390 Quick Answers 29 January


Time for credit for opening meetings is dwindling.  It ends 5 February.  Curiously that’s the day that your research project topic is due.  i would be happy to both get to know you and talk about your topic.

For any who don't know - Conventions in personal dates:  b. “born” d. “died”, ca. “circa” (about), fl. “flourished” (was around and doing stuff)


Lecture

360° is entirely arbitrary.  It cannot be prove or justified.  We could instead decide that there are 100 in a right angle for those who like percents (this system is called gradians).  The only two reasons that we have 360° are 1. 6x60 works well with base 60 and 2.  Degrees are (still, in some settings) divided into 60 minutes and then 60 seconds.   Why not use base 10 for time?  Come back in a few millennia - we'll see that idea. 

Unsurprisingly, decoding artifacts can take a very wide range of time.  Some are quite apparent and others quite obscure … from … hours to decades?

Why is the multicultural right triangle theorem (MCRTT) called something else?  Western chauvinism, coupled with earlier ignorance. 

I very much believe the Babylonians reasoned geometrically with diagrams as I presented to you about completing the square. 

Someone said learning the history makes learning the mathematics feel much more significant.  I hope you can all feel this way sometimes. 

Bablyonian (we call them such even if they were more technically Mesopotamian; Jeff explains this) numerals are read left to right.  At least with higher on the left, although this is also mostly an unnatural way to read that is a carryover from the history of our base ten numerals. There isn’t really a “why”, but it is true.  I’m glad that you noticed this is different.  This was most easily seen in the √2 work. 

Trust me, we will see a lot of the MCRTT*.  It could be a topic you might use on your exam (same with π and circle formulas).  Plimpton 322 is a mysterious artifact, but it’s the oldest clue we have.   Similarly the square root of two tablet suggests at least some connection with this theorem.  

*A colleague who specialises in school-teaching tells me that there are too many places it is called otherwise to not call it otherwise sometimes.  I think it is important to emphasise that it was known by many cultures.  I will continue to call it the multicultural right triangle theorem.  



Reading

Thales' propositions were more observations than theorems.  He probably thought that checking them was good enough and didn't think to prove. 

The Pythagoreans did state and likely prove the MCRTT, but just not _first_. 

After incommensurables (irrationals) were discovered, the Pythagoreans, coincidentally, were trying to influence local politics, which lead to them being shut down. 

Euclid organised the elements, but almost none of it was his original work, with the possible exception of results in number theory (e.g. Euclidean algorithm for greatest common divisor, and infinitude of primes).  Euclid's parallel postulate is the original, as it was stated (and the awkward language contributes to some of the challenge it presents).  It is equivalent to any parallel postulate you know, not to any theorems you know. 

Archytas' platagi:  https://www.namuseum.gr/en/monthly_artefact/sleep-little-baby-quot/

I think Eratosthenes being second best to someone as great as Apollonius is a great compliment.  Isn't second best still pretty great?  And does it need to be a competition? 

old questions:

How do we know about lost works?  It’s a natural question with a simple answer - we have records of someone else (often times many others) mentioning it.  Yes, books were being copied at this time.

The Greeks did build their mathematics on the backs of slaves.  “Liberal arts” are the study of free people - but in contrast to what the slaves did.  There is no such thing as “liberal arts” without someone taking care of the manual labour.  This is the basis of their laws and following the laws encouraged a reliance upon proof.  Aside from that, if you find logistics of computing areas of parallelograms more interesting that geometry of dissections (rearranging parallelograms), then you are likely to be disappointed in the direction this course takes.