Poesis of Mathematics

Poesis of Mathematics : Poesis of Mathematics Sam Stueckle Trevecca Nazarene University Jeremy Case Ken Constantine Taylor University Troy Riggs Union University

Mathematics:Where does it come from? : Mathematics: Where does it come from? Discovered? Created?

Platonic:a priori : Platonic: a priori Traditional: Mathematics exists, where? “God has a book containing the most elegant proofs of all mathematical theorems. You don’t have to believe in God, but you must believe in the book.” Paul Erdos

Slide4 : Christian: Mathematics exists in the mind of God Where is human creativity?

Naturalist:empirical : Naturalist: empirical Mathematics: human reason discovering nature. Usually described as a product of the evolutionary process. Then why does it work so well for things removed from experience?

Theological premises : Theological premises There is no realm from which God is absent, in particular there is no such thing as purely human reason. Both reason and revelation are aspects of participation of the mind of God. Thus, “to reason truly one must be already illumined by God, while revelation itself is but a higher measure of such illuminations.” John Milbank

Slide7 : Our relationship to God is participatory. God allows for truly independent action on the part of his creation by limiting his control. God’s self limitation is not a lack of sovereignty; it is God’s choice of how to exercise his sovereignty. This empowers his creation with life. For humanity, this includes intellectual and spiritual life, carrying the impression of God’s image, with the ability to gain knowledge, and the freedom and responsibility to make choices. Hence, we participate with God in the ongoing development of his creation and in the working out of his purposes in the world.

Slide8 : Hence, from these first two premises we conclude that neither God nor the human is absent in the development of mathematics.

Slide9 : Poesis, or the art of making. God is a creative God and is continually creating. In developing the realm of mathematics, we are co-creating with God. Just as an artist, musician or author may be seen to be in partnership with God in the creation of beauty, so mathematicians may be seen as participating with God in creating mathematics.

Aesthetic : Aesthetic Given the assumption that every creative endeavor has at its core guiding aesthetic principles, we may then conclude that the heart of the mathematical endeavor is aesthetic. - Axioms: Geometry, Set theory, Probability, etc.

Slide11 : These aesthetic considerations are not individual considerations but are defined by the mathematical community. This creative process is not unconstrained.

Slide12 : The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics… It may be very hard to define mathematical beauty, but that is just as true of beauty of any kind – we may not know quite what we mean by a beautiful poem, but that does not prevent us from recognizing one when we read it. G.H. Hardy

Slide13 : Aspects of aesthetics in mathematics; symmetry, clarity, economy, unity, usefulness, representation, or harmony with observed data, elegance in mathematical proofs and theories, unexpected connections and even synergy between apparently disparate concepts, beauty in mathematical patterns, the intricate structure of mathematical systems.

Mimesis (representation, models)“Unreasonable effectiveness of mathematics” : Mimesis (representation, models) “Unreasonable effectiveness of mathematics” Linguistic Features Limits in closed rational systems Incommensurables Completeness vs. consistency in axiomatic systems Physical theories Beyond Limits Paradoxes, context Apeiron (openness) principle of potential, as opposed to limited rational.

Further directions : Further directions Interactions between mathematics and other artistic endeavors Parallels between the development of mathematics and the development of other artistic endeavors Problems with non-theistic approaches Pedagogical implications