# How the Martians Discovered Algebra

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All the known bugs in set theory have been resolved. By suitably restricting set theory we get rid of some of the whoppers like the Russell Frege paradox. The Zormelo Frankel axioms can be used safely as far as has been determined.

After the patches and the repairs Cantor's theory of transfinite numbers still holds up.

Yes, Roger, cardinality of the set of integers is the same as the cardinality of the set of even integers, a proper subset of the set of integers.

Mathematics runs on deduction, not on empirical induction. (BTW mathematical induction is no the least bit like empirical induction or causal abduction which was discovered by Charles Peirce in the 19th century.

And the "mind numbing logic" of Boolean algebra are still used to construct descrete logical circuits for computers. And the Godel incompleteness theorems are still as true as the day they were published in 1931.

Mathematics has been flourishing quite well since logic and the theory of real variables were cleaned up properly in the mid 19 th century. Set theory is the basic furniture of nearly all mathematical systems from arithmetic to topology to infinite dimension spaces.

Ba'al Chatzaf

P.S. Algebra (Arabic al - Jabir or rearrangement) was invented by Muslim mathematicians not Martians. Omar Kayyam, he of the Rubiyat , was one of the early contributors to the field.

P.P.S for a non zero number x 1 = x ^ a / x ^ a = x ^ (a -a) = x ^ 0. A simple application of the law that exponents add.

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I hope to read Roger’s book soon.

Bob, on sets and topology, you may be interested in a book recently in my library.

New Foundations for Physical Geometry

The Theory of Linear Structures (Look at Table of Contents.)

Tim Maudlin (Oxford 2014)

From the back cover:

Topology is the mathematical study of the most basic geometrical structure of a space. Mathematical physics uses topological spaces as the formal means for describing physical space and time. This book proposes a completely new mathematical structure for describing geometrical notions such as continuity, connectedness, boundaries of sets, and so on, in order to provide a better mathematical tool for understanding space-time. This is the initial volume in a two-volume set, the first of which develops the mathematical structure and the second of which applies it to classical and Relativistic physics.

The book begins with a brief historical review of the development of mathematics as it relates to geometry, and an overview of standard topology. The new theory, the Theory of Linear Structures, is presented and compared to standard topology. The Theory of Linear Structures replaces the foundational notion of standard topology, the open set, with the notion of a continuous line. Axioms for the Theory of Linear Structures are laid down, and definitions of other geometrical notions developed in those terms. Various novel geometrical properties, such as a space being intrinsically directed, are defined using these resources. Applications of the theory to discrete spaces (where the standard theory of open sets gets little purchase) are particularly noted. The mathematics is developed up through homotopy theory and compactness, along with ways to represent both affine (straight line) and metrical structure.

Oh yeaaaaaaah...

Go Roger!

Just got it.

Michael

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Great, guys! (I mean: the two guys immediately preceding. ;-)

Michael, I'm sure you will notice in the Preface that I gave a nod to Objectivist Living. (How many books have done that? :-) Chapters 7-9 certainly wouldn't have happened without the stimulation/provocation of our discussions here.

Stephen, I encourage you to keep working on your projects and bring them to "print" so we can read them under one "umbrella," as it were. Keep up the excellent work, in any case!

Now that I've seen first-hand how incredibly easy and fun it is to put out an e-book, I'm going to blaze on and do more and more. I certainly don't expect to get rich, or even to be able to pay my gas and electric bill with my writings, and I'm not going to bore people with repeated reports of my sales. But it's really encouraging that in just two days, and with very modest promotional efforts, I've sold 10 copies of the e-book, and I figure I will sell a bunch more. Since my costs are all time and energy, it's "pure profit," such as it is. Next April, I'll have to fill out a new tax schedule to report the royalties. :-)

It's so easy to be a critic, whether in one-shot sniping Internet posts or in published critiques in magazines and journals, but quite another to try to do something constructive and coherent. It's very challenging, but also very satisfying seeing it come together to the point you can feel comfortable sharing it with others. Stephen is very masterful at this. I'm sure others of you are, too. Come on in. The water's fine. :-)

REB

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It's so easy to be a critic, whether in one-shot sniping Internet posts or in published critiques in magazines and journals, but quite another to try to do something constructive and coherent. It's very challenging, but also very satisfying seeing it come together to the point you can feel comfortable sharing it with others. Stephen is very masterful at this. I'm sure others of you are, too. Come on in. The water's fine. :-)

Roger,

I'm proud of both you and Stephen.

I mean it.

Michael

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Thanks, Michael - for this, and for all the years of appreciation and encouragment. :-)

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Roger, My first thought upon seeing the title of your new book, How The Martians Discovered Algebra, was Coudn't they keep it to themselves? I mean, really.

Seriously, I have ordered your book from Amazon Kindle. Congratulations!

You might gather from my visceral reaction that I did not enjoy algebra, or any of its offspring, when in (what Rush Limbaugh would call) "screwal". :blush:

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Jerry, I feel your pain. :-)

I tried to make the math user-friendly and especially thinker-friendly. Unavoidably, though, the material on the Pythagorean equation just sprawls and sprawls with pages of my lengthy process of discovering and then validating what I have laid claim to.

It's not easy, discovering the matter-energy equivalence - let alone retaining that discovery - without algebra. As the cover of my book hints, good old Albert had something to do with that, so he gets part of the credit, blame, too. :-)

The Martians discovered what we (Einstein) knew, and we (Einstein) discovered what they knew - at least, that's the playful premise of my short story (the only fiction in the book).

It might be that the world would be better off without what the Martians knew. The St. Lawrence Seaway might have been built without nuclear explosions speeding along the digging process. My dad might have survived the anticipated invasion of the Japanese mainland.

As for what Einstein taught the Martians, that has been known to scare or frustrate young people, even more than the spectre of atom bombs going off at the nearby air force base. At least part of that is the fault of the schools, IMO.

In any case, Jerry and Michael, thank you for buying my book! There is a lot of reader-friendly, non-geek stuff to read and enjoy. Also, you'll learn quite a bit more about me than you might care to know. (Surreptitious autobiography for no extra charge!)

REB

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Does lack of math skills preclude reading this book?

--Brant

I've got a lot of lack

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Brant, there is enough variety in the book so that both math geeks and philosophy freaks will (hopefully) get their money's worth from it.

I'd say that the (lengthy) introduction - which is largely autobiographical - and chapters 6 through 9 are easily accessible to those for whom math is not their cup of tea. (Assuming interest and familiarity with more specifically philosophical issues, anyway.)

Now would be a good time to thank the good folks at Objectivist Living who tried to pound some sense into me during our lengthy discussion about the validity of mathematical operations using zero. Thank you!

I did eventually realize there were a couple of errors in my thinking. However, the basic premise of zero as an "operation-stopper" proved to be highly fruitful once I corrected the errors.

Also, I'd say that anyone who struggled with exponents in high school algebra will find the chapter 3 on x-to-the-zero to be very helpful. (By the way, the standard "proof" that x-to-the-zero = 1 is based on a logical fallacy. I show that not in chapter 3, but in chapter 7.)

All of that said, Brant, I don't know whether you'd like the book or not. Unfortunately, since I'm only publishing it as an e-book, there won't be any used copies available for \$0.01 plus \$3.99 shipping - but it's only \$3.99 as a "new" e-book, so what the heck!

REB

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Brant, there is enough variety in the book so that both math geeks and philosophy freaks will (hopefully) get their money's worth from it.

I'd say that the (lengthy) introduction - which is largely autobiographical - and chapters 6 through 9 are easily accessible to those for whom math is not their cup of tea. (Assuming interest and familiarity with more specifically philosophical issues, anyway.)

Now would be a good time to thank the good folks at Objectivist Living who tried to pound some sense into me during our lengthy discussion about the validity of mathematical operations using zero. Thank you!

I did eventually realize there were a couple of errors in my thinking. However, the basic premise of zero as an "operation-stopper" proved to be highly fruitful once I corrected the errors.

Also, I'd say that anyone who struggled with exponents in high school algebra will find the chapter 3 on x-to-the-zero to be very helpful. (By the way, the standard "proof" that x-to-the-zero = 1 is based on a logical fallacy. I show that not in chapter 3, but in chapter 7.)

All of that said, Brant, I don't know whether you'd like the book or not. Unfortunately, since I'm only publishing it as an e-book, there won't be any used copies available for \$0.01 plus \$3.99 shipping - but it's only \$3.99 as a "new" e-book, so what the heck!

REB

Logical fallacy?????????? x^0 = x^(a - a) = x^a/x^a = 1. QED. Find a logical fallacy there!

I really get semi-steamed when people who know virtually nothing about modern mathematics comment on it as though they knew something.

(What are your mathematical qualifications. Do you know anything about the theory of integers, rational number, real numbers or the analysis of functions of real variables. Do you know any calculus. Do you know what a proof is.) ?

Ba'al Chatzaf

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Brant, there is enough variety in the book so that both math geeks and philosophy freaks will (hopefully) get their money's worth from it.

I'd say that the (lengthy) introduction - which is largely autobiographical - and chapters 6 through 9 are easily accessible to those for whom math is not their cup of tea. (Assuming interest and familiarity with more specifically philosophical issues, anyway.)

Now would be a good time to thank the good folks at Objectivist Living who tried to pound some sense into me during our lengthy discussion about the validity of mathematical operations using zero. Thank you!

I did eventually realize there were a couple of errors in my thinking. However, the basic premise of zero as an "operation-stopper" proved to be highly fruitful once I corrected the errors.

Also, I'd say that anyone who struggled with exponents in high school algebra will find the chapter 3 on x-to-the-zero to be very helpful. (By the way, the standard "proof" that x-to-the-zero = 1 is based on a logical fallacy. I show that not in chapter 3, but in chapter 7.)

All of that said, Brant, I don't know whether you'd like the book or not. Unfortunately, since I'm only publishing it as an e-book, there won't be any used copies available for \$0.01 plus \$3.99 shipping - but it's only \$3.99 as a "new" e-book, so what the heck!

REB

Logical fallacy?????????? x^0 = x^(a - a) = x^a/x^a = 1. QED. Find a logical fallacy there!

I really get semi-steamed when people who know virtually nothing about modern mathematics comment on it as though they knew something.

(What are your mathematical qualifications. Do you know anything about the theory of integers, rational number, real numbers or the analysis of functions of real variables. Do you know any calculus. Do you know what a proof is.) ?

Ba'al Chatzaf

Gee Bob. That was fast. What else was in Chapter 7?

--Brant

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Brant, there is enough variety in the book so that both math geeks and philosophy freaks will (hopefully) get their money's worth from it.

I'd say that the (lengthy) introduction - which is largely autobiographical - and chapters 6 through 9 are easily accessible to those for whom math is not their cup of tea. (Assuming interest and familiarity with more specifically philosophical issues, anyway.)

Now would be a good time to thank the good folks at Objectivist Living who tried to pound some sense into me during our lengthy discussion about the validity of mathematical operations using zero. Thank you!

I did eventually realize there were a couple of errors in my thinking. However, the basic premise of zero as an "operation-stopper" proved to be highly fruitful once I corrected the errors.

Also, I'd say that anyone who struggled with exponents in high school algebra will find the chapter 3 on x-to-the-zero to be very helpful. (By the way, the standard "proof" that x-to-the-zero = 1 is based on a logical fallacy. I show that not in chapter 3, but in chapter 7.)

All of that said, Brant, I don't know whether you'd like the book or not. Unfortunately, since I'm only publishing it as an e-book, there won't be any used copies available for \$0.01 plus \$3.99 shipping - but it's only \$3.99 as a "new" e-book, so what the heck!

REB

Logical fallacy?????????? x^0 = x^(a - a) = x^a/x^a = 1. QED. Find a logical fallacy there!

I really get semi-steamed when people who know virtually nothing about modern mathematics comment on it as though they knew something.

(What are your mathematical qualifications. Do you know anything about the theory of integers, rational number, real numbers or the analysis of functions of real variables. Do you know any calculus. Do you know what a proof is.) ?

Ba'al Chatzaf

Gee Bob. That was fast. What else was in Chapter 7?

--Brant

x^0 = 1 when x != 0 is not based on a fallacy. To say so is to not comprehend what a proof is. It is no more a fallacy than to assert a - a = 0. I used to give my pupils big red "F" s when they wrote such silly things.

Ba'al Chatzaf

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You don't have any students here I know of. As for the proof you embrace, it looks self-contained or tautological. I don't know mathematical proofs, however, but Roger merely states his work is in chapter 7 in his book and all you do is dump on him without even the courtesy of asking him how it might be considering the simplicity of the proof you proferred. Even I can understand that proof. I would be very interested if you had read his chapter 7 at least and then you had jumped all over him. As it is all the value was in the proof you put up, not your bad manners in demanding he come up with an argument from authority in lieu of his work--or a logical fallacy, for God's sake. If he had you'd be no more accepting of anything he said about his claim so the argument from authority rests on you. Now, let's say that somehow Euler were alive and functioning and you knew Euler was Euler, one of the absolutely greatest mahematicians of all time (so I've read) and he posted here what Roger did, would that satisfy you? If yes, then shame on you for what you wrote. If no, then you shouldn't have demanded what you demanded from Roger so shame on you for that. You were entitled to one request from Roger, to come with a little actual substance concerning his claim to justify someone like yourself, who is well conversant in mathematics, reading his book and especially his book's chapter 7. Since you didn't do that that places you into Ellen's "dummkoff" category so far in this thread--that is, you put yourself there. I know you have the brains to take yourself out, just not the basic manners,

--Brant

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You don't have any students here I know of. As for the proof you embrace, it looks self-contained or tautological. I don't know mathematical proofs, however, but Roger merely states his work is in chapter 7 in his book and all you do is dump on him without even the courtesy of asking him how it might be considering the simplicity of the proof you proferred. Even I can understand that proof. I would be very interested if you had read his chapter 7 at least and then you had jumped all over him. As it is all the value was in the proof you put up, not your bad manners in demanding he come up with an argument from authority in lieu of his work--or a logical fallacy, for God's sake. If he had you'd be no more accepting of anything he said about his claim so the argument from authority rests on you. Now, let's say that somehow Euler were alive and functioning and you knew Euler was Euler, one of the absolutely greatest mahematicians of all time (so I've read) and he posted here what Roger did, would that satisfy you? If yes, then shame on you for what you wrote. If no, then you shouldn't have demanded what you demanded from Roger so shame on you for that. You were entitled to one request from Roger, to come with a little actual substance concerning his claim to justify someone like yourself, who is well conversant in mathematics, reading his book and especially his book's chapter 7. Since you didn't do that that places you into Ellen's "dummkoff" category so far in this thread--that is, you put yourself there. I know you have the brains to take yourself out, just not the basic manners,

--Brant

The assertion that x^0 = 1 when x != 0 is a fallacy is just plain wrong. false. not true. end of issue.

Ba'al Chatzaf

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You don't have any students here I know of. As for the proof you embrace, it looks self-contained or tautological. I don't know mathematical proofs, however, but Roger merely states his work is in chapter 7 in his book and all you do is dump on him without even the courtesy of asking him how it might be considering the simplicity of the proof you proferred. Even I can understand that proof. I would be very interested if you had read his chapter 7 at least and then you had jumped all over him. As it is all the value was in the proof you put up, not your bad manners in demanding he come up with an argument from authority in lieu of his work--or a logical fallacy, for God's sake. If he had you'd be no more accepting of anything he said about his claim so the argument from authority rests on you. Now, let's say that somehow Euler were alive and functioning and you knew Euler was Euler, one of the absolutely greatest mahematicians of all time (so I've read) and he posted here what Roger did, would that satisfy you? If yes, then shame on you for what you wrote. If no, then you shouldn't have demanded what you demanded from Roger so shame on you for that. You were entitled to one request from Roger, to come with a little actual substance concerning his claim to justify someone like yourself, who is well conversant in mathematics, reading his book and especially his book's chapter 7. Since you didn't do that that places you into Ellen's "dummkoff" category so far in this thread--that is, you put yourself there. I know you have the brains to take yourself out, just not the basic manners,

--Brant

The assertion that x^0 = 1 when x != 0 is a fallacy is just plain wrong. false. not true. end of issue.

Ba'al Chatzaf

The issue is your insufferableness. I've no qualifications for a discussion of mathematical proofs.

--Brant

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The issue is your insufferableness. I've no qualifications for a discussion of mathematical proofs.

--Brant

There is a sure cure for ignorance: learn something about proofs. Mathematics is mostly about proofs.

I do not suffer the ignorance of others gladly. I suffer my own ignorance with even less joy, but I try to do something about it. When I am ignorant of something important I immediately try to learn about that something.

Ba'al Chatzaf

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The issue is your insufferableness. I've no qualifications for a discussion of mathematical proofs.

--Brant

There is a sure cure for ignorance: learn something about proofs. Mathematics is mostly about proofs.

I do not suffer the ignorance of others gladly. I suffer my own ignorance with even less joy, but I try to do something about it. When I am ignorant of something important I immediately try to learn about that something.

Ba'al Chatzaf

It was okay to put up the proof and challenge Roger to show his stuff--some stuff--considering his claim. No book purchase/read required. You didn't have to lard everything up with mathematical self-righteousnesss and demands for credentials. There was a math PhD on another thread that got your ire. That he had the degree didn't offer any protection from you. I wasn't going to watch it anyway, especially not for its over an hour length.

Roger was specifically addressing me, a non-mathematician, in a courteous and generous posting and you showed up-jumped in shovelling dirt with a moral howl of mathematical righteousness. You may not suffer from the ignorance of others, "gladly," but think of yourself--your ignorance, not necessarily in this case but in cases generally. We all suffer from ignorance--it's a necessary part of the human condition. The trick is making ourselves less so respecting what we are interested in.

--Brant

sometimes we learn things--about people--that hurt us and we know/learn how in special cases truth can be a disvalue: that truth doesn't have to be told with ill-intent beating all the lies you can invent (Blake), but it can be

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The issue is your insufferableness. I've no qualifications for a discussion of mathematical proofs.

--Brant

There is a sure cure for ignorance: learn something about proofs. Mathematics is mostly about proofs.

I do not suffer the ignorance of others gladly. I suffer my own ignorance with even less joy, but I try to do something about it. When I am ignorant of something important I immediately try to learn about that something.

Ba'al Chatzaf

It was okay to put up the proof and challenge Roger to show his stuff--some stuff--considering his claim. No book purchase/read required. You didn't have to lard everything up with mathematical self-righteousnesss and demands for credentials. There was a math PhD on another thread that got your ire. That he had the degree didn't offer any protection from you. I wasn't going to watch it anyway, especially not for its over an hour length.

Roger was specifically addressing me, a non-mathematician, in a courteous and generous posting and you showed up-jumped in shovelling dirt with a moral howl of mathematical righteousness. You may not suffer from the ignorance of others, "gladly," but think of yourself--your ignorance, not necessarily in this case but in cases generally. We all suffer from ignorance--it's a necessary part of the human condition. The trick is making ourselves less so respecting what we are interested in.

--Brant

sometimes we learn things--about people--that hurt us and we know/learn how in special cases truth can be a disvalue: that truth doesn't have to be told with ill-intent beating all the lies you can invent (Blake), but it can be

None the less he asserted that the provable proposition x^0 where x is not 0 = 1 is false. That is flat out wrong, wrong, wrong.. The proof is not difficult.

Ba'al Chatzaf

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None the less he asserted that the provable proposition x^0 where x is not 0 = 1 is false. That is flat out wrong, wrong, wrong.. The proof is not difficult.

Re-read #11. He doesn't say the proposition is false. He says the standard proof of it is fallacious.

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None the less he asserted that the provable proposition x^0 where x is not 0 = 1 is false. That is flat out wrong, wrong, wrong.. The proof is not difficult.

Re-read #11. He doesn't say the proposition is false. He says the standard proof of it is fallacious.

The standard proof given in every math book is correct. He is asserting a false statement.

Standard proof. Let x != 0 then x^a/x^a = 1. But x^a/x^a = x^(a - a)( = x^0

so x^0 = 1. Q.E.D. That is a correct proof based on the fact that exponents add in multiplication. It is the basis for logarithms.

Wrong is Wrong is Wrong.

Ba'al Chatzaf

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Well, this is very entertaining and amusing - but also very heartening. Thanks, Brant, for your kind words. I'm grateful any time someone takes a stand in favor of simple courtesy and justice - especially in the face of intransigent condescension and argument from authority.

Yes, Merlin, you have it right. I am *not* denying the truth of x^0 = 1, just the validity of the *standard proof* of it, which is indeed based on a fallacy. I don't think there's anyone else on earth who has come up with either the proof I offer nor the refutation of the standard proof I offer, so that alone should be reason for curiosity and not dogmatic rejection. I mean, it's not as if I've turned all Graham Priest and am rejecting the Principle of Non-Contradiction.

However, I'm not going to post the material here. For anyone who wants to see it first-hand, it's available for \$3.99 from Amazon Kindle. And what better place to smite such an ignorant, unqualified heathen as I than by writing a well-informed, righteously indignant review on Amazon!

But if Ba'al wants to persist in his belief that I have nothing to offer and that hurling insults and loudly repeating the conventional wisdom are a substitute for intellectual engagement, that's on him. I don't reward my children's tantrums, and I'm not going to reward his.

I will make this offer: if Ba'al is willing to pony up the \$3.99 to Kindle and actually *read* chapter 7 of my book, and then show me where my argument is wrong, I will cheerfully admit it here on OL and just as cheerfully refund his \$3.99.

REB

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Well, this is very entertaining and amusing - but also very heartening. Thanks, Brant, for your kind words. I'm grateful any time someone takes a stand in favor of simple courtesy and justice - especially in the face of intransigent condescension and argument from authority.

Yes, Merlin, you have it right. I am *not* denying the truth of x^0 = 1, just the validity of the *standard proof* of it, which is indeed based on a fallacy. I don't think there's anyone else on earth who has come up with either the proof I offer nor the refutation of the standard proof I offer, so that alone should be reason for curiosity and not dogmatic rejection. I mean, it's not as if I've turned all Graham Priest and am rejecting the Principle of Non-Contradiction.

However, I'm not going to post the material here. For anyone who wants to see it first-hand, it's available for \$3.99 from Amazon Kindle. And what better place to smite such an ignorant, unqualified heathen as I than by writing a well-informed, righteously indignant review on Amazon!

But if Ba'al wants to persist in his belief that I have nothing to offer and that hurling insults and loudly repeating the conventional wisdom are a substitute for intellectual engagement, that's on him. I don't reward my children's tantrums, and I'm not going to reward his.

I will make this offer: if Ba'al is willing to pony up the \$3.99 to Kindle and actually *read* chapter 7 of my book, and then show me where my argument is wrong, I will cheerfully admit it here on OL and just as cheerfully refund his \$3.99.

REB

The proof I give is three steps wrong. Here and now, tell me which step you don't like. Adding exponents? Dividing a number by it self and getting 1? Or substracting a from a and getting 0. Spit it out. I suspect you are a mathematical incompetent and here is your chance to prove I am wrong.

Ba'al Chatzaf

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Sorry, I'm going to play the Will Thomas card.

If you want to know what I say in the book, you're going to have to buy and read the book.

However, here is a hint: the conventional view's attempted proof of x^0 = 1 is just as unsound as the one used for x*0 = 0.

Here's a better hint: if zero is an operation-blocker for addition, then it is also an operation-blocker for addition *of exponents*.

Here's the best hint of all: the fact that [n + (-n] = 0 does *not* imply that 0 = [n + (-n)]. (The fact that the addition of three chairs to an empty room and the removal of those chairs leaves a net of 0 chairs does *not* imply that a room in which there are 0 chairs means that three were added and three were removed.) This mistaken implication, which is built into the standard proof, is the fallacy I allude to.

REB

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