Russell's and Frege's Definition of Number

  • Thread starter Barbie
  • Start date
  • Tags
    Definition
In summary, Russell wrote "Definition of Number" as a response to Frege's earlier definition and was motivated by the confusion surrounding the definition of number. His definition states that the number of a class is the class of all classes that are similar to it, and that a number is anything which is the number of some class. This definition has been criticized by other mathematicians, such as Hausdorff, who believe that the behavior of numbers is more important than their definition. However, Russell's definition is in line with how sets model logic and is related to Gödel's work on the Principia Mathematica and the assertion that number theory cannot be both complete and consistent.
  • #1
Barbie
24
0
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? :confused:

Also, I am 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 ( :rolleyes: ), but I want to hear opinions of others as well. What do you think of this definition? How do you understand it?
 
Physics news on Phys.org
  • #2
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:
  • #3
I see where you are coming from. Russell and Frege are clearly essentialists.
 
  • #4
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:
  • #5
Yeah, Russell did not work to earn money.

I personally don't understand why he felt the need to publish this when it was already published by Frege??
 
  • #6
Russell's definition is like saying that a property is defined simply by collecting together all those things that have that property.

Well, that is how sets model logic. :tongue2: Each unary relation of the language is modeled as the set of all objects satisfying the relation.


"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

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
Yes, the class of classless classes...all this leads to Goedel, doesn't it?
 
  • #8
guitry said:
Yes, the class of classless classes...all this leads to Goedel, doesn't it?

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
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. :wink:
 
  • #10
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, that's all.
 
Last edited:
  • #11
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 pursue 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
mathwonk said:
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.

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.

mathwonk said:
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}.

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.
 
  • #14
guigus said:
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.

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
AlephZero said:
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".

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

AlephZero said:
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".

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.
 
  • #17
"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

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:
  • #18
guigus said:
Are you saying that the "one" in "a set with one element" is not the number one? What is it, then?
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. [itex]\left( \exists x: x \in X \right) \wedge \left( \forall x \in X: \forall y \in X: x = y \right)[/itex] (or some equivalent thereof)
 
  • #19
Jarle said:
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.

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:
  • #20
Hurkyl said:
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. [itex]\left( \exists x: x \in X \right) \wedge \left( \forall x \in X: \forall y \in X: x = y \right)[/itex] (or some equivalent thereof)

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.
 
  • #21
guigus said:
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.

The definition does not use the concept of number to define a number. He first defines, for any class, the "number of a class" as the class of all classes that are similar to it (presumably bijectively), and then goes on defining a "number" as anything that is a "number of a class". Succinctly; he states that a number is an equivalence class (under bijection) of classes. There is no circularity here (nor pedagogy, but I'll leave it at that).

Sure it is rigorous insofar his system of axioms is, but it is not a definition of what a number is. It is a definition of what he calls "numbers" with respect to his axiomatic system of classes, but it does not encapsulate the general concept of a number. It was proposed, as I understood OP, as a universal definition of what it means to be a number. That simply cannot be done.

Compare with the concept of a function. It has many definitions in different axiomatic settings, but they don't contradict each other, nor are they battling for the status as the "correct" definition of a function. They simply reflect how a function can be treated in the particular setting one find oneself.

The concept of a number, and a function, remains a part of language, not logic.
 
Last edited:
  • #22
guigus said:
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.
My apologies. When I said "cannot be parsed into smaller pieces", I was assuming we were not considering parsing things in an incorrect fashion -- such as parsing (as (possibly meaningless) English) the English phrase that consists of the same letters in that order.
 
  • #23
Hurkyl said:
My apologies. When I said "cannot be parsed into smaller pieces", I was assuming we were not considering parsing things in an incorrect fashion -- such as parsing (as (possibly meaningless) English) the English phrase that consists of the same letters in that order.

It is the word "parsing" that is incorrect: we are just taking the words of an English expression separately, and they do make sense as any separate words do, otherwise dictionaries would be impossible (and the expression meaningless). The word "one" in the referred expression means the number one, since it means the number of elements of a set -- the set with one element -- and if it stops meaning that, then the sentence becomes meaningless.
 
  • #24
Jarle said:
The definition does not use the concept of number to define a number. He first defines, for any class, the "number of a class" as the class of all classes that are similar to it (presumably bijectively), and then goes on defining a "number" as anything that is a "number of a class". Succinctly; he states that a number is an equivalence class (under bijection) of classes. There is no circularity here (nor pedagogy, but I'll leave it at that).

There is circularity if you use a number to define a number, as in "a number is the number of a class," although there is no circularity in "a number is an equivalence class." The same definition can be stated circularly or not, despite staying correct.

Jarle said:
Sure it is rigorous insofar his system of axioms is, but it is not a definition of what a number is. It is a definition of what he calls "numbers" with respect to his axiomatic system of classes, but it does not encapsulate the general concept of a number. It was proposed, as I understood OP, as a universal definition of what it means to be a number. That simply cannot be done.

It is "just saying" that "simply cannot be done": you must prove what you say, by showing a number that doesn't fit Russell's definition.

Jarle said:
Compare with the concept of a function. It has many definitions in different axiomatic settings, but they don't contradict each other, nor are they battling for the status as the "correct" definition of a function. They simply reflect how a function can be treated in the particular setting one find oneself.

The concept of a number, and a function, remains a part of language, not logic.

Logic itself goes far beyond "logic," and it certainly can, just as language can, define a number. And Russell's definition goes far beyond what Russell himself has imagined, as I can show you, if you wish. And yes, it defines what a number is, you just have to really understand it.
 
  • #25
guigus said:
It is the word "parsing" that is incorrect: we are just taking the words of an English expression separately, and they do make sense as any separate words do, otherwise dictionaries would be impossible (and the expression meaningless). The word "one" in the referred expression means the number one, since it means the number of elements of a set -- the set with one element -- and if it stops meaning that, then the sentence becomes meaningless.
But what AlephZero defined wasn't an English phrase. The English phrase you want to consider and the technical mathematical term AlephZero referred to are merely homonyms.

Of course, the name was chosen to be suggestive -- the technical term is meant to be logically equivalent to something we might mean by the English phrase.

The technical term doesn't define "set with ____ element(s)" where ___ is replaced by some sort of number. It only defines "set with one element".

English dictionaries are not mathematical reference books. :tongue:



(moderator's hat on) Anyways, this discussion had been dead for 5 years, is in the wrong section, and the necromancer seems to just want to be contrary, so I don't see any reason to leave the thread open.
 

What is Russell's and Frege's Definition of Number?

Russell's and Frege's Definition of Number is a philosophical theory that attempts to define the concept of number in a precise and rigorous way. It was developed separately by Bertrand Russell and Gottlob Frege in the late 19th and early 20th centuries.

What are the key principles of Russell's and Frege's Definition of Number?

The key principles of Russell's and Frege's Definition of Number include the idea that numbers are logical objects that exist independently of human minds, and that they can be defined in terms of sets or collections of objects.

How does Russell's and Frege's Definition of Number differ from other theories of number?

Russell's and Frege's Definition of Number differs from other theories of number, such as the traditional view that numbers are mental constructs, in that it seeks to provide a purely logical and objective account of numbers.

What are the main criticisms of Russell's and Frege's Definition of Number?

Some of the main criticisms of Russell's and Frege's Definition of Number include the difficulty of defining numbers in terms of sets, the problem of the "third man" argument, and the lack of a satisfactory account of the relationship between numbers and physical objects.

How has Russell's and Frege's Definition of Number influenced the field of mathematics?

Russell's and Frege's Definition of Number has had a significant influence on the field of mathematics, particularly in the development of set theory and the foundations of mathematics. It has also sparked debates and discussions about the nature of numbers and their role in mathematical reasoning.

Similar threads

Replies
3
Views
639
  • Linear and Abstract Algebra
Replies
9
Views
2K
  • Quantum Physics
Replies
22
Views
408
  • Linear and Abstract Algebra
Replies
1
Views
2K
  • Programming and Computer Science
Replies
32
Views
3K
  • Linear and Abstract Algebra
Replies
6
Views
6K
  • Set Theory, Logic, Probability, Statistics
Replies
24
Views
3K
Replies
1
Views
2K
  • Classical Physics
Replies
4
Views
2K
  • Special and General Relativity
Replies
5
Views
910
Back
Top