CB: I remember when I was in elementary school there was some "New Math" or something, SMSG, and they were teaching us elementary set theory, associative, distributive, etc. What's ur opinion of the Russsell-Whitehead project, Chuck ? I get the impression that Goedel's proof makes it something of a failure.
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I don't know CB, that some difficult stuff to think on.
I came across new math from my little sister who was starting junior high school, while I was second year at CSUN. I had just taken an inductive logic class that introduced the set-logic version. My sisters arithmetic book had some Venn diagrams. I thought wow. It was so cool, I bought a little book on sets that used a lot of Venn diagrams along with the set symbolism. Her elementary school book was my first insight that there was a lot more in mathematics than I ever dreamed.
So, I forgot about all that when student politics was heating up.
But getting back. Didn't that presentation of the laws of algebra give you some insight into the structure of algebras? I later found out those laws form a metamathematical hierarchy. The bottom line is studying these laws in a more general context like sets, opens the door to modern abstract algebra---the very subject I would sure advocate be taught to elementary and secondary teachers at a level where the ideas are accessible to them. These are also the laws of arithmetic embedded in the elementary operations of addition, substraction, multiplication, and division. This is the reason some of new math sets ended up in my baby sister's book. Personally I prefer the modern geometry route because I think it is easier to just see the relationships.
The R-W project? Goedel made the grand idealist project a futile dream at the metamathematic level, a world based on number is a perfect reasoned world. It takes down Plato, Descartes, Kant, probably Hegel, along with much of the analytic tradition at the philosophy level. But it leaves most of empiricism alone. That's not a bad deal, except for advanced work on inductive proof. Near certainty, probable certainty is close enough.
R-W works was re-worked by a series of others mostly over in Germany that culminated in the 1930s. I am glad they did and adopted a different notation. I tried to look at Russell's technical work and it was pretty illegible because of his notation. Standard set notation was much better.
That later work became the set theory you and I were exposed to in the early 1960s. Regardless of the metamathematical work of Godel, elementary sets remain a very useful tool to get at some mathematical thinking. While they may not be universally true in the logical and idealistic sense, they remain great for other uses like computers. Computers don't know about Goedel.
Remember the old Star Trek? Give the computer a problem that leads to a condradiction and its starts smoking and blows up screaming error, error...?
Later thinking about the new math sets, I suspect it was considered a failure for several reasons. Education establishment resistance. Poor teacher background, since they probably hadn't been taught why it might help and didn't understand it themselves. And of course parents who definitely didn't know what it was, and what is was doing in their kids' math books..
The failure of that program to last much passed the mid-70s got me to thinking about how to use another branch in geometry to get at the same ideas in a much more direct way. Everybody knows how to draw a triangle, except the blind. Part of their learning deficit comes from their troubles with identifying shapes. So use the set version with symbols in special ed.
I have to go check in at work and make them think I am doing something...
CG