Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano, was an Italian mathematician from the Republic of Pisa, considered by some to be the most talented Western mathematician of the Middle Ages. In those times contests where quadratic and/or cubic equations had to be solved were quite popular.

Using pure reason, he solved this set of equations in a few minutes  back around 1202:

a² + 5 = b²
a² - 5 = c²

Now  a,b and c have to be fractions (or rational numbers).  Examples of rational numbers are 4/5, 23/45, 13/73 etc. The numerator and the divisor must be whole numbers.

The challenge is - what are the values of a,b and c?

I'll leave this as a challenge to anyone who wants to take it on. I'm pretty sure you can't Google the answer.

It might have taken Fibonacci (as he was known much later) a few minutes, but it took me about half an hour with the aid of a computer.

John_Taverner wrote on Sep 4^th, 2022 at 10:53am:

Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano, was an Italian mathematician from the Republic of Pisa, considered by some to be the most talented Western mathematician of the Middle Ages. In those times contests where quadratic and/or cubic equations had to be solved were quite popular. Using pure reason, he solved this set of equations in a few minutes  back around 1202: a2 + 5 = b2 a2 - 5 = c2 Now  a,b and c have to be fractions (or rational numbers).  Examples of rational numbers are 4/5, 23/45, 13/73 etc. The numerator and the divisor must be whole numbers. The challenge is - what are the values of a,b 5 c? I'll leave this as a challenge to anyone who wants to take it on. I'm pretty sure you can't Google the answer. It might have taken Fibonacci (as he was known much later) a few minutes, but it took me about half an hour with the aid of a computer.

Freedom, love,  wisdom.  :P

(30 seconds)

Frank wrote on Sep 4^th, 2022 at 11:22am:

John_Taverner wrote on Sep 4th, 2022 at 10:53am: Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano, was an Italian mathematician from the Republic of Pisa, considered by some to be the most talented Western mathematician of the Middle Ages. In those times contests where quadratic and/or cubic equations had to be solved were quite popular. Using pure reason, he solved this set of equations in a few minutes  back around 1202: a2 + 5 = b2 a2 - 5 = c2 Now  a,b and c have to be fractions (or rational numbers).  Examples of rational numbers are 4/5, 23/45, 13/73 etc. The numerator and the divisor must be whole numbers. The challenge is - what are the values of a,b 5 c? I'll leave this as a challenge to anyone who wants to take it on. I'm pretty sure you can't Google the answer. It might have taken Fibonacci (as he was known much later) a few minutes, but it took me about half an hour with the aid of a computer. Freedom, love,  wisdom.  :P (30 seconds)

Very good answer Frank

Back in those days, freedom to think for yourself was beneficial, real practical progress could be made, and names became prominent

Not so the Tasmanian 14 year old who cracked the 50c coin code the other day, no prominent name for him as yet. Instead, in this media video he had a sort of stand-in by the name of Nick Cavanetta, about 28 years of age, he reckons he also cracked the code. What's going on?


https://www.bing.com/videos/search?pc=MOZI&q=Secret+code+hidden+on+coin+solved+by+a+14-year-old+boy&ru=%2fsearch%3fform%3dMOZSBR%26pc%3dMOZI%26q%3dSecret%2bcode%2bhidden%2bon%2bcoin%2bsolved%2bby%2ba%2b14-year-old%2bboy&view=detail&mmscn=vwrc&mid=0788112AC24B8ABCE0C60788112AC24B8ABCE0C6&FORM=WRVORC

Hmmm rusty maths brains. . .

a2 + 5 = b2
a2 - 5 = c2

so

2a² + 5 - 5 = b² + c²

Then substitute a² = b² - 5

So 2b² - 10 = b² +c²

So b² = c² + 10

Can now substitute a for b and a for c and calculate a then b then c.

Jovial Monk wrote on Sep 4^th, 2022 at 3:48pm:

Hmmm rusty maths brains. . . a2 + 5 = b2 a2 - 5 = c2 so 2a2 + 5 - 5 = b2 + c2 Then substitute a2 = b2 - 5 So 2b2 - 10 = b2 +c2 So b2 = c2 + 10 Can now substitute a for b and a for c and calculate a then b then c.

I got as far as you then got lost with all the substituting.
I think there's a trick as normally 3 unknowns
require 3 equations and there are only 2 equations.

That is why I created the third equation.

Should be able to work it out—I am too tired and stressed right now to continue it. Maybe write it on paper tomorrow.

I doubt they had simultaneous equations back in the Renaissance—in fact I am wondering if it was done geometrically.

Dear God! It was Father's Day yesterday and you guys were in HERE?? 

😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳

Conversations: with John Lennox, Emeritus Professor of Mathematics at Oxford University
AI, Man & God | Prof. John Lennox

https://www.youtube.com/watch?v=17bzlWIGH3g&t=400s

By the way, the name Fibonacci comes Figlio Bonacci (The son of Bonacci)

The Fibonacci sequence is arrived at by adding the two previous numbers. It's a sequence that is found in nature a great deal. For example in plants.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144....

https://www.treehugger.com/how-golden-ratio-manifests-nature-4869736


Quote:

When the golden ratio is applied as a growth factor (as seen below), you get a type of logarithmic spiral known as a golden spiral.

Oh yes.... Does anyone want an answer yet?

Frank wrote on Sep 4^th, 2022 at 11:22am:

Freedom, love,  wisdom.  :P (30 seconds)

Nice, but you can't design a microwave oven with those three alone.

https://www.youtube.com/watch?v=LfduUFF_i1A

;D ;D
Never seen that one before.

Lisa Jones wrote on Sep 5^th, 2022 at 12:46pm:

Dear God! It was Father's Day yesterday and you guys were in HERE??  😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳

And?

Lisa Jones wrote on Sep 5^th, 2022 at 12:46pm:

Dear God! It was Father's Day yesterday and you guys were in HERE??  😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳

Isn't that some kind of US celebration?  We've never really caught on to these commercially inspired events. Even Christmas only really became important in Victorian times.  Just a couple of hundred years before that, there were instances of people being accused of witchcraft for celebrating Yule, although it remained a strong tradition in Central Europe.

Ok, I'll post the solution on Saturday if  anyone is still interested.

Can I take forcible possession of this Forum and rename it to History or Languages or something more interesting? The owner seems to have gone walkabout and has not been seen since May.

The idea appeals to the Viking in me. 

Maybe I'll just squat here and post things that have nothing to do with Atheism until somebody notices. Atheism is a meaningless term anyway. You can no more unite atheists than you can herd cats.

Yes, please post the solution.

I have a niggling feeling I am missing a clue re the “rational numbers” thing.

Jovial Monk wrote on Sep 8^th, 2022 at 10:49am:

Yes, please post the solution. I have a niggling feeling I am missing a clue re the “rational numbers” thing.

A rational number is any whole number or any number that can be expressed as a fraction, so 14/67 is a rational number, whereas 14.1/67 is not. A Rational Number can be made by dividing an integer by an integer.

The square root of minus 1 or √(−1) is a very useful imaginary number, but not a rational number. The square root of 2 is not a rational number because it can't be expressed as a fraction.

John_Taverner wrote on Sep 6^th, 2022 at 10:33am:

Frank wrote on Sep 4th, 2022 at 11:22am: Freedom, love,  wisdom.  :P (30 seconds) Nice, but you can't design a microwave oven with those three alone. https://www.youtube.com/watch?v=LfduUFF_i1A

In foxholes nobody is cooking with microwaves.

John_Taverner wrote on Sep 8^th, 2022 at 8:24am:

Ok, I'll post the solution on Saturday if  anyone is still interested...

Yes please... I've filled a lined A5 pad with substitutions LOL.

OK, here is the solution:

a = 41/12
b = 49/12
c =  31/12


This is the very crude method  I used:

I set up a spreadsheet with a row from 1 to 1000.
in the second row I used (A1/12)^2 and filled up to 1000
In the third row, I used B 1+5 and filled
In the fourth row, I used B1+10 and filled

Then in row 5 and 6, I used something like:
=IF(ISERROR(MATCH(C1,B:B, 0)), "", "Match on row " & MATCH(A1,B:B, 0))

I also used conditional formatting to colour it red.

So, I'm looking for instances where B1+ 5 and B1+10 both exist in the row.

I could also have used nested loops to do the same thing using a small BASIC program.

Fibonacci must have had a more elegant mathematical way of getting to the solution though.

To get the spreadsheet to work, I had to multiply the values by the square of the denominator first to obtain whole numbers. Spreadsheets find it difficult to compare infinitely repeating numbers.

I have an incredible respect for these Renaissance mathematicians.

I did not expect that studying a childhood discipline would lead me to wonder about divine matters, but the possibility of a divine entity is threaded throughout mathematics, which, in its essence, so far as I can tell, is a mystical pursuit, an attempt to claim territory and define objects seen only in the minds of people doing mathematics. Why do I care about abstract possibilities and especially about God, when I have no idea what such a thing might be? A concept? An actual entity? Something hidden but accessible, or forever out of reach? Something once present and now gone? Something that ancient people appear to have experienced at close hand?

https://www.newyorker.com/culture/culture-desk/how-mathematics-changed-me

John_Taverner wrote on Sep 10^th, 2022 at 9:07pm:

OK, here is the solution: a = 41/12 b = 49/12 c =  31/12 This is the very crude method  I used: I set up a spreadsheet with a row from 1 to 1000. in the second row I used (A1/12)^2 and filled up to 1000 In the third row, I used B 1+5 and filled In the fourth row, I used B1+10 and filled Then in row 5 and 6, I used something like: =IF(ISERROR(MATCH(C1,B:B, 0)), "", "Match on row " & MATCH(A1,B:B, 0)) I also used conditional formatting to colour it red. So, I'm looking for instances where B1+ 5 and B1+10 both exist in the row. I could also have used nested loops to do the same thing using a small BASIC program. Fibonacci must have had a more elegant mathematical way of getting to the solution though. To get the spreadsheet to work, I had to multiply the values by the square of the denominator first to obtain whole numbers. Spreadsheets find it difficult to compare infinitely repeating numbers. I have an incredible respect for these Renaissance mathematicians.

That's not a mathematical solution -
it's an iterative solution.
I suspected it was a trick question.


Bobby,

Quote:

I think there's a trick as normally 3 unknowns require 3 equations and there are only 2 equations.

Also - polynomials normally have complex roots.
They are seen in a mathematical solution.

Bobby. wrote on Sep 11^th, 2022 at 7:10pm:

That's not a mathematical solution - it's an iterative solution. I suspected it was a trick question. Bobby, Quote: I think there's a trick as normally 3 unknowns require 3 equations and there are only 2 equations. Also - polynomials normally have complex roots. They are seen in a mathematical solution.

Nevertheless it’s a unique solution. There is obviously a more elegant solution possible. Nobody, including myself came up with that method, but the result would be the same. My method could be described as brute force.


No, Bobby. It wasn’t  a trick question. A mathematical problem posed to Fibonacci in 1202 was a trick question? Give me strength! How pusillanimous can you get?

Frank wrote on Sep 11^th, 2022 at 7:06pm:

I did not expect that studying a childhood discipline would lead me to wonder about divine matters, but the possibility of a divine entity is threaded throughout mathematics, which, in its essence, so far as I can tell, is a mystical pursuit, an attempt to claim territory and define objects seen only in the minds of people doing mathematics. Why do I care about abstract possibilities and especially about God, when I have no idea what such a thing might be? A concept? An actual entity? Something hidden but accessible, or forever out of reach? Something once present and now gone? Something that ancient people appear to have experienced at close hand? https://www.newyorker.com/culture/culture-desk/how-mathematics-changed-me

In this case, if you substitute the values given into the original equations, blind Freddy can see that the solution is valid. It’s about as mystical as a lump of cheese.

However, I think along the same lines as the quotation (apart from the claiming territory crap) and I agree that there is a spark of the divine in mathematics. In a basic sense, that “divine” equates with intelligent life. Whether you actually call it divine or otherwise is not really relevant. As soon as you define something with a word, the word itself changes the nature of that something:

If you represent existence with a totally calm, totally clear pond and you throw a small pebble into that pond, you create ripples which make the pond easier to discern, but changes the nature of the pond. That pebble is the word and the person who threw it, the observer.

Going further, the universe has an innately complex mathematical foundation. Is it a “divine” foundation? Whatever you call it, it’s beautiful and awe inspiring.

( and yes, I get up early. It’s part of growing old)

Quote:

The German mathematician Georg Cantor, the inventor of set theory, in the late nineteenth century, wrote that what surpasses all, the infinite set, was “the single, completely individual unity in which everything is included, which includes ‘the Absolute,’ incomprehensible to the human understanding. This is the ‘Actus Purissimus,’ which by many is called ‘God.’ ”

The final paragraph of that New Yorker article.

John_Taverner wrote on Sep 12^th, 2022 at 4:47am:

Bobby. wrote on Sep 11th, 2022 at 7:10pm: That's not a mathematical solution - it's an iterative solution. I suspected it was a trick question. Bobby, Quote: I think there's a trick as normally 3 unknowns require 3 equations and there are only 2 equations. Also - polynomials normally have complex roots. They are seen in a mathematical solution. Nevertheless it’s a unique solution. There is obviously a more elegant solution possible. Nobody, including myself came up with that method, but the result would be the same. My method could be described as brute force. No, Bobby. It wasn’t  a trick question. A mathematical problem posed to Fibonacci in 1202 was a trick question? Give me strength! How pusillanimous can you get?

Nevertheless - if you don't understand what I wrote
then you're a mathematical dunce.

No. I understood perfectly.  You explained the method you used in order to fail to solve it.

John_Taverner wrote on Sep 12^th, 2022 at 5:10am:

Going further, the universe has an innately complex mathematical foundation. Is it a “divine” foundation? Whatever you call it, it’s beautiful and awe inspiring. ( and yes, I get up early. It’s part of growing old)

So who are YOU calling 'old man', then?

"By the way, logarithmically you are half way between the Universe and an atom."
(40 mins 20 secs.)



Dawkins's argument is that if HE were god he wouldn't do it this way.

Frank wrote on Sep 17^th, 2022 at 5:52pm:

John_Taverner wrote on Sep 12th, 2022 at 5:10am: Going further, the universe has an innately complex mathematical foundation. Is it a “divine” foundation? Whatever you call it, it’s beautiful and awe inspiring. ( and yes, I get up early. It’s part of growing old) So who are YOU calling 'old man', then?

Did I really say that?  It's a hangover from Amateur Radio.  ;D

I avoid Dawkins and Hitchens. They are too evangelical for my taste.

John_Taverner wrote on Sep 17^th, 2022 at 6:00pm:

Frank wrote on Sep 17th, 2022 at 5:52pm: John_Taverner wrote on Sep 12th, 2022 at 5:10am: Going further, the universe has an innately complex mathematical foundation. Is it a “divine” foundation? Whatever you call it, it’s beautiful and awe inspiring. ( and yes, I get up early. It’s part of growing old) So who are YOU calling 'old man', then? Did I really say that?  It's a hangover from Amateur Radio.  ;D I avoid Dawkins and Hitchens. They are too evangelical for my taste.

Lennox is after my own heart although my convictions are a lot weaker than his. He does make it clear, as does Dawkins despite himself, that god is NOT a scientific question (like so many other, lesser things aren't).


https://www.youtube.com/watch?v=OVEuQg_Mglw
55.20 esp 55.55

It is a delight to to listen to him and he brings the best out of Dawkins who cannot talk down to Lennox.


Today's word: paradigm.

Frank wrote on Sep 17^th, 2022 at 6:11pm:

Lennox is after my own heart although my convictions are a lot weaker than his. He does make it clear, as does Dawkins despite himself, that god is NOT a scientific question (like so many other, lesser things aren't). https://www.youtube.com/watch?v=OVEuQg_Mglw 55.20 esp 55.55 It is a delight to to listen to him and he brings the best out of Dawkins who cannot talk down to Lennox. Today's word: paradigm.

Yes. There has been a paradigm shift with regards to religiosity, and I  agree that god (lower case acknowledged) is not a scientific question, except maybe in the field of psychology.

When I talk about god, I'm not talking about the Abrahamic God that has dominated the last two millennia.

My previous point (maybe on the other thread) was about the decline in established religion in recent decades. I don't think that "scientism" has played a significant role in the decline. I think it's more about a  shift in cultural paradigm.  It started long before the end of the Second World War, but the real avalanche where more and more people started to question established ideas and principles, was catalysed by WWII.

First you had John Osborne and his angry young men, but then there was an enormous influence on the arts and popular music. Some of it came from new found prosperity, but with the 60s and 70s came a music revolution and a powerful youth rebellion against social norms. Something similar happened in the 90s, but the 60s and 70s were the beginning.

So while atheists often use science as a lame justification, I think that science and religion are if anything complementary, and frankly I don't want to waste time in denigrating somebody else's worldview.

So what next in this Atheism forum? I might start a thread about Ancient Celtic languages and the myth of mass migrations. Maybe something about cultural migration.  That might be fun.

Maybe that will wake up the moderator.

John_Taverner wrote on Sep 17^th, 2022 at 7:35pm:

Frank wrote on Sep 17th, 2022 at 6:11pm: Lennox is after my own heart although my convictions are a lot weaker than his. He does make it clear, as does Dawkins despite himself, that god is NOT a scientific question (like so many other, lesser things aren't). https://www.youtube.com/watch?v=OVEuQg_Mglw 55.20 esp 55.55 It is a delight to to listen to him and he brings the best out of Dawkins who cannot talk down to Lennox. Today's word: paradigm. Yes. There has been a paradigm shift with regards to religiosity, and I  agree that god (lower case acknowledged) is not a scientific question, except maybe in the field of psychology. When I talk about god, I'm not talking about the Abrahamic God that has dominated the last two millennia. My previous point (maybe on the other thread) was about the decline in established religion in recent decades. I don't think that "scientism" has played a significant role in the decline. I think it's more about a  shift in cultural paradigm.  It started long before the end of the Second World War, but the real avalanche where more and more people started to question established ideas and principles, was catalysed by WWII. First you had John Osborne and his angry young men, but then there was an enormous influence on the arts and popular music. Some of it came from new found prosperity, but with the 60s and 70s came a music revolution and a powerful youth rebellion against social norms. Something similar happened in the 90s, but the 60s and 70s were the beginning. So while atheists often use science as a lame justification, I think that science and religion are if anything complementary, and frankly I don't want to waste time in denigrating somebody else's worldview. So what next in this Atheism forum? I might start a thread about Ancient Celtic languages and the myth of mass migrations. Maybe something about cultural migration.  That might be fun. Maybe that will wake up the moderator.

Agree.

Cultural migration is interesting. Religio is means binding, bond, obligation.
With the loss of religion in the West but with massive immigration by non-Western peoples who are not minded to lose their religion- what will bind us together? What will be the basis of cultural and societal bonds, solidarity, fraternity, mutuality?

Frank wrote on Sep 18^th, 2022 at 12:53pm:

I don't think that "scientism" has played a significant role in the decline. I think it's more about a  shift in cultural paradigm.  It started long before the end of the Second World War,

Long before the second world war? You can say that again! You mean the Enlightenment?

The American founding fathers were already either atheists or deists at best.

And, apologies to Nietzsche.

MeisterEckhart wrote on Sep 18^th, 2022 at 1:02pm:

Frank wrote on Sep 18th, 2022 at 12:53pm: I don't think that "scientism" has played a significant role in the decline. I think it's more about a  shift in cultural paradigm.  It started long before the end of the Second World War, Long before the second world war? You can say that again! You mean the Enlightenment? The American founding fathers were already either atheists or deists at best. And, apologies to Nietzsche.

And yet the US is the most religious Western country. And a scientific powerhouse. Science and the Enlightenment both grew from a Christian soil, as did human rights, anti-slavery, emancipation of all sorts.

As JT says, rightly, religion and science are complimentary.

Frank wrote on Sep 18^th, 2022 at 1:47pm:

MeisterEckhart wrote on Sep 18th, 2022 at 1:02pm: Frank wrote on Sep 18th, 2022 at 12:53pm: I don't think that "scientism" has played a significant role in the decline. I think it's more about a  shift in cultural paradigm.  It started long before the end of the Second World War, Long before the second world war? You can say that again! You mean the Enlightenment? The American founding fathers were already either atheists or deists at best. And, apologies to Nietzsche. And yet the US is the most religious Western country. And a scientific powerhouse. Science and the Enlightenment both grew from a Christian soil, as did human rights, anti-slavery, emancipation of all sorts.

Those New England puritans and quakers weren’t having any of that secularism. Industrialisation as a result of scientific progress was fine. Atheism, not so much.

The US almost stands alone as a highly successful country while full to the brim with godism. Nearly every other state where godism is involved is backward.

MeisterEckhart wrote on Sep 18^th, 2022 at 2:48pm:

The US almost stands alone as a highly successful country while full to the brim with godism. Nearly every other state where godism is involved is backward.

Well it's successful financially, but highly dysfunctional. I could tell you some horror stories about how industry operates over there.

6 hours for this one.
livescience.com/nasa-perseverance-rover-parachute-secret-code.html

Schrödinger's Plates: They are both broken and not broken until you open the door.


https://pbs.twimg.com/media/FdWvkh9aMAAmvx8?.jpg

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