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

The truth of progress is somewhere between "social forces" and "the great man." Obviously, someone invents new ideas. But many of the seeds never took, apparently having died for lack of fertile soil.

Quote

Baal wrote: If only Archimedes had the zero (sigh). We would be traveling about in Star Ships rather than jet propelled airplanes.

I point to the discussion of the Antikythera Device here in OL, especially my comments, not Baal's. I got my information from the best lecture, not Nova or Discovery. And I referenced the papers of the best discovery for those who want to explore the details for themselves. The point here is that the Antikythera Device, attributable perhaps to Archimedes or perhaps to a collaboration among Euclid, Apollonius, and Archimedes, was not a singular creation. It was obviously the result of a long development and could not have existed in isolation. Yet, where are the other evidences? We know from references about coin-operated prayer machines, and the steam engine of Heron. But we have not much else... I point also to a reference that the First Citizen ("emperor") Claudius wrote a multivolume history of the Etruscans. He may have included a grammar of the language. All of that, too, is lost, and it came from the center of power of the time.

I am not one of the doomsayers here an in O-land who seem to look forward to the coming collapse of civilization. (See The Future and Its Enemies by Virginia Postrel.) Civilization is not collapsing. The end is not in sight. ... no matter what they want to believe. That said, however, if you consider the Bronze Age Collapse, and the decline of Rome, it is clear that all of this is very fragile. It deserves respect and protection.

If Archimedes had had the zero, we would likely be pretty close to where we are today. See The Invention of Enterprise by Joel Mokyr, ed. and Against the Gods by Peter L. Bernstein. Every civilized society - and many other cultures - has "merchants" but capitalism only came from the confluence of Renaissance and the invention of statistics. The Romans of Archimedes' time had merchants. But their society valued conquest more. Successful merchants turned their enterprises over their freedmen and slaves, and retired to the countryside to live as gentleman farmers. You could have given them Alan Turing, and it would not have made any difference.

(And I apologize for sidetracking the discussion of Roger's book. I will make up for it in a later post. The book is due here Monday.)

The truth of progress is somewhere between "social forces" and "the great man." Obviously, someone invents new ideas. But many of the seeds never took, apparently having died for lack of fertile soil.

I point to the discussion of the Antikythera Device here in OL, especially my comments, not Baal's. I got my information from the best lecture, not Nova or Discovery. And I referenced the papers of the best discovery for those who want to explore the details for themselves. The point here is that the Antikythera Device, attributable perhaps to Archimedes or perhaps to a collaboration among Euclid, Apollonius, and Archimedes, was not a singular creation. It was obviously the result of a long development and could not have existed in isolation. Yet, where are the other evidences? We know from references about coin-operated prayer machines, and the steam engine of Heron. But we have not much else... I point also to a reference that the First Citizen ("emperor") Claudius wrote a multivolume history of the Etruscans. He may have included a grammar of the language. All of that, too, is lost, and it came from the center of power of the time.

I am not one of the doomsayers here an in O-land who seem to look forward to the coming collapse of civilization. (See The Future and Its Enemies by Virginia Postrel.) Civilization is not collapsing. The end is not in sight. ... no matter what they want to believe. That said, however, if you consider the Bronze Age Collapse, and the decline of Rome, it is clear that all of this is very fragile. It deserves respect and protection.

If Archimedes had had the zero, we would likely be pretty close to where we are today. See The Invention of Enterprise by Joel Mokyr, ed. and Against the Gods by Peter L. Bernstein. Every civilized society - and many other cultures - has "merchants" but capitalism only came from the confluence of Renaissance and the invention of statistics. The Romans of Archimedes' time had merchants. But their society valued conquest more. Successful merchants turned their enterprises over their freedmen and slaves, and retired to the countryside to live as gentleman farmers. You could have given them Alan Turing, and it would not have made any difference.

(And I apologize for sidetracking the discussion of Roger's book. I will make up for it in a later post. The book is due here Monday.)

You are aware, I assume, that what we know of the Device is based on reverse engineering. Using the most advanced scanning technology the innards of the device were made visible. The scholars studying the device could see which gears connect to which gears and how many teeth were on each will. By working out the gear ratios they were able to guess that the device did astronomical calculations. Perhaps it took a team to make the Device in the first place. It surely took a very talented team to come up with a hypothesis based reconstruction of the Device.

In a way, the reconstruction of the Device invoked one of Aristotle's Four Causes. The most subtle of the cause, the Final Cause says a thing exists in order to achieve an end, purpose or goal.

The scholars studying the A.K. Device concluded that it was a hand carried astronomical simulator and astronomical calendar. Its purpose was to imitate the observed positions of observable astronomical bodies and moon phases. Once the scholars knew or postulated what the device -was for- they were able to fill in gaps in their knowledge of the total device, because not all the pieces survived corrosion. So final cause was bounded back to formal cause and material cause a bounced forward for rechecking. Cool! Score one for the Philosopher and the A.K. Device team.

If the Greek mathematicians had the zero, then their successors would have come up with the mathematics of motion much sooner. Think of having differential equations one thousand years sooner. Advances in physics were driven by both new facts discovered by observation and by knowing the underlying mathematical symmetries of cosmos. O.K., I was exaggerating a bit about the Star Ships but I am certain we would have had heat engines and flying machines much sooner than we actually did. It is almost certain that optics would have been advanced with better mathematics, which in term means we would have had telescopes and microscopes sooner.

As it was Archimedes did come up with something very close to integral calculus (look uj Archimedes Codex). But his math did not "catch on". He did not establish a school. There is a case of the Great Man not achieving his full potential because the society around him was not ready for it.

Your observations of the rise of capitalism out of mercantilism appear to be on the mark. For one Great Man to succeed a lot of little pieces and a lot of smaller men have to be ready to carry the matter through. In a sense, the times have to be right and ripe for the advance.

How the Martians Discovered Algebra by Roger E. Bissell arrived yesterday. It is nicely done. CreateSpace did a good job.

I ordered the book on July 18, 2017, and got a programmed response to that.

I received a confirmation on July 22 with a promise of July 31 arrival.

The book arrived via UPS on July 26.

Just reading a random page for now, it reminds me of George Boole. We are taught to think of syllogisms as reductions. Boole presented his algebra as a way to expand statements. So, too, does Bissell (as far as I have read) expansively explain that 0 + 5 is not "five added to zero," but starting with no action, you begin with the existence of five items.

I am about 20 pages into the book and enjoying it. I read at bedtime, so I do not work the problems, but I will work through your method for generating Pythagorean Triples. I recently read an old book that presented 300 proofs of the Pythagorean Theorem. (Reviewed on my blog, here: http://necessaryfacts.blogspot.com/2017/04/elisha-s-loomis-and-pythagorean.html )

I am about 20 pages into the book and enjoying it. I read at bedtime, so I do not work the problems, but I will work through your method for generating Pythagorean Triples. I recently read an old book that presented 300 proofs of the Pythagorean Theorem. (Reviewed on my blog, here: http://necessaryfacts.blogspot.com/2017/04/elisha-s-loomis-and-pythagorean.html )

^ means exponentiate, i.e. raise to a power

(M^2 + N^2)^2 - (M^2 - N^2)^2 = 4M^2N^2 = (2MN)^2

so, (M^2 - N^2)^2 + (2MN)^2 = (M^2 + N^2)^2

The interesting part comes in proving that -all- the integral Pythagorean triples are produced by this formula.

The ancient Egyptian stone cutters long before Pythagoras and his friends used a knotted rope with equally spaced knots in the pattern 3 knots, 4 knots, 5 knots to "square" off the blocks of marble and granite that they quarried. 3,4,5 is the simplest Pythagorean integer triple.

I am about 20 pages into the book and enjoying it. I read at bedtime, so I do not work the problems, but I will work through your method for generating Pythagorean Triples. I recently read an old book that presented 300 proofs of the Pythagorean Theorem. (Reviewed on my blog, here: http://necessaryfacts.blogspot.com/2017/04/elisha-s-loomis-and-pythagorean.html )

You Rational Empiricists are all alike - a quick 7 or 8 theorems and you're off with the boys!

I just received in the mail today Eli Maor's 2007 book The Pythagorean Theorem: A 4,000 Year History (Princeton University Press). It's a very nice looking book, and I can't wait for bedtime to read it!

Last week, I received a mug and tee-shirt celebrating what appears to be the fourth and final Pythagorean Theory Day in the 21st century. It's coming right up on August 15. (08/15/17, which is a Pythagorean triple) (The other three were (03/04/05, 06/08/10, and 05/12/13.)

REB

P.S. - It's fascinating to me that Leonard Peikoff, 45 years ago, claimed that before the ancient Greeks, there was only "primitive knowledge" in areas like mathematics and astronomy. Our beloved Pythagorean theorem actually comes from not the Greeks, but the Babylonians about 1000 years prior to Euclid et al. (Some speculate the ancient Egyptians knew of it, too, but I haven't seen any conclusive evidence for the claim.)

You Rational Empiricists are all alike - a quick 7 or 8 theorems and you're off with the boys!

I just received in the mail today Eli Maor's 2007 book The Pythagorean Theorem: A 4,000 Year History (Princeton University Press). It's a very nice looking book, and I can't wait for bedtime to read it!

Last week, I received a mug and tee-shirt celebrating what appears to be the fourth and final Pythagorean Theory Day in the 21st century. It's coming right up on August 15. (08/15/17, which is a Pythagorean triple) (The other three were (03/04/05, 06/08/10, and 05/12/13.)

REB

P.S. - It's fascinating to me that Leonard Peikoff, 45 years ago, claimed that before the ancient Greeks, there was only "primitive knowledge" in areas like mathematics and astronomy. Our beloved Pythagorean theorem actually comes from not the Greeks, but the Babylonians about 1000 years prior to Euclid et al. (Some speculate the ancient Egyptians knew of it, too, but I haven't seen any conclusive evidence for the claim.)

The right triangle theorem was also known in China quite independent of the Greeks. It seems that in every culture which develops mathematics (particularly geometry) the right triangle theorem is discovered.

Last week, I received a mug and tee-shirt celebrating what appears to be the fourth and final Pythagorean Theory Day in the 21st century. It's coming right up on August 15. (08/15/17, which is a Pythagorean triple) (The other three were (03/04/05, 06/08/10, and 05/12/13.)

Fourth and final? 04/03/05, 08/06/10, 12/05/13, 9/12/15, and 12/9/15 also were, and 12/16/20 is yet to come.

Last week, I received a mug and tee-shirt celebrating what appears to be the fourth and final Pythagorean Theory Day in the 21st century. It's coming right up on August 15. (08/15/17, which is a Pythagorean triple) (The other three were (03/04/05, 06/08/10, and 05/12/13.)

Fourth and final? 04/03/05, 08/06/10, 12/05/13, 9/12/15, and 12/9/15 also were, and 12/16/20 is yet to come.

Those were the dates on the mug. I'm gonna ask for a refund!

Those were the dates on the mug. I'm gonna ask for a refund!

REB

If l,m,n is a pythagorean integer triple, so is kl, km, kn for any positive integer k. All those examples you responded to were generated from 3,4,5 which is the basic triple. Count only those integer pythagorean triples where l,m,n have no common factor other than 1. For example 5,12,13.

I woke up at 2:00 AM and have to be in the office early anyway... so, I worked the Bissell Algorithm for Pythagorean Triples and it did not come out right for me. I want to check my arithmetic, of course. In the mean time, I am disappointed that no one else here bought the book. Let me tell you a story...

When I was about 8 or 10, one summer, a neighbor kid from the next block that some of the other kids knew had family over from out of town. They went to an Indians game. The out of town cousin caught a foul ball that was hit into the stands. He brought to to show us. No one believed him. Many years later, it occurred to me that the problem was the Dead End Kids. There was an implicit sense of life assumption that nothing great would happen to any of us. We all knew the Abraham Lincoln myths about America. Everyone's family was there seeking something better. But this was far deeper than that. And I have seen this elsewhere in life and it is operative here.

Major Premise: I am an idiot.

Minor Premise: Bissell hangs out with me.

Conclusion: Therefore, Bissell is an idiot.

I am not going to stomp off mad and never come back, just to come back anyway, but I can discuss this with Roger offline.

I am disappointed that no one else here bought the book. Let me tell you a story...

This is not just a special malady afflicting posters to Objectivist Living, but people posting to Objectivist fora in general, and in regard to any book they think they will disagree with. They will mock and criticize when they see certain cue words and phrases, and they will close their eyes and ears and shout "lalalalalala," and not bother to read and understand the arguments. I call it the "James Taggart don't-bother-me Virus." There is no known cure.

You Rational Empiricists are all alike - a quick 7 or 8 theorems and you're off with the boys!

I just received in the mail today Eli Maor's 2007 book The Pythagorean Theorem: A 4,000 Year History (Princeton University Press). It's a very nice looking book, and I can't wait for bedtime to read it!

Last week, I received a mug and tee-shirt celebrating what appears to be the fourth and final Pythagorean Theory Day in the 21st century. It's coming right up on August 15. (08/15/17, which is a Pythagorean triple) (The other three were (03/04/05, 06/08/10, and 05/12/13.)

REB

P.S. - It's fascinating to me that Leonard Peikoff, 45 years ago, claimed that before the ancient Greeks, there was only "primitive knowledge" in areas like mathematics and astronomy. Our beloved Pythagorean theorem actually comes from not the Greeks, but the Babylonians about 1000 years prior to Euclid et al. (Some speculate the ancient Egyptians knew of it, too, but I haven't seen any conclusive evidence for the claim.)

Like those "primitive" Babylonians from whom the Greeks learned astronomy. By the way the Babylonians had a decent logarithmically based number system. It is positional in nature and was base 60. The number system we use is positional and is base 10. The computer geeks use positional numbers base 16 (hexadecimal). The major contribution of the Greeks was axiomatic mathematics, which consists of finding a manageable set of principles (axioms or postulates) from which all other propositions can be inferred by logic. That was the greatest and most revolutionary breakthrough the Greeks made. Unfortunately the Greeks did not have zero (as did the Babylonians) so their arithmetic was the same clunky thing as Roman Numerals. Try multiplying or dividing with Roman Numerals and you will see what I mean. If the Greeks had the zero the Eudoxus could have derived the real number system 2000 years before it was devised in Europe.

Unfortunately the Greeks did not have zero (as did the Babylonians) so their arithmetic was the same clunky thing as Roman Numerals. Try multiplying or dividing with Roman Numerals and you will see what I mean. If the Greeks had the zero the Eudoxus could have derived the real number system 2000 years before it was devised in Europe.

Not only is the Roman numeral system clunky for doing math, it's lame for even making a numbered list. I was labeling some files in a folder recently, and I made the mistake of using Roman numerals, and I kept wondering where the 5th file was. Finally, I saw it way at the bottom of the folder's list, underneath files whose name started with S and T and U. Yikes. Then I realized the first four were only together because the computer interpreted I, II, III, and IV as starting with the *letter* I. Double yikes. Well, all I can say is: thank God and Bill Gates that MS Word's indexing function doesn't work that way when using Roman numerals!

Like those "primitive" Babylonians from whom the Greeks learned astronomy. By the way the Babylonians had a decent logarithmically based number system. It is positional in nature and was base 60. The number system we use is positional and is base 10. The computer geeks use positional numbers base 16 (hexadecimal). The major contribution of the Greeks was axiomatic mathematics, which consists of finding a manageable set of principles (axioms or postulates) from which all other propositions can be inferred by logic. That was the greatest and most revolutionary breakthrough the Greeks made. Unfortunately the Greeks did not have zero (as did the Babylonians) so their arithmetic was the same clunky thing as Roman Numerals. Try multiplying or dividing with Roman Numerals and you will see what I mean. If the Greeks had the zero the Eudoxus could have derived the real number system 2000 years before it was devised in Europe.

I assume you mean the Babylonians didn't have the zero too.

"Although [the Babylonians] understood the idea of nothingness, it was not seen as a number—merely the lack of a number."

There is a metaphysical interpretation Roger might like. But I wouldn't regard 0 degrees on a thermometer nor a pH=0 (very acidic) as "nothingness."

zero is NOT nothingness. It is a definite thing with algebraic properties. 1. 0 + x = x + 0 = x for all x. 2. 0 X a = a X 0 = 0 for all a. 3. given x there exists -x such that x + (-x) = 0. 0 is too busy to be nothing. 0 also hold a column open to receive a carried digit. 0 is how we tell 220 and 22 and 202 apart. That is the trick the Babylonian "primitives" had that the Greeks did not. The closest 0 come to being "nothing" is being the cardinal number of the empty set.

In a binary circuit 0 is low voltage and 1 is high voltage. In a spintronic circuit (the kind of circuit the quantum computers will have) 0 is spin-down and 1 is spin--up. These are the quantified values of angular momentum for Fermions.

Only a philosopher or a jester could confuse 0 with Nothing.

zero is NOT nothingness. It is a definite thing with algebraic properties.

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.

It depends on what zero is attached to. All numbers are epistemological. Zero by itself is not a number but is still epistemological. Now 20 or 20,000 is as much something as 19 or 7. "020" is nothing at all, it doesn't even have an epistemological place. (Maybe somebody can invent a place.) The invention of zero ranks with the invention of the wheel or harnessing fire. Exactly what, mathematically speaking, has been more important?

## Recommended Posts

## Top Posters In This Topic

25

16

11

10

## Popular Days

Jul 16

15

Dec 14

7

Aug 16

6

Aug 8

6

## Top Posters In This Topic

BaalChatzaf25 postsRoger Bissell16 postssyrakusos11 postsBrant Gaede10 posts## Popular Days

Jul 16 2017

15 posts

Dec 14 2017

7 posts

Aug 16 2017

6 posts

Aug 8 2017

6 posts

## Popular Posts

## 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

## syrakusos 1

The truth of progress is somewhere between "social forces" and "the great man." Obviously, someone invents new ideas. But many of the seeds never took, apparently having died for lack of fertile soil.

I point to the discussion of the Antikythera Device here in OL, especially my comments, not Baal's. I got my information from the best lecture, not Nova or Discovery. And I referenced the papers of the best discovery for those who want to explore the details for themselves. The point here is that the Antikythera Device, attributable perhaps to Archimedes or perhaps to a collaboration among Euclid, Apollonius, and Archimedes, was not a singular creation. It was obviously the result of a long development and could not have existed in isolation. Yet, where are the other evidences? We know from references about coin-operated prayer machines, and the steam engine of Heron. But we have not much else... I point also to a reference that the First Citizen ("emperor") Claudius wrote a multivolume history of the Etruscans. He may have included a grammar of the language. All of that, too, is lost, and it came from the center of power of the time.

I am not one of the doomsayers here an in O-land who seem to look forward to the coming collapse of civilization. (See

The Future and Its Enemiesby Virginia Postrel.) Civilization is not collapsing. The end is not in sight. ... no matter what they want to believe. That said, however, if you consider the Bronze Age Collapse, and the decline of Rome, it is clear that all of this is very fragile. It deserves respect and protection.If Archimedes had had the zero, we would likely be pretty close to where we are today. See

The Invention of Enterpriseby Joel Mokyr, ed. andAgainst the Godsby Peter L. Bernstein. Every civilized society - and many other cultures - has "merchants" but capitalism only came from the confluence of Renaissance and the invention of statistics. The Romans of Archimedes' time had merchants. But their society valued conquest more. Successful merchants turned their enterprises over their freedmen and slaves, and retired to the countryside to live as gentleman farmers. You could have given them Alan Turing, and it would not have made any difference.(And I apologize for sidetracking the discussion of Roger's book. I will make up for it in a later post. The book is due here Monday.)

## Link to post

## Share on other sites

## BaalChatzaf 60

You are aware, I assume, that what we know of the Device is based on reverse engineering. Using the most advanced scanning technology the innards of the device were made visible. The scholars studying the device could see which gears connect to which gears and how many teeth were on each will. By working out the gear ratios they were able to guess that the device did astronomical calculations. Perhaps it took a team to make the Device in the first place. It surely took a very talented team to come up with a hypothesis based reconstruction of the Device.

In a way, the reconstruction of the Device invoked one of Aristotle's Four Causes. The most subtle of the cause, the Final Cause says a thing exists in order to achieve an end, purpose or goal.

The scholars studying the A.K. Device concluded that it was a hand carried astronomical simulator and astronomical calendar. Its purpose was to imitate the observed positions of observable astronomical bodies and moon phases. Once the scholars knew or postulated what the device -was for- they were able to fill in gaps in their knowledge of the total device, because not all the pieces survived corrosion. So final cause was bounded back to formal cause and material cause a bounced forward for rechecking. Cool! Score one for the Philosopher and the A.K. Device team.

If the Greek mathematicians had the zero, then their successors would have come up with the mathematics of motion much sooner. Think of having differential equations one thousand years sooner. Advances in physics were driven by both new facts discovered by observation and by knowing the underlying mathematical symmetries of cosmos. O.K., I was exaggerating a bit about the Star Ships but I am certain we would have had heat engines and flying machines much sooner than we actually did. It is almost certain that optics would have been advanced with better mathematics, which in term means we would have had telescopes and microscopes sooner.

As it was Archimedes did come up with something very close to integral calculus (look uj Archimedes Codex). But his math did not "catch on". He did not establish a school. There is a case of the Great Man not achieving his full potential because the society around him was not ready for it.

Your observations of the rise of capitalism out of mercantilism appear to be on the mark. For one Great Man to succeed a lot of little pieces and a lot of smaller men have to be ready to carry the matter through. In a sense, the times have to be right and ripe for the advance.

## Link to post

## Share on other sites

## syrakusos 1

How the Martians Discovered Algebraby Roger E. Bissell arrived yesterday. It is nicely done. CreateSpace did a good job.I ordered the book on July 18, 2017, and got a programmed response to that.

I received a confirmation on July 22 with a promise of July 31 arrival.

The book arrived via UPS on July 26.

Just reading a random page for now, it reminds me of George Boole. We are taught to think of syllogisms as reductions. Boole presented his algebra as a way to

expandstatements. So, too, does Bissell (as far as I have read) expansively explain that 0 + 5 isnot"five added to zero," butstarting with no action,you begin with the existence of five items.(More later...)

## Link to post

## Share on other sites

## syrakusos 1

I am about 20 pages into the book and enjoying it. I read at bedtime, so I do not work the problems, but I will work through your method for generating Pythagorean Triples. I recently read an old book that presented 300 proofs of the Pythagorean Theorem. (Reviewed on my blog, here: http://necessaryfacts.blogspot.com/2017/04/elisha-s-loomis-and-pythagorean.html )

## Link to post

## Share on other sites

## BaalChatzaf 60

^ means exponentiate, i.e. raise to a power

(M^2 + N^2)^2 - (M^2 - N^2)^2 = 4M^2N^2 = (2MN)^2

so, (M^2 - N^2)^2 + (2MN)^2 = (M^2 + N^2)^2

The interesting part comes in proving that -all- the integral Pythagorean triples are produced by this formula.

The ancient Egyptian stone cutters long before Pythagoras and his friends used a knotted rope with equally spaced knots in the pattern 3 knots, 4 knots, 5 knots to "square" off the blocks of marble and granite that they quarried. 3,4,5 is the simplest Pythagorean integer triple.

## Link to post

## Share on other sites

## Roger Bissell 15

AuthorYou Rational Empiricists are all alike - a quick 7 or 8 theorems and you're off with the boys!

I just received in the mail today Eli Maor's 2007 book The Pythagorean Theorem: A 4,000 Year History (Princeton University Press). It's a very nice looking book, and I can't wait for bedtime to read it!

Last week, I received a mug and tee-shirt celebrating what appears to be the fourth and final Pythagorean Theory Day in the 21st century. It's coming right up on August 15. (08/15/17, which is a Pythagorean triple) (The other three were (03/04/05, 06/08/10, and 05/12/13.)

REB

P.S. - It's fascinating to me that Leonard Peikoff, 45 years ago, claimed that before the ancient Greeks, there was only "primitive knowledge" in areas like mathematics and astronomy. Our beloved Pythagorean theorem actually comes from not the Greeks, but the Babylonians about 1000 years prior to Euclid et al. (Some speculate the ancient Egyptians knew of it, too, but I haven't seen any conclusive evidence for the claim.)

## Link to post

## Share on other sites

## BaalChatzaf 60

The right triangle theorem was also known in China quite independent of the Greeks. It seems that in every culture which develops mathematics (particularly geometry) the right triangle theorem is discovered.

## Link to post

## Share on other sites

## merjet 84

Fourth and final? 04/03/05, 08/06/10, 12/05/13, 9/12/15, and 12/9/15 also were, and 12/16/20 is yet to come.

## Link to post

## Share on other sites

## Roger Bissell 15

AuthorThose were the dates on the mug. I'm gonna ask for a refund!

REB

## Link to post

## Share on other sites

## BaalChatzaf 60

If l,m,n is a pythagorean integer triple, so is kl, km, kn for any positive integer k. All those examples you responded to were generated from 3,4,5 which is the basic triple. Count only those integer pythagorean triples where l,m,n have no common factor other than 1. For example 5,12,13.

Please see: http://www.friesian.com/pythag.htm

## Link to post

## Share on other sites

## syrakusos 1

I woke up at 2:00 AM and have to be in the office early anyway... so, I worked the Bissell Algorithm for Pythagorean Triples and it did not come out right for me. I want to check my arithmetic, of course. In the mean time, I am disappointed that no one else here bought the book. Let me tell you a story...

When I was about 8 or 10, one summer, a neighbor kid from the next block that some of the other kids knew had family over from out of town. They went to an Indians game. The out of town cousin caught a foul ball that was hit into the stands. He brought to to show us. No one believed him. Many years later, it occurred to me that the problem was the Dead End Kids. There was an implicit sense of life assumption that nothing great would happen to any of us. We all knew the Abraham Lincoln myths about America. Everyone's family was there seeking something better. But this was far deeper than that. And I have seen this elsewhere in life and it is operative here.

Major Premise: I am an idiot.

Minor Premise: Bissell hangs out with me.

Conclusion: Therefore, Bissell is an idiot.

I am not going to stomp off mad and never come back, just to come back anyway, but I can discuss this with Roger offline.

## Link to post

## Share on other sites

## Brant Gaede 219

Roger's book is a foul ball?

--Brant

seriously, math is a tool and my needs are sundry arithmetical calculations only

## Link to post

## Share on other sites

## Roger Bissell 15

AuthorNo, it is more like a bat. If used improperly, it will *produce* foul balls.

REB

## Link to post

## Share on other sites

## Roger Bissell 15

AuthorThis is not just a special malady afflicting posters to Objectivist Living, but people posting to Objectivist fora in general, and in regard to any book they think they will disagree with. They will mock and criticize when they see certain cue words and phrases, and they will close their eyes and ears and shout "lalalalalala," and not bother to read and understand the arguments. I call it the "James Taggart don't-bother-me Virus." There is no known cure.

REB

## Link to post

## Share on other sites

## BaalChatzaf 60

Like those "primitive" Babylonians from whom the Greeks learned astronomy. By the way the Babylonians had a decent logarithmically based number system. It is positional in nature and was base 60. The number system we use is positional and is base 10. The computer geeks use positional numbers base 16 (hexadecimal). The major contribution of the Greeks was axiomatic mathematics, which consists of finding a manageable set of principles (axioms or postulates) from which all other propositions can be inferred by logic. That was the greatest and most revolutionary breakthrough the Greeks made. Unfortunately the Greeks did not have zero (as did the Babylonians) so their arithmetic was the same clunky thing as Roman Numerals. Try multiplying or dividing with Roman Numerals and you will see what I mean. If the Greeks had the zero the Eudoxus could have derived the real number system 2000 years before it was devised in Europe.

## Link to post

## Share on other sites

## Roger Bissell 15

AuthorNot only is the Roman numeral system clunky for doing math, it's lame for even making a numbered list. I was labeling some files in a folder recently, and I made the mistake of using Roman numerals, and I kept wondering where the 5th file was. Finally, I saw it way at the bottom of the folder's list, underneath files whose name started with S and T and U. Yikes. Then I realized the first four were only together because the computer interpreted I, II, III, and IV as starting with the *letter* I. Double yikes. Well, all I can say is: thank God and Bill Gates that MS Word's indexing function doesn't work that way when using Roman numerals!

REB

## Link to post

## Share on other sites

## Brant Gaede 219

I assume you mean the Babylonians didn't have the zero too.

--Brant

## Link to post

## Share on other sites

## merjet 84

Correct. Babylonian numerals.

## Link to post

## Share on other sites

## BaalChatzaf 60

positional numbers imply a zero.

## Link to post

## Share on other sites

## merjet 84

"Although [the Babylonians] understood the idea of nothingness, it was not seen as a number—merely the lack of a number."

There is a metaphysical interpretation Roger might like. But I wouldn't regard 0 degrees on a thermometer nor a pH=0 (very acidic) as "nothingness."

## Link to post

## Share on other sites

## BaalChatzaf 60

zero is NOT nothingness. It is a definite thing with algebraic properties. 1. 0 + x = x + 0 = x for all x. 2. 0 X a = a X 0 = 0 for all a. 3. given x there exists -x such that x + (-x) = 0. 0 is too busy to be nothing. 0 also hold a column open to receive a carried digit. 0 is how we tell 220 and 22 and 202 apart. That is the trick the Babylonian "primitives" had that the Greeks did not. The closest 0 come to being "nothing" is being the cardinal number of the empty set.

In a binary circuit 0 is low voltage and 1 is high voltage. In a spintronic circuit (the kind of circuit the quantum computers will have) 0 is spin-down and 1 is spin--up. These are the quantified values of angular momentum for Fermions.

Only a philosopher or a jester could confuse 0 with Nothing.

## Link to post

## Share on other sites

## BaalChatzaf 60

Roger, did any of your mathematical investigations turn this up:

## Link to post

## Share on other sites

## Roger Bissell 15

AuthorThis is a false alternative. Zero is not

absolutenothingness. But that doesn't mean it issomething.It is theabsenceof something. Not the absence of anything whatsoever (that would be absolute nothingness), but the absence ofsomething 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 ofattemptingto 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

arereasons 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

ONLYthe claim of amethodological limitationon simultaneous measurement of position and momentum of particles, and not ametaphysical lawthat such particles do not simultaneouslypossessposition 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

## Link to post

## Share on other sites

## Brant Gaede 219

It depends on what zero is attached to. All numbers are epistemological. Zero by itself is not a number but is still epistemological. Now 20 or 20,000 is as much something as 19 or 7. "020" is nothing at all, it doesn't even have an epistemological place. (Maybe somebody can invent a place.) The invention of zero ranks with the invention of the wheel or harnessing fire. Exactly what, mathematically speaking, has been more important?

--Brant

you may now return to intelligent discussion

## Link to post

## Share on other sites

## Create an account or sign in to comment

You need to be a member in order to leave a comment

## Create an account

Sign up for a new account in our community. It's easy!

Register a new account## Sign in

Already have an account? Sign in here.

Sign In Now