Harold Walsby Design Research Project
Paper No. 1
Communicating mathematical ideas is a problem even among mathematicians. Many leading mathematicians are distressed over a style of mathematical writing that has become commonplace in the last decade or two. Mathematical papers are compressed to the limit, until all intuitive ideas are squeezed out. As a result, one mathematician complains, ‘Most papers are read only three times – once by the author, once by the editor, and once by the reviewer.’
Some mathematicians today fear that their subject is becoming almost a pure exercise in manipulating symbols. It is true that skill in symbolism can cover up and embellish trivial ideas. ‘There is less to this than meets the eye,’ is the comment that one leading mathematician applies to impressive-looking papers that are low on ideas. While he disparages only the content of the papers, other mathematicians are concerned about the soaring abstraction of their subject. They argue that while mathematics has always derived its most fertile inspirations from the physical world, most creative mathematicians today are getting completely out of touch ‘with physical reality.”
The New World of Mathematics, G.A.W. Boehm.
If we marvel at the patience and the courage of the pioneers, we must also marvel at their persistent blindness in missing the easier ways through the wilderness and over the mountains. What human perversity made them turn east to perish in the desert, when by going west they could have marched straight through to ease and plenty? …
The very crudities of the first attack on a significant problem… are more illuminating than all the pretty elegance of the standard texts which has been won at the cost of perhaps centuries of finicky polishing.
Mathematics, E.T. Bell.
To Independent Thinking
This paper contains the beginnings of a revolution. Whether or when that revolution develops depends much on others. It rests a lot on those who are independent enough to see a new valid viewpoint beyond that of the main current, the orthodox. However, if the content of this essay is revolutionary, so is its form. It is no brittle blossom of today’s fashionable school of ultra-formalistic mathematics. Rather, it is a revolt against that overspecialistic trend. Its approach is holistic rather than specialistic, unitive rather than separative, since this is vital to its reconciliation of form with content. “Content” or meaning – temporarily ignored and submerged in the present obsession with empty “form” – belongs to that “inward eye,” the source of our intuitions, without which there would be no mathematics. Moreover, in the weary human struggle for better things, it is the evolution of content which furnishes the great theoretical and inspirational challenges, not only of our time but of all time.
continue reading The Paradox Principle by Harold Walsby (1967):
Dedication | Aristotle’s Principle | The Role of Logic | Do Self-Contradictions Exist? | Three Types of Contradictions | Meaningful Self-Contradictions | Infinity and Self-Contradictions | Models for Self-Contradiction | The Paradox Principle and Applications | Appendix