The thought behind it
The most recent issue of New Scientist has an article called Thoughts: The inside story. In it, philosopher Tim Bayne begins with a survey of all of the things we mean by the word ‘thought’ – the mental activity that accompanies perceptions, problem solving, the integration of various perceptions, the uncontrolled associative train of connected concepts, the organization of these associations, possibilities that are not perceived but fully imagined, and so on. Early on, Bayne makes the following remark:
Although the distinction between perception and thought is intuitive, no one has been able to characterise it unequivocally.
I believe that this particular observation is worthy of more attention than it was given. And mathematics, given its role in the development of science, may be in a good position to contribute to the discussion. Mathematics is, in some sense, the conscious application of thought to perception. So how does the body make thoughts and what is it doing?
Bayne spends some time outlining why a “physicalist conception of thought” is more easily defended than an immaterial ‘mind’ or ‘soul’ conception, largely because of our observations of the significance of the brain’s role in thought. But, I believe even more to the point, he argues that “the materialist account of thought does justice to the continuity of nature,” and our relationship to other living creatures. The significance of the role that language plays in the presence of thought is somewhat undermined by studies that demonstrate thoughtful activity in non-human species – the chimpanzees’ ability to compare quantities and grasp simple fractions, or the baboon’s awareness of social hierarchies or the monkey’s ability to assess the difficulty of a task . While he didn’t mention it, the ability to discern quantity has been observed in fish as well. We are only beginning to grasp the complexity of non-human lives.
Language and symbol are here understood as facilitators of thought, perhaps providing for the unique sophistication of human thought, and the shared or social nature of our cognitive breakthroughs. Bayne quotes philosopher Andy Clark:
As the philosopher Andy Clark has remarked, “experience with external tags and labels thus enables the brain itself… to solve problems whose level of complexity and abstraction would otherwise leave us baffled.”
But Bayne seems to accept the usual bias about mathematical thoughts:
Sometimes thought is controlled by the application of a rule. Mathematical and logical operations, for example, are rule-based, and philosophers have invented many other systematic “thinking tools” to help them think more clearly (see “Tools for thought”). But this is an unusual kind of activity, and most episodes of thinking involve no rule.
It is certainly true that mathematics is used to organize thoughts as well as perceptions, but mathematics doesn’t emerge as a ‘tool,’ it emerges as a ‘thought.’ And, I would argue that the birth of many mathematical thoughts is an action of the body, and not solely directed by our conscious will. Perhaps mathematics itself is an act of perception.
As the foundation of many of the sciences, it would seem to be in a unique position to reveal something about the relationship between our thoughts and the world in which we live.
Bayne also makes this observation:
Until recently, undirected thought was seen as a useless and wasteful aspect of our internal mental lives. But research now suggests that it is a normal and even necessary aspect of thought. Brain activity during mind-wandering is reminiscent of that seen when people are deliberately engaged in creative thinking. It may be that, paradoxically, undirected thought is when we get our best thinking done.
The great pleasure that mathematicians feel about their discipline may be due to the fact that the source of mathematical thoughts, lies deep within the layers of our perceptive and cognitive processes, and their rise to consciousness is, in fact, not willed. Yet they invite us to make more of them, with very carefully directed reflection.
I appreciated Bayne’s consideration of the limits of thoughts.
Given that the machinery of human thought is part of our biology, there is every reason to suspect that it suffers from the kinds of bugs and blind spots that constrain other biological systems. It is doubtful whether chimpanzees possess the ability to think about quantum mechanics, for example. Perhaps that is one of the limitations of lacking language. But if there are parts of reality that are inaccessible to other thinking species, why should we assume that no part is inaccessible to us?
The role that mathematics plays in science demonstrates the extent to which our images and ideas can exceed the limits of our perceptive abilities. While mathematics opens the door to, what still appear to be, limitless imagined structures, it also opens the door to the inaccessible reaches of our physical world. So what is thought, as an action of the body, designed to accomplish? And what might mathematics tell us about that?