Sam Stueckle

Trevecca Nazarene University

Beauty in Mathematics

I am interested in the concept of beauty as a foundation for mathematics. The foundation of mathematics has generally been tied to an axiomatic system through the use of pure reason. Whether mathematics is seen as the discovery of abstract truth in a Platonic sense, just playing games with symbols in a formal sense, a construction of the human mind in appropriate cultural contexts, or as the discovery of aspects of the way God’s mind works, it seems that it always reduces to different views of the axiomatic system.

While it is true that mathematics is built on an axiomatic foundation, it seems to me that a strong case could be made for the ultimate foundation of mathematics being its beauty. In fact, mathematicians are so enamored of the beauty of mathematical systems and structures that if inconsistencies were found in the axiomatic foundations of mathematics, most mathematicians would probably prefer to change the axiomatic foundations than to give up the beauty of the body of mathematics. When asked to explain mathematics, many mathematicians will resort to a discussion about discovering the beauty of the mathematical landscape, and not generally to talk of the axiomatic foundations of the subject. Even though many of us give lip service to the beauty of mathematics we do not generally place beauty as central to what mathematics is.

I believe that when we see beauty as central to mathematics we can put mathematics in its proper place alongside the study of nature and the creation of art in our search for understanding and experiencing God. In fact, I think we also underestimate beauty as a central characteristic of the nature of God Himself. While we may speak of mathematical objects as being in the mind of God, and they may well be, and we boldly claim to discover such objects, this is not how we usually speak of experiencing God. We usually experience God in the awesome beauty of nature, or in the sublime beauty of music or a work of art, or in the intimate beauty of relationship with another, or in the peaceful beauty of silence and meditation. I believe that if we were to put beauty at the center of mathematics we would discover the same sense of experiencing God in the discovery or contemplation of mathematical structures and systems. We could experience the beauty of God in the awesome beauty of the transfinite, or in the sublime beauty and elegance of a “book proof”, or in the surprising beauty of unexpected relationships, such as Wiles’ proof of Fermat’s Last Theorem, with the connection between modular forms and elliptical equations.

I also think that viewing beauty as foundational to mathematics would change our pedagogy in teaching mathematics. Too often it seems that we pursue education in mathematics from either a structural or application point of view. From a structural point of view we insist on building up all of the tools one may need in a sequential, logical order, building all of the smaller pieces before we can build any of the larger ideas. An analogy for this would be if in building a house we forced someone to study all of the nails, screws, bolts, and tools before we would ever let them even see the plans for the house. I think this is one of the main reasons most students graduate from 12 years of schooling still thinking that mathematics is a pointless exercise in playing with formulas that has no significance further than balancing a checkbook. The application point of view leads to making up stupid “word problems” that appear to be about the real world but everyone knows they are really artificial. It also leads to focusing at higher levels on only the applications. Hence in calculus we spend a lot of time plodding through various physical applications, without letting students see the bigger picture. Or we spend time in “liberal arts math” talking about such things as linear programming, which give great applications, but are generally tedious and don’t give most students much added appreciation for mathematics.

If we view beauty as central to mathematics we would be introducing ideas that, while they may have application or may build some tools, would certainly let students see a much bigger picture of mathematics at a much earlier stage in their mathematical development. Students could discover the amazing number patterns in nature, such as the Fibonacci sequence in the spirals on pinecones and pineapples. Or they could broaden their minds by seeing the surprising differences that arise when we move to non-Euclidean geometries. Fractals and such things as the Mandelbrot set can be introduced with a minimum of background and give rise to amazingly beautiful images and ideas. Even such deep and thought provoking ideas and countable and uncountable infinities can be understood by students having primarily curiosity, creativity, and an open mind, but very few mathematical tools. All of these things give students a sense of the expanse and beauty of mathematics. I believe that with exposure to more beautiful areas of mathematics students will have much less hatred for mathematics and a greater appreciation for God’s creation.

I am not really a philosopher of mathematics, but I would like to have the chance to discuss these ideas with others who could help me either flesh them out in a more complete form, or discover the weakness of this viewpoint.