'A trio of men, for example, is an instance of the number 3, and the number 3 is an instance of number;?

Question

but the trio is not an instance of number'. This is a statement which I found in Bertrand Russel's 'Intro. to mathematical philosophy'.

Can anyone explain this to me as in the way I see it, if a trio is an instance of the no. 3 and the no. 3 is an instance of number, then automatically a trio to me seems like an instance of number.

Answer 1

"Trio" is an "ordinal measurement" according to this link:



"ordinal measurement: A variable is ordinally measurable if ranking is possible for values of the variable. For example, a gold medal reflects superior performance to a silver or bronze medal in the Olympics, or you may prefer French toast to waffles, and waffles to oat bran muffins. All variables that are cardinally measurable are also ordinally measurable, although the reverse may not be true." http://www.unc.edu/depts/econ/byrns_web/Economicae/Essays/CardOrdinal.htm



Ordinals (1st,2nd,3rd,etc.) and cardinals (1,2,3,etc) are methods of categorizing what sort of number a number is. "Trio" is cardinal according to the above because its has "measurable ranking", being above "duet" and below "quartet".

Answer 2

Here is a Parallel: The Belgian Parliament is an instance of a Goverment. A Goverment is an Instance of Governance (anarchy is another). But the Belgian Parliament is not an instance of Governance. Likewise, a trio is not an instance of Number even if it is an instance of a number.

Answer 3

You and Russell are thinking about two different things and there is no objective "who is right and who is wrong" on the matter.



Russell defined particular numbers as the sum of all their instances. What 2 was was all the two-sets or pairs, 3 was all the three-sets or triples, 4 was the quadruples, etc. He then defined another object, which you might call "natural-numerosity" or just "number" as the set of all the previously defined particular numbers. You and Russell coincide on the first level of abstraction but diverge on the second level..



Here's a graphic way of showing what Russell has in mind: When I bracket something I'm abstracting away from it as an individual to it as a "group" or set.



trio (the name for the triple sets)

3 = {trio} (the set of all triple sets)

4 = {tetra}

number = {{},{uno},{duo},{trio},{tetra},...} = {0,1,2,3,4,...}



So number does not have trio as a member or "instance". Instead it has only 3, and 4, and so on as a member or "instance".



And here's what you have in mind:



trio

3 = {trio}

4 = {trio,{trio}} = {trio,3}

5 = {trio, {trio},{trio, {trio}}} = {trio,3, 4}

6 = similarly formed

number = {0,1,2,3,4,...}



So in this case number does have trio (and also 3, and 4, and so on) as a member or "instance".



If you're having difficulty understanding what you just read then look back to the definition of 4, 5, etc..



It turns out that your way of thinking about numbers is a generally preferred way among set theoreticians, offers certain symbolic advantages, and it even has a name: "Kuratowsi numbers". But Russell's {{{}}} way is, in one obvious respect, simpler: It's simpler to draw. Ultimately it's not really any worse than Kuratowski's {{},{{}}} way. You just have to be clear on which you're using and thinking about.



The formal way of stating the difference in Russell's and Kuratowski's numbers is to say that with Kuratowski the condition for membership in the set is *inclusion* in a previously-formed set. For Russell the condition for membership in the set is *membership* in a previously-formed set.



So, simply put, you are thinking about inclusion and Russell was thinking about membership. There is no objective right or wrong here but it is important to keep the difference straight.



Some basic definitions:



- membership is a listing or gathering up of things into sets of things: the things can be sets themselves or nonsets. membership is a nontransitive relation. You're a member when you get gathered up into a set, and you're a set when you have some members down to and including no members ... the set with no members has a special name: "null". You're a nonset if you neither have members nor are null.



- partial inclusion is when one set has some of the members as another.



- complete inclusion is when one set has all the members of another. complete inclusion is a transitive relation.



- complete proper inclusion (or just proper inclusion) is when one set has all the members of another and not vice-versa.



Here are some further examples:



A: {w,x,y}

B: {w,x,y,{w,x}}

C: {w,x,{w,x,y}}

D: {w,x{w,x}}

E: {z}

F: {{z}}

G: {z,{{z}}}

H: {z,{z}}



Now, notice the following things:



1) B includes all of the members of A but does not have A itself as a member.

2) C has A itself as a member but does not include all of the members of A.

3) D neither includes A's members nor has it as a member.

4) E is not the same thing as z even though z is E's only member.

5) F is neither the same thing as E nor as z. F's sole member is E.

6) G has F but not E as a member. It also includes all the members of E but none of F.

7) H has E but not F as a member. It also includes all the members of E and of F.



If you want to begin to understand why it's so very important to keep the difference between inclusion and membership clear you can review 1-8 and maybe come up with your own examples. If you want a stumper you can also ask yourself "What is included in E? in F?". Remember: this is not the same question as "What is E's member? F's?". The answer to the first question is very different from the answer to the second.



There are important logical and philosophical issues wrapped up in an individual's preference for thinking in terms of inclusion vs. membership. I tend to prefer inclusion.



The study of the inclusion relation and the development of alternative notions of inclusion is called "mereology" and has also gone by other names such as "region connection calculus" and "protothetic". The study of the membership relation and the development of alternative notions of membership is called "set theory" or sometimes "class theory".



These ideas are very simple to grasp once you get the hang of them despite their initial strangeness. If you can grasp the difference then you will have improved your critical thinking and grasped some of the most important but often-neglected ideas developed throughout the 19th and 20th centuries.



Happy Thinking.

Answer 4

You're overthinking it.



It's a distinction between mathematical sets and linguistic sets.

A trio is the same thing as 3, but you can't use 'trio' in mathematical language. It's just not practical.



If we combine math and linguistic convenience, our math would be entirely illogical.



12 can be a dozen. 13 can be a baker's dozen. Is 'baker's' therefore equal to '1'?



So yes, a trio is an instance of three occurring in the real world, but we can;t plug the real world into the abstract world of math and not get illogical word salad.



Hope that helps.