Is an algebraic structure a set?

[bentley4]

A set is defined as a collection of 'objects'. Does there exist something that is not considered an object in mathmatics?
The reason why I ask is because when I look up the definition of a ring, it says a ring is an 'algebraic structure' existing of a set and 2 binary operations.
Why can't one say a ring is a set that includes 2 binary operations?
Do they give this definition because depicting an object of a set neglects semantics?
If that was true then we wouldn't be able to calculate anything with numbers, because numbers are just an agreed representation of a concept communicated through informal human language, right?

 

[micromass]

A set is defined as a collection of 'objects'.

This is not the exact mathematical definition of a set. Sets would be ill defined if we just said that it's a "collection" of "objects". Sets are entities that fulfill certain axioms (=ZF axioms). For practical purposes, most mathematicians indeed see a set as a collection of objects. But this is the informal meaning of set.

Does there exist something that is not considered an object in mathmatics?

The things that are the elements of sets, are again sets. In (standard) mathematics, all is sets!

The reason why I ask is because when I look up the definition of a ring, it says a ring is an 'algebraic structure' existing of a set and 2 binary operations.
Why can't one say a ring is a set that includes 2 binary operations?

So, you would say that a ring R is a set such that $$+\in R$$, why in Earth would you want that? Or did you mean something else?

Anyway, it doesn't really matter what a ring is, it matters what the properties of rings are.

 

[Fredrik]

Several different structures can have the same underlying set. These are some examples from a post I wrote a few months ago:

ℝxℝ is a set.
(ℝxℝ,addition) is an abelian group.
(ℝxℝ,addition,scalar multiplication) is a vector space over ℝ.
(ℝxℝ,addition,multiplication) is a field. (To be more specific, it's the field of complex numbers).
(ℝxℝ,addition,scalar multiplication,inner product) is a Hilbert space over ℝ.
(ℝxℝ,addition,scalar multiplication,norm) is a Banach space over ℝ.
(ℝxℝ,addition,scalar multiplication,norm,multiplication) is a Banach algebra over ℝ.

However, each of these structures is defined as an ordered n-tuple, for some integer n, and ordered n-tuples are sets too. For example, (x,y) is defined as {{x},{x,y}}, and ordered n-tuples with n>2 are defined similarly. In fact, in ZFC set theory, which is usually taken as the foundation of mathematics, every "thing" that's a member of a set is as set.

There are however other set theories, and even things that aren't set theories, that can also be thought of as "foundations" of mathematics. So the question of whether everything in mathematics is a set doesn't have a definitive answer. It depends on what you choose to think of as the foundation of it all.

 

[bentley4]

This is not the exact mathematical definition of a set. Sets would be ill defined if we just said that it's a "collection" of "objects". Sets are entities that fulfill certain axioms (=ZF axioms). For practical purposes, most mathematicians indeed see a set as a collection of objects. But this is the informal meaning of set.

The only axiom in the ZFC that I found that tells something more about the characteristics of the elements of a set is the well-ordering theorem(axiom). This is the basis of the usefulness of numbers compared to words in math(I think). The axioms don't tell anything on how we should treat words.
Ah, ok. I think I understand now, the fact that we are using human language anyway in the axioms implies sufficiently that we don't need to axiomise that we suppose the current semantics of natural language and that the basis of mathmetics is subjected to a minimal arbitrariness. This is strictly theoretically speaking because I can't really think of essential words used in definitions or axioms that have a clear arbitraryness in them.(for example I can't imagine anyone having a different interpretation of the words 'number' or 'if'. Or not having an idea of which number is smaller then another one(this is supposed by the zfc)).

In wiki: an operation ω is a function of the form ω : V → Y, where V ⊂ X1 × … × Xk.
But that uses a multiplication in its definition which is again an operator!(but maybe they don't mean an operation as in multiplication or addition). So ok, an operation can be expressed as a function apparently.(I would like to understand everything in math using only the concepts of set, relation, a minimum of natural language concepts(approx. first order logic) and the axioms.)

Anyway, it doesn't really matter what a ring is, it matters what the properties of rings are.

I agree that knowing its properties is most important yet I think it is interesting to know what it is because I want to be able to strictly classify things which makes it far easier to understand and remember. I also crave for conceptual dichotomy.

 

[micromass]

The only axiom in the ZFC that I found that tells something more about the characteristics of the elements of a set is the well-ordering theorem(axiom).

Hmm, you won't find any axiom that actually tells you what the element of a set is. What you do find (in standard ZF set theory) is that everything is a set. And what is a set? Any entity that behaves as demanded in the axioms. We can't tell you exactly what a set is, but we can tell you how a set behaves, which is all we need really.

This is the basis of the usefulness of numbers compared to words in math(I think).

I think you might interpreted the well-ordering theorem wrong. Numbers are useful in math even without well-ordering In fact, well-ordering is such a controversial axiom, that ZF specifically doesn't allow it. You'll need ZFC for it.

The axioms don't tell anything on how we should treat words.

Words don't exist in mathematics. They have no meaning and no existence. Only sets exist.

Ah, ok. I think I understand now, the fact that we are using human language anyway in the axioms implies sufficiently that we don't need to axiomise that we suppose the current semantics of natural language and that the basis of mathmetics is subjected to a minimal arbitrariness. This is strictly theoretically speaking because I can't really think of essential words used in definitions or axioms that have a clear arbitraryness in them.(for example I can't imagine anyone having a different interpretation of the words 'number' or 'if'. Or not having an idea of which number is smaller then another one(this is supposed by the zfc)).

Like I said, there can not be arbitraryness in mathematics. That is why words are not allowed. The real game of mathematics is played with symbols and well-formed formulas...

In wiki: an operation ω is a function of the form ω : V → Y, where V ⊂ X1 × … × Xk.
But that uses a multiplication in its definition which is again an operator!(but maybe they don't mean an operation as in multiplication or addition). So ok, an operation can be expressed as a function apparently.(I would like to understand everything in math using only the concepts of set, relation, a minimum of natural language concepts(approx. first order logic) and the axioms.)

How is multiplication used in the definition of an operator?? That isn't the case, otherwise, it wouldn't be a good definition...

I agree that knowing its properties is most important yet I think it is interesting to know what it is because I want to be able to strictly classify things which makes it far easier to understand and remember. I also crave for conceptual dichotomy.

Well, I understand your point. But the reason why they don't do this from the start is because most students are not ready for it. They simply don't see the reason for set theory and stuff.

A suggestion: buy the book "Introduction to set theory" from Hrbacek and Jech. It'll answer many questions for you! Maybe a logic-book wouldn't hurt too...

 

[bentley4]

We can't tell you exactly what a set is, but we can tell you how a set behaves, which is all we need really.

Thnx, that is a good answer.

Words don't exist in mathematics. They have no meaning and no existence. Only sets exist.

Can you drop all words in theorems so that they still make sense, even in their strictest representation? (Not that I'm aware of at least). Anyway, words are also symbols, agreed representations for a concept.
I think this might depend on your view of what math is. If you regard the description of a theorem as metamath and not math itself then I agree with you.

Like I said, there can not be arbitraryness in mathematics. That is why words are not allowed. The real game of mathematics is played with symbols and well-formed formulas...

All content of those symbols is only possible through the existence of natural language.
More accurately, through the existence of what are called 'semantic primes'.
"a basic set of innate 'concepts', or perhaps more precisely, a non-conscious propensity and eagerness to acquire those concepts and encode them in sound-forms (words). The words that those concepts become encoded in what is called semantic primes, or alternatively, semantic primitives — 'semantic' because linguists have assigned that word in reference to the meaning of words (=linguistic symbols). Words that qualify as semantic primes need no definition in terms of other words. In that sense, they remain undefinable."
(Anna Wierzbicka, Semantics: Primes and Universals, 1996) see also wiki for the list of semantic primitives(+- 60 words). Mathmatical descriptions don't need all of these but they do require some complex composites.
So math symbols are just other, maximally uniform and very efficient symbols based on natural language. Since words are prone to different interpretations, some people might initially misinterpret certain math symbols. But the more they use this language the better it will be. (but depends again if you see the descriptions of theorems itself not as math or not, or only their concepts. If you see math as having totally no arbitrariness then you must suppose that pure math only exists in on's thoughts and not on paper.)

How is multiplication used in the definition of an operator?? That isn't the case, otherwise, it wouldn't be a good definition...

Srry, I understood those small x's denoted multiplication here:
" ω : V → Y, where V ⊂ X1 × … × Xk."

A suggestion: buy the book "Introduction to set theory" from Hrbacek and Jech. It'll answer many questions for you!

Thnx for the hint.

Maybe a logic-book wouldn't hurt too...

Lol

 

[micromass]

Mathematics is done without using words. See http://us.metamath.org/ to see how mathematics should really be done: all symbols, no words.
The only reason why we use words is to facilitate the reading. It's easier to see a written proof, then just reading a series of symbols. BUT we should be aware that the words are not the real kind of mathematics, just like the words in a cookbook are not cooking. It is just the description of how you should do the proof. The real proof and the real theorems uses only symbols, and no words.

All content of those symbols is only possible through the existence of natural language.
More accurately, through the existence of what are called 'semantic primes'.
"a basic set of innate 'concepts', or perhaps more precisely, a non-conscious propensity and eagerness to acquire those concepts and encode them in sound-forms (words). The words that those concepts become encoded in what is called semantic primes, or alternatively, semantic primitives — 'semantic' because linguists have assigned that word in reference to the meaning of words (=linguistic symbols). Words that qualify as semantic primes need no definition in terms of other words. In that sense, they remain undefinable."
(Anna Wierzbicka, Semantics: Primes and Universals, 1996) see also wiki for the list of semantic primitives(+- 60 words). Mathmatical descriptions don't need all of these but they do require some complex composites.
So math symbols are just other, maximally uniform and very efficient symbols based on natural language. Since words are prone to different interpretations, some people might initially misinterpret certain math symbols. But the more they use this language the better it will be. (but depends again if you see the descriptions of theorems itself not as math or not, or only their concepts. If you see math as totally no arbitrariness then you must suppose that pure math only exists in on's thoughts and not on paper.)

Like I said, math needs no words and needs no interpretation. So the problem is evaded. There is of course a standard interpretation of mathematics, but that is not mathematics but some kind of metamathematics.

Srry, I understood those small x's denoted multiplication here:
" ω : V → Y, where V ⊂ X1 × … × Xk."

No, these x's denote the cartesian product of sets. It certainly doesn't denote multiplication of elements in V...

 

[bentley4]

Mathematics is done without using words. See http://us.metamath.org/ to see how mathematics should really be done: all symbols, no words.
The only reason why we use words is to facilitate the reading. It's easier to see a written proof, then just reading a series of symbols. BUT we should be aware that the words are not the real kind of mathematics, just like the words in a cookbook are not cooking. It is just the description of how you should do the proof. The real proof and the real theorems uses only symbols, and no words.

The parrallel of cooking you give is correct I think. The act of cooking is equivalent to the act of performing mathematics in your head. Thnx for the link. I think I understand what you mean.(but I still don't agree about the symbols but nm)

No, these x's denote the cartesian product of sets. It certainly doesn't denote multiplication of elements in V...

A product is the result of 'a' multiplication right?

 

[Fredrik]

A product is the result of 'a' multiplication right?

The Cartesian product of two sets X and Y is another set. It's denoted by X×Y. Its members are all the ordered pairs (x,y) such that x is a member of X and y is a member of Y. (Ordered pairs are defined as in my previous post). There's clearly no multiplication involved in this definition.

 

[bentley4]

Thnx Fredrik,

Now I understand that the cartesian product does not use multiplication in its definition. The term product is a bad choice of words here or its definition is broader then for what is it generally used.
http://en.wikipedia.org/wiki/Multiplication

 

[Stephen Tashi]

The term product is a bad choice of words here or its definition is broader then for what is it generally used.
http://en.wikipedia.org/wiki/Multiplication

One of the meanings discussed in that article is that a product is multiple copies of something. That's an apt description of a Cartesian product of sets. In same spirit, there are also "direct products" and "tensor products".

 

[Stephen Tashi]

Mathematics is done without using words. See http://us.metamath.org/ to see how mathematics should really be done: all symbols, no words.

Some sort of orderly manipulation of symbols can be done without using words ( if we don't count words as symbols) , but I wouldn't equate that to Mathematics. It's not clear how much of Mathematics can be put into an orderly symbolic system. If someone claims they have put the Mathematics of some subject like ring theory into a symbolic system, it isn't possible to verify this claim without getting into semantics and semantics goes beyond symbols and syntax.

To say that Mathematics "really should be done" as a orderly manipulation of symbols is an extreme position. I'm somewhat sympathetic to it. The tangled and ambiguous way that beginners use words when they talk about mathematics is motivation for a teachers to avoid the verbal problem altogether. Also, what is called "the power of mathematics" often amounts to the ability to get correct results by manipulating symbols and not having to think about the details of what one is doing.

However, I doubt that it is possible to teach even secondary school math to human beings as a completely symbolic system. People that are interested in teaching computers to do mathematics are recognized type of specialist in the fields of Computer Science and Mathematics, but it isn't accurate to characterize their work as the essence of Mathematics.

 

[bentley4]

Some sort of orderly manipulation of symbols can be done without using words ( if we don't count words as symbols) , but I wouldn't equate that to Mathematics.

Thank you stephen, I think this is the root of most of my disagreement/confusion.

It's not clear how much of Mathematics can be put into an orderly symbolic system. If someone claims they have put the Mathematics of some subject like ring theory into a symbolic system, it isn't possible to verify this claim without getting into semantics and semantics goes beyond symbols and syntax.

To say that Mathematics "really should be done" as a orderly manipulation of symbols is an extreme position. I'm somewhat sympathetic to it. The tangled and ambiguous way that beginners use words when they talk about mathematics is motivation for a teachers to avoid the verbal problem altogether. Also, what is called "the power of mathematics" often amounts to the ability to get correct results by manipulating symbols and not having to think about the details of what one is doing.

However, I doubt that it is possible to teach even secondary school math to human beings as a completely symbolic system. People that are interested in teaching computers to do mathematics are recognized type of specialist in the fields of Computer Science and Mathematics, but it isn't accurate to characterize their work as the essence of Mathematics.

This is exactly the kind of explanation I was looking for, great!

 

[micromass]

Some sort of orderly manipulation of symbols can be done without using words ( if we don't count words as symbols) , but I wouldn't equate that to Mathematics. It's not clear how much of Mathematics can be put into an orderly symbolic system. If someone claims they have put the Mathematics of some subject like ring theory into a symbolic system, it isn't possible to verify this claim without getting into semantics and semantics goes beyond symbols and syntax.

To say that Mathematics "really should be done" as a orderly manipulation of symbols is an extreme position. I'm somewhat sympathetic to it. The tangled and ambiguous way that beginners use words when they talk about mathematics is motivation for a teachers to avoid the verbal problem altogether. Also, what is called "the power of mathematics" often amounts to the ability to get correct results by manipulating symbols and not having to think about the details of what one is doing.

However, I doubt that it is possible to teach even secondary school math to human beings as a completely symbolic system. People that are interested in teaching computers to do mathematics are recognized type of specialist in the fields of Computer Science and Mathematics, but it isn't accurate to characterize their work as the essence of Mathematics.

Hi Stephen

You are of course correct. I was probably a bit too fast and I should have mentioned that there are other point-of-views of mathematics. The point-of-view that I mentioned is the formalism of Hilbert. I wouldn't exactly call it an extreme point-of-view as there are many mathematicians who adhere to it. The benifit of formalism is that it eliminates discussion on what "words" are and what the "meaning" of mathematics is. However, all mathematicians DO give meaning to mathematics, and thus formalism is not the philosophical right way of viewing mathematics. But I still like it, and it my opinion, it's the only philosophy that let's one do mathematics without being annoyed by questions of "meaning"...

I never said that teachers should explain mathematics as a manipulation of symbols, or that textbooks should write mathematics that way. That would be madness

The OP might be interested in this wikipedia article: http://en.wikipedia.org/wiki/Philosophy_of_mathematics

 

[bentley4]

Hi Stephen

You are of course correct. I was probably a bit too fast and I should have mentioned that there are other point-of-views of mathematics. The point-of-view that I mentioned is the formalism of Hilbert. I wouldn't exactly call it an extreme point-of-view as there are many mathematicians who adhere to it. The benifit of formalism is that it eliminates discussion on what "words" are and what the "meaning" of mathematics is. However, all mathematicians DO give meaning to mathematics, and thus formalism is not the philosophical right way of viewing mathematics. But I still like it, and it my opinion, it's the only philosophy that let's one do mathematics without being annoyed by questions of "meaning"...

I never said that teachers should explain mathematics as a manipulation of symbols, or that textbooks should write mathematics that way. That would be madness

The OP might be interested in this wikipedia article: http://en.wikipedia.org/wiki/Philosophy_of_mathematics

Thank you, so next time I ask a question touching the foundations of math I should ask for the formalism you use to be sure our disagreement is not rooted because we use a different point of view. The Hilbert formalism you use, do you mean what is called the http://en.wikipedia.org/wiki/Hilbert%27s_program" ?

 

[bentley4]

How did you expect me to know you were using the Hilbert formalism when I'm clearly not as well versed in math as you?

 

[micromass]

Thank you, so next time I ask a question touching the foundations of math I should ask for the formalism you use to be sure our disagreement is not rooted because we use a different point of view. The Hilbert formalism you use, do you mean what is called the http://en.wikipedia.org/wiki/Hilbert%27s_program" ?

Yes, I mean Hilbert's program. Although Hilbert's program was essentially destroyed by Godel's incompleteness theorem, I still think that Hilbert's philosophy is still very attractive nowadays.

I'm very much interested in set theory and logic, and you'll see many people in those fields adhering to Hilbert's philosophy. But I understand that outside the foundations of mathematics, other philosophies are used.

 

[bentley4]

To say that Mathematics "really should be done" as a orderly manipulation of symbols is an extreme position. I'm somewhat sympathetic to it. The tangled and ambiguous way that beginners use words when they talk about mathematics is motivation for a teachers to avoid the verbal problem altogether. Also, what is called "the power of mathematics" often amounts to the ability to get correct results by manipulating symbols and not having to think about the details of what one is doing.

Do you know how this point of view is generally called?

 

[micromass]

How did you expect me to know you were using the Hilbert formalism when I'm clearly not as well versed in math as you?

I didn't expect you to use anything. I just explained you the point-of-view of set theorists. I saw that you disagreed with this point-of-view, which is fine since it's only a philosophy.
I did make the mistake not to say that there were other philosophies out there, so I'm sorry for that. But I didn't expect the discussion to turn into a philosophical one, so I just presented the philosophy that is commonly used in discussing set theoretic issues...

 

[bentley4]

Yes, I mean Hilbert's program.

Thnx

But I understand that outside the foundations of mathematics, other philosophies are used.

You mean 'outside' of the agreed/shared framework of mathematics there are different interpretations which are the foundations of math.(But this again depends on what you mean by foundation/ aaargh language)

 

[bentley4]

I didn't expect you to use anything. I just explained you the point-of-view of set theorists. I saw that you disagreed with this point-of-view, which is fine since it's only a philosophy.
I did make the mistake not to say that there were other philosophies out there, so I'm sorry for that. But I didn't expect the discussion to turn into a philosophical one, so I just presented the philosophy that is commonly used in discussing set theoretic issues...

Thnx, that is kind of you to say!

 

[micromass]

You mean 'outside' of the agreed/shared framework of mathematics there are different interpretations which are the foundations of math.(But this again depends on what you mean by foundation/ aaargh language)

Yes, that is correct. There is a general idea on what mathematics should contain: it should contain analysis on the real numbers, it should contain group theory, probability, etc. But there is no consensus on what the foundations on mathematics really are. You have people suggesting other interpretations of mathematics (like formalism, platonism, realism, constructivism,...) and you also have many kinds of foundational theories (like ZF, which is by far the most popular, but you also have NBG and NF, for example).

All these foundations and interpretations have their advantages and disadvantages. You should look into some logic books, set theory books and philosophy of mathematics books to really get an explanation of what's going on.