three characters "one" doesn't have any meaning of its own in that phrase.

The definition of "X is a set with one element" is pretty simple -- it is the conjunction of:
• There exists x such that x is in X
• For all x and y such that x is in X and y is in X, x = y

i.e. $\left( \exists x: x \in X \right) \wedge \left( \forall x \in X: \forall y \in X: x = y \right)$ (or some equivalent thereof)

19. ### guigus

9
Russell's definition, in the form you cited, is just a pedagogical one. You cannot use the concept of a number to define a number, for obvious reasons. Despite that, the definition is rigorously correct. One just needs to reformulate it in a number-free way. I already gave a first formulation: a class of similar classes. That is, a number is a class containing only classes that are similar to each other. Then, you have to define similarity without resorting to the concept of a number. By which the one-to-one correspondence between the elements of any two classes will not suffice, as it still depends on the numbers two and one. Meanwhile, Russell's definition remains correct. And if you repute it as wrong, then you must find a number that does not fit it.

Last edited: Feb 2, 2011
20. ### guigus

9
You must be joking... of course "one" means the number one in that phrase. All you have to do to see that is replace those three letters by the three letters "two," and you will see how your "unparseable" sentence -- which I have just parsed -- changes.