John H. Conway on Weird Programming Languages

My daughter used to enjoy doing what we called "Mathematical Experiments". These were just ways we used to come up with for interpreting arithmetic as constructions that could be made with some sort of building bricks, which could be just beads or squares of coloured paper. It was good fun because it gives a kind of left-brain-right-brain balance to arithmetic and algebra which otherwise can end up very "left-brained," which tends to put many people off subjects like number theory. Now, here is a fairly "left-brained" description of a small part of the idea behind the rather long and complicated proof of a long-standing conjecture that was only settled a few years ago.

Now listen to a former Cambridge lecturer on an interesting correspondence between computation and a certain simple form of sequence construction. This lecture, FRACTRAN a Ridiculous Logical Language was given at UC Berkeley in 1996.  This might suggest a novel way to look at sequences like the Catalan numbers. It might also make an interesting programming project, to create tools for exploring sequences this way. You could call it a "Blackboard Management System": cf comment at 28 minutes 33 secs. See Genesis for an extended discussion on metalanguage semantics. See this post for more vague ideas on how to treat computability: Logic, which also intersect with Conway's work in the 60s on Regular Languages and Finite Automata.

At 29 minutes 29 seconds, there is a horrifying story of an attempt on Conway's life made whilst he lectured at Durham! 😂

Here are some more "right-brained" ways to look at algorithms and sequences

On Fibonnaci numbers in the Mandelbrot set:

It seems to me that the most interesting part of the explanation has been left out, which is the part which explains why the antennae have that "symmetry", ...

And on Pi in the Mandelbrot set:

Again, the most interesting part of the explanation is why the number of iterates converges to the decimal approximation of pi, ... and she doesn't say anything about that, ... Listen Conway at 56 mins 47 secs, ...

Here are some suggestions if you are interested in these questions:

First, think about the effect of multiplication by a complex number as a rotation, and a scaling of the modulus:

Here is a more detailed look at this complex exponential form:

Here is a nice intuitive explanation of how the points of the set are generated. This shows the value of programming in studying mathematics:

Now you will be able to make more sense of this brief description of the Mandelbrot set:

Now watch this:

And then look at Continued Fraction Representation and Hurwitz's Theorem.

These are sorts of things that the teaching/research software discussed here would be good for: Genesis See Feynman on Patents and the Value of an Idea.

Now think about Fractran and what sort of infinitely expandable Analytical Engine Ada Lovelace might have been thinking about when she wrote about it in the 1840s: see Sketch of the Analytical Engine.

This paper gives some idea of the depth and subtlety of Ada Lovelace's thought: Lovelace & babbage and the creation of the 1843 'notes' - Annals of the History of Computing, IEEE.