Roger Bissell 2 Report post Posted August 16, 2017 On 8/14/2017 at 8:03 PM, BaalChatzaf said: Roger, did any of your mathematical investigations turn this up: 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 Share this post Link to post Share on other sites

Roger Bissell 2 Report post Posted August 16, 2017 On 8/9/2017 at 1:53 PM, BaalChatzaf said: Only a philosopher or a jester could confuse 0 with Nothing. Only a computer science geek could confuse relative nothingness (the absence of something in particular) with absolute Nothingness. Otherwise, why talk about adding zero, as though zero were some actual quantity, rather than the absence of a quantity? REB Share this post Link to post Share on other sites

BaalChatzaf 8 Report post Posted August 16, 2017 14 hours ago, Roger Bissell said: This is a false alternative. Zero is not absolute nothingness. But that doesn't mean it is something. It is the absence of 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. https://www.sciencedaily.com/releases/2017/07/170712145654.htm What is being the identity element of the additive semi-group of counting numbers the nothingness of? Share this post Link to post Share on other sites

BaalChatzaf 8 Report post Posted August 16, 2017 13 hours ago, Roger Bissell said: Only a computer science geek could confuse relative nothingness (the absence of something in particular) with absolute Nothingness. Otherwise, why talk about adding zero, as though zero were some actual quantity, rather than the absence of a quantity? REB Nothing is absolute,. not relative. One either has nothing or something. There is no middle ground and there is no degree. Share this post Link to post Share on other sites

syrakusos 1 Report post Posted December 11, 2017 How the Martians Discovered Algebra: Explorations in Induction and the Philosophy of Mathematics by 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.Full review here:https://necessaryfacts.blogspot.com/2017/12/how-martians-discovered-algebra.html Share this post Link to post Share on other sites

regi 0 Report post Posted December 11, 2017 On 7/15/2017 at 8:26 PM, BaalChatzaf said: 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? Why aren't these definitions? Just curious? Randy Share this post Link to post Share on other sites

BaalChatzaf 8 Report post Posted December 11, 2017 2 hours ago, regi said: 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. Share this post Link to post Share on other sites

anthony 23 Report post Posted December 12, 2017 "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. Share this post Link to post Share on other sites

BaalChatzaf 8 Report post Posted December 12, 2017 8 hours ago, anthony said: "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. Share this post Link to post Share on other sites

BaalChatzaf 8 Report post Posted December 12, 2017 On 12/10/2017 at 7:44 PM, syrakusos said: How the Martians Discovered Algebra: Explorations in Induction and the Philosophy of Mathematics by 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.Full review here:https://necessaryfacts.blogspot.com/2017/12/how-martians-discovered-algebra.html 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 Share this post Link to post Share on other sites

syrakusos 1 Report post Posted December 14, 2017 On 12/12/2017 at 8:59 AM, anthony said: 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/ Share this post Link to post Share on other sites

syrakusos 1 Report post Posted December 14, 2017 On 12/12/2017 at 5:19 PM, BaalChatzaf said: 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. Share this post Link to post Share on other sites

syrakusos 1 Report post Posted December 14, 2017 On 12/12/2017 at 5:35 PM, BaalChatzaf said: Let A = (M^2 + N^2), B = 2*M*N and C = (M^2 - N^2) where M > N > 0 ... Feel free to use this formula --- no charge. Ba'al Chatzaf The thing is though that this is well known. Bissell's proof was his own invention. Share this post Link to post Share on other sites

anthony 23 Report post Posted December 14, 2017 Thanks, Michael! Most memorable line: "...you need to care about the stars..." And all that's under them then goes without saying. Share this post Link to post Share on other sites

BaalChatzaf 8 Report post Posted December 14, 2017 9 hours ago, syrakusos said: 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. Share this post Link to post Share on other sites

BaalChatzaf 8 Report post Posted December 14, 2017 9 hours ago, syrakusos said: 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. Share this post Link to post Share on other sites

BaalChatzaf 8 Report post Posted December 14, 2017 On 8/15/2017 at 11:45 PM, Roger Bissell said: 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. Share this post Link to post Share on other sites

syrakusos 1 Report post Posted December 15, 2017 13 hours ago, BaalChatzaf said: 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 Erdős The 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. Share this post Link to post Share on other sites

BaalChatzaf 8 Report post Posted December 15, 2017 12 hours ago, syrakusos said: 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 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? Share this post Link to post Share on other sites

BaalChatzaf 8 Report post Posted December 15, 2017 12 hours ago, syrakusos said: 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 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. Share this post Link to post Share on other sites

syrakusos 1 Report post Posted December 16, 2017 9 hours ago, BaalChatzaf said: 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. I think that just about covers it. Share this post Link to post Share on other sites

Roger Bissell 2 Report post Posted December 28, 2017 On 12/15/2017 at 9:55 AM, BaalChatzaf said: 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. REB Share this post Link to post Share on other sites

Roger Bissell 2 Report post Posted December 28, 2017 On 12/14/2017 at 8:14 AM, BaalChatzaf said: 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.). REB Share this post Link to post Share on other sites

Roger Bissell 2 Report post Posted December 28, 2017 On 12/15/2017 at 9:53 AM, BaalChatzaf said: 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. REB Share this post Link to post Share on other sites