In his previous books How mathematicians think and The blind spot, Byers introduced his basic vision on the progress of science. Most people consider science to be a haven of certainty. It provides the best solutions to precise problems. Mathematics with its formal logical system is the prototype of science where such a guarantee can be provided. However, it is Byer's conviction that progress and creativity is the result of conflicting situations, ambiguity, and uncertainty, not only in mathematics, but for science in general and it is even the essence of being, the motor of evolution in general.
To elaborate on this idea, Byers recurrently uses numbers as an example. The basic concept of counting numbers is known to an infant at a certain level of a conceptual system (CS). This is different from the mathematical definition of a positive integer. Hence the difficulty to consider 0 or negative numbers as numbers too. There is a difference between ratios of integers and the rational number it represents. So to step up from one concept of a counting number to, say a rational number is far from obvious. It is obvious for someone who is familiar with rational numbers, but not for somebody who is yet to reach this level. All our experiences are relative to the window through which we make our observation, and that is different for everybody. This makes all concepts ambiguous. This ambiguity, or even contradiction, can only be resolved to step up to a next conceptual system. The original CS remains true, but it is absorbed into the new CS. This step requires what Byers calls deep thinking and creativity. This is different from analytic thinking within the original system, following the rules and logic that is known within that system. But following these rules, and these rules only, it will be impossible by definition to migrate to the next level where other, more general rules are in order. In other words, what is true for Newtonian mechanics absorbed in relativity theory, also plays at the most elementary levels of everybody's daily life. This step from one CS to the next one is a discontinuity. The prevailing rules have to be broken. It entails uncertainly, since one does not exactly know where the next CS will bring you. It requires a leap of faith.
If this viewpoint is accepted, then this has important consequences for example on how society should be managed and how we should teach. For one thing, artificial intelligence, which is defined by existing rules, can only result in continuous evolution. It will never be able to make this discontinuous jump to the next CS. Also top-down management in science kills creativity, forcing to work within the existing framework and this will prevent evolution. Creativity is not separable from deep thinking. It is the result of keeping two different, even conflicting conceptual visions in mind at the same time, which creates an uncomfortable situation, and creativity is the only way out to bring you to the next CS.
An important consequence is also on how and what we should teach. All too often, students are drilled in analytic thinking, following the rules of logic. However, this will not educate them to leave the system at some point and move to the next level. If the student is at the level CS1 of integers and the teacher is at the level CS2 of rational numbers, then the teacher can give the objective definition and computational rules for rational numbers, and the student can apply the definition and the computational rules for rational numbers, but as long as he keeps doing this from his CS1, he will never really understand what rational numbers are, there is no 'aha' moment, and as long as the teacher is not able to leave his CS2 level, he will find things obvious and will not understand that the student doesn't 'get' it. Learning, i.e., deep learning only can happen when both concepts of a rational number from CS1 and CS2 are present in the mind of the student. Similarly if the teacher is able to keep both concepts in mind, he will learn to teach.
This whole vision is so obviously represented by the (human) brain. Learning the skills within one CS is analytic and focused and is an activity mainly governed by the left hemisphere. It is called flashlight consciousness as opposed to a broader, less focussed view or lantern consciousness residing in the right hemisphere. Babies and young children are much less focussed than adults, which makes them excel in learning. The right hemisphere is looking out for the 'new', the 'unknown', what is 'out there', while the left hemisphere is focussing on the 'known', the 'expected'. Once the 'new' is assimilated, it is shifted to the left half of the brain. Traditional mathematics teaching overemphasizes the left hemisphere and neglects the right one. 'Learn how to learn' should be the educational paradigm. Since we are living in a constantly evolving world, 'lifelong learning' is the key concept.
Byers gives some specific examples to teach undergraduate mathematics (real numbers, calculus and analysis, and linear algebra). Formal mathematics is the body of formal definitions and rules, but true mathematics is conceptual, a body of concepts whose understanding requires deep thinking. A formal definition is a concept in formal mathematics but that is quite different from a concept in a conceptual view on mathematics. Some concepts have taken centuries before if was accepted by the mathematics community, so be not surprised that a student does not manage to do it in a few weeks.
The subtitle of the book 'What mathematics can teach us about the mind' is discussed in the last chapter. Here Byers summarizes once more what he discussed in the previous chapters. The working of the human mind has been illustrated almost solely with examples from mathematics. This resulted in the penultimate chapter in a reflection of what mathematics really is. In the last chapter he generalizes these conclusions: what holds in a mathematical context is also true in a broader context.
In many aspects Byers warns us that we are not teaching mathematics as we should and that mathematics is not just the formal system that most people identify mathematics with and it is not representing the unconditional truth. He gives some quite strong arguments and illustrations of the contrary. This is certainly applicable and with important consequences for mathematics, but it may well be generalized to science and to the evolution of the human race, and perhaps not even restricted to us humans.
It is philosophy, yes, but of a very understandable kind, certainly for mathematicians, who will recognize from their own teaching experience many of the problems exposed by the examples given in the book.
As a side remark, just one flow: on page 52-53 the golden mean is placed in the same category as pi of important numbers that are not constructible, but the golden mean is constructible in the Greek sense with straightedge and compass.