The Power of Zero
The Power of Zero
Category: School, College, Greek
Samuel Thomas Poling, Blog 124, The Power of Zero
This is an essay I turned into my college math instructor last week. She has said "it's an enjoyable read," but hasn't finished it yet, nor commented further.
Samuel Thomas Poling
October 30, 2006 A.D.
Math
The Power of Zero: The Mistake
Math is a world of logic, nothing more. Many contend that logic and math are different things, but math is just a certain area of logic to do with amounts, values, and things around that area. To navigate any mathematical problem you must rely on reasoning. Every theorem and shortcut must be provable within the world of logic. This is math. Anything else, anything else at all, can never be used to support a mathematical idea. Not desire, not how many people believe it, not by who said it to be true – only logic can say what works and what doesn't.
Another thing one must realize about mathematics is that it's simply a language. The sequence 1 + 1 = 2 appears as words in our minds. They are symbols used to represent values and processes. There are not actually any 1s floating around, nor are there any 2s. There isn't a great plus sign in the sky whose power you summon – to satisfy the deity of equals. They don't actually exist, they are just ideas used to make logical statements about amounts and so forth. With another luck we might have evolved to use the label of division in place of addition, or the word three in place of the word four. It doesn't matter; these numbers and sequences are just ways of explaining our reasoning.
One such method of explanation has been the mathematical method of powers. Exponents. You have a base number, like any other number, with another smaller number floating next to it, top right. What does this say to the human race in mathematical language? What did we decide to make that mean? The exponent, the power, the smaller number floating above, is meant to tell you how many times the base number should be multiplied by itself. That's what it means, clear and simple. It's a shorter way of writing something out. Instead of 3 * 3 * 3, you can simply write 3^3. As I type this, I couldn't find the ability to make the exponent float up next to the base on Microsoft word, but fortunately the "^" symbol works in mathematical language as well. 3^3 means 3 to the power of 3. Which means 3 multiplied by itself 3 times. If it were to the power of 4, then it would be 3 multiplied by itself 4 times, or 3^4. It's quite simple.
So following what powers have been made by our human race to represent, we can figured out what 3^3 would be. 3 * 3 * 3 = 27. Ergo, 3^3 = 27. What about 3^2? Easy, it is telling you the number 3 twice, multiplied. 3 * 3. And that is equivalent to 9. How about 3 to the power of 1? Easy again, that's three one time. Having nothing to multiply to, it is just left as three. 3^1 = 3. This is how it is contended by all.
What is a number to the power of zero equal to? Currently mathematicians hold it to be true that it is equal to one. No matter what the base number is, if the exponent is zero, then it is equal to one. However, if you follow the same reasoning of meaning we have given exponents to possess, then we see a different picture. 3^0 would be 3 zero times. Having nothing to multiply to, we are left with zero, and not with one. Following the same pattern, the same meaning, the same logic as we have with 3^2 = 9, and 3^1 = 3, we arrive at 3^0 = 0. Why? 3^2 is two 3s. 3^1 is three 3s. And 3^0 would be no 3s. "To the power" means you multiply the base by itself the number of times shown by the exponent. If you have 3^0, the exponent is telling you that you have no 3s to multiply together. This is just using common sense. To refute this, you need some pretty special and pretty solid evidence.
The math world thinks it has found the evidence. I have scoured online and off, hearing several supposed "solid" proofs for 3^0 = 1, but I have seen flaw in all of them so far. A common flaw at that. It's difficult to explain, but it's there. To show you what I mean, I'll have to first express to you the supposed proofs they have mustered together.
The world of exponents has several short cuts. Short cuts can be found in math all of the time. However, initially they are risky. You see, you have to have proof the shortcut works. You must solve the problems the long, hard, usual, certain way as well as the short cut way, and check them together to make sure they arrive at the same solution. You must not only check these short cuts in many different areas (positive, negative, fraction, decimal, zero, even, odd, ect.) to make sure the cut holds true, but you must also be able to explain why it is a short cut, and how it works the way it does. You must mathematically prove it. This is pretty much been done well with the exponent short cuts, now called exponent "rules."
This leads me directly to the first problem with the evidence asserting a number to the zero power equaling one instead of zero. You must prove a short cut with the long, normal way of doing things, not the other way around. Although the thinking, reasoning, long-cut way of 3^0 claims that it equals zero with common sense and definition of exponent, the short cut says otherwise. It is in the shortcuts where 3^0 will equal one. And only in the short cuts. I would normally expect you to then realize that there is something incomplete with the description of the short cut! However the math world made a mistake and instead of fixing their short cuts, they used the faulty short cuts to change reality.
Let's say I say to you there is a faster way of solving a division problem. I just switch the denominator with the numerator and subtract, and then claim it was just a short cut. You do the long-cut, more difficult (yet more trustworthy) way of dividing something, and you notice we get different answers. But instead of me changing my short cut, I change the usual laws of division. What is my proof to back up my insanity? My short cut. This is, of course, rediculous backwards reasoning.
Yet this is exactly how the mathematicians are trying to prove 3^0 = 1.
They have faulty short cuts, and when it is discovered these short cuts don't work in the case of zero, which many short cuts in math do not, instead of adding it to the rules of exponents, the change the reality.
Here is the most common "proof" of 3^0 equaling 1:
There is an otherwise proven short cut exponent "rule" that says when you are multiplying two numbers of the same base, you can just add the exponents. For example, if you have (3^3) * (3^2). They have the same base, it is 3. Ergo, says the short cut, you can easily add up their exponents and just put it over the base. So (3^3) * (3^2) = 3^5, because 3 + 2 = 5. Would this trick work in the case of (3^3) * (4^3)? No, because, as you can clearly see, the bases are not the same. For this rule to work, the bases must be the same. Base 3 is not the same as base 4; the shortcut does not apply here.
Understand? Alright, so lets get right to it and see how an exponent of zero will play the part. (3^3) * (3^0) will be our problem. It has the same base, doesn't it? They both have the base of three. Ergo, the shortcut says we can go ahead and add the exponents. So we will end up with 3^3, right? Wait, if what I said is true, then we should end up with zero! (3^3) * (3^0) should equal zero, if what I said is true, because 3^0 = 0. And anything by zero will be zero. The shortcut proves me wrong. Or does it? It proves me wrong the same way my example division "short cut" proves division wrong. But let's see, if any number to the zero power does, in fact, equal one, then the short cut isn't broken, because any number, including 3^3, by one should equal itself, as it was shown above to actually happen. The short cut really does work against me well, doesn't it? No wonder so many math experts believe a number to the power of zero equals one.
But again, I assert there is an exception with zero being in one of the powers. There are several mathematical rules and short cuts with exceptions to zero, many of which are other exponent rules themselves. For example, 0^0 can't equal one, can it? There's an exception there, isn't there? The shortcut is incomplete; it needs to add an exception to zero being in one of the powers.
Although, it isn't so much incomplete. Just saying there is an exception to zero would make it easier for students to learn this. The rule, as it stands, already says that (3^3) * (3^0) = 0. You just have to look more closely. Pay attention, this is my trump card. When multiplying two numbers with exponents, you can only use the short cut if both the bases are the same, as I explained earlier. However, in the case of (3^3) * (3^0), the bases are not actually the same. It's an illusion.
Here is the part that is the most difficult to explain. Once this is a shortcut, it can't really prove one way or the other, 3^0 = 1 or 0. Circular reasoning will enter into this if you try to use it to prove one way or the other. But I'm not trying to prove 3^0 = 0 right now, I'm only trying to disprove 3^0 = 1. Show another possibility. So pay attention as I bring this to even ground. If I am right, then 3^0 = 0. In which case the base numbers are not the same and you cannot use the short cut, but the short cuts own rules! "Oh, but both base numbers are 3!" No, they're not. It just appears that way. If I am right, then 3^0 = 0, which is not at all the same base number as 3. Writing it in 3^0 to begin with was a stupid way. If I am right, then 3^0 = 4^0 and any other number you want to put in front of the zero power! (3^3) * (3^0) is really (3^3) * (X^0), X being any integer. Ergo, they do not, in reality, have the same base. And if that isn't good enough for you, then look at another mathematical process. You can write 3^3 out nicely. By the definition of powers, 3^3 is telling you 3 multiplied by itself 3 times. So 3 * 3 * 3. Looking at the definition of powers, 3^0 means 3 multiplied by itself never. So nothing. So multiplying 3 * 3 * 3 by nothing gives you nothing. Ergo, the short cut still doesn't work, in the case of the raw definitions and meanings of powers. What exponents mean testifies against 3^0 = 1, and testifies for what I say to be true.
"But 3^0 doesn't equal 0, it equals 1!" Shouts the mathematician, falling into circular reasoning. They put in (3^3) * (3^0) and get 1 due to the short cut. They take apart the numbers as 3 * 3 * 3 * 1, again getting the solution of 3^3, meaning 3^0 equaling one works! They're using circular reasoning. They are using what they are trying to prove as part of the proof for that thing. They are assuming 3^0 = 1 being true as they are trying to prove it. When you ask them to prove to you 3^0 = 1, and they, in the proof, use 3^0 equaling one, then they are guilty of circulus in demonstrando. Circular reasoning, circular argument. Until they have proven it to you, assert and reassert that as of now 3^0 = ?. And they have to deal with that until they've proven otherwise. They'll slap 3^0 = 1 somewhere in their "proof" but you just snap their attention back to the fact that right now, until they've proven it, 3^0 = ?, not yet 1. Not until they've proven it.
But isn't circular reasoning what I was doing above? I said, "If I'm right," a few times up there, that's true. But as I said, I wasn't trying to prove 3^0 = 0, I was only trying to show that there is not yet any proof that 3^0 = 1. If they can, in their proof, slap in 3^0 = 1, then I can, in my proof, slap in 3^0 = 0. If they attack me on that, I can correctly accuse them of special pleading and hypocrisy.
Where, then, do the tables turn? In the common sense. Where I did in fact prove 3^0 = 0 was much earlier on, with the very simple description of what powers are created to mean. How many of the base numbers are being multiplied together. If there is none, then there is none. There is zero. Following the same reasoning and definitions of all the other exponents, a number to the zero power will be equivalent to zero. This is where the tables turn.
The others "proofs" I've looked at for 3^0 = 1 are other variations of the same one I just destroyed, as common sense can clearly see. Instead of putting in 3^0 = 1 in any of their proofs, go ahead and try 3^0 power, apply my same logic, and you'll see it could work that way to. There is that exception to zero, there is that fact that it isn't actually the same base, and so on. If they say that any number over another number is equal to one, and then put 3^0 over 3^0 on a fraction bar, then just write zero over zero next to that, and ask them if that equals one. (3^0) / (3^0) = 0 / 0, which does not equal 1. Hey, they could assume 3^0 = 1 in their proof, why can't I assume 3^0 = 0 in mine? I also have the bonus of the common sense and the actual meaning of exponents on my side. I'm actually more justified.
There may be evidence that a number to the power of zero is truly one, but my main point here today is that I have not seen it. I heard dozens of experts, viewed several sites, heard my math instructor, herself, explain it to me. However, their evidence is not conclusive. If they are right, they haven't proven it yet. If it does not equal zero, which I very well may not, then that doesn't mean this essay was wrong. I'm pointing out the flaws currently being made. And also pointing out that, as of now, my side is more justified. I also understand that this would change everything greatly, so close mindedness to my entire argument here by mathematicians is unfortunately likely. But their complaining would change anything until they have some conclusive proof.
So, in closing, there is a big mistake. A mistake of circular reasoning, using what you are trying to prove as part of the proof for that thing. A mistake of backwards reasoning, using short cuts to change reality, instead of reality to fix short cuts. There is no proof for 3^0 = 1, that isn't fallacious. My math instructor once told me to take 3^0 = 1 on faith. Only error needs the assistance of faith. If, in math, it's true, it can be proven logically, and, as it is so far, it has not yet been proven to me. Much we can learn from our teachers, more from our colleagues, but the most from our students.
Doubt everything.
Samuel Thomas Poling, Blog 124, The Power of Zero
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I think I need to address
I think I need to address this argument in more detail:
Let's follow your definition of multiplying B by A^0, which is multiplying B by A 0 times.
According to your argument against Samuel, the result of this multiplying of B by A 0 times is B. Indeed, unless this is the result, your argument against Samuel that A^0 is equal to 1, falls to the ground. Your argument requires that multiplying B by A 0 times is equal to B. That's how you are attempting to argue that A^0 is equal to 1. Your reasoning is that since B times 1 is equal to B and multiplying B by A 0 times is equal to B, therefore A^0 is equal to 1. I believe that this is a fair representation of the "gist" of your argument against Samuel.
My argument is based on the basic assumption that performing some operation zero times is exactly equivalent to not doing it, ie just another way of saying the same thing, AND the first IF statement below.
Here is yet another version of my argument:
IF multiplying B by A^0 is equivalent to multiplying B by A 0 times;
AND multiplying B by A 0 times is equivalent to not performing any operation on B, which I maintain is self-evident;
THEN multiplying B by A^0 leaves B unchanged
THEREFORE B x (A^0) = B
Which for non-zero B, dividing both sides by B, leaves us with
(A^0) = 1
I do not start with "since B times 1 is equal to B", that comes at the end.
And I keep acknowledging that the potential hole here is in the first IF statement, which is a reasonable if not mathematically defined extrapolation from the definition that A^n for positive n is a repeated multiplication expression with n factors all equal to A, and the observation that removing a factor in a multiplication is equivalent to replacing it by 1.
So, now let's let B be equal to A. Now, we have agreed that multiplying B by A zero times is equal to B. Therefore, it follows, that multiplying B, A, by A 0 times is equal to B, A.
Now according to the Wikepedia article, A^n is the result of multiplying A repeatedly by A n times. With this definition, it follows that A^0 is the result of multiplying A by A 0 times.
Which is the chief error on your part, as I said in the previous post - that is not what Wiki says. There is no multiplication for any n less than 2. A naive extrapolation of the Wiki definition to n = 0 would be that A^0 is the result of 'multiplying' A by A '-1' times, which could plausibly be interpreted as reverse multiplication, ie division by A, which result is 1.
By your observation about the result of multiplying B (in this case, A) by A 0 times, I demonstrated, in the second to last paragraph, that the result of multiplying A by A 0 times is equal to A.
After all, if it's "good" for any B, then it's "good" for B, when B is equal to A.
Which is based on the assumption that "A^n is the result of multiplying A repeatedly by A n times.", which neither I, nor Wiki, nor you accept as correct.
Hence, we have just proven, on the basis of your observation about the effect of multiplying B by A 0 times and the Wikepedia's definition of a6n, that A^0 is equal to A.
We have just proved that an invalid definition can lead to an invalid conclusion.
Your continued incorrect assumption about Wikipedia's definition of a^n is troubling, I have pointed this out to you several times now.
I think that I much prefer the mathematician's definition of A^0 to the Wikepedia article's definition of A^0, with which latter definition you apparently agree.
The definition of A^0, in both cases, is simply A^0 = 1.
Are you referring to the justification for that definition? Or did you mean to refer to Wikipedia's definition of A^n? Because you have a persistent misconception of what Wiki's definition of A^n is.
This is a second response to
This is a second response to BobSpence1:
In your attempted
Submitted by BobSpence1 on November 9, 2009 - 5:05pm.You write:
"For someone who comes across as very insistent on precise definitions, to use such a sloppy and downright wrong definition for exponentiation displays very poor thinking skills."
I guess that I am just repeating myself here, but what makes you think that I was using "such a sloppy and downright wrong definition for exponentiation"? I was doing no such thing. I was taking as a premise this "sloppy and downright wrong definition for exponentiation" for an argument which shows the logical consequences of taking this "sloppy and downright wrong definition for exponentiation" as a definition of exponentiation.
(With a little more work at defining what it means to multiply A by itself n times (for n greater than 1, n = 1 and n = 0) (the kind of work that is needed to turn your definition involving repeated multiplication by A and Wikepedias definition into a definition that works for integer n = 1 and n = 0, as well as for any integer n greater than 1) this "sloppy and downright wrong definition could be turned into a correct definition of A^n. Indeed, depending how one defines multiplication of A by itself 0 times, I'm sure that Samuel could come up with a definition for which A^0 = 0. Having such a definition would not make Samuel either sloppy or wrong. It would just mean that he is using a different definition of exponentiation (i.e. a different concept of exponentiation) than mathematician's use. I think that he would probably have a hard time convincing people to study his concept, but he certainly would not be a target for a charge of being sloppy and downright wrong. One cannot be wrong about a technical definition. In making a technical definition one is simply telling everyone how one is using a term. Sloppiness can only be charged when the definition is not determinative of a meaning.)
Again, I don't see how you make the step from my using a premise for an argument to getting that I was using the premise as a definition. Apparently, you think that you can make this step; otherwise why would you be carrying on about how sloppy, downright wrong, and even inconsistent I was being?
Sincerely,
Dr. John D. McCarthy
Re definition:I apologise I
Re definition:
I apologise I quoted where I was considering the definition of exponentiation, rather than A0 itself.
I should have referred to this post:
Exponentiation of A by zero is defined as corresponding to no operation at all, making it consistent with both above definitions, ie, neither multiplication or division is actually involved.
Wikipedia wrote:IOW, the definitions for the cases where n = 1, 0, and negative, (and non-integer, of course), are chosen so as to be as mathematically consistent as possible with the basic definition for non-zero exponents greater than 1.
Favorite oxymorons: Gospel Truth, Rational Supernaturalist, Business Ethics, Christian Morality
"Theology is now little more than a branch of human ignorance. Indeed, it is ignorance with wings." - Sam Harris
The path to Truth lies via careful study of reality, not the dreams of our fallible minds - me
From the sublime to the ridiculous: Science -> Philosophy -> Theology
You and Samuel are the only
You and Samuel are the only ones who have used the expression "A^n is the result of multiplying A by itself n times.", and it is not valid.
Using that to demonstrate a fallacy is a straw man, and irrelevant to anything I have written.
Favorite oxymorons: Gospel Truth, Rational Supernaturalist, Business Ethics, Christian Morality
"Theology is now little more than a branch of human ignorance. Indeed, it is ignorance with wings." - Sam Harris
The path to Truth lies via careful study of reality, not the dreams of our fallible minds - me
From the sublime to the ridiculous: Science -> Philosophy -> Theology
(No subject)
Oops! I left out something
Oops!
I left out something from my definition of a^n (the mathematician's definition) which I had included in my earlier posts.
What I should have written was what I wrote in an earlier post:
DEFINITION: (i) a^0 = 1 for every nonzero real number a
(ii) a^1 = a
(iii) a^{n + 1} = (a^n)(a) for every positive integer n
(iv) a^{-m} = 1/(a^m) for every positive integer m and every nonzero real number a
I mistakenly left off the last clause "every nonzero real number a".
I apologize for this oversight.
Sincerely,
Dr. John D. McCarthy
This is a response to
This is a response to BobSpence1:
Re definition:I apologise I
Submitted by BobSpence1 on November 9, 2009 - 9:05pm.You quote yourself earlier:
"Exponentiation of A by zero is defined as corresponding to no operation at all, making it consistent with both above definitions, ie, neither multiplication or division is actually involved."
How does this phrase "corresponding to no operation at all" allow you to say "Exponentiation of A by zero is defined as ..."?
I can understand how someone could say something like "The definition of exponentiation of A by zero shows that exponentiation of A by zero corresponds to no operation at all." Call me slow, if you wish, but I don't understand how one can say that "Exponentiation of A by zero is defined as corresponding to no operation at all".
The best interpretation which I can think of for this statement is "Exponentiation of A by zero is defined so that it corresponds to no operation at all."
Perhaps, you have a different interpretation of this statement than I have been able to get. I would be interested in hearing it.
Since I am unable to see how "Exponentiation of A by zero is defined as corresponding to no operation at all", nor how that makes it "consistent with both above definitions, neither multiplication of division is actually involved.", it is my distinct impression that by this statement "Exponentiation of A by zero is defined as corresponding to no operation at all, making it consistent with both above definitions, ie, neither multiplication or division is actually involved." you have not defined A^0.
As I said, perhaps you have an interpretation of this statement than I have been able to get. Again, I would be interested in hearing it.
By the way, I suppose that I am showing how out of touch I am, but I don't know what "IOW" means.
I can see, as I indicated in my earlier post, that the Wikepedia article was careful to define x^0. I already apologized for not recognizing the care with which Wikepedia handled the definition of x^0.
Sincerely,
Dr. John D. McCarthy
This is a response to
This is a response to BobSpence1:
Submitted by BobSpence1 on November 9, 2009 - 9:11pm.You and Samuel are the only
You write:
"You and Samuel are the only ones who have used the expression "A^n is the result of multiplying A by itself n times.", and it is not valid."
Fair enough. As I said, the mistake I made was in assuming that, since you did not challenge Samuel's definition of A^n, but only went on to give your argument for A^0, I assumed that you were willing to live with Samuel's definition.
I apologize for making this assumption.
I did not assume that when Samuel said that A^n is the result of multiplying A by itself n times that he meant anything other than that to calculate A^n you take n factors of A and multiply them. It is pretty obvious that that is what Samuel meant since he pointed out that 3 * 3 * 3 = 3^3 and 3 * 3 = 3^2, even though he said "The exponent, the power, the smaller number floating above, is meant to tell you how many times the base number should be multiplied by itself." I gave him the credit that by "how many times the base number should be multiplied by itself" he meant how many times you take the base number as a factor of your product.
I agree with you that this is not using the term "multiplied" correctly.
You write;
"Using that to demonstrate a fallacy is a straw man, and irrelevant to anything I have written."
I don't believe that I demonstrated a fallacy. The best that I can get out of what I demonstrated is that the premises of my argument have a certain conclusion; a conclusion which is inconsistent with the desirable definition: A^0 = 1 (a definition to which Samuel is apparently not attracted). Perhaps, I'm missing something about my own argument.
I have already indicated that the mistake I made was in trying to guess what your definition of A^0 was. Hence, my demonstration is not relevant to what you have written. I have already apologized for having made this guess.
Now I wish to return to another issue which I was addressing in my recent posts.
How do you get from my using Samuel's definition as a premise for an argument that I was being sloppy, downright wrong, and inconsistent with my own definition of A^n?
How do you get from my using Samuel's definition as a premise for an argument that I was defining A^n as being the result of multiplying A by itself n times?
I made it very clear that that was not my definition of A^n.
So, as they say, "what gives"?
Sincerely,
Dr. John D. McCarthy
BTQ (= By the Way), IOW =
BTW (= By the Way), IOW = In other words.
And now for your 'argument':
===================
ARGUMENT: Suppose that n is a non-negative integer and two assumptions hold:
(i) A^n is the result of multiplying A by itself n times. ==> Error: A^n is the result of multiplying A by A n-1 times, where n > 1, eg A x A is multiplying A by itself 1 times and is equivalent/equal to A^2. The cases for n <= 1 are what is being investigated
(ii) B * A^n is the result of multiplying B by A n times
(iii) multiplying B by A zero times is equivalent to not executing the operation of multiplication of B by A.
Then A^0 = A.
PROOF OF ARGUMENT. Let:
(a) B = A.
By assumption (i):
(b) A^0 is the result of multiplying A by itself zero times. ==> Error: not valid, since n < 1 for this definition - it would require A to be multiplied by A -1 times, which is certainly not defined, although it could be defined as dividing A by A 1 time.
By assumption (ii),
(c) B * A^0 is the result of multiplying B by A zero times.
By (a) and (c):
(d) A * A^0 is the result of multiplying A by A zero times.
Since A is equal to itself, (d) implies that:
(e) A * A^0 is the result of multiplyiing A by itself zero times.
By (b) and (e):
(f) A^0 = A * A^0. ==> Error: (b) is not valid for n == 0, so (f) is not valid
Now by (iii):
(f) the result of multiplying A by itself zero times is A:
By (e) and (f):
(g) A * A^0 = A. ==> Error: (f) is not valid, so (g) is not valid
END OF ARGUMENT So I have pointed out the original error, and its propagation through the rest of the 'argument'.By (f) and (g) and transitivity of equality:
(h) A^0 = A. ==> Error: (f) and (g) are not valid
Favorite oxymorons: Gospel Truth, Rational Supernaturalist, Business Ethics, Christian Morality
"Theology is now little more than a branch of human ignorance. Indeed, it is ignorance with wings." - Sam Harris
The path to Truth lies via careful study of reality, not the dreams of our fallible minds - me
From the sublime to the ridiculous: Science -> Philosophy -> Theology
Ok, but your argument
Ok, but your argument produced a 'wrong' result for A0, because one of the initial assumption was explicitly invalid, and all you had to do was point out to Samuel that point, as I spelled out in my comment on that assumption.
To construct an argument with that initial assumption is like making one of your initial assumptions (in some other argument) something like " 1 == 2". The rest of the 'argument' is a waste of space.
It was certainly not relevant to me, since I never used it.
Favorite oxymorons: Gospel Truth, Rational Supernaturalist, Business Ethics, Christian Morality
"Theology is now little more than a branch of human ignorance. Indeed, it is ignorance with wings." - Sam Harris
The path to Truth lies via careful study of reality, not the dreams of our fallible minds - me
From the sublime to the ridiculous: Science -> Philosophy -> Theology
Arrgh - I find it a fun
Arrgh - I find it a fun challenge to try and get people like this to get it...
Favorite oxymorons: Gospel Truth, Rational Supernaturalist, Business Ethics, Christian Morality
"Theology is now little more than a branch of human ignorance. Indeed, it is ignorance with wings." - Sam Harris
The path to Truth lies via careful study of reality, not the dreams of our fallible minds - me
From the sublime to the ridiculous: Science -> Philosophy -> Theology
This is a response to: BTQ
This is a response to:
BTQ (= By the Way), IOW =
Submitted by BobSpence1 on November 10, 2009 - 12:33am.
You write:
"ARGUMENT: Suppose that n is a non-negative integer and two assumptions hold:
(i) A^n is the result of multiplying A by itself n times. ==> Error: A^n is the result of multiplying A by A n-1 times, where n > 1, eg A x A is multiplying A by itself 1 times and is equivalent/equal to A^2. The cases for n <= 1 are what is being investigated"
You've gotten off to a poor start here. Once again, you are attempting to show that my argument is erroneous by challenging the premise. This is an attempt to do something which is not possible. The premise of an argument does not have to be valid in order for the argument to be valid. Demonstrating that the premise "x + 1 = x" of the argument "If x + 1 = x, then x + 2 = x + 1." is an erroneous premise in no way undermines the validity of the argument that "If x + 1 = x, then x + 2 = x + 1.". As I pointed out in my previous post, there is a very simple completely logically sound argument that "If x + 1 = x, then x + 2 = x + 1.". Pointing out that x + 1 is not equal to x, in no way undermines the validity of this completely logically sound argument.
Apparently, you think that it does. As I pointed out in my previous email, whether you think that it does or not I don't know (but it certainly appears that you do), does not represent good "thinking skills".
I have already conceded that the premise of my argument is not relevant to what you have written.
I also agree that the premise "(i) A^n is the result of multiplying A by itself n times." has not been defined clearly. But, if this is the problem which you are having with the argument, you could simply ask me how I am defining multiplication of A by itself n times. I really have not attempted to define this phrase, but from what Samuel said about it I assume that he meant that you take 0 factors of A and "multiply" them (whatever that means). Apparently, whatever Samuel understands this to mean something; and, whatever he means by it, I'm wiling to assume that it gives him 0.
Again, I understand that this is not relevant to what you have said. I'm sorry to waste your time and mine by giving this argument. I guess that I should have addressed it to Samuel. At least it serves the purpose of showing that two assumptions (i) that A^0 is the result of multiplying A by itself 0 times and (ii) that B * A^0 is the result of multiplying B by A 0 times yield the conclusion that A^0 = A. Hopefully, this would show Samuel that if he wants to have A^0 to be equal to 0 he has to explain how he is interpreting statements (i) and (ii) so that they don't together imply that A^0 = A.
Sincerely,
Dr. John D. McCarthy
This is a response to
This is a response to BobSpence1:
Ok, but your argument
Submitted by BobSpence1 on November 10, 2009 - 12:48am.You write:
"To construct an argument with that initial assumption is like making one of your initial assumptions (in some other argument) something like " 1 == 2". The rest of the 'argument' is a waste of space."
Apparently, you don't appreciate the power of a proof that starts with a questionable (or even false) assumption and derives a consequence from it. Such an argument is not "a waste of space". It serves the purpose of demonstrating where the assumption leads.
Such arguments are frequently used in proving some of the most famous theorems of mathematics (theorems which you apparently would regard as "a waste of space"
.
Now I wish to return to another issue which I was addressing in my recent posts.
How do you get from my using Samuel's definition as a premise for an argument that I was being sloppy, downright wrong, and inconsistent with my own definition of A^n?
How do you get from my using Samuel's definition as a premise for an argument that I was defining A^n as being the result of multiplying A by itself n times?
I made it very clear that that was not my definition of A^n.
So, as they say, "what gives"?
I must admit, I guess I also am interested in the challenge of trying to get people who make erroneous arguments to get it.
It's one of the things that I am frequently engaged in when teaching my students how to make a rational argument.
Sincerely,
Dr. John D. McCarthy
This is a response to
This is a response to BobSpence1:
Arrgh - I find it a fun
Submitted by BobSpence1 on November 10, 2009 - 12:53am.You write:
"Arrgh - I find it a fun challenge to try and get people like this to get it..."
Please don't waste my time and yours with trying to get people like me to "get it".
I think my abilities to get it speak for themselves.
I think that the judgement of my peers has much more weight than your judgement has shown to have by your reasoning (e.g. the use of a premise in a proof gives you space to say that the person who used the premise in the proof was contradicting himself when elsewhere he accepted somethiing contradictory to that premise).
One of my peers, who was given the highly distinguished honor of giving a main address to the International Congress of Mathematicians by demonstrating his ability to get it, said of me that I was the most logical person he had ever met.
Another one of my peers, who has received equally prestigious honors for his ability to get it, said of me that "If McCarthy accepts a proof, then it must be correct.". This peer's opinion of me formed part of the basis for my being promoted to Professor of Mathematics.
My student comments bear testimony to my ability to get it.
Sincerely,
Dr. John D. McCarthy
The logical validity of an
The logical validity of an argument does mean it is telling us anything useful. This is clearly the case in the example you give.
All that really demonstrates is that if you start with a false premise you can come to a false conclusion. I think we already knew that.
The situation with respect to the premise "A^n is the result of multiplying A by itself n times." is not just a matter of it being not defined clearly.
The more I look at it, and taking your description of multiplication as a binary operation, it is actually defined quite well enough, and it is in direct conflict with the basic definition of exponentiation on at least two points.
1. It does not restrict its domain of validity with respect to the values of n, which at the very least should be for positive integers > 0.
2. It conflicts with the standard basic definition of exponentiation, which requires n factors, not n multiplications.
You have demonstrated that this particular definition allows you to 'conclude' that A0 = 0, which is showing the same thing Samuel did, perhaps more rigorously. Which is of no real assistance in clarifying for Samuel just what his error was.
Ny argument:
is the sort of argument that would be useful to show his error far more directly. NOTE: this is the sort of thing you manifestly failed to "get".
Your approach is just as likely to reinforce his erroneous assumption, depending how he reads it.
Favorite oxymorons: Gospel Truth, Rational Supernaturalist, Business Ethics, Christian Morality
"Theology is now little more than a branch of human ignorance. Indeed, it is ignorance with wings." - Sam Harris
The path to Truth lies via careful study of reality, not the dreams of our fallible minds - me
From the sublime to the ridiculous: Science -> Philosophy -> Theology
OK, so you were using that
OK, so you were using that to show that that his erroneous assumption lead to his conclusion. But as I pointed out in my previous post, even if this has some mathematical validity, it is probably worse than useless in pointing out his error.
So, whatever your math credentials, you appear to fail miserably in understanding how to approach correcting such misconceptions. Arguing in such subtle and pedantic ways, especially when there is a much more straightforward way to point out the error, as I showed, is not the way to go.
I suspect now that the level at which you study math is actually hampering you in addressing errors at this level. Such a style is simply going to go straight over the head of the mathematically naive, who are precisely the people who will make such errors.
It looks like this is at the heart of our disagreements here.
Do you address students who are starting with very little math background? Your approach may well work if your students are already at a relatively advanced level.
Favorite oxymorons: Gospel Truth, Rational Supernaturalist, Business Ethics, Christian Morality
"Theology is now little more than a branch of human ignorance. Indeed, it is ignorance with wings." - Sam Harris
The path to Truth lies via careful study of reality, not the dreams of our fallible minds - me
From the sublime to the ridiculous: Science -> Philosophy -> Theology
This is a response to
This is a response to BobSpence1:
OK, so you were using that
Submitted by BobSpence1 on November 10, 2009 - 3:42pm.You write:
"I suspect now that the level at which you study math is actually hampering you in addressing errors at this level. Such a style is simply going to go straight over the head of the mathematically naive, who are precisely the people who will make such errors."
Apparently, from what you tell me about your original argument, I missed what you were doing in your original arguments in response to Samuel. Even now, when I go back and read what you actually wrote, I find it very difficult to believe that you were not doing what I thought you were doing. The arguments which you gave at that point are not presented in the form of a motivation for defining A^0 to be equal to 1. They are presented as an argument that A^0 is equal to 1.
For instance:
"Multiplying any number A by
Submitted by BobSpence1 on November 6, 2009 - 7:42am.Multiplying any number A by any number B zero times leaves A unchanged:
A * (B0) = A
Therefore (B0) = A/A = 1."
This argument has all the hallmarks of a deduction, not a motivation. The colon indicates an equivalence; what comes before the colon is presented as implying what comes after the colon. It appears as if it is saying:
STATEMENT: Multiplying any number A by any number B zero times leaves A unchanged. That is to say, A * (B0) = A.
I can believe that this is not what you intended to do; but I think that it is fair to say that it certainly appears that way. If this is not what is being done, then how did you get from what comes before the colon to what comes after the colon?
I think that since this is what this argument appears to be saying, that it does not challenge Samuel at all. I am pretty sure that his reaction to this argument would be the same as his reaction to other "proofs" that A^0 = 1 that he was getting from his instructors and other people he went to in order to find out why A^0 = 1. Namely, he would see this argument of yours as circular reasoning. In order to justify what comes after the colon as following from what comes before the colon you have to assume that multiplication of any number A by B^0 is the same thing as "Multiplying any number A by any number B zero times". I'm pretty confident that Samuel would ask himself the same question that he asked himself about the "proofs" which people offered him. Namely, how are you getting the thing which comes after the colon from the thing which comes before the colon. I'm pretty sure that Samuel would say that this is done by assuming that a pattern which holds for positive integers also holds for the integer 0; an assumption which Samuel rightly points out does not have to be valid. This is one of the germs of truth in Samuel's argument; pattern which hold for positive integers don't have to hold for positive integers and 0. He rightly points out that there are patterns which have exceptions. So why shouldn't the pattern that:
(1) A * (B^3) is the same thing as multiplying A by B three times,
(2) A * (B^2) is the same thing as multiplying A by B two times,
(3) A * (B) is the same thing as multiplying A by B one time,
break down when we get to multiplication of A by B^0? This is actually a very good question that Samuel asks. Indeed, why shouldn't it break down? Why should we expect anything different than that it might break down? Samuel had seen a number of fallacious attempts at showing that A^0 = 1, all of which were based upon an unwarranted assumption that a certain pattern (which Samuel refers to as "the shortcut"
were sustained when we get to the power of zero. Samuel rightly saw that these arguments were circular reasoning.
I don't think that Samuel would have seen your argument as anything different. I think that he would have seen your argument as just another example of circular reasoning; just another example of making an unwarranted assumption about a pattern being sustained.
Hence, unlike your evaluation of my response to Samuel, I think that my response to Samuel was much more to the point. I cleared the mathematician of the false charge by Samuel of circular reasoning, by pointing out that mathematicians do not attempt to prove that A^0 = 1; and explicitly offered a motivation for why mathematicians define A^0 to be equal to 1.
In my estimation, my answer to Samuel was much more likely to have a chance of persuading Samuel of the benefits of defining A^0 to be equal to 1; while at the same time clearing mathematicians of Samuel's false charges of circular reasoning.
I'm sorry. I simply don't agree with your analysis of whose approach to Samuel's post stood a better chance of convincing Samuel that the more desirable course is to define A^0 to be equal to 1.
I agree that had you couched your argument in the form of a motivation rather than in the form of a proof then you would have come across as doing essentially the same thing I was doing; namely, providing a motivation for defining A^0 to be equal to 1. You may not agree with this, but I think that what you actually said comes across as a proof that A^0 is equal to 1, not as a motivation for defining A^0 to be equal to 1.
You write:
"Do you address students who are starting with very little math background? Your approach may well work if your students are already at a relatively advanced level."
The lowest level at which I deal with students is at the level of calculus. But some of these students come into my class with poor algebra skills, never mind skills with, say, geometric proofs. (I think that the sort of two-column proofs which I learned as a junior high student have been left out of at least some of these student's pre-college education, if not all of them.)
For the last several semesters, I have been teaching first semester freshman calculus. I require the students to justify all of their solutions by giving sound arguments. They are not required to write out detailed arguments, but only to indicate what needs to be done to get a sound argument. Some of my students have told me that this has really helped them to get a hold on the material. Others I think are just bothered by this requirement.
So, for the past several semesters, I would not say that I am dealing with students at an advanced level. But I teach them about such things as the direct method of proof of a conditional statement (assume the hypothesis and then deduce the conclusion); the logical fallacy of backwards reasoning (claiming that one has established a conditional statement by assuming the conclusion and deducing the hypothesis); and how to use a theorem (verify the hypotheses of the theorem and then appeal to the theorem to get its conclusion).
I have given them challenges to find the error in an argument. For example, I asked them to tell me what is wrong with the following argument:
PROBLEM: Solve the equation x/(x - 1) = 1/(x - 1).
ARGUMENT: Suppose that x/(x - 1) = 1/(x - 1). Then, by multiplying both sides of this equation by (x - 1) it follows that
(x/(x _ 1))(x - 1) = (1/(x - 1))(x - 1). That is to say, x = 1. Hence, x = 1 is the solution of the equation x/(x - 1) = 1/(x - 1).
END OF ARGUMENT
With such examples the students are taught the fallacy of backwards reasoning. They find such examples to be quite surprising, since they have been taught in pre-college to use this fallacy to solve equations. I myself was taught to use this fallacy in my earlier education. It is quite common to use this fallacy in arguments.
Some of the students just continue to stick to their old habits. Others see the point and change their methodology.
Sincerely,
Dr. John D. McCarthy
This is a second response to
This is a second response to BobSpence1:
OK, so you were using that
Submitted by BobSpence1 on November 10, 2009 - 3:42pm.I want to add one more thought about the advantage of my direct approach to responding to Samuel.
I was right up front about how mathematicians define A^0 and why they define A^0 this way.
I didn't try to claim, as some of those who responded to Samuel have apparently claimed, that Samuel had made a mistake in his argument. I think that Samuel actually showed himself to be an astute observer in seeing the circular reasoning that was involved in the proofs which his instructors and others had used in their attempts to prove that A^0 = 1.
The only mistake which I have been able to find in Samuel's post is that he never defined what he meant by multiplying A by itself n times when n is equal to 0 (or even what he meant by this when n is equal to 1). (He did indicate what he meant by this when n is an integer greater than 1; namely, that he meant that one takes n factors of A and multiplies them. This is clear from his examples of 3 * 3 * 3 = 3^3 and 3 * 3 = 3^2. Samuel may not have been using correct terminology here, but he made himself clear as to how he was using the terminology.) As I have said in an earlier post, I am willing to grant that Samuel has a definition of multiplying A by itself 0 times which in fact implies that A^0 is equal to 0.
. Hopefully, this would get Samuel to examine the intelligibility of his own approach to exponentiation. This would not necessarily lead Samuel to give up on his course of having A^0 to be equal to 0, but at least it would guide him into being self conscious about his own thoughts and the need to make them precise. If I can get a student to just learn this habit of thought, then I think that I have helped that student, even if he chooses to define exponentiation in a different way than mathematicians define it.
If Samuel were my student and submitted to me a paper containing the gist of his argument, then the first thing that I would do is I would ask him what does he mean by multiplying A by itself 0 times? If he told me something like what he said in his article, having no A's to multiply we get zero, I would then ask him what makes him think that we get anything by not doing anything to nothing (i.e. having no A's to multiply, not multiplying "them"
Of course, as long as the student is in my class and being tested on exponentiation I will require him to interpret any problem involving exponentiation using the mathematician's definition of exponentiation. After all, that is the concept of exponentiation which we talk about in my class, not whatever is Samuel's concept of exponentiation.
I actually think that this approach is treating Samuel with the respect which he deserves for not accepting circular reasoning as a justification for a position.
I should add that at least one person who responded to Samuel's argument by showing that Samuel's definition leads to the conclusion that some huge positive number (I don't recall what was this number) is equal to 0, is convinced that he has properly interpreted what Samuel actually said. This person may be correct in his interpretation of Samuel's meaning. I, for one, have not been able to see how to get from what Samuel actually said about exponentiation to a string of equations like:
(I) 3 * 3 * 3 = 3 * 3 * 3 * (13)^0 (since there are no factors of 13 in 3 * 3 * 3)
= 3 * 3 * 3 * (0) (by Samuel's conclusion about 13^0 being 0)
= 0.
This is a distillation of the argument which this person gave. The step in the argument which I have not been able to justify from what Samuel actually said is the first step:
(I) 3 * 3 * 3 = 3 * 3 * 3 * (13)^0 (since there are no factors of 13 in 3 * 3 * 3)
If this step is in fact a logical consequence of what Samuel actually said, then this argument shows that Samuel's argument is fallacious. As I indicated, perhaps this is a correct response to Samuel. I just have not been able to see this step. Perhaps, I am just missing something.
Sincerely,
Dr. John D. McCarthy
pedantic
Dr, John D. McCarthy
It is a deduction, and was meant to be a deduction. I don't quite know what you mean by a "motivation" here. It does require acceptance of the idea that "Multiplying any number A by any number B zero times leaves A unchanged", which in ordinary terms is a perfectly comprehensible statement, whatever technical problems you may have with in the strictest formal mathematical terms.
I then showed that this leads to the conclusion that A0 = 1.
I have already conceded that there some sloppiness there, and presented more carefully expressed arguments in subsequent posts.
I was not trying to address Samuel on the details of his argument, just showing that, unless he could find a logical flaw in that STATEMENT, which there isn't, he has to accept that there must be some flaw in his argument.
The basic 'assumption' in the STATEMENT is that performing some operation zero times is exactly equivalent to not doing it, just another way of saying the same thing.
The pattern actually breaks down at (3), for n = 1, because the the definition of exponentiation in terms of multiplication operations.
Both of which are incorrect on two counts:
1. Exponentiation is only defined in terms of repeated multiplication for the case of positive integers n > 1, not applicable for n <= 1;
2. For such values of n, the number of multiplication operations implied is n - 1, NOT n.
I will also remind you of this
I pointed out then, and I will repeat, that that is not an accurate description of what that Wikipedia article said.
If the Wiki statement was as you describe it, it would indeed lead to that conclusion, but it does not say that. It says explicitly that An is the result of repeated multiplication, and uses diagrams to show that this involves n instances of A, NOT multiplying n times, for n an integer greater than 0. This conflation is at the core of Samuel's error.
Let me repeat: you are making errors, and you have yet to acknowledge this.
He would have no justification for such a conclusion, because I did not base my later argument on any thing about patterns - I based it on that STATEMENT above, that performing an operation zero times is equivalent to not performing it, which I still maintain is logically self-evident. Plus the repeated acknowledgement that ultimately it has been decided to define A0 = 1, chosen because it is consistent with various deductive arguments, and fits the pattern. This could be expressed as the definition being chosen to make that pattern consistent, rather than a unjustified extrapolation from the pattern
You write:
"Do you address students who are starting with very little math background? Your approach may well work if your students are already at a relatively advanced level."
The lowest level at which I deal with students is at the level of calculus. But some of these students come into my class with poor algebra skills, never mind skills with, say, geometric proofs. (I think that the sort of two-column proofs which I learned as a junior high student have been left out of at least some of these student's pre-college education, if not all of them.)
For the last several semesters, I have been teaching first semester freshman calculus. I require the students to justify all of their solutions by giving sound arguments. They are not required to write out detailed arguments, but only to indicate what needs to be done to get a sound argument. Some of my students have told me that this has really helped them to get a hold on the material. Others I think are just bothered by this requirement.
So, for the past several semesters, I would not say that I am dealing with students at an advanced level. But I teach them about such things as the direct method of proof of a conditional statement (assume the hypothesis and then deduce the conclusion); the logical fallacy of backwards reasoning (claiming that one has established a conditional statement by assuming the conclusion and deducing the hypothesis); and how to use a theorem (verify the hypotheses of the theorem and then appeal to the theorem to get its conclusion).
I have given them challenges to find the error in an argument. For example, I asked them to tell me what is wrong with the following argument:
PROBLEM: Solve the equation x/(x - 1) = 1/(x - 1).
ARGUMENT: Suppose that x/(x - 1) = 1/(x - 1). Then, by multiplying both sides of this equation by (x - 1) it follows that
(x/(x _ 1))(x - 1) = (1/(x - 1))(x - 1). That is to say, x = 1. Hence, x = 1 is the solution of the equation x/(x - 1) = 1/(x - 1).
END OF ARGUMENT
With such examples the students are taught the fallacy of backwards reasoning. They find such examples to be quite surprising, since they have been taught in pre-college to use this fallacy to solve equations. I myself was taught to use this fallacy in my earlier education. It is quite common to use this fallacy in arguments.
Some of the students just continue to stick to their old habits. Others see the point and change their methodology.
Sincerely,
Dr. John D. McCarthy
Curious. I see no "fallacy of backwards reasoning" in that example.
Multiplication of both sides by the same factor is valid, as long as that factor is non-zero. Since it turns out to be zero for the apparent solution, that shows that in this case that solution is not valid.
If you reduce the equation by subtracting both sides by 1/(x - 1), you get (x - 1)/(x - 1) = 0 which implies that (x - 1), and therefore x, is what we describe in computing as NaN (Not a Number).
Favorite oxymorons: Gospel Truth, Rational Supernaturalist, Business Ethics, Christian Morality
"Theology is now little more than a branch of human ignorance. Indeed, it is ignorance with wings." - Sam Harris
The path to Truth lies via careful study of reality, not the dreams of our fallible minds - me
From the sublime to the ridiculous: Science -> Philosophy -> Theology