Notes on Formal Constructivism

Abstract

Our aim is to sketch some ideas related to how we (as in, we two) think we (as in, we humans) think. "That theory is useless. It isn't even wrong." Wolfgang Pauli. Our hope in this paper is to provide a theory, admittedly somewhat vague, of how we think about mathematics. We also hope our ideas do not cause the reader to be reminded of Pauli's quote above. These notes were motivated by the interesting book by Changeaux and Connes [CC]. REALISM VS CONSTRUCTIVISM Realism: Mathematical objects exist independently of experience (or "physical reality") which we process using our senses (smell, touch, sight, ... ) and interpret using our brain. For example, Descartes speaks of a triangle as an "immutable and eternal" figure whose existence is independent of the mind which imagines it. Similar statements are made regarding God by many religious experts. Constructivism: Mathematical objects exist solely in the mind as a certain electro-chemo-biological pattern of neurons, synapses, chemicals, ... in the brain. For an extreme example, Hume believed that ideas are merely copies of sense impressions. Examples: Alain Cannes (and probably most mathematicians) are realists. For example, the famous quote of Kronecker's, "The integers are made by God, all else is made by man," indicates a realist point-of-view. On the other hand, the biologist Jean Pierre Changeux and philosopher David Hume are constructivists (though Hume is the more extreme). Poincare was possibly a constructivist in this sense (see [D], chapter 9). The realist position might be roughly summarized as 28 follows: The physical world is modeled as much as possible by mathematics. Mathematicians merely discover what is already in existence. The constructivist position might be summarized as follows: Models for the physical world are constructions of the mind (only) and all such mental constructs exist solely as electro-chemo-biological patterns of neurons, ... in the brain. To the question, "Why is mathematics so well-suited to the description of physics?", the constructivist might counter that physicists tend to examine reproducible phenomena which tend to have "universal" characteristics. Hence mathematics, which is also universal, is admirably suited for physical description. POINTS OF AGREEMENT • Mathematics provides a "universal language", i.e., a grammar and set of terms which can be understood by anyone (sufficiently trained), independently of their cultural background. • There is a "physical world" independent of our mind (which, however, we sense using our brain and sensory organs). • Mathematical objects can be represented as a certain electro-chemo-biological pattern of neurons, synapses, chemicals, ... in the brain. • A given mathematical construction can be represented as a program in a "Turing machine". (Using an over-simplification, these representations are modeled using neural networks, which are related to Turing machines [M].) FORMAL CONSTRUCTIVISM AssoRTED THOUGHTS OF OuR OwN • Though mathematics may indeed be universal, its development and current state is inspired and influenced by culture and the human experience. Humanistic Mathematics Network Journal #26