lecture-1.3.md

Lecture 1.3

Summary of what we learned so far:

• computation as rewriting to normal form
  • build terms from operations (terms are trees)
  • rewrite terms by applying equations using pattern matching
  • equations are also called rewrite rules in this context
  • computation terminates if no rule applies
  • a term to which no rule applies is called a normal form

(we talked about how it is important in research to simplify a problem in the right way; we learned some basic ideas: computation as rewriting to normal form; now let us make things more complicated again and see how far we can go ... there are many ways to do this: negative numbers, binary numbers (or decimals), exponentiation, square roots, calculus, if-then-else, etc etc ... but we will look at something else first, namely how to go from numbers to algebra, or from primary school to secondary school)

Algebra

How far did we go in our quick tour of school mathematics? Roughly primary school, computing with numbers.

Algebra induces a radical new big idea: instead of thinking of variables as place holders for terms/numbers: variables as "first class citizens"

Historically, this did not come easy. The first time the method of algebra appears fully developed in most of its basic ideas is a turning point in history. Please have a look at Descartes' Geometry, first published in 1637. (It is always worth looking at Wikipedia so read up on Descartes who led a very interesting live that is full of amusing anecdotes and his books, the Discourse on the Method and its appendix, the Geometry, are, in my opinion, arguably the most important publications in philosophy and mathematics of all time. A great resource on philosophy and logic and some areas of mathematics is the Stanford Encyclopedia of Philosophy which also has an article on Descartes mathematics.) Even if you don't read French, just by browsing through the pages, you see that Descartes, who just escaped the middle ages, did write in a style that is still readable today and that looks like modern mathematics. You can also look at a facsimile of the original. Even without trying to understand the maths in detail, I found for example page 301 of the original worth looking at. We see that he didnt use "=" (which, in fact, was introduced already earlier by Robert Recorde in 1557 but not widely used yet), but that otherwise all the basic ideas of algebra are already there.

	Little research project: Why did Descartes use a symbol for "=" that is not symmetric? 
	Could there be a connection to the idea of rewriting that we mentioned above?

Putting Descartes in historical context can help to appreciate his importance. He was a generation younger then Galileo and a generation older than Newton. Looking at Galileo's theorems of motion as formulated by himself in the celebrated Discourses and Mathematical Demonstrations Relating to Two New Sciences, we see that he needs 6 theorems with complicated proofs in order to express the simple equation d=v*t because he does not have the algebra of Descartes. (Thanks to Andrea DiSessa, Changing Minds (2002) for pointing this out.) But just a generation later, Leibniz and Newton were able to extend Descartes' algebra of numbers by an algebra of functions including operations of differentiation and integration.

Ok, after this historic excursion, let us go back to calculating with terms containing variables.

	do we need new equations?
		what, for example, about (x+y)+x = 2x+y ? Get out pen and paper.
	needs commutativity
	write out all equations we have so far (could do this together at the whiteboard)
	innocent but important question: how do we know that we have all equations?

The answer to the last question, leads to the next topic, the title of which contains the four big ideas of syntax, semantics, soundness and completeness.

But first, we could use this opportunity to get used to consult research literature. The problem we have been discussing has an interesting history and drew the attention of some real heavyweights. Read up to (and excluding) Section 1.1 of the article by Burris and Yeats.

Also read the Wikipedia article on Tarski's High School Algebra Problem.

Homework

Read up to (and excluding) Section 1.1 of the article by Burris and Yeats. Can you summarise its contents? (I don't expect you to understand all of it and we will explain everything in detail ... but if you try to understand the main ideas now, you will understand better what follows.)

Summary of big ideas

• terms are trees (maybe only a small idea? But it is of fundamental importance)
• variables as first class citizens (ideas can seem small in hindsight, that is why I emphasised Descartes)
• syntax (syntax has no meaning, just given by "naked" rules)

Summary of jargon

• rewrite rule
• normal form
• pattern matching
• syntax
• abstract syntax
• BNF
• syntactic sugar
• ...