Breakthrough mathematics, fundamental research and ideas

During my time on the train last week, I read some of the book ‘God created the integers: the mathematical breakthroughs that changed history’ by Stephen Hawking and several free hotel newspapers: the Globe and Mail, the Toronto Star and the National Post. This served as a supplement to my general musings on how to be more imaginative in my research and the innovation agenda.

The book title is based on a quote by the nineteenth century mathematician, Leopold Kronecker; in full, ‘God created the integers. All the rest is the work of Man.’ The quote speaks to the fact that modern mathematics is a magnificent outgrowth of the most humble beginnings: the whole numbers. The book starts with quoting Euclid as writing that “The Pythagoreans… having been brought up in the study of mathematics, thought that things are numbers… and the whole cosmos is a scale and a number.” In the first chapter, what caught my interest was the Pythagorean cult and their treatment of mathematical results such as the square root of 2 being irrational:

 The Pythagoreans carefully guarded this great discovery (irrational numbers) because it created a crisis that reached to the very roots of their cosmology. When the Pythagoreans learned that one of their members had divulged the secret to someone outside their circle they quickly made plans to throw the betrayer overboard and drown him while on the high seas. -p3

Next, I read the Intellectual Property supplement to the National Post, and in reading about intellectual property, I noted that priority to developing new technologies such as Google Glasses* are protected by patents, yet throwing people overboard to protect new advances in fundamental research is no longer appropriate. In fact, amongst scientists insights and new results are freely shared. Arguably, as a consequence, advances in fundamental research then have no market value – if they are keenly given away to anyone, of any company, of any country (or so my reasoning goes).

Back to the book: the next chapters covered Archimedes, Diophantus, Rene Descartes, Isaac Newton and Leonhard Euler. Despite making advances in fundament research, some of these mathematicans also worked on very applied projects: Archimedes on identifying counterfeit coins and Euler on numerous projects including how to set up ship masts, on correcting the level of the Finow canal, in advising the government on pensions, annunities and insurance, in supervising work on a plumbing system, and on the Seven Bridges of Konigsberg Problem. With regard to the Seven Bridges of Konigsberg Problem,

Euler quickly realized he could solve the problem of the bridges simply by enumerating all possible walks that crossed bridges no more than once. However, that approach held no interest to him. Instead, he generalized the problem… – p388.

On the shoulders of Giant's - perhaps (perhaps necessary but not sufficient). Irrespective of the boost: uncommonly brilliant and arguably unmatched. The photo is sourced from Andrew Dunn (http://www.andrewdunnphoto.com/)

… and perhaps that quote speaks to the tension in advancing applied research at the expense of fundamental research.

In reading the book, so far I’m most impressed by Newton**. How on earth did he think of that? By studying pendulums on earth he arrives at a mechanistic model of planetary motion? Swinging pendulums and falling apples? Swinging and thudding? This doesn’t naturally evoke ideas of elliptical motion for me, let alone that these events over such small distances are generalizable to a cosmic scale. Setting that aside, and continuing to generalize: every object I have ever pushed has… stopped. Yet, Newton’s first law, when it comes to objects in motion, earthly observations are the exception to the rule (not generalizable) and it takes an extra twist (external forces) to explain why, on earth, things always stop. Generalize for the universal theory of gravity; don’t generalize for the first law. I find it so not-obvious! And consequently, I’m so very impressed.

Footnotes

*Google is amazing **and Newton, much moreso.

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About Amy Hurford

I am a theoretical biologist. I became aware of mathematical biology as an undergraduate when I conducted an internet search to learn about the topic. Now, twelve years later, I want to know, what is it that makes great models great? This blog is the chronology of my thoughts as I explore this topic.

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