Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations

Mathematics and the brain

A possible reason for some constructs to be more common in mathematics is that they are somehow easier for human brains to manipulate. Typical human experience makes small positive integers and simple shapes familiar—so that all human brains are at least well adapted to such constructs. Yet of the limited set of people exposed to higher mathematics, different ones often seem to think in bizarrely different ways. Some think symbolically, presumably applying linguistic capabilities to algebraic or other representations. Some think more visually, using mechanical experience or visual memory. Others seem to think in terms of abstract patterns, perhaps sometimes with implicit analogies to musical harmony. And still others—including some of the purest mathematicians—seem to think directly in terms of constraints, perhaps using some kind of abstraction of everyday geometrical reasoning.

In the history of mathematics there are many concepts that seem to have taken surprisingly long to emerge. And sometimes these are ones that people still find hard to grasp. But they often later seem quite simple and obvious—as with many examples in this book.

It is sometimes thought that people understand concepts in mathematics most easily if they are presented in the order in which they arose historically. But for example the basic notion of programmability seems at some level quite easy even for young children to grasp—even though historically it appeared only recently.

In designing Mathematica one of my challenges was to use constructs that are at least ultimately easy for people to understand. Important criteria for this in my experience include specifying processes directly rather than through constraints, the explicitness in the representation of input and output, and the existence of small, memorable, examples. Typically it seems more difficult for people to understand processes in which larger numbers of different things happen in parallel. (Notably, FoldList normally seems more difficult to understand than NestList.) Tree structures such as Mathematica expressions are fairly easy to understand. But I have never found a way to make general networks similarly easy, and I am beginning to suspect that they may be fundamentally difficult for brains to handle.

Image Source Notebooks:

From Stephen Wolfram: A New Kind of Science [citation]