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 occured 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).