If A = 0 and B = 0, are A & B eaual to each other?
I’ve seen it as part of the definitioon of an equivalence relation in mathatmatics, i.e., “~” is an equivalance relation means that (A~C and B~C imply A~B). Other parts of the definition are that (for all A, A~A) and (A~B implies B~A).
Equality is an equivalence relation.
Yes. I believe it’s the transitive rule. http://en.wikipedia.org/wiki/Transitive_relation
Under the normal usage of the = operator, that does follow by the transitive property of an equivalence relation. Note that two objects can be equal without being identical, so A = 0 and B = 0 doesn’t imply that A and B are the same object.
What if A and B are different types of variables? What if one of them = 0 the number (i.e. the natural number, or the real number, or the complex numer) and the other one = the zero vector (of some size), or the zero function, or the zero matrix (of some size), or something like that?
Then they’re not identical, but they may be equal, depending on how your equality operator is defined.
Doesn’t the transitive property live within certain defined types of mathematics? IOW, it’s not intended to be a universal truism, but a defined type of relationship inside a larger world of rules.
As long as you’re not doing anything that requires second- or higher-order logic, equality behaves the way you’d expect. It may well also work in higher-order logics; I’m just not familiar enough with them to say.
I would second this falling under the transitive relation, but then again, such a vague question warrants a vague answer…
If A = 0 and B = 0 then A = B. Equality denotes identity, which is obviously an equivalence relation (see elsewhere in this thread).
On the other hand, what do you mean by A = 0? What is 0? What is our universe of discourse? Is “=” an approximation arrived at by identifying isomorphism classes? Might A and B be isomorphic, yet not equal? There’s a lot of wiggle room in your question as stated.
Wiggle room is right. Another possibility is that the equality is being used in the context of a physical measurement. So then “A=0” really means “A=0 plus or minus margin of error”. If the margin of error is, say, 3, then you could have A “=” 0 (but really A = 3) and B “=” 0 (but really B = -3). Therefore, since A and B differ by more than the margin of error, you do not have A “=” B (much less A=B).
The transitive property works for real numbers. If A and B are merely numbers not relating to anything, or numbers relating to the same thing, then A = B. However, if A is the temperature in degrees Fahrenheit (for example) and B is temperature in degrees Celsius, then A ≠ B. You must watch your units. As John Allen Paulos put it, 1 cup of water plus 1 cup of popcorn does not equal 2 cups of soggy popcorn.
Nah, there are all kinds of notions of equality that don’t imply identity. Consider for a moment the standard construction of Z from N, where we take ordered pairs of naturals and say that (a, b) = (c, d) iff a + d = b + c. So (3, 2) = (2, 1), even though those aren’t identical.
The thing you have to remember is that identity is necessarily a very strictly defined relation, but anything that’s reflexive and allows substitution works for equality. See section 2.8 of Mendelson’s Introduction to Mathematical Logic, 4e, for more details.
No, you do have identicality of equal objects. By writing “(3, 2) = (2, 1)”, you are indicating that you have passed to the set of equivalence classes, where (3, 2) and (2, 1) are just two different symbols for one and the same element. Therefore, they are identical, not just equal.
Similarly, you have that 0.999… is identical to 1 (not just equal) because “0.999…” does not designate a geometric series. It designates an *equivalence class of * geometric series, the same class designated by “1”.
It indicates no such thing. = does not denote identity, and I meant the objects (3, 2) and (2, 1). All that = denotes is a relation which is reflexive and allows substitution of unquantified variables (with a couple extra caveats). See pp. 99-100 of Mendelson’s book for an explicit discussion of this.
I’m not sure where you got this, but it’s just wrong. A decimal representation of a real number is defined to be a single geometric series, and the number it represents is the sum of that series. See chapter 3 of Rudin’s Principles of Mathematical Analysis for a more detailed discussion.
No, you’re oversimplifying here. The construction does not say that integers are pairs of natural numbers. It says that integers are equivalence classes of pairs of natural numbers under a certain equivalence relation. The classes of (3,2) and (2,1) are equal because (3,2) and (2,1) are equivalent.
Mendelson is decategorifying. He passes without mention from discussing equivalence classes do discussing representatives, which is amazingly, hideously bad form in the long run. You can kinda sorta get away with it at the level of 2-categories, and maybe even 1-monoidal 2-categories, but when you hit 3-categories you can not strictify in general and treat isomorphism as identity.
Thanks for more than I expected to learn.
Things that appear simple may not be quite so.
:eek: 
:eek: :eek: :eek: :eek:
and that, folks, is why I will leave maths to the mathematicians.
Si
“Decatgorifying” is a fancy word for “treating equivalent things as if they were actually equal”, which is the real philosophical nub here.
Consider finite sets of things. For each one we can count the number of elements in it and get a number. We can say that two sets have the same number of things if we can pair them off.
For instance, if I have some apples on a table, I look at them in turn and raise a finger each time i move from one to the next, and at the end I have three fingers up, then there must be three apples on the table. What I’ve done is describe a function from the set of apples to the set of raised fingers, and at the end the two sets must have the same number of elements: three.
When we just talk about counting numbers (fancy word: “cardinals”) and forget about the explicit function pairing off the elements of two sets, we are “decategorifying” finite sets into cardinal numbers. We can do arithmetic on cardinal numbers by forgetting everything about a set but the number of elements – its “cardinality” – and treating equivalent sets (same cardinality) as if they were equal. Nobody would say that the set of apples and the set of raised fingers are “equal” – one is made of apples and the other of fingers – but they are equivalent in this sense.
What I went on to say is that we just do this sort of thing naturally because most mathematical structures we’re concerned with are actually what we call “categories”, in which we can define a notion of equivalence. If we forget all about on object in a category (a set of three apples) but which equivalence class it belongs to (“three”) we are “decategorifying”. Ultimately for simple categories this doesn’t really cause much of a problem. However, there are more intricate structures called “higher categories” where we can talk about things, how those things are equivalent, how those equivalences are equivalent, and so on. Once you get a few steps up this ladder, you lose essential information about the structure when you decategorify, so it becomes extremely important to keep track of what things are really equal, and what things are merely equivalent.