Ian Stewart wrote a popular book, In Pursuit of the Unknown: 17 Equations That Changed the World, published by Basic Books in 2013. Stewart is one of the 20th century’s most well‑known and creative mathematicians, so I am a bit arrogant to go up against him. However, the more I thought about his list, helped by a glass of wine, I decided that he has the stick by the wrong end.
Mathematics is best described as an organized collection of ideas that capture patterns with precision so that they can be manipulated reliably. The key words in this description are: ideas, patterns, precision, and reliably. An equation is one way to record a pattern. For example, the Pythagorean Theorem gives us an equation that captures the relation between the lengths of the sides of a right triangle.
But there are other ways to record a pattern. During the 19th Century, a number of mathematicians noticed that the symmetries of a geometric figure could be thought of as operations that could be combined in sequence to produce other symmetries. They identified a set of rules for these operations that record a pattern now called a group. Since then the idea of group has become fundamental for progress in mathematics and has spread into physics and chemistry to enable progress in those fields. For example, the study of the crystalline structure of materials would be nowhere without the idea of group.
Here is my list of the most important mathematical ideas that changed the world. The dates approximate when an idea began development in Western Civilization. I need not apologize for considering only Western Civilization because, although some of these foundational ideas began elsewhere, the people of the Greek, Roman, and West European societies brought precision and reliable thinking to the ideas. That is the way you change the world.
- [4th century BCE] The idea of proof enabled reliable reasoning about objects and patterns.
- [13th century CE] The decimal system for natural numbers extended numerical calculation to new fields and a wider class of people.
- [16th century] Algebra substituted letters for words and phrases in problem expression, enabling the study of complicated relations between quantities.
- [17th century] Coordinates combined the powers of algebra and geometry, vastly extending the reach of both.
- [17th century] The calculus brought precision and reliable thinking to dynamic processes, such as motion and growth.
- [mid 19th century] The idea of vector added direction to the study of magnitudes. For example, vectors model the behavior of electricity and magnetism.
- [mid 19th century] The idea of manifold extended geometry beyond the ancient models of Euclid. For example, Einstein expressed his theory of gravitation in the language of manifolds.
- [late 19th century] The idea of group extended algebra to the study of more complex static phenomena, such as symmetries and permutations.
I struggled mightily to reduce the number to 7 and almost achieved it by omitting the idea of complex numbers which facilitates the study of periodic quantities. Why 7? Because a list that is too long becomes a jumble in the mind, rather than a coherent list of related items. Many people who have thought about such things have observed that "too long" means roughly larger than 7. It may be OK to have 8, but all these observers would agree that 17 is too many.
Can you suggest an idea to either omit, add, or replace?
Friday, December 3, 2021
Crescat scientia, vita excolatur.