Mathematics as an Ontological Science and its Implications in Teaching
This article is dedicated to the 80th Anniversary of the Incompleteness Theorem by Kurt Gödel
The discovery of Gödel’s incomplete theorem in logic had a profound impact on the philosophy of mathematics and mathematics education. It single-handedly uprooted Hilbert’s formalism, threatened the British school of analytic philosophy championed by Russell. But most of all, the incompleteness theorem suggests that mathematics is a positive science. The term “positive” refers to a positum, a given being that is partially known in the pre-scientific stage. To borrow a page from Heidegger’s Sein und Zeit, what is scientifically knowable is given in advance by a “truth” which is not graspable by positive sciences.
2011 marks the eightieth anniversary of the incompleteness theorem. The debate between the structuralists and traditionalists still persists. Ironically, 2010 is the seventy-fifth anniversary of Nicolas Bourbaki, the renowned intellectual illuminati that gave chase to the illusive structure hidden in mathematics (Munson, 2010). If internal logical coherence within even a modestly complex mathematical system is impossible, what merits the emphasis on structure in mathematics education? Can educators give a reason beyond the cognitive expediency of a superstructure that lurks ghostly behind the vast arrays of formulae?
All educators are familiar with the power of structures in mathematics. Structures illuminate, inform, and intrigue the learner. But more importantly, the Incompleteness Theorem needs not be taken as the demise of a logically complete superstructure in mathematics. It certainly implies that most structures in discussion should be open ended. This realization confers true power upon the learner. Imagine, if the Incompleteness Theorem is not valid, then mathematics educators might face a desolate landscape of self-complete but disconnected mini-structures, each with little explanatory power. With Gödel’s result, a learner is always prompted to contemplate outside – at a hefty intellectual premium, one that this learner welcomes.
Take symmetry for example, from Galois theory to supersymmetry in particle physics, it is a powerful idea in mathematics and physics. Even after Gödel’s result, many ideas using symmetry continue to flourish in science. What happens when symmetry is fractured by the introduction of new evidence? What occurs when such philosophical comfort is vexed by the uncanny? This is the moment in which Heidegger’s Dasein is transformed: “[mathematical] knowledge has a transformative power: it transforms the object of knowledge and in the course of knowing and learning, the subject is itself transformed. There is a dialectical relationship between subject and object that can be better understood by saying that learning is a process of objectification (knowing) and subjectification (or agency), that is a process of being.”(Radford, 2008)
Educators and mathematicians alike should turn to the other side of Gödel’s theorem. The incompleteness theorem is not an end, it is an beginning, similar to a phoenix rising out its ashes, for behold, “all my devils become my angels.”
Muson, A.(2010). Bourbaki at Seventy-Five: Its Influence in France and Beyond Journal of Mathematics Education at Teachers College. Vol 1, No 2 (2010)
Radford, L. (2008). The ethics of being and knowing: towards a cultural theory of learning. In L. Radford, G. Schubring & F. Seeger (Eds.), Semiotics in Mathematics Education: Epistemology, History, Classroom, and Culture (pp. 215–234). Rotterdam: Sense Publishers.