Old Babylonian Mathematics and Plimpton 322

[Ferocitus]

This new small video series from the University of New South Wales might be of
interest to very nerdiest of Civ fanatics.

It examines the famous Babylonian clay cuneiform tablet "Plimpton 322", the
mathematics (and errors!) contained in the tablet, the unique mathematical
culture developed by the Babylonians circa 1900 - 1600 B.C.E., and how it
offers some new approaches to topics that are not immediately obvious to
present-day mathematicians.

The first video is:
Old Babylonian Mathematics and Plimpton 322: A new perspective (introduction)

 

[Kyriakos]

I was (and am) reluctant to post, due to not being very knowledgeable on babylonian math, yet the prevalent view (since ancient times, and apparently correct?) is that pre-greek math was not theorem-based. There was no actual proof, although there was knowledge of something being true (as in the pythagorean theorem, known to be true for previous cultures, but not proven to be true rigorously). Going by the history of philosophy and math, Thales is attributed with being the first person to use an axiom set as basis to arrive to conclusive proof, with his eponymous theorem argued to be the first such, by Euclid, Aristoteles and others.

I mostly posted cause your thread should have discussion, and possibly it now will

 

[Ferocitus]

I was (and am) reluctant to post, due to not being very knowledgeable on babylonian math, yet the prevalent view (since ancient times, and apparently correct?) is that pre-greek math was not theorem-based. There was no actual proof, although there was knowledge of something being true (as in the pythagorean theorem, known to be true for previous cultures, but not proven to be true rigorously). Going by the history of philosophy and math, Thales is attributed with being the first person to use an axiom set as basis to arrive to conclusive proof, with his eponymous theorem argued to be the first such, by Euclid, Aristoteles and others.

I mostly posted cause your thread should have discussion, and possibly it now will

Very kind, K., but there's not much to discuss yet because the two profs haven't
started giving details. I'm interested in what the new points of view the
Babylonian "style" might offer the thoroughly modern nerd!

I agree that the Greek approach, starting with Thales, and in Euclid's superb
work, is the crowning glory of Ancient Greek civilization. The strict reliance
on proof is, I believe, unique for those times and dominates current maths.

However, there are many aspects to "mathematical culture", and to the many
diverse ways in which maths is, and has been, practiced.
A very recent long article addressing that interesting topic is:
Karine Chemla,
"The Diversity of Mathematical Cultures: One Past and Some Possible Futures",
Newsletter of the European Mathematical Society, No 214, June 2017, pages 14--24
http://www.ems-ph.org/journals/newsletter/pdf/2017-06-104.pdf

You can download the pdf of just the article itself from:
http://www.ems-ph.org/journals/show_pdf.php?issn=1027-488X&vol=6&iss=104&rank=3

Maths, and the way it is practised, is also changing now. Maths is not really
considered a science, per se. Recently, however, there has emerged a branch that
is very much closer to the scientific method than most others. It is called by
some "Experimental Computational Mathematics". An example might give you a
better idea.

Suppose I calculate some quantity more and more accurately, and get a particular
sequence of numbers, for example:
1.64, 1.644, 1.6449, 1.64493, 1.644934, 1.6449340, 1.64493406, ...

I can form the hypothesis that the true result is "pi squared divided by 6", and
then use other, independent mathematical methods to rigorously prove that
result.

That is much more like the classical scientific method of: experiment,
hypothesis, verification, (apply for further funding!), repeat.

I suspect that the Babylonians might have had a more pragmatic, "scientific"
slant to their mathematical culture than the (later) Greeks, Egyptians and
others.
We'll see if my guess is correct soon!

 

[Kyriakos]

Again noting i am not really the person to post insight on this, from my understanding of ancient greek philosophy and discussion of math there, math was already defined as being far closer to the mental world than "physics" (byword for any examination of the external world, by greek thinkers). Ultimately, though, i have to suspect that anything we examine is not an examination of an external thing, but an examination (if unwittingly) of what it triggers in our own mental world, and not just through form and traits gathered by sense, but also due to any mathematical qualities that object has. I suppose that we are looking at our own outer shell of how we think, and below that there isn't much of any tie between us as physical stuff, while at least math is inherently part of human thinking (albeit not below the surface either; the relative surface, anyway).

What i am trying to get at is that it is one phenomenon to prove the pythagorean theorem (in a set axiom basis), and another phenomenon to examine what it means to prove it as a human, using human thought. That theorem (and any other math logic) are not their own thing, but probably outer layers of how we can form notions and go about examining stuff, while we are not near what our mental world is about.
I am sure it is not surprising that i approach it from the (generally useless physically) philosophical standpoint

 

[Ferocitus]

I don't know that it's "useless".
There's a saying that "physics without engineering is just philosophy",
but there are a lot of useful, interesting ideas to discuss without being
practical or building things.

Engineers can hammer and chainsaw away at the world; I prefer the
gentler intellectual way, the one where at the end of the day one goes
home with all of one's fingers and toes.

 

[Ferocitus]

I've never thought of maths as anything but a human construct, purely a product
of human thought and imagination. Sometimes useful, sometimes just a past-time
without any other purpose but to amuse the player.

I don't believe that there is an ideal (Platonic, is that the word I'm grasping
for?) "5" out there in the universe somewhere.

One painting I first saw as a child has intrigued me ever since.
If there is such an ideal "5", is it just a colourless idea, the way I think of
it, or is it different for everyone?

I Saw the Number 5 in Gold - Charles Demuth ( 1883–1935 ).

 

[Kyriakos]

@Ferocitus , i now read an article on this due to some new OT thread (i know, i am lazy) and am posting my question here as well


I gather this is not a theorem (proof) of the pythagorean a^2+b^2=c^2, and the tablet had been theorized in the past to just have some such pythagorean triplets, albeit large numbers. And it uses a base60 numbering system.
But: given there was no theorem on this in Babylonian times, what is the new discovery/importance about? The article mentions that the approach seems to be about ratios (of what?) instead of "angles or circles", but again where is the theorem allowing for this to be used? Afaik the first theorem (which was a theorem of ratios moreover...) is attributed to Thales, and is the one about ratios of parts of sides of triangles when created by the same line cutting them all. So where is the babylonian math basis for using such ratios? (and if it isn't using those ratios, what is it using, and with what proof? cause otherwise it was already known for a long time that this tablet had pythagorean triplets ).

 

[Ferocitus]

@Ferocitus , i now read an article on this due to some new OT thread (i know, i am lazy) and am posting my question here as well


I gather this is not a theorem (proof) of the pythagorean a^2+b^2=c^2, and the tablet had been theorized in the past to just have some such pythagorean triplets, albeit large numbers. And it uses a base60 numbering system.
But: given there was no theorem on this in Babylonian times, what is the new discovery/importance about? The article mentions that the approach seems to be about ratios (of what?) instead of "angles or circles", but again where is the theorem allowing for this to be used? Afaik the first theorem (which was a theorem of ratios moreover...) is attributed to Thales, and is the one about ratios of parts of sides of triangles when created by the same line cutting them all. So where is the babylonian math basis for using such ratios? (and if it isn't using those ratios, what is it using, and with what proof? cause otherwise it was already known for a long time that this tablet had pythagorean triplets ).

You heard about it here first!

See my reply in the other thread:
https://forums.civfanatics.com/threads/for-the-mathematicians.621647/