Supermodelling

Given the formal purity of mathematics and thus (one would think) its perfect suitability for machine representation and manipulation, why are there no decent Automated Mathematicians?

Mathematicians, it turns out, do a lot more than use rules of inference to produce true statements from a database of axioms and other true statements.

It appears to me as if the formal mechanics of mathematical proof constitute only a small proportion of the activity of mathematics as practiced by general intelligences. Constructing informal frameworks in which mathematical constructs can be developed and expressed is much more important. More specifically:

• invention of terms and relationships between them
• generalizing or otherwise modifying mathematical ideas
• development of methods, processes and arguments (e.g. mathematical induction, numerical approximation techniques)

Trouble is, these tasks would seem to be hardly different for mathematics than for any other area of intellectual endeavor. With the exception of the mechanical details of proofs, mathematical concepts are  exemplars for at least some of the things that concepts in general must be and do. This may make math an interesting lens for viewing general intelligence — but mathematics would seem to be an “AGI Complete” subject overall. No surprise that it remains “solved” in only the most trivial and superficial senses.

Math is interesting, though! As far as I can tell, what we call “progress” in mathematics comes from two different approaches:

• Build a mathematical model of some phenomena. Perhaps later generalize the model or modelling technique to cover more phenomena. I guess this would count as applied mathematics.
• Build an abstract structure with interesting properties. What exactly are the properties that make the structure interesting? This counts as pure mathematics or maybe abstract mathematics.

AM and similar efforts have always acknowledged these kinds of points, at least in part. They are often referred to as conjecture generators or something similar. Unfortunately, this strikes me as a rather shallow metamathematical approach, resulting in surface imitation at best. I have the same reaction to simple formal approaches to mimicking analogy, such as Douglas Hofstadter’s Copycat. We must dig deeper for the source of mathematics and mind.