I started today by taking a look at what might be the latest on what cognitive scientists were saying about mathematics. The broad scope of cognitive science includes the investigation of what Mark Turner calls (in the title of one of his books) “the riddle of human creativity.” When exploring the origins of conceptual systems, the rise and effectiveness of mathematical conceptual systems is often one of the more mysterious, despite recent attempts to demystify it (like the work of Rafael Nunez). There are some papers to be found at The Cognitive Science Network. Research in these areas provides valuable insights into how our image-filled worlds are formed. I am particularly interested in the relationship between gesture and thought and will follow this up in future posts. But today I was a little side-tracked by an essay I found by Paul Bernays written in 1959. He is responding to Ludwig Wittgenstein’s Remarks on the Foundations of Mathematics.
I decided to write a bit about this essay because I found the mathematician’s response to Wittgenstein’s critique of mathematics contains important insights into what cognitive scientists explore or have yet to explore.
Early on, Bernays makes the following observation (without characterizing Wittgenstein a behaviorist). He says:
Two problematic tendencies, however, appear throughout. The one is to dispute away the proper role of thinking – reflective intending – in a behavioristic manner.
Bernays remarks that Wittgenstein allows only one kind of ‘factuality,’ namely concrete reality. Quoting Wittgenstein:
I can calculate in the imagination, but not experiment.
To this Bernays adds:
An engineer or technician has doubtless just as lively an image of materials as a mathematician has of geometrical curves, and the image which any one of us may have of a thick iron rod is no doubt such as to make it clear that the rod could not be bent by a light pressure of the hands.
Bernays makes the argument that “concept-formations” are “an extract from experience.” and observes that Wittgenstein points to this when he says “Imagination teaches us it.” But Bernays then makes an observation of the role of intuition in mathematics that crystallizes one of its most provocative characteristics. While mathematics can always be shown to follow reason, or to make sense, it can lead to a fact that seems unreasonable, yet is true. Here’s what Bernays says:
In considering geometrical thinking in particular it is difficult to distinguish clearly the share of intuition from that of conceptuality, since we find here a formation of concepts guided so to speak by intuition, which in the sharpness of its intentions goes beyond what is in a proper sense intuitively evident, but which separated from intuition has not its proper content.
In other words, mathematics brings thought beyond what is immediately apparent to the intuition, but which only has meaning because of it. This is the hallmark of the discipline. And, in this way, mathematics liberates us from the limits of our perceptions. Again from Bernays:
The strictness of the logical and the exact does not limit our freedom. Our very freedom enables us to achieve precision through thought in a perceptive world of indistinctness and inexactness.
Despite its history of foundational uncertainties and philosophical disputes, the mathematician’s intimacy with his or her subject reveals its nature, however difficult it may be to justify it. The indistinctness of perception is resolved by a disciplined use of the imagination. This is what the body manages with mathematics. And this is what I hope will continue to inform research efforts in cognitive science.