Real Number Operations Caveat

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I too think we are.

I don't profess that my above analogy is particularly accurate(?). I constructed my "why" sentence purely by taking the sentence "The cat sat on the mat" and replacing the word "mat" with an alternative that was... shall we say less immediately comprehensible.

I was well aware that one could approach what I came up with in a similar way to how one can drag semantics out of "Time flies like an arrow" (a minimum of three "meanings"). One can similarly look at "0^0" and come up with reasons(sic) why it should be 0, 1 or whatever. However, saying that it is "indeterminate" is generally more useful. Note that I do not say "correct".

Mathematics (or logic - as they are in some senses the same thing) is always about challenging "accepted wisdom". For instance, The Axiom of Choice is just soooo... obviously true - or is it?

I'll still contend that mathematics has to rely on some form of linguistics in order to try to be communicable, but is not actually dependent on using language at all.

P.S. I've just noticed that I chose to mention the Axiom of Choice. That axiom really only comes into its own when you are dealing with infinite sets. Does linguistics have even a concept of such?

I don't know that linguistics proper has such a concept. There are so many branches, even off the branch that is theoretical. Much work is being done in AI and Minimalism (syntax). I have no interest in either. I am also sure that Formal Logic, which I am interested in, has such a concept. But I am not interested in infinite sets.

I am a problem solver, a troubleshooter by nature. I seek to simplify things and so find the difficulty and its solution. I have seen several excellent theories discarded because of problems at a high level of analysis. Invariably, there was a flaw or two in the fundamental premises of the theory, which, if tweaked, would have resolved those problems. But there was a mindset among theorists that the premises in question were valid. I question (doubt) everything a la DesCartes, especially the fundamental premises. Whence our discussion re 0^0=1.

My interest is in the mental representation of a proposition and its decomposition into Logical Form and beyond into the mental states which support it. I follow Thoreau: "Life is frittered away with detail. Simplify, simplify."

I thought I'd give a brief sketch of my proof that 0^0 = 1. During the break it occurred to me that the problem everyone was having with my objection to 0^0 being 0^1/0^1 is the primary issue that needs to be resolved.

I never had a problem in grade school math with the idea that one cannot divide by zero. But even then I was not satisfied with the explanation for x^0 = 1. The text books say that it is 1 because it acts like 1. The context in which the problem arose and was first resolved has determined and limited its analysis. I had the beginning of my resolution to the problem when studying Wittgenstein, and another major component in linguistic theory.

Mathematics arose out of ordinary language. It is an abstract language, with practically every component of language. Mathematics is not a set of Platonic ideas in the ether; it is a mental construct, a rule-governed system that is represented in the mind before it is translated to the blackboard. It is this mental representation that requires 0^0=1.

Let me be as clear as I can be. I understand and still reject the "proof" that 0^0 = 0^1/0^1 and is ergo undefined. This "proof" is illegitimate. Every response to my claim fails to understand my objection and only repeatedly asserts the "proof" as if repeating it several times makes it true.

Before exponents were invented as a convenient notation for multiplication, we already had a proscription on division by zero. How does violating a prior restraint in a proof prove anything except the validity of the restraint? The only explanation I can come up with is that it is believed that is the underlying representation of 0^0 because that is the context within which the problem was discovered and reputedly resolved. This is backward.

I liken the method of analysis to a mathematics office receiving a box of positive exponents to be sorted and put on the shelf. Everyone understood positive exponents, or, as they were called, exponents, and the task was easy.

Next came the box of negative exponents. These were a bit harder to figure out, but some bright office clerk noticed after playing with them for awhile that they were the equivalent of division by positive exponents.

Next came 0 as an exponent, more difficult to understand than negative. After a while someone reasoned that since x^y/x^y = 1 = x^(y-y) = x^0, all x^0 = 1. Job complete. But then someone noticed 0^0 still on the table and questioned where on the shelf it was to go. What sort of creature is this? "It looks like 0" said one worker." "It looks like 1" said another." "I know!" said a third. "Since n-n = 0, then let n = 1 and voila!

0^0 = 0^1-1 = 0^1/^1 = undefined. Q.E.D.! Finis! Pats on the back for everyone!

This is a functional analysis, not a theoretical one. A stop-gap measure because they didn't know what else to do with it. My favorite philosopher said "In theory, there's no difference between theory and practice. In practice, there is." (Yogi Berra) In mathematics, at least, there shouldn't be any difference. The practical analysis is faulty.

What they failed to notice was that as a consequence of that analysis, all 0^n is undefined. For if you can invoke an error deliberately to "prove" anything, then you can invoke again it in any similar circumstance. Thus, x^7 = x^7+0 = x^ 7+(1-1) = x^8/x^1 = undefined. So, 0^n = 0^n+(1-1) = 0^n+1/0^1 = undefined. Thus, 0^n is undefined for all n. Q.E.D. This proof has yet to be addressed. Thus far it has been ignored or skirted.

If you restrict the analysis to 0^0, you demonstrate clearly that it is a stop-gap measure. If you don't, you must accept that all 0^n is undefined. Neither is acceptable. The circularity of the reasoning becomes even clearer when you have to analyze negative exponents in terms of positive ones to explain where in

x^-1 = 1/x

the second 1 comes from. So, x^-1 = x^2-3 = x^2/x^3 = x^2/(x^2 * x^1) = 1/x^1.

A definition ought not define a term in more complex terms, but simpler and in terms other than itself.

In ancient Greek education, there were 7 liberal arts divided into 2 tiers; the trivia, or 3 ways, and the quadrivia, 4 ways. The trivia were Arithmetic, Logic, and Grammar. The quadrivia were Geometry, Rhetoric, Astronomy and Music. The four are based on and built on the three. A good theory is always a bottom up one.

My proof is quite simple once the underlying principles are demonstrated. It neatens the paradigm and gives it a principled explanation. But first, I need to know if any one is still interested in the topic.

Mathematics came from comparing lengths and comparing heaps of stuff. There are human societies that have language and virtually no mathematics.

And if you insist that 0^0 be defined and be defined as 1, then you require that 0/0 be defined.

O.K. We assume 0/0 = 1. but 0 = 0 x N so 0/0 = N hence N = 1 for any N. Contradiction. The cost of defining 0^0 is inconsistency. As Wallace Shawn said in -Princess Bride- Incontheivable!

ruveyn

I don't understand why you can't understand my claim. 0/0 is undefined. I have never said otherwise. Please don't accuse me of this again. One of my fundamental claims here is that 0^0 is not equal to 0/0. Address this please: if you insist that 0^0 is division by zero, then you must believe all 0^n is undefined.

There are no cultures that have math but have no language.

There are cultures which have language and no mathematics. So mathematics does not necessarily flow from language. Every human culture has language. Humans are born to blab.

0^N / 0*N = 0^ ( N - N ) by the law of exponents. But 0 ^ N = 0 for N ~= 0. Hence 0/0 = 0^0, if either of them are defined. To avoid contradiction, neither may be defined.

So neither can be defined, else there would be a contradiction.

P.S. Are you capable of following a simple proof?

ruveyn

You can have language without math, but you can't have math without language. Language is necessary to math. Can you follow that?

This proof is what I object to as illegitimate. You appeal unthinkingly to the law of exponents in which is a violation of a prior rule and imported for the sole purpose of getting rid of a troublesome concept. This is an unprincipled aspect of the paradigm and the one I object to here. Then you simply state something I have never objected to as final proof and still you have not addressed the issue. You have merely spouted back at me the catechism of arithmetic. Thinking men of old wrestled with the ideas you so readily accept as dogma from on high.

Can you follow/write a proof? Can you answer this question? Why is not all 0^n = 0^n+1/0^1 and therefore undefined? If you can do that with 0^0, why not with any 0^n? To be clear,

Let n=0; 0^n = 0^n+1/0^1 = undef.
Let n=1; 0^n = 0^n+1/0^1 = undef.
Let n=2; 0^n = 0^n+1/0^1 = undef.
Etc., etc.

Restrict the rule allowing division by zero to 0^0 to get rid of the problem and you are unprincipled. Do not restrict it and you have an untenable result. There is your dilemma.

I have to give you an A for consistency, though. So far not one of your answers have addressed the issue.

for n > 0 0^n = 0*0 .... * 0 (n terms) = 0. Done and Done.

there is no division by 0 here.

To be more rigorous I will prove the above by induction:

For n = 1 0^n = 0^1 = 0.

Now assume 0 ^n = 0. 0^(n + 1) = 0^n * 0^ 1 = 0 ^ 0 = 0. QED. Did you see a single division by 0? No you did not.

You apparently do no know how to prove a theorem.

ruveyn

Another fallacious proof. Multiply x by 0 to prove x = 0. Wow! And you still haven't answered my question. We're done.

I cannot deal with Invincible Ignorance. You do not comprehend mathematical induction. You are hopeless. Stick to literature. Mathematics is three levels above your intellectual pay-grade.

ruveyn

Nope. You have still given no proof. You have made some arbitrary assertions that do result in a number system that is self-contradictory.

Maybe you need to have the statement that "0^0 (or 0/0) is undefined" reworded as "0^0 (or 0/0) is going to cause you heartache if you insist on attaching any specific meaning or value to it." Mathematicians are happy to be more succinct.

Nope.


It occurs to me that you will not like the gamma function. Again, that is a "completion" of the factorial function, but done in an elegant way - which is another of the essential qualities of good mathematics.

So far as your overarching insistence about mathematics versus language - I'm willing to concede that natural languages serve reasonably well as metalanguages to convey understanding of mathematical concepts. They are far from ideal - as they generally encourage the meek acceptance of "obviously true" statements, which turn out to be neither obvious nor true. E.g. "parallel lines never meet" or "through a point, there exists a single line that is parallel to any specified line"

Nope. I'm done. You both seem incapable of even understanding what I am saying and neither of you ever addressed my questions. You, Lau, have been gracious enough, but I am tired of not having my questions and issues addressed and told repeatedly that I am asserting something I am not. And now you reject the principles I say I will demonstrate before I even assert them. You reject the notions that math is an abstract language with its own special syntax, function and content categories, logical operators, etc., that it is a mental construct first and foremost, and that math relies fundamentally on linguistic principles.

I looked at several math textbooks today to see how exponents were introduced. One totally ignored the zero power, the other 3 explained it almost exactly as I described above and simply declared 0^0 as undefined with no explanation. Mathematical dogma, wisdom received from on high, accepted uncritically. I want to discuss the fundamental, you appeal to the complex. I want to define the simple in terms as simple as possible to avoid the errors I see, you insist on defining the simple in terms of the complex. This is what defining 0 as 1-1 amounts to. If 0^0 is indeed 0/0, then it is so for reasons other than what you assert.

We can get nowhere because all you do is answer questions and assertions I neither asked nor asserted and fail to address what I do.

Thank you for your input. I was willing to have my views challenged, but to do so meaningfully entails that they be understood. I will look elsewhere.

There is a good reason for that. From a mathematical point of view you have spouted nonsense and gibrish.

ruveyn

No. You're just too dense. And "gibrish'? Don't you mean "gibberish"? But don't get bogged down in details, you might have to actually think about something.

I'm afraid you have that backwards. You have defined 0^0. I have not.


I have never said that "0^0 is indeed 0/0". I have repeated said that they are not defined.[/quote]


The only assertion you seem to have made is that you want to define 0^0 as unity. I have responded to that by asking what you then will do with a self-contradictory number system.


The mathematics textbooks you have looked at are obviously not at a level that addresses your concerns. A good place to look would be in a mathematical analysis text.
http://en.wikipedia.org/wiki/Mathematical_analysis