# How the Martians Discovered Algebra

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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

notice I get hints, not straight answers.

By the way, equality is a symmetric relation. a = b then b = a. Once again a sure sign of not knowing what your are talking about. If a and b are numbers and a = b then a, b are one and the same number.

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

notice I get hints, not straight answers.

By the way, equality is a symmetric relation. a = b then b = a. Once again a sure sign of not knowing what your are talking about. If a and b are numbers and a = b then a, b are one and the same number.

In the case of numbers where + is defined a + b = b + a + is a commutative operation. subtraction is, in general not commutative.

a - b in general does not equal b - a unless a and b are equal in which case a - b = b - a = 0.

Ba'al Chatzaf

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Which leads me to ask Roger: "Roger, what possessed you to write such nonsense?" Was it a joke or a satire. Were you serious when you wrote what you wrote?

Ba'al Chatzaf

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No satire, though the first half of chapter 4 is a short story, which qualifies as fiction, I guess.

In addition to being an operation blocker, zero is also a symmetry blocker. There is no symmetry in the absence of anything to be symmetrical.

That's the last tip I'm providing on this.

REB

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Roger,

I did a little housekeeping in your opening post. The links were side-by-side and not clickable.

I also very cunningly snuck in the OL Amazon affiliate link.

Now OL will get about 16 cents per book off the sweat of your brow, but you'll still get the same either way.

But don't worry. I won't buy any old car with it. I'm a Mercedes Benz man, myself...

Michael

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Well - aren't you the enterprising young feller! And helpful, too.

Naturally I hope for you and Kat to retire wealthy from your share of my Amazon Kindle sales.

Thanks again for everything!

Best, REB

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I have been getting some very encouraging reports from Amazon.com about the sales of my Kindle book. I am pleased to report that it has already sold dozens of copies, and the royalties generated have nearly covered my production costs.

I've also gotten a nice Amazon review from a satisfied reader, as well as some very welcome feedback from several readers, including the suggestion/request that I re-issue the book sometime as a paperback. We'll see about that...

In the meantime, I'm hard at work on my next Kindle book, "What They Didn't Teach Me in Music School: A Guide for Making Music and Living Life." I'm hoping to finish and upload it late this fall.

REB

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Martian Algebra has continued to sell briskly into the New Year. I've made back my costs and actually had to file a Schedule E for 2014!

Meanwhile, I discovered another philosopher, a paid professional yet, who holds a similar view about zero. Here is an excerpt from a review of his book, with a link to the full review.

REB

https://ndpr.nd.edu/news/24450-more-kinds-of-being-a-further-study-of-individuation-identity-and-the-logic-of-sortal-terms/

E. J. Lowe
More Kinds of Being: A Further Study of Individuation, Identity, and the Logic of Sortal Terms

Published: August 12, 2010E. J. Lowe, More Kinds of Being: A Further Study of Individuation, Identity, and the Logic of Sortal Terms, Wiley-Blackwell, 2009, 227pp., \$ 99.95 (hbk), ISBN 9781405182560.

Reviewed by Gary S. Rosenkrantz, University of North Carolina, Greensboro

[...]

Chapter 4: Number, Unity, and Individuality

Unlike the majority of philosophers of mathematics, Lowe is unwilling to accept that zero is a number. On pp. 53-4, Lowe argues that

Taking a single object to be the limiting or degenerate case of a plurality, this allows us to say that every single object possesses the number one. But since the notion of a 'null' plurality is a manifest absurdity, we are not committed to the existence of 'the number nought'.

Lowe is apparently seeking to convince the reader that numbers are fundamentally properties of pluralities -- with the number one being an exception. He motivates his argument by arguing that it is unintuitive to think of zero as a number, writing on p. 52 that

Most ordinary folk would consider it at best a bad joke to be told that, say, there is a number of pound notes in a sealed envelope that has just been given to them, when in fact the envelope is empty.

But Lowe gives us little reason to believe that these ordinary folk would be irritated because they do not intuitively accept that zero is a number. Their irritation would most likely have another source. They were led to falsely believe, by a sentence deliberately constructed to be unclear within the context of ordinary discourse, that they were about to receive a plurality of pound notes. The collective noun, number, when preceded by a, is most often used as a plural and the plurality of notes accentuates that plurality. Also note that, suspiciously, the use of the singular is, while essential for Lowe's purposes, is actually at variance with that ordinary pattern of usage of number. Even mathematicians who do intuitively accept that zero is a number would likely be no less irritated than so-called ordinary folk. Lowe observes that most ordinary folk would not be pacified by being told that zero is a number. But it is likely also true that our mathematicians would not be pacified by being told this. Lowe's observation hardly shows that ordinary folk [plural noun] do not intuitively accept that zero is a number.

I further counter that, intuitively, numbers are quantitative entities. Moreover, a quantitative entity necessarily pertains to how many there are of something of some sort. For example, "How many socks are in a pair?", "How many stars are there in our solar-system?", "How many centaurs are there on the Earth?" My intuitions are that the answer to the first question is two, the answer to the second question is one, and the answer to the third question is zero. Hence, in my view, it is intuitive to think of zero as a number, and thus, zero's being a number is part of the intuitive data for the ontology of elementary arithmetic. To the extent that such a viewpoint is legitimate, Lowe's argument that numbers are fundamentally properties of pluralities is unmotivated. (Arguably, numbers are quantitative properties of abstract sets, e.g., the null set in the case of zero.)