Confused about the concept of equality

[turin]

Ever since I took a "high-level" math class (grad level topology), I have been baffled. This baffling has been further confounded by a couple of math-methods courses in which the professor was a highly gifted and recognized mathematical physicist, and so very careful (and picky) about the mathematics. This was a dramatic mental transition for me which occurred from pre-masters degree to post-masters degree. All through my life up until this point I viewed myself as better at math than perhaps everyone that I knew (except my professors). Now, my basic conclusion is that I am terrible at math, and I literally didn't even know what math was until after my masters degree.

OK, enough about me, now let's talk about math, with whom I maintain a love/hate relationship. The speific issue that came to my mind today is the concept of equality. Also, closely related to that is the concept of identity, that is, some element (of a set) that is referred to as an identity w.r.t. some operation. I have no idea where to begin the discussion, so the rest of this post will probably appear quite clumsy. I will just throw out some more specific confusions. I invite any comments regarding the general concept of equality (and identity), as well as any affirmation that you also are confused. I wouldn't be surprised to find that all of these specific confusions are just different labels for the same confusion. Please understand that my terminology is probably sloppy.

functions vs. equivalence classes (equality vs. equivalence relation):
Basically, there seems to be a weaker notion of equality than what I "grew up with", and it allows things to be "equal" that I would not declare as equal. Furthermore, there can be different equivalence relations under which completely different objects are "equal" and "unequal". I think that this comes down to the set theory concept of a partition. As far as I know, I can arbitrarily partition a set. At least, there is more than one way to do it. And sometimes (for uncountable sets), there may be more than one way to achieve what I will call "maximum fine-graining" (that is, the maximum number of equivalence classes for the most nontrivial equivalence relation). This is the kicker, and it leads me to my next specific confusion ...

0.999... = 1:
I don't really have anything to say about this. Of course, it makes sense from a practical standpoint; I just don't get it (mathematically), which leads me to my next specific confusion ...

rationals vs. irrationals (and uncountibility of the reals):
I just can't get my mind around this, especially in light of the previous confusion. It almost seems as if efficient/practical predictibility has something to do with it. That is, we have an efficient algorithm that we can run to determine any arbitrary digit in 0.999...: it is always 9. This is extremely efficient. However, as far as I know, there is no efficient algorithm to determine an an arbitrary digit, say the 1-trillionth, of, say, √2. Apparently, it is on this basis that we distinguish between rationals and irrationals. That seems more of a practical issue than an ontological one, but I think that the distinction is intended to be ontological.

uniqueness of the null vector:
I guess this confusion arose from physics (especially QM), rather than pure math. I just don't understand why a null vector can't have an additional label. To put it mathematically, I don't understand what is wrong with the concept of an equivalence class of null vectors, or why this equivalence class must contain exactly one element. I understand equivalence classes as basically ignoring the features that you don't care about (or better yet, the features that you don' even know about). So, for instance, you might say that 0 is the null vector for the integers, and it is unique, because no other integer can be the additive identity. OK, but what if for instance there was another dimension. Then, the null vector for one dimension would be (0,n). Different n's would lead to different null vectors. They are equivalent under the equivalence class of first coordinates, but they are ontologically different. The "extra feature" could just be something that I artificially add, or it could actually represent something physical that I either don't know about or want to ignore. Anyway, it all mathematically amounts to the same thing.

automorphism, as opposed to isomorphism (in group theory):
I just don't understand how automorphism is any more specific that isomorphism (in terms of abstract groups). To me, a group begins with an abstract set, that is, a set whose only characteristic is the shear number of elements in it. Then, the set is equipped with a combination rule under which the necessary group criteria are satisfied. So, for a given set (i.e. number of elements), I can perceive of more than one valid combination rule, thus distinguishing different groups. What confuses me is how the same set can be equipped with the same combination rule, but produce two different groups in the first place, which leads me to my next specific confusion ...

abstract group vs. its representation (esp. its defining repn.):
I became confused when I learned about Lie groups, and I started to dig into the meaning of SU(2) as an abstract group, rather than just the rotation group. I cannot separate in my mind the definition of the group from its fundamental/defining representation.

I guess that's enough for now. Please tear me apart (so that I can rebuild myself).

 

[willem2]

0.999... = 1

It's just application of axioms and definitions. 0.999... is defined to be a certain limit, and that
limit is easily shown to be equal to 1.

rational vs. irrationals

a number r is rational if two integers m, n exist such that r = m/n. you can prove that rational numbers have repeated decimal expansions, but that isn't the definition, nor is it the only way to prove that a number is rational.

null vectors
you can't add vectors if the vector spaces they are in aren't both a part of the same vector space. In this vector space there is only 1 null-vector. Saying that a null vecor of a 1-dimensional space, is equivalent to the null-vector of a 2-dimensional space makes no sense if you can't add a vector in the 1-dimensional space to a vector in the 2-dimsional space.

 

[slider142]

functions vs. equivalence classes (equality vs. equivalence relation):
Basically, there seems to be a weaker notion of equality than what I "grew up with", and it allows things to be "equal" that I would not declare as equal. Furthermore, there can be different equivalence relations under which completely different objects are "equal" and "unequal". I think that this comes down to the set theory concept of a partition. As far as I know, I can arbitrarily partition a set. At least, there is more than one way to do it. And sometimes (for uncountable sets), there may be more than one way to achieve what I will call "maximum fine-graining" (that is, the maximum number of equivalence classes for the most nontrivial equivalence relation). This is the kicker, and it leads me to my next specific confusion ...

The concept of an equivalence class generalizes the concept of equality in that we use the most important algebraic concepts of arithmetic equivalence of numbers: transitivity of the relation, reflexivity, and symmetry, to put various types of objects which may not be numbers into similar types of classes. Ie., in geometry, we may want to view two objects as equivalent if they can be rigidly moved into each other where the rigid motions are defined by some functions. In number theory, it is frequently useful to define two numbers as equivalent if their difference is divisible by some fixed number. Ie., when adding two times together, 1200h and 1500h, you will want to add them to the more useful 300h, not the useless 2700h. In topology, your classes were defined by homeomorphisms, where two topological spaces were equivalent if there existed a homeomorphism between them. This allows you to classify surfaces into very useful subclasses, as well as work more generally with solutions of differential equations on manifolds. In mathematics, definitions that are useful stick around, and those that were not disappear. Equivalence classes are very useful; they allow us to do algebra with more than just numbers.

0.999... = 1:
I don't really have anything to say about this. Of course, it makes sense from a practical standpoint; I just don't get it (mathematically), which leads me to my next specific confusion ...

This is a purely mathematical proof and follows immediately from either definition of the system of real numbers. Which definition are you most comfortable with ? Dedekind cuts or Cauchy sequences? You can even prove this using a very general idea of a real number being a supremum of a set that is bounded above (the property that the set of rational numbers is a subset of the real numbers plus the property that every set of numbers in the real number system that has an upper bound also has a unique least upper bound that is a real number).

rationals vs. irrationals (and uncountibility of the reals):
I just can't get my mind around this, especially in light of the previous confusion. It almost seems as if efficient/practical predictibility has something to do with it. That is, we have an efficient algorithm that we can run to determine any arbitrary digit in 0.999...: it is always 9. This is extremely efficient. However, as far as I know, there is no efficient algorithm to determine an an arbitrary digit, say the 1-trillionth, of, say, √2. Apparently, it is on this basis that we distinguish between rationals and irrationals. That seems more of a practical issue than an ontological one, but I think that the distinction is intended to be ontological.

Newton's method superconverges to the square root of 2 from any starting point, so I'm not sure what gave you this impression of "efficiency". Irrational numbers are simply numbers that cannot be represented by p/q where both p and q are integers. That is all there is to know about them (in terms of defining properties). That is the basis on which they are distinguished; there is no other.
The uncountability of the reals comes from ingenious diagonal slash argument[/url], which was quite controversial when first presented. Please click on the link to read about the precise details; the method of proof used is very important to any mathematician and applies to more than just uncountability of the reals.

uniqueness of the null vector:
I guess this confusion arose from physics (especially QM), rather than pure math. I just don't understand why a null vector can't have an additional label. To put it mathematically, I don't understand what is wrong with the concept of an equivalence class of null vectors, or why this equivalence class must contain exactly one element. I understand equivalence classes as basically ignoring the features that you don't care about (or better yet, the features that you don' even know about). So, for instance, you might say that 0 is the null vector for the integers, and it is unique, because no other integer can be the additive identity. OK, but what if for instance there was another dimension. Then, the null vector for one dimension would be (0,n). Different n's would lead to different null vectors. They are equivalent under the equivalence class of first coordinates, but they are ontologically different. The "extra feature" could just be something that I artificially add, or it could actually represent something physical that I either don't know about or want to ignore. Anyway, it all mathematically amounts to the same thing.

You have answered this one yourself. Unique always means "unique up to equivalence class". If you want to talk about other features, such as other geometries homeomorphic to the 2-sphere, you have to exit topology and enter another category, perhaps you are interested in diffeomorphisms within that homeomorphism class and you enter differential topology, or perhaps geometry, then you enter differential geometry; each category has its own focus on the objects.

automorphism, as opposed to isomorphism (in group theory):
I just don't understand how automorphism is any more specific that isomorphism (in terms of abstract groups). To me, a group begins with an abstract set, that is, a set whose only characteristic is the shear number of elements in it. Then, the set is equipped with a combination rule under which the necessary group criteria are satisfied. So, for a given set (i.e. number of elements), I can perceive of more than one valid combination rule, thus distinguishing different groups. What confuses me is how the same set can be equipped with the same combination rule, but produce two different groups in the first place, which leads me to my next specific confusion ...

Consider the isomorphism between (Z₆, +) and (Z₂ X Z₃, +), illustrated here. This is not something you can consider an automorphism. Since there are isomorphisms that are not automorphisms and every automorphism is an isomorphism, the set of automorphisms is "smaller" or more specific, and we say it is a stronger term.

abstract group vs. its representation (esp. its defining repn.):
I became confused when I learned about Lie groups, and I started to dig into the meaning of SU(2) as an abstract group, rather than just the rotation group. I cannot separate in my mind the definition of the group from its fundamental/defining representation.

A representation is a very specific embodiment of a group as set of linear operators and thus set of matrices with composition occurring under matrix multiplication. This makes working with the group very simple, as knowledge of matrix theory and linear algebra is quite extensive. The group itself is independent of any particular representation; this type of isomorphism is just very useful and thus has garnered a name for itself.

 

[turin]

willem and slider, I do appreciate both of your comments. Thank you.

I'm afraid that the definition of real numbers must be too sophisticated for me. I have never heard of Dedekind cuts, and I do not appreciate the difference between upper bound and supremum. This is going to take me a while.

Regarding rationals, I would say that p and q can be made arbitrarily large so that r=p/q is as close as required to whatever number, rational or irrational. I have read proofs -- for instance the proof that √2 is irrational -- I just don't understand these proofs. In particular, I have seen a (the?) diagonalization proof, but I don't understand it.

I do not understand the null vector in the context of topology, only vector space. Is a vector space necessarily a topological space? I know that a vector space with a metric suggests a (default?) topology, but I didn't think that an abstract vector space (in particular before an inner product is defined on it) suggests any topology (unambiguously). My topology professor told me that dimensionality is a topological property, but he didn't even try to explain it to me, saying that it was too advanced for me. The dimensionality in my example was not intended to have a topological sense, but a sense of vector independence. Perhaps I am not allowed to separate these two notions. I suggest that I can work with the 1-D vectors as equivalence classes of 2-D vectors, so that I can add a 1-D vector to a 2-D vector by adding it to the 1-D equivalence class of the 2-D vector in order to obtain a resulting 1-D vector (which is also an equivalence class of 2-D vectors). Also, the result is an arbitrary (but restricted) 2-D vector. (Is an indeterminate result a problem? I don't recall determinacy being an axiom of vector spaces.)

@slider:

Are you saying that, once I identify the equivalence class of null vectors, this immediately suggests some very simple topology at least, e.g. the existence of at least one open set (of vectors, which are all null)?

I consider ℤ₆ and ℤ₂xℤ₃ as two different representations (a 1-D repn and a 2-D repn, respectively) of what is actually just the same group to begin with. At any rate, I certainly don't understand what prevents the automorphism of these two "groups". They have the same number of elements and the same combination structure (as they must in order to be isomorphic). In what (group theory) sense are they different? This probably means that I also don't understand group/repn theory (in addition to not understanding equivalence). I thought that the definition of a group was one of the few interesting mathematical structures that I can understand, but I suppose not.

 

[VeeEight]

I consider ℤ6 and ℤ2xℤ3 as two different representations (a 1-D repn and a 2-D repn, respectively) of what is actually just the same group to begin with.

If m and n are relatively prime, then Z_mn is isomorphic to Z_mxZ_n.

My topology professor told me that dimensionality is a topological property, but he didn't even try to explain it to me, saying that it was too advanced for me. The dimensionality in my example was not intended to have a topological sense, but a sense of vector independence. Perhaps I am not allowed to separate these two notions.

The dimension of a vector space is the number of elements in a basis for the space.
For topology, see here: http://www.math.okstate.edu/mathdept/dynamics/lecnotes/node36.html

Regarding rationals, I would say that p and q can be made arbitrarily large so that r=p/q is as close as required to whatever number, rational or irrational. I have read proofs -- for instance the proof that √2 is irrational -- I just don't understand these proofs. In particular, I have seen a (the?) diagonalization proof, but I don't understand it.

An irrational cannot be expressed as a fraction of two integers.
For the proof of '√2 is irrational', see here: http://www.homeschoolmath.net/teaching/proof_square_root_2_irrational.php.

I do not appreciate the difference between upper bound and supremum.

If A is a set, then x is an upper bound for A provided that a≤x for all a in A.
The supremum for a set is the least upper bound.
The concept for a supremum is used in analysis for things like metrics, norms, etc.