390 Quick Answers 31 January


Time for credit for opening meetings is dwindling.  It ends 7 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.  Of note:  I am keeping topics nonoverlapping.  The best way to get the one you want is to submit early. 

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

Oh, I notice this again - those who submit their reactions earlier are more likely to be mentioned here. 

General comment:  I see occasionally "I hope we come back to this."  It's fair to trust aside from my lecture discussion this will basically never happen.  We are constantly moving forward.  Human history is vast, and we will find new before repeating. 


Lecture

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

We only know what we have records of.  There could be endlessly many things that someone figured out in history that we don't know about.  It is why we are careful to say "there is no evidence" and never say "this didn't happen", because we just have no way of knowing. 

The base of 60 is not nearly as important as the fact that the Babylonians had a _place value_ system.  That allowed them to express any number using two symbols.  That is power, and it made arithmetic more possible.  That being said, 60 is better than 10 because more useful fractions, 1/3, 1/12, have terminating expansions. 

We will say more about types of quadratic equations, but the Babylonians knew one type.  We will see later that (and why) there are many others. 

Reading mathematics is different than almost anything else, because we can read it and know what we expect would be there, so we can fill in steps from damaged artifacts, but knowing what we know about how to get from one step to another.  This relates to a fact that reading mathematics written in different modern languages is far easier than reading anything else. 




Reading

All the reading today and Monday is what I would call Greco-Roman culture.  You may keep track more precisely of where people are, but for our exam purposes, that suffices for "where". 

Remember that Greek culture is built on the back of a slave culture.  "Liberal arts" are those of free people - in contrast to the slaves.  The rules were the rules to separate the two.  I believe it is important to keep this in mind - that Greek culture had the opportunity to consider philosophical questions because there were slaves handling the menial tasks.  I believe it is also important to not overly laud the creation of rules designed for these purposes.  This fits with the context that people are studying more what is interesting than what is useful, which is a privilege that is possible because others are doing the hard work. 

Thales' propositions were more observations than theorems.  He probably thought that checking them was good enough and didn't think to prove.  However there is also belief that he proved some of them. 

Right angles are more fundamental than any notion of degrees, it is a quarter of a circle, or two of them make a straight angle.  This is preferred for angle reference over degrees. 

Be careful about cute stories.  Often they're not true.  I'd say it's ok to share them _if_ you say "it's probably not true, but … " There is one we'll get to later that I would say "stop sharing this".  I'm thinking about Pythagoras and hammers here, and about royal roads to geometry, or profiting from mathematics.  Just, be careful.   We know almost nothing about Pythagoras for sure about him personally. 

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. 

The quadratrix (which we will discuss) is a curve used for measuring area.  The process of measuring area is quadrature, because area is two dimensional.  This has no significant connection with quadratric equations beyond the fact that those are equations involving squares. 

I saw two comments one said "we have no evidence" and another said "nothing was proven".  I'm not quite sure what either one meant, but we do have plenty of sources, some passed down, not first hand.  And _most_ of what we're discussing in Greek culture was proven (not Thales, probably).  Oh, this is a common question - how do we know about things for which we don't have sources?  The simple answer is that we often have many sources that mention the source that we don't have, including sometimes describing what was in it. 

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?