Thank you for referencing the thread "The Opposite of Nothing Is/Isn't Everything" I think that thread was OL at its very, very best. It is a shame that we are so hung up on matters political a

Couldn't agree with you more, BC. And the next time you want to condescendingly, sneeringly put down my announcements about my work, at least give a passing nod to objectivity by actually quoting me i

This is a false alternative. Zero is not absolute nothingness. But that doesn't mean it is something. It is the absenceof something. Not the absence of anything whatsoever (that would be absolute nothingness), but the absence of something in particular.

The phrase "zero apples" does not mean that there is some number of apples, and that number is zero. It means that there are not any apples, that any attempt to count the apples does not produce any results, and by convention, we say that we have "counted zero apples," when in fact we have not counted any apples. All of the so-called "algebraic properties" of zero are actually just the results of attempting to perform calculations in the absence of any quantity that one would normally be able to perform such calculations.

Some say this is "a difference without a difference." By the same token, quantum mechanical equations produce the same results regardless of whether one adopts the Copenhagen interpretation or a more realistic interpretation. And perhaps there are not now any reasons for preferring one interpretation of the metaphysics of quantum mechanics or the metaphysics of zero over another. But I'm confident that there are reasons for preferring a realistic interpretation over one that reifies non-existence, even the relative or particular non-existence captured in how we use the concept of "zero" in mathematics.

Even now, we have recently seen some Danish students who have found a method of measuring the position and momentum of subatomic particles, and who have thus proved that Heisenberg's Uncertainty Principle is ONLY the claim of a methodological limitation on simultaneous measurement of position and momentum of particles, and not a metaphysical law that such particles do not simultaneously possess position and measurement. For decades, the anti-Identity modern philosophers were pushing the former interpretation. But Aristotle has had the last laugh. And I'm chuckling along with him.

How the Martians Discovered Algebra: Explorations in Induction and the Philosophy of Mathematicsby Roger E. Bissell delivers an algorithm for generating Pythagorean Triples. Central to the thesis of the work, Bissell explains how he discovered this by means of induction, not deduction. From there, Bissell takes the reader into number theory in order to validate his new explanation of the proper understanding of multiplication, and to challenge widespread assumptions about the empty set and infinity.

The relationships between music and mathematics go back to Pythagoras. So, this set of essays by musician Roger Bissell enjoys a solid foundation. Bissell also dabbles at mathematics and has several philosophical explorations to his credit, published in The Reason Papers and the Journal of Ayn Rand Studies.

Objectivism is an integration of rationalism and empiricism. Objectivism rejects the false dichotomies of Descartes, Hume, James, and the myriad other philosophers before and after. Consequently, Bissell and other Objectivists provide logically consistent, reality-based and practicable methods for understanding the universe including our inner selves. Objectivism is what the scientific method was intended to be: a guide to living.

That said, this book failed to convince me on several points with which I was pre-disposed to agree. And I concede that the ultimate failure may be mine, not the author’s.

Bissell begins with some techniques in speed math. These discoveries from his senior year in high school demonstrate his inductive method. They also provide an introduction to his algorithm for discovering Pythagorean Triples. That alone is worth the price of the book. It is easy enough to explain, though hard to show with the typesetting available here. Basically, you want three integers such that a^2 + b^2 = c^2. Easily, there must be some number, x to begin with. The other number must be some number added to x that can be expressed as x+a, and the result of adding their squares must be some (x+b)^2. It all follows from there.

But I had a hard time following it. I tend to read at bedtime. So, I filled my notebook with pages with arithmetic when I was tired. I told Roger that his algorithms did not work. He asked me to send him PDF scans. I did. He corrected my homework. So, I agree that the Bissell Algorithm will, indeed, generate Pythagorean Triples.

The central essay, “How the Martians Discovered Algebra” (Chapter 4) is a parable to demonstrate induction in mathematics as the doorway that opened to the world of algebra.

Bissell's original algorithm for generating Pythagorean Triples is worth the price of the book. If you have any interest in epistemology, mathematics, or the problem of induction, then Roger Bissell's book delivers more for the money.

And try defining the real and complex numbers without set theory.

Aren't the real numbers all numbers that have decimal representations that have a finite or infinite sequence of digits to the right of the decimal point, positive, negative, or zero?

Aren't complex numbers all the real numbers, imaginary numbers, and sums and differences of real and imaginary numbers?

Aren't the real numbers all numbers that have decimal representations that have a finite or infinite sequence of digits to the right of the decimal point, positive, negative, or zero?

Aren't complex numbers all the real numbers, imaginary numbers, and sums and differences of real and imaginary numbers?

Why aren't these definitions?

Just curious?

Randy

These facts about real numbers tell us nothing of the topology of the real number line.

"Objectivism is what the scientific method was intended to be: a guide for living". (MEM)

Well, there's something cart before horse in that. Do I read it right? The scientific method can establish ~what we know~ of reality, and even so, its method is dependent on (metaphysics and) epistemology - how we know - to which Objectivism (uniquely) and the axioms of existence, identity and identification is integral and fundamental. (Nothing new, straight from the O'ist canon). So, knowledge, as the prerequisite for man's life is primarily answered by the O'ist identity/ identification. Second, I can't see the scientific method coming up with "a guide for living", or how to live; discovering the laws of nature alone, has no means for discovering the 'good' in nature, or what is best or better or bad or worse, i.e. - value - to life. I'd put it extremely, that all scientific knowledge has equivalence - if there's no corresponding theory of value, and an objective one, at that. Also nothing new to you.

Michael: From past debate you'll know I am always interested about Rationalism in contrast to Empiricism (also the effects on O'ism when they each, largely rationalism, re-appear from Objectivists - not to rule out myself either...). Basically I don't believe the two poles of this false dichotomy can be integrated (or combined, matched, etc.) as you suggest. ("Objectivism is an integration of rationalism and empiricism"). I sense by their particular identities there would unavoidably be in the mind a varying cognition-gap between empiricism and rationalism, a clash, then to a resurgence of one over the other, in turn. Even as a rough explanation for Objectivism, the observed facts of reality "without recourse to concepts", Empiricism -- and 'a priori' abstractions without facts, Rationalism, can't share a methodological middle ground - I think. Simplest is to discard both concepts: radically outside the dichotomy, the system of Objectivist methodology starting at the senses and percepts, to identification and integration into one's concepts, gives us the fundamental path to individual knowledge. That establishes THE process and structure within which, most significantly too, all scientific knowledge (found and learned) can be integrated. Therefore, there's no contradiction here with science - as the special, empirical disciplines - either. Sorry to be pedantic or maybe picky about all that, but I think this subject is core to Objectivism.

"Objectivism is what the scientific method was intended to be: a guide for living". (MEM)

Well, there's something cart before horse in that. Do I read it right? The scientific method can establish ~what we know~ of reality, and even so, its method is dependent on (metaphysics and) epistemology - how we know - to which Objectivism (uniquely) and the axioms of existence, identity and identification is integral and fundamental. (Nothing new, straight from the O'ist canon). So, knowledge, as the prerequisite for man's life is primarily answered by the O'ist identity/ identification. Second, I can't see the scientific method coming up with "a guide for living", or how to live; discovering the laws of nature alone, has no means for discovering the 'good' in nature, or what is best or better or bad or worse, i.e. - value - to life. I'd put it extremely, that all scientific knowledge has equivalence - if there's no corresponding theory of value, and an objective one, at that. Also nothing new to you.

Michael: From past debate you'll know I am always interested about Rationalism in contrast to Empiricism (also the effects on O'ism when they each, largely rationalism, re-appear from Objectivists - not to rule out myself either...). Basically I don't believe the two poles of this false dichotomy can be integrated (or combined, matched, etc.) as you suggest. ("Objectivism is an integration of rationalism and empiricism"). I sense by their particular identities there would unavoidably be in the mind a varying cognition-gap between empiricism and rationalism, a clash, then to a resurgence of one over the other, in turn. Even as a rough explanation for Objectivism, the observed facts of reality "without recourse to concepts", Empiricism -- and 'a priori' abstractions without facts, Rationalism, can't share a methodological middle ground - I think. Simplest is to discard both concepts: radically outside the dichotomy, the system of Objectivist methodology starting at the senses and percepts, to identification and integration into one's concepts, gives us the fundamental path to individual knowledge. That establishes THE process and structure within which, most significantly too, all scientific knowledge (found and learned) can be integrated. Therefore, there's no contradiction here with science - as the special, empirical disciplines - either. Sorry to be pedantic or maybe picky about all that, but I think this subject is core to Objectivism.

Physical science can offer very little in the domain of values. The best physics can say is whether an ethical. code is physically realizable or whether it is physically impossible. Nature does not give a hoot as to what we think is right or wrong.

How the Martians Discovered Algebra: Explorations in Induction and the Philosophy of Mathematicsby Roger E. Bissell delivers an algorithm for generating Pythagorean Triples. Central to the thesis of the work, Bissell explains how he discovered this by means of induction, not deduction. From there, Bissell takes the reader into number theory in order to validate his new explanation of the proper understanding of multiplication, and to challenge widespread assumptions about the empty set and infinity.

The relationships between music and mathematics go back to Pythagoras. So, this set of essays by musician Roger Bissell enjoys a solid foundation. Bissell also dabbles at mathematics and has several philosophical explorations to his credit, published in The Reason Papers and the Journal of Ayn Rand Studies.

Objectivism is an integration of rationalism and empiricism. Objectivism rejects the false dichotomies of Descartes, Hume, James, and the myriad other philosophers before and after. Consequently, Bissell and other Objectivists provide logically consistent, reality-based and practicable methods for understanding the universe including our inner selves. Objectivism is what the scientific method was intended to be: a guide to living.

That said, this book failed to convince me on several points with which I was pre-disposed to agree. And I concede that the ultimate failure may be mine, not the author’s.

Bissell begins with some techniques in speed math. These discoveries from his senior year in high school demonstrate his inductive method. They also provide an introduction to his algorithm for discovering Pythagorean Triples. That alone is worth the price of the book. It is easy enough to explain, though hard to show with the typesetting available here. Basically, you want three integers such that a^2 + b^2 = c^2. Easily, there must be some number, x to begin with. The other number must be some number added to x that can be expressed as x+a, and the result of adding their squares must be some (x+b)^2. It all follows from there.

But I had a hard time following it. I tend to read at bedtime. So, I filled my notebook with pages with arithmetic when I was tired. I told Roger that his algorithms did not work. He asked me to send him PDF scans. I did. He corrected my homework. So, I agree that the Bissell Algorithm will, indeed, generate Pythagorean Triples.

The central essay, “How the Martians Discovered Algebra” (Chapter 4) is a parable to demonstrate induction in mathematics as the doorway that opened to the world of algebra.

Bissell's original algorithm for generating Pythagorean Triples is worth the price of the book. If you have any interest in epistemology, mathematics, or the problem of induction, then Roger Bissell's book delivers more for the money.

Let A = (M^2 + N^2), B = 2*M*N and C = (M^2 - N^2) where M > N > 0

Then A^2 = M^4 + 2*M^2*N^2 + N^4, B^2 = 4*M^2*N^2 and C^2 = M^4 - 2*M^2*N^2 + N^4 from which we get

A^2 = B^2 + C^2 by simple algebra. Let M > N >0 and M and N range over positive integers that satisfy the inequality and you get all the pythagorean triples.

Feel free to use this formula --- no charge.

Let''s try out one set to see how it works. Let M = 2, N = 1 than A = 5, B = 4 and C= 3 which gives the famous 3,4,5 triangle, the simplest Pythagorean right triangle.

Well, there's something cart before horse in that. Do I read it right? ... Basically I don't believe the two poles of this false dichotomy can be integrated (or combined, matched, etc.) as you suggest. ... Therefore, there's no contradiction here with science - as the special, empirical disciplines - either. Sorry to be pedantic or maybe picky about all that, but I think this subject is core to Objectivism.

Thanks for the reply. I have no problem with your pedantry. I think that you are right: "... the system of Objectivist methodology starting at the senses and percepts, to identification and integration into one's concepts, gives us the fundamental path to individual knowledge." I just take a broader view and place Objectivism into the historical matrix of philosophy. http://www.coppelia.io/2012/06/graphing-the-history-of-philosophy/

Physical science can offer very little in the domain of values. The best physics can say is whether an ethical. code is physically realizable or whether it is physically impossible. Nature does not give a hoot as to what we think is right or wrong.

You are wrong. Moral choices have real consequences and to be pro-life, they must be based on identifications of reality. True enough: the stars do not care if you live a good life, but in order to live a good life, you need to care about the stars, i.e, to whatever extent possible, you must understand the world you are in. The scientific method, rational-empiricism, or Objectivism, can and does lead to a workable realistic morality. That morality is the source of individual happiness.

The statement "You cannot get an ought from an is" is false. You can. In truth, that is the only way to discover a pro-life moral code.

You are wrong. Moral choices have real consequences and to be pro-life, they must be based on identifications of reality. True enough: the stars do not care if you live a good life, but in order to live a good life, you need to care about the stars, i.e, to whatever extent possible, you must understand the world you are in. The scientific method, rational-empiricism, or Objectivism, can and does lead to a workable realistic morality. That morality is the source of individual happiness.

The statement "You cannot get an ought from an is" is false. You can. In truth, that is the only way to discover a pro-life moral code.

Physical law -constrains- morality. It does not -determine- morality. Any moral code that contradicts the natural physical laws is doomed to failure because it cannot be maintained.

The thing is though that this is well known. Bissell's proof was his own invention.

When I was 14 years old I thought this up all by myself. It turns out that hundreds of mathematicians got to it hundreds of years before I did. Every mathematician or wanna-be mathematician goes through a Pythagorean phase.

Every metric geometrical space which is smooth enough to be differentiable with continuous first derivatives is locally Euclidean. If you take a very small patch of the space, very very small it is nearly flat and Euclidean. So the geometry of planes is a kind of limiting case for all differential manifolds. The space may be bumpy or curved globally (like the surface of a sphere) but locally it desparately wants to be Euclidean. A Euclidean space is one where you can have arbitrarily large right triangles that obey Pythagoras' Theorem.

This video is great fun, Ba'al. Thanks for sharing it!

I confess that I did not go through any complicated logic in order to discover my method for generating Pythagorean triples. I just made a table showing various values that worked, and eventually I saw some suspicious looking patterns. I generalized from those patterns, tried some more variations, generalized a bit further, then realized I had a method that seemed always to work. Then I realized that I could solve the Pythagorean equation for x (though with difficulty, since it required completing a rather messy, unwieldy square), and then I found that I could plug any rational number less than -1 or greater than 0 into my solution for x and generate a Pythagorean triple. It's all in the book, for anyone who wants to see both the inductive jungle I hacked my way through, or the rather straightforward, though difficult deductive mountain I scaled in order to validate the inductive result. (The Einstein/Martians essay was supposed to have illustrated in a briefer, more enjoyable way the two paths to knowledge that my Pythagorean triple essay rather long-windedly illustrated, but I'm not sure that the message has gotten through.)

REB

You got to Pythagoras' Theorem inductively. That is clever. The ancient Egyptian stone cutters and surveyors got to the special case 3-4-5 which was handy to make a "T" square out of knotted rope. All mathematics starts out life as an inductive enterprise. Only later on does it morph into a deductive way of thinking. The Greeks learned their geometry from Egyptian surveyors and stone cutters. In fact the word "geometry" is derived from the Greek word for surveying (earth measure ---- geo metrein.) Historically all mathematics started out from answering two questions --- how many and how big.

You got to Pythagoras' Theorem inductively. ... All mathematics starts out life as an inductive enterprise. Only later on does it morph into a deductive way of thinking. ...

Well, therein lies the brilliant exposititon. Roger's thesis is that mathematics is inductive first. To say that it "morphs into a deductive way of thinking" is a sloppy way to identify something complicated and perhaps not well examined. From what little I know, number theory is inductive. (I reviewed a biography of Paul ErdősThe Man Who Loved Only Numbers by Paul Hoffman on my blog.) Yes, when you publish, your proof must be deductive. Also, Objectivist epistemology shows that abstractions can be treated as concretes and from them, wider abstraction can be identified.

Well, therein lies the brilliant exposititon. Roger's thesis is that mathematics is inductive first. To say that it "morphs into a deductive way of thinking" is a sloppy way to identify something complicated and perhaps not well examined. From what little I know, number theory is inductive. (I reviewed a biography of Paul ErdősThe Man Who Loved Only Numbers by Paul Hoffman on my blog.) Yes, when you publish, your proof must be deductive. Also, Objectivist epistemology shows that abstractions can be treated as concretes and from them, wider abstraction can be identified.

Advanced number theory (analytic number theory) is beyond induction. It uses the full machinery of real and complex function analysis. Many of Gauss prime number theorems are deducible from the properties of the zeta function. There are also probabilistic extensions of number theory which are derived deductively. When a mathematical subject is deep enough it has to be developed deductively. Also any mathematics involving infinities or infinite sets must be developed deductively because there are no infinite sets or quantities in the physical universe -- they are purely abstract and idealistic.

A puzzle for you. What is the cardinal number of the set of first rate mathematicians who are adherents to Ayn Rand's philosophy?

Well, therein lies the brilliant exposititon. Roger's thesis is that mathematics is inductive first. To say that it "morphs into a deductive way of thinking" is a sloppy way to identify something complicated and perhaps not well examined. From what little I know, number theory is inductive. (I reviewed a biography of Paul ErdősThe Man Who Loved Only Numbers by Paul Hoffman on my blog.) Yes, when you publish, your proof must be deductive. Also, Objectivist epistemology shows that abstractions can be treated as concretes and from them, wider abstraction can be identified.

Have a look at this: Do you think you can get this inductively? If so, please indicate how.

Have a look at this: Do you think you can get this inductively? If so, please indicate how.

As the narrator said at the beginning, we have Pythagorean Triples on clay tablets. The Cartesian coordinate system was a logical leap from the meridians and longitudes used 2000 years earlier; it also can be laid to the checkerboard which is descended from the Roman abacus. We use, adapt, extend, invent... Some of us do, at any rate. (And, yes, a nice nod to you for having found that Pythagorean Formula on your own.) But you are stealing a concept here. Mathematics depends on deduction, of course. It also depends on natural language. But it is not solely any one mode to the exclusion of all others. And no one said it was. That is a straw man of your own construction.

As for the number of great mathematicians who are Objectivists, the number may be zero, the same as the number who are Platonists, Humeans, or followers of DeLuze or Rorty. However, if you are counting great minds in order to decide what you should believe, then you should be a communist. As I understand it, of all the mathematicians of the 20th century and of all of the philosophies known at least to 1989, the only consistent correspondence was the great mathematicians of the USSR.

On the other hand, it might also be said, more truthfully, that every mathematician, whatever their stature in your mind, is an Objectivist to the extent that they recognize reality, conform to reason, and place no one else's judgment above their own... which suggests that all of those communist mathematicians were lying to save their skins -- also a logical course of action based on the facts of reality.

Have a look at this: Do you think you can get this inductively? If so, please indicate how.

This is like pointing to quantum mechanics or relativity theory and asking if you can "get" either of them through sense perception. The reply is: how in the world ELSE do you think you can "get" them - or this marvelous display of all the Pythagorean triples? You cannot get abstract conceptual products without lower-level concepts based on sense perception - and you cannot get deductive conclusions without inductively gained information and conclusions on which all deduction is necessarily based.

As I stated in my book, "Not that deduction is insignificant. It is the engine of proof, after all. However, without induction, mathematics - like any other discipline - simply could not get off the ground" (How the Martians Discovered Algebra, p. 51).

And thank you, Michael, for pointing out the Stolen Concept involved in Ba'al's question.

You got to Pythagoras' Theorem inductively. That is clever.

It was induction based on observation of numerical relations aligned in columns that made certain additional relationships observable. Not all that different from being a medical examiner or forensic scientist: you lay out the data in a helpful way and look for connections. But in induction, you look for *repeated* connections that have a necessary foundation, and then you figure out what that foundation is and express it as a generalization. And then you try to link it to your other knowledge and give a deductive validation of it.

But I didn't think the inductive part of my work was all that clever. It was just a lot of hard work and attention to detail and to patterns in that detail. It was when I hit upon the idea of validating my method by solving the Pythagorean equation that I thought I was particularly clever. It was mind-crunchingly difficult to actually solve it, but I figured what the hell, put on your big boy pants and do it. And the payoff was that I validated my method. If the Pythagorean equation was true, then so was my method. Q.E.D.

And actually, I didn't get to Pythagoras's equation by induction. I got my Pythagorean triple method by induction and I validated it by solving Pythagoras's equation by completing the square. I took Pythagoras's equation as a given and deduced what it would mean about the relationships between all the terms in the equation - which exactly matched what I had developed inductively from the columns of numbers that in no way depended on the Pythagorean equation (except that I worked with squares and the like, instead of cubes or square roots etc.).

A puzzle for you. What is the cardinal number of the set of first rate mathematicians who are adherents to Ayn Rand's philosophy?

1

I'm more interested in: what is the cardinal number of the set of talented and productive amateur mathematicians who are adherents to Ayn Rand's philosophy. I'm pretty sure that cardinal number is greater than 1.

I was looking at old letters and found my choice for most scathing yet factually true letter of all time, in my dusty files. Insightful. Rational. Intuitive. Relevant. I knew the story but I never made his connections before and it’s perfect for Halloween! All of this is found in the following letter from REB, closed up for brevity. Enjoy. Peter

From: RogerEBissell To: atlantis Subject: ATL: Speaking of "Passover"... (was Psychology of Religion) Date: Tue, 25 Sep 2001 21:47:06 EDT

Dear List Members: We have been discussing whether or not there is some strong, inherent tendency in religion to occasionally/often devolve into advocacy/support of the kind of bloody conflicts and barbarism seen during the Crusades, the Spanish Inquisition and invasion of Latin America, the second World War, and the attack on the WTC and Pentagon -- to name a few.

Debbie Clark has stated that such acts or processes are not actually a result of the essence of religion but, instead, a ~perversion~ of religion, which preaches love and peace. God / Allah / Whoever does not believe in terror and butchery, and those who commit such in His/Her name are twisting religion in the direction of evil. Yet, Debbie doesn't address some of the....morally questionable.... acts of the very God that such evildoers are supposedly blaspheming with their violent deeds.

Ironically, Debbie reminded me of one of the most pertinent such acts in this comment of hers: > I don't have a bible in front of me at the moment andrarely read it anymore so my memory is rusty -- but I think in the book of Acts, the apostle Paul is admonishing the church members about what their conduct should be when they come together to eat the Passover, i.e. the bread and wine symbolizing the body of Christ (commonly referred to as communion by Catholics and Protestants, but as originally established, it was an annual event at Passover as instituted by Christ, not a weekly event).He said something to them along the lines of, "What?Don't you have homes to eat in?" and the reason he said that, to my understanding, is because they were not being reverent during the Passover ceremony and understanding the meaning of it.The purpose of it was not for the sensual pleasure of eating or to nourish the physical body -- those are things which everyone does in the course of normal physical lives.But eating the bread and wine at Passover, which Christ instituted as symbols of his body, i.e. taking the place of the lamb that previously was slaughtered and eaten at Passover, was something that was supposed to have spiritual meaning, not done for the purpose of sensual pleasure or nourishment of the body.Get it?

Ah, yes, the Passover! Does anyone ~not~ know the original context of this celebration? The Hebrews were in captivity in Egypt, and the Pharaoh refused Moses' demands that they be allowed to leave, so God "visited" various plagues and....misfortunes....upon the Egyptians, in an attempt to get Pharaoh to change his mind. The ultimate "act of God" was the killing by "the angel of the Lord" (or some such flunky) of all firstborn sons of the Egyptian families -- the Hebrew firstborn children being "passed over", because their folks had been warned to smear lamb's blood over their doorway (a signal agreed upon in advance by Moses and God).

Now, doesn't this suggest to anyone that the Ultimate Terrorist in history was not just religion, but the very Object of Religion? Not only the Jews, but also the Christians ~and~ the Muslims accept this as part of their religious history and heritage. It's OK for Big G to carry out such butchery of THE INNOCENT, but ~not~ OK for Muslim Fundamentalists? At least the freaking MF's are consistent. There seems also to be a huge papering over of the massive amount of genocide that the Hebrews carried out in their long process of taking over the "Promised Land" after leaving Egypt. I once did the arithmetic and found that the total number ~said~ to have been slain by the Hebrews in those campaigns totaled about SIX MILLION. Sound familiar? This ain't Nostradamus, folks; it's the good old King James, Torah, and Koran -- and God OK'd it (hell, he ~cheered them on~).

Of course, I don't believe the above-described fairy tales for a moment. But Judeo-Christians and Muslims ~do~, or ~say~ they do, and Debbie seems to be among the many, many folks who conveniently....forget?.... more than a little of the..."ruling style"....of the Supreme Being they claim to worship.

Rand really said all that needed to be said about the nature of the diabolical union of faith and force in her "For the New Intellectual" characterizations of the Witch Doctor and Attila, and her Ford Hall Forum lecture "Faith and Force: Destroyers of the Modern World." And all that needs to be ~done~ about the most recent incarnation of this union is to meet it with overwhelming force, grab it firmly by the throat, and choke it to a much-deserved death -- which I fervently hope GWB et al are able to do. Roast in hell, Osama & Company, roast in hell.

Be grateful that this post was written by the ~mellow~ Roger Bissell. No ad hominems (that I can detect). Just terminal weariness for apologetics for religion and its supposed essence of love and peace. Best regards to all, Roger Bissell

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## BaalChatzaf

Thank you for referencing the thread "The Opposite of Nothing Is/Isn't Everything" I think that thread was OL at its very, very best. It is a shame that we are so hung up on matters political a

## Roger Bissell

Couldn't agree with you more, BC. And the next time you want to condescendingly, sneeringly put down my announcements about my work, at least give a passing nod to objectivity by actually quoting me i

## Roger Bissell 15

AuthorOnly a computer science geek could confuse relative nothingness (the absence of something in particular) with absolute Nothingness.

Otherwise, why talk about

addingzero, as though zero were some actual quantity, rather than the absence of a quantity?REB

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## BaalChatzaf 60

What is being the identity element of the additive semi-group of counting numbers the nothingness of?

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## BaalChatzaf 60

Nothing is absolute,. not relative. One either has nothing or something. There is no middle ground and there is no degree.

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## syrakusos 1

How the Martians Discovered Algebra: Explorations in Induction and the Philosophy of Mathematicsby Roger E. Bisselldelivers an algorithm for generating Pythagorean Triples. Central to the thesis of the work, Bissell explains how he discovered this by means of induction, not deduction. From there, Bissell takes the reader into number theory in order to validate his new explanation of the proper understanding of multiplication, and to challenge widespread assumptions about the empty set and infinity.The relationships between music and mathematics go back to Pythagoras. So, this set of essays by musician Roger Bissell enjoys a solid foundation. Bissell also dabbles at mathematics and has several philosophical explorations to his credit, published in

The Reason Papersand theJournal of Ayn Rand Studies.Bissell's original algorithm for generating Pythagorean Triples is worth the price of the book. If you have any interest in epistemology, mathematics, or the problem of induction, then Roger Bissell's book delivers more for the money.

Full review here:

https://necessaryfacts.blogspot.com/2017/12/how-martians-discovered-algebra.html

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## regi 0

Aren't the real numbers all numbers that have decimal representations that have a finite or infinite sequence of digits to the right of the decimal point, positive, negative, or zero?

Aren't complex numbers all the real numbers, imaginary numbers, and sums and differences of real and imaginary numbers?

Why aren't these definitions?

Just curious?

Randy

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## BaalChatzaf 60

These facts about real numbers tell us nothing of the topology of the real number line.

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## anthony 172

"Objectivism is what the scientific method was intended to be: a guide for living". (MEM)

Well, there's something cart before horse in that. Do I read it right? The scientific method can establish ~what we know~ of reality, and even so, its method is dependent on (metaphysics and) epistemology - how we know - to which Objectivism (uniquely) and the axioms of existence, identity and identification is integral and fundamental. (Nothing new, straight from the O'ist canon). So, knowledge, as the prerequisite for man's life is primarily answered by the O'ist identity/ identification. Second, I can't see the scientific method coming up with "a guide for living", or how to live; discovering the laws of nature alone, has no means for discovering the 'good' in nature, or what is best or better or bad or worse, i.e. - value - to life. I'd put it extremely, that all scientific knowledge has equivalence - if there's no corresponding theory of value, and an objective one, at that. Also nothing new to you.

Michael: From past debate you'll know I am always interested about Rationalism in contrast to Empiricism (also the effects on O'ism when they each, largely rationalism, re-appear from Objectivists - not to rule out myself either...). Basically I don't believe the two poles of this false dichotomy can be integrated (or combined, matched, etc.) as you suggest. ("Objectivism is an integration of rationalism and empiricism"). I sense by their particular identities there would unavoidably be in the mind a varying cognition-gap between empiricism and rationalism, a clash, then to a resurgence of one over the other, in turn. Even as a rough explanation for Objectivism, the observed facts of reality "without recourse to concepts", Empiricism -- and 'a priori' abstractions without facts, Rationalism, can't share a methodological middle ground - I think. Simplest is to discard both concepts: radically outside the dichotomy, the system of Objectivist methodology starting at the senses and percepts, to identification and integration into one's concepts, gives us the fundamental path to individual knowledge. That establishes THE process and structure within which, most significantly too, all scientific knowledge (found and learned) can be integrated. Therefore, there's no contradiction here with science - as the special, empirical disciplines - either. Sorry to be pedantic or maybe picky about all that, but I think this subject is core to Objectivism.

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## BaalChatzaf 60

Physical science can offer very little in the domain of values. The best physics can say is whether an ethical. code is physically realizable or whether it is physically impossible. Nature does not give a hoot as to what we think is right or wrong.

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## BaalChatzaf 60

Let A = (M^2 + N^2), B = 2*M*N and C = (M^2 - N^2) where M > N > 0

Then A^2 = M^4 + 2*M^2*N^2 + N^4, B^2 = 4*M^2*N^2 and C^2 = M^4 - 2*M^2*N^2 + N^4 from which we get

A^2 = B^2 + C^2 by simple algebra. Let M > N >0 and M and N range over positive integers that satisfy the inequality and you get all the pythagorean triples.

Feel free to use this formula --- no charge.

Let''s try out one set to see how it works. Let M = 2, N = 1 than A = 5, B = 4 and C= 3 which gives the famous 3,4,5 triangle, the simplest Pythagorean right triangle.

Ba'al Chatzaf

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## syrakusos 1

Thanks for the reply. I have no problem with your pedantry. I think that you are right: "... the system of Objectivist methodology starting at the senses and percepts, to identification and integration into one's concepts, gives us the fundamental path to individual knowledge." I just take a broader view and place Objectivism into the historical matrix of philosophy. http://www.coppelia.io/2012/06/graphing-the-history-of-philosophy/

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## syrakusos 1

You are wrong. Moral choices have real consequences and to be pro-life, they must be based on identifications of reality. True enough: the stars do not care if you live a good life, but in order to live a good life, you need to care about the stars, i.e, to whatever extent possible, you must understand the world you are in. The scientific method, rational-empiricism, or Objectivism, can and does lead to a workable realistic morality. That morality is the source of individual happiness.

The statement "You cannot get an ought from an is" is false. You can. In truth, that is the only way to discover a pro-life moral code.

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## syrakusos 1

The thing is though that this is well known. Bissell's proof was his own invention.

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## anthony 172

Thanks, Michael!

Most memorable line: "...you need to care about the stars..."

And all that's under them then goes without saying.

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## BaalChatzaf 60

Physical law -constrains- morality. It does not -determine- morality. Any moral code that contradicts the natural physical laws is doomed to failure because it cannot be maintained.

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## BaalChatzaf 60

When I was 14 years old I thought this up all by myself. It turns out that hundreds of mathematicians got to it hundreds of years before I did. Every mathematician or wanna-be mathematician goes through a Pythagorean phase.

Every metric geometrical space which is smooth enough to be differentiable with continuous first derivatives is locally Euclidean. If you take a very small patch of the space, very very small it is nearly flat and Euclidean. So the geometry of planes is a kind of limiting case for all differential manifolds. The space may be bumpy or curved globally (like the surface of a sphere) but locally it desparately wants to be Euclidean. A Euclidean space is one where you can have arbitrarily large right triangles that obey Pythagoras' Theorem.

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## BaalChatzaf 60

You got to Pythagoras' Theorem inductively. That is clever. The ancient Egyptian stone cutters and surveyors got to the special case 3-4-5 which was handy to make a "T" square out of knotted rope. All mathematics starts out life as an inductive enterprise. Only later on does it morph into a deductive way of thinking. The Greeks learned their geometry from Egyptian surveyors and stone cutters. In fact the word "geometry" is derived from the Greek word for surveying (earth measure ---- geo metrein.) Historically all mathematics started out from answering two questions --- how many and how big.

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## syrakusos 1

Well, therein lies the brilliant exposititon. Roger's thesis is that mathematics is inductive first. To say that it "morphs into a deductive way of thinking" is a sloppy way to identify something complicated and perhaps not well examined. From what little I know, number theory is inductive. (I reviewed a biography of Paul Erdős

The Man Who Loved Only Numbersby Paul Hoffman on my blog.) Yes, when you publish, your proof must be deductive. Also, Objectivist epistemology shows that abstractions can be treated as concretes and from them, wider abstraction can be identified.## Link to post

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## BaalChatzaf 60

Advanced number theory (analytic number theory) is beyond induction. It uses the full machinery of real and complex function analysis. Many of Gauss prime number theorems are deducible from the properties of the zeta function. There are also probabilistic extensions of number theory which are derived deductively. When a mathematical subject is deep enough it has to be developed deductively. Also any mathematics involving infinities or infinite sets must be developed deductively because there are no infinite sets or quantities in the physical universe -- they are purely abstract and idealistic.

A puzzle for you. What is the cardinal number of the set of first rate mathematicians who are adherents to Ayn Rand's philosophy?

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## BaalChatzaf 60

Have a look at this: Do you think you can get this inductively? If so, please indicate how.

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## syrakusos 1

As the narrator said at the beginning, we have Pythagorean Triples on clay tablets. The Cartesian coordinate system was a logical leap from the meridians and longitudes used 2000 years earlier; it also can be laid to the checkerboard which is descended from the Roman abacus. We use, adapt, extend, invent... Some of us do, at any rate. (And, yes, a nice nod to you for having found that Pythagorean Formula on your own.) But you are stealing a concept here. Mathematics depends on deduction, of course. It also depends on natural language. But it is not solely any one mode to the exclusion of all others. And no one said it was. That is a straw man of your own construction.

As for the number of great mathematicians who are Objectivists, the number may be zero, the same as the number who are Platonists, Humeans, or followers of DeLuze or Rorty. However, if you are counting great minds in order to decide what you should believe, then you should be a communist. As I understand it, of all the mathematicians of the 20th century and of all of the philosophies known at least to 1989, the only consistent correspondence was the great mathematicians of the USSR.

On the other hand, it might also be said, more truthfully, that every mathematician, whatever their stature in your mind, is an Objectivist to the extent that they recognize reality, conform to reason, and place no one else's judgment above their own... which suggests that all of those communist mathematicians were lying to save their skins -- also a logical course of action based on the facts of reality.

I think that just about covers it.

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## Roger Bissell 15

AuthorThis is like pointing to quantum mechanics or relativity theory and asking if you can "get" either of them through sense perception. The reply is: how in the world ELSE do you think you can "get" them - or this marvelous display of all the Pythagorean triples? You cannot get abstract conceptual products without lower-level concepts based on sense perception - and you cannot get deductive conclusions without inductively gained information and conclusions on which all deduction is necessarily based.

As I stated in my book, "Not that deduction is insignificant. It is the engine of proof, after all. However, without induction, mathematics - like any other discipline - simply could not get off the ground" (How the Martians Discovered Algebra, p. 51).

And thank you, Michael, for pointing out the Stolen Concept involved in Ba'al's question.

REB

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## Roger Bissell 15

AuthorIt was induction based on observation of numerical relations aligned in columns that made certain additional relationships observable. Not all that different from being a medical examiner or forensic scientist: you lay out the data in a helpful way and look for connections. But in induction, you look for *repeated* connections that have a necessary foundation, and then you figure out what that foundation is and express it as a generalization. And then you try to link it to your other knowledge and give a deductive validation of it.

But I didn't think the inductive part of my work was all that clever. It was just a lot of hard work and attention to detail and to patterns in that detail. It was when I hit upon the idea of validating my method by solving the Pythagorean equation that I thought I was particularly clever. It was mind-crunchingly difficult to actually solve it, but I figured what the hell, put on your big boy pants and do it. And the payoff was that I validated my method. If the Pythagorean equation was true, then so was my method. Q.E.D.

And actually, I didn't get to Pythagoras's equation by induction. I got my Pythagorean triple method by induction and I validated it by solving Pythagoras's equation by completing the square. I took Pythagoras's equation as a given and deduced what it would mean about the relationships between all the terms in the equation - which exactly matched what I had developed inductively from the columns of numbers that in no way depended on the Pythagorean equation (except that I worked with squares and the like, instead of cubes or square roots etc.).

REB

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## Roger Bissell 15

AuthorI'm more interested in: what is the cardinal number of the set of talented and productive amateur mathematicians who are adherents to Ayn Rand's philosophy. I'm pretty sure that cardinal number is greater than 1.

REB

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## Peter 237

I was looking at old letters and found my choice for most scathing yet factually true letter of all time, in my dusty files. Insightful. Rational. Intuitive. Relevant. I knew the story but I never made his connections before and it’s perfect for Halloween! All of this is found in the following letter from REB, closed up for brevity. Enjoy. Peter

From: RogerEBissell To: atlantis Subject: ATL: Speaking of "Passover"... (was Psychology of Religion) Date: Tue, 25 Sep 2001 21:47:06 EDT

Dear List Members: We have been discussing whether or not there is some strong, inherent tendency in religion to occasionally/often devolve into advocacy/support of the kind of bloody conflicts and barbarism seen during the Crusades, the Spanish Inquisition and invasion of Latin America, the second World War, and the attack on the WTC and Pentagon -- to name a few.

Debbie Clark has stated that such acts or processes are not actually a result of the essence of religion but, instead, a ~perversion~ of religion, which preaches love and peace. God / Allah / Whoever does not believe in terror and butchery, and those who commit such in His/Her name are twisting religion in the direction of evil. Yet, Debbie doesn't address some of the....morally questionable.... acts of the very God that such evildoers are supposedly blaspheming with their violent deeds.

Ironically, Debbie reminded me of one of the most pertinent such acts in this comment of hers: > I don't have a bible in front of me at the moment and rarely read it anymore so my memory is rusty -- but I think in the book of Acts, the apostle Paul is admonishing the church members about what their conduct should be when they come together to eat the Passover, i.e. the bread and wine symbolizing the body of Christ (commonly referred to as communion by Catholics and Protestants, but as originally established, it was an annual event at Passover as instituted by Christ, not a weekly event). He said something to them along the lines of, "What? Don't you have homes to eat in?" and the reason he said that, to my understanding, is because they were not being reverent during the Passover ceremony and understanding the meaning of it. The purpose of it was not for the sensual pleasure of eating or to nourish the physical body -- those are things which everyone does in the course of normal physical lives. But eating the bread and wine at Passover, which Christ instituted as symbols of his body, i.e. taking the place of the lamb that previously was slaughtered and eaten at Passover, was something that was supposed to have spiritual meaning, not done for the purpose of sensual pleasure or nourishment of the body. Get it?

Ah, yes, the Passover! Does anyone ~not~ know the original context of this celebration? The Hebrews were in captivity in Egypt, and the Pharaoh refused Moses' demands that they be allowed to leave, so God "visited" various plagues and....misfortunes....upon the Egyptians, in an attempt to get Pharaoh to change his mind. The ultimate "act of God" was the killing by "the angel of the Lord" (or some such flunky) of all firstborn sons of the Egyptian families -- the Hebrew firstborn children being "passed over", because their folks had been warned to smear lamb's blood over their doorway (a signal agreed upon in advance by Moses and God).

Now, doesn't this suggest to anyone that the Ultimate Terrorist in history was not just religion, but the very Object of Religion? Not only the Jews, but also the Christians ~and~ the Muslims accept this as part of their religious history and heritage. It's OK for Big G to carry out such butchery of THE INNOCENT, but ~not~ OK for Muslim Fundamentalists? At least the freaking MF's are consistent. There seems also to be a huge papering over of the massive amount of genocide that the Hebrews carried out in their long process of taking over the "Promised Land" after leaving Egypt. I once did the arithmetic and found that the total number ~said~ to have been slain by the Hebrews in those campaigns totaled about SIX MILLION. Sound familiar? This ain't Nostradamus, folks; it's the good old King James, Torah, and Koran -- and God OK'd it (hell, he ~cheered them on~).

Of course, I don't believe the above-described fairy tales for a moment. But Judeo-Christians and Muslims ~do~, or ~say~ they do, and Debbie seems to be among the many, many folks who conveniently....forget?.... more than a little of the..."ruling style"....of the Supreme Being they claim to worship.

Rand really said all that needed to be said about the nature of the diabolical union of faith and force in her "For the New Intellectual" characterizations of the Witch Doctor and Attila, and her Ford Hall Forum lecture "Faith and Force: Destroyers of the Modern World." And all that needs to be ~done~ about the most recent incarnation of this union is to meet it with overwhelming force, grab it firmly by the throat, and choke it to a much-deserved death -- which I fervently hope GWB et al are able to do. Roast in hell, Osama & Company, roast in hell.

Be grateful that this post was written by the ~mellow~ Roger Bissell. No ad hominems (that I can detect). Just terminal weariness for apologetics for religion and its supposed essence of love and peace. Best regards to all, Roger Bissell

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