Math Discussion One

From ShiftingTheBurdenOfProof, where somebody picked a not so good example and somebody else jumped in to show off his math prowess.

For instance, in the system of numerical algebra, it is an indisputable fact that any number multiplied by zero is equal to zero, so it follows that if ab=0 then a or b is equal to zero; This logic is indisputable for this system of mathematics because the rules used to justify it are part of the definition of the very system. It is possible to define a system where 5+5 is not equal to 10; but in that case the rules are different and a subsequent proof of something is different. This form of logic is really the same as the everyday kind, but in the real world we don't have the privilege of any laws that are set in stone (even Newton's laws are based on empirical evidence), so we have to rely on observing the reactions to determine the causes instead of extrapolating outcomes from a set of rules. -- TimothySeguine

I wish you knew more about math before tossing examples based on it.

2 * 5 = 0 mod 10. Just because ab = 0 doesn't imply that a = 0 or b = 0.

If ab=0 and both a and b are non-zero, the ring is said to have zero divisors. A ring which has no zero divisors is an integral domain. Not all rings are integral domains, so the no zero divisors axiom is in addition to all the other axioms of rings. Your statement "so it follows" is simply false.

Had you meant the natural numbers under ordinary addition and multiplication, you should've said so. But even if you had, your concluding an axiom about the natural numbers is ludicrous.

Furthermore, zero is DEFINED as a*0 = 0 for every a, since not all systems have a 0, so saying that it is an "indisputable fact" is ludicrous. [nonsense]

Someone who knows so little about math really couldn't be expected to know anything about logic. Like the difference between deductive and inductive logic.

I specifically mentioned numerical algebra (to distinguish it from linear algebra, matrix math, vector math, etc.) My wording was an attempt to use non-technical terms for the description. I don't know why you brought up modular math, because it was irrelevant and I already mentioned it (I said it is possible to implement a situation in which 5+5 does not equal 10; the 5+5mod10 example you gave). Of course I mean normal addition because that is implied in my wording (while still attempting to be non-technical). I was using the mathematical subspace that is defined in layman's terms as standard arithmetic. I seriously think you should have actually read my argument as you are focusing on the example I chose to use, which was perfectly fine if you take it in context. I am well aware of the many systems of mathematics available and it was very rude of you to imply that I wasn't. This is a wiki page about rhetorical logic, not mathematical logic, and therefore I could not assume that the readers would know the things you mentioned. I could show various mathematical proofs at this point to show the reality of the ab=0 assumption with regards to different mathematical systems, but that would bore everyone and miss entirely the point I was trying to make.

To clarify what you misunderstood, I will define my example more clearly (I shouldn't have had to). For starters, we are in real number space, in the interval (-infinity,infinity). Standard arithmetic rules apply. Now given the equation ab=0 where a and b are any real numbers, we have this: Given the commutative property of multiplication, we have ab=ba=0, so in case 1 a is not equal to zero: if we multiply both sides by 1/a we get b=0*(1/a), by the zero product property b=0. case 2 b is not equal to zero: similar to the previous case, except results in a=0. case 3 a and b are both =0: this case is trivial because 0*0 is equal to zero. So we come to the conclusion that a or b must be zero. Alternatively, I could have disproved the converse (I think that is the right word, I never remember) that given ab=0 both a and b are non negative, but that is unnecessary. There - was that necessary? Regardlesss, my point holds. The fact you ignored about my post (which was the reason for posting) was that the logic of math itself was the supporting evidence for a proof. A given type of math is defined by the way addition, subtraction, the inner product (when referring to geometric spaces), etc., are defined. I think your comments were in poor taste.

That's all just so much BS. My arguments stand and you haven't addressed any of them.

You made a statement, that the definition of zero implies the absence of zero divisors, which is clearly bullshit. As an implication, your statement is technically true for the field of natural numbers, but if you had intended to be understandable to laypeople, you would not have used the technical meaning of 'implication'. In general usage, an implication means a logical cause. Yet the absence of zero divisors in the field of natural numbers does not logically follow from the definition of zero. And on the flip side, if you had meant to be technical, you would not have restricted yourself to N, R, and C. I already explained all of this when I said that your statement was either flat out false or ludicrous.

And to repeat myself yet again, your claim that all logic is alike is yet more bullshit.

It's not rude for me to claim that you don't know math and logic, because it's obvious that it's true. Or at the least, you certainly don't understand it. Just looking at what you consider to be a proof that there are no non-zero divisors in R (verbose BS that obscures, and even completely misses, the essence of the proof) ought to be sufficient to convince most people. See also UnderstandingVsKnowledge.

Beyond not understanding proofs and math, as if that weren't enough, you also don't understand any mathematical logic. If you did, you wouldn't be saying BS like "a given type of math... blah blah blah". Modular arithmetic is not a "kind of math". It's part of standard math. You could make that statement about an algebra or a language but not a "math" because there is no such thing as a "math". Unless you start playing with the axioms of mathematics, but I'm pretty sure you don't know what that term would refer to since you resort to BS like "the logic of math itself".

And you want to know what poor taste is? It's defending yourself by claiming you want to be understood by laypeople when you don't know what you're talking about. Oh, and here's a hint: don't talk down to the lowest common denominator. That's my half-dozen years experience on C2 wiki talking.

Non-zero divisors do follow from the definition of zero: http://mathworld.wolfram.com/Zero.html and http://mathworld.wolfram.com/DivisionbyZero.html. Division by zero is alway undefined, and unless you are considering limits for some purpose (in which case the limit approaches infinity), you can conclude that dividing by zero gives an undefined and irrelevant result. Modular math is a separate system from standard algebra (not talking about computer algebra here). Integral Calculus is a separate system from Linear Algebra. That is what I meant by "kind of math".

Group theory and ring theory are part and parcel of algebra. The same algebra to which the integers belong. And in group and ring theory, there are matrices and modular arithmetic. There's no such thing as "modular math" distinct from "standard algebra".

If you knew more about algebra, you'd know to interpret the definition at wolfram in a way that leaves open the possibility of divisors of zero.

I really don't care what you meant by "kind of math". You don't know what you're talking about and that's all there is to it. The fact that you believe "integral calculus" and "linear algebra" to be "systems" only underpins the depth of your ignorance. Integral calculus is no more a system than arithmetic. Same with linear algebra.

Peano set theory is a system. ZF set theory is a system. Euclid [EuclidOfAlexandria] and Reimannian geometries are systems. Linear algebra is nothing at all to mathematics, it's a kind of arithmetic. And arithmetic is nothing more than a bunch of symbols and notations useful to refer to mathematical objects, not mathematical in themselves. Math is about axioms, statements, languages and theories. It's not about the symbols you use to refer to these things. It's not about the symbol "0".

Again, you really don't understand mathematics. I highly recommend you take a course, sit in on some lectures, or just read a book on mathematical logic. You'll learn what math is really about. But if you don't understand axiomatic algebra or analysis, then you'll have a very difficult go of it.

Oh, and about defining zero. You can define it in a group. You can define it in a ring. You can define it using sums. You can define it using plums. You can eat it with eggs and ham. And I ain't no Dr Seuss, am I?

[Some of this sort of thing was discussed a few months ago in InfinityConsideredHarmful. Math is interesting; if this discussion were less confrontational, it might lead to some substantive issue(s). It doesn't ultimately matter whether someone does or does not know a topic, only whether what is said is true or not, and counter-examples (e.g. in the form of URLs) typically suffice to settle that for mathematical issues.]

I did some research on rings (honestly I never dealt with the term before) [http://en.wikipedia.org/wiki/Integral_domain], and found that the set of all real numbers is an integral domain (i.e. the product of two non-zero elements is never zero). So given ab=0, either or both has to be zero (regardless of the proof I gave earlier). I also found you were correct about modular math being arithmetic (my bad). I still think the insults were uncalled for. The wolfram site also regularly refers to the words math and algebra in the plural sense (implying different types). It also refers to all but modular math (I apologized) as a branch of mathematics. The point is I have now given you all of the things I didn't realize I needed to state before (because I was referring to elementary school algebra), and all you have done is insult me. All you did was write a rhyme when I asked you to explain a situation where dividing by zero gives a real (read real number) result.

Actually, the rhyme was in response to someone else, who is an evil-minded SOB.

I'm glad you looked things up and it seems like you have a much better basis for understanding mathematics. A first course on algebra, typically only deals with linear algebra. After that comes an axiomatic approach to algebra. You're only introduced to groups and rings in subsequent courses. And mathematical logic is considered an advanced subject.

In R, dividing by zero is not defined. Even in *R, only dividing by an infinitesimal around zero is defined. (Hyperreals are better than delta-epsilons.) In modular rings, dividing by zero is still not defined. In the ring Z mod 10, you can compute 2/3= 2*7=4 (mod 10), because 7 is the inverse of 3, but 0/2 is undefined because 2 doesn't have an inverse element. There's no x such that x*2 = 1 (mod 10). In general, a/0 is undefined because in order to define it you'd have to find an x so that x*0 = 1.

I guess I should have given you more credit earlier, because I am only an undergrad. Hey no harm done after we let things cool down. This thread should probably get deleted though; pretty OT.

Not to mention that many claims are still incorrect because somebody's too proud to correct his own errors.

[Oh dear, oh dear. It's best to avoid examples at the limits of your knowledge - it's just too easy to make mistakes. The mathematics is on the side of the more critical contributor, but his way of putting it was unnecessarily blunt and aggressive. Abstract algebra may seem approachable enough to the trained mathematician, but it is very tough for the layman. For most people, "taking a course" would be largely a waste of time.]

The problem with being blunt and aggressive is when one makes a fool of oneself. The blunt contributor has admitted that his (0/2=5 mod 10) was incorrect, but has yet to recognize that 0 is most definitely not defined as [a*0=0 for all a]; he has also to admit that the fact a*b=0 => a=0 \/ b=0 does follow from the definition of 0, contrary to his claims.

There's more than one definition of zero. a+0=a is the standard definition in an additive group, of which rings are a special case; a*0=0 is the standard definition in a multiplicative semigroup, of which rings are also a special case. I don't see where he claimed a=0 v b=0 follows; it looks like he claimed the opposite.

Speaking of such technicalities, the definition of a semigroup does not imply the existence of an element "0" such as a*0=0 forall a, therefore 0 cannot be possibly defined as claimed above (the element in a semigroup for which a*e=e for all a). A multiplicative semi-group is just a conventional name for a semigroup whose binary operation is denoted by the symbol "*".
True, the definition of a semigroup doesn't imply such elements exist. However, that doesn't change for one moment that when they do exist, they're called zeros, or at least the term is defined this way in several texts. The multiplicative is partly just notation, but rings are semigroups under two different operations, and this specifies which way they're to be included.
But speaking to the substance, the axioms of a group (or semigroup, ring, etc) do not define a "0". They define a theory wherein particular symbols like '0','+' are nothing but symbols denoting any number of different entities in different settings. For example, there's nothing preventing one from saying that 0 stands for the identity endomorphism, + stands for composition, and voila, we have the group of endomorphisms, where 0 stands in for anything but what one would expect (i.e. it has nothing of the characteristics ordinary people associate with 0, whereas 0 in say a vector space would be much more recognizable as 0). Only a concrete model for group/ring etc. will give a proper definition for a particular 0, and depending on the formalism, 0 can be properly defined for example as the cardinality of the empty set. It was obvious from the context that the initial contributor meant 0 as defined as an integer number, and not 0 as stand-in symbol for any number of elements in groups/rings, etc.
In groups described using additive notation, the identity may be called the zero; once again, the notation relates to how rings are considered. In rings, the additive identity serves as a common definition for zero, which happens to match the ones for both groups and semigroups. You will often hear mathematicians talk about zeros in modular arithmetics, field extensions, and yes, even endomorphism rings, although none of those have much to do with cardinalities. It's not reasonable to say you need a concrete formalism before you decide what zero is, it's defined by its properties. Whether the context of the original writer was clear or not, I don't know; obviously not everyone interpreted him as restricting his attention to the integers.
Then speaking of 0 as a natural (integer/rational ... etc.) number, it is then pretty obvious for somebody with a background in math that the statement a*b=0 => a=0 \/ b=0 derives from the very definition of natural numbers and therefore from the very definition of 0. [But not directly from a*0=0, as was claimed.]

The "critical guy" makes a funny confusion between the definition of 0 in abstract algebra, which is merely a notational convention for the neutral element in an additive group (and respectively rings, fields, etc) and the symbol 0 used to denote the starting element in the standard axiomatization of numbers. For 0 as a number (rather than a conventional notation) its definition together with the definitions for N, Z, Q imply that (Z,+,*), ( Q,+,*), (R,+,*) are integral domains, and it is derived as a theorem that 0 as a number in the axiomatization of arithmetics maps to 0 as the neutral element for addition in the abstract algebra definition of ring/field, etc. So the first contributor was technically correct in asserting that "a*b=0 => a=0 \/ b=0 follows from the definition of 0", because he meant the definition of 0 as a natural number.

That wasn't his assertion. He claimed a*0=0 implies "a*b=0 => a=0 \/ b=0".

All in all much ado about nothing, just because somebody wanted to show off and he didn't have much to show for it.

[If anyone reading this page believes they are competent enough in abstract algebra to do more than quote from textbook definitions, I suggest they have a go at contributing to either AlgebraicHoop or AlgebraicHoopConstruction.]

Enough flip flapping your jaws and hyperactively extending your fingers on the keyboard. The original poster is correct that any form of math is based on rules and definitions and that is the main point of the story. If you just follow them rules and definitions without proof, you may end up falling over the edge of a cliff due to a miscalculation. Some rules are in place just because they sound good. I'm not gonna get real technical because people who babble too much are boring and don't think for themselves; they just follow the leader and never lead. So anyway, add any two negative numbers and the result for some is a negative number due to rules and definitions. Multiply any two negative numbers and the result for some is a positive number due to rules and definitions. So after that rather primitive example, if you believe in the additive inverse rule you are a follower. On the other side, if you multiply any two negative numbers and get a negative number then you are a leader. The additive inverse rule is false. Try working that one out with logic. If you reply, please don't bore me with mathematical rules and definitions that are false or have no evidence to prove they are correct.