Mathematics as a Cognitive Activity

{The document is in process of making. Also, since I have no formal training in cognitive science/cognitive linguistics, there may be errors of understanding on my part.}

Research in cognitive science have given rise to `Embodied Mind Theories' (EMT) in Philosophy of Mathematics(PoM). This is a recent development (2000). `Cognitive Science of Mathematics' (CSM) is the study of mathematical ideas using the techniques of cognitive science. Specifically, it is the search for foundations of mathematics in human cognition.

Some of the implications of EMT are:

1) Mathematics as we know it is a product of the human body and brain;
it is not part of the objective structure of the universe - this or any other . (Lakoff)
2) The physical world is not based on mathematics ...
rather mathematics evolved to explain the physical world the way that the eye evolved to provide sight.

The most important observation in this context is: "Our brain seems to be equipped from birth with a number sense. Elementary arithmetic appears to be a basic, biologically determined ability inherent in our species (and not just our own, since we share it with many animals). Furthermore it has a specific cerebral substrate, a set of neuronal networks that are similarly localized in all of us and that hold knowledge of numbers and their relations." (Dehaene)

From a mathematician's point of view, the main radical idea in CSM is the anti-platonist position it favors. The fact(!) that understanding & using mathematics is essentially a human activity may be evident to anybody; but not to the most of the mathematicians. As they climb up the tree of mathematics, they start developing a feeling that mathematics is transcendentally real & mathematical truth is indepedent of human thought. Most of the working mathematicians are ignorant about the philosophy of mathematics they are attached to. These are generally platonists; who believe they are discoverers, not inventors. Now, these great mathematicians whose discoveries are used successful in every scientific discipline, cannot be completely wrong.

And yet the experiences - ranging from those that we encounter when we learn multiplication tables to when we apply different strategies to solve a difficult problem ; or a sudden flash of insight which leads us in the correct direction - indicate that there is something profoundly psychological about the nature of mathematics. Though it is not the aim of the book (1) to explain this apparent contradiction, we can do that based on the data & techniques provided in the book. Add to this the investigations carried out by developmental psychologists on infants & higher primates. Even a quick look at the findings of this research will convince almost anybody that mathematics is a basically a cognitive activity.

Of course, care must be taken by anyone reading the book, who is not familiar with philosophy. Because understanding CSM can take place at two levels: philosophical & pedagogical. As a pedagogical tool CSM will undisputedly of immense importance. But as a PoM, I don't know where CSM stands. This great contribution to PoM by a linguist & a psychologist was not intended only as a contribution to PoM, but its chief purpose was to put forth more evidence in regard to the 'embodied cognition' (opposed to computational approach) view that Lakoff & his co-authors advocate in Cognitive Science proper. To use a term by Lakoff, this book is an illustration of the 'Philosophy in the Flesh'.

There are some interesting parallels between the two revolutionary books 'Principia Mathematica' of Whitehead & Russell and 'Where Mathematics Comes From' of Lakoff & Núñez, like both were inspired originally by the junior author and both written by non-mathematicians (philosophers in the first case, cognitive scientists in the later.)

DISCLAIMER: I am not at all an expert in this field. I am what can be described as an 'immature amateur'. Moreover, the above writing is pretty loosely & improperly organized to convey what embodied theory of mathematics has really to say. I personally don't like the words 'emodied mind'(by Lakoff), 'mind', 'philosophy' because of their sometimes mystical overtones, but unfortunately this terminology is quite usual in the literature. I apologize for such hurried description of a very serious concept. I will soon improve on it & will also add the approriate links.