Several people have raised the question of whether the discovery process is better described as “linear” or “spiral.” Is there a straight progression of one discovery leading to the next, and the next, and so on? Or is there more circling around, where the scientist partially grasps one point, which helps lead to the next point, which makes possible a fuller understanding of the first point, which then leads to a more advanced point that sheds further light on the earlier points, and so on?
My answer is that we do not have to choose between the linear and spiral models of knowledge; both are true, and the emphasis depends on the depth at which we describe the discovery process. As an example, consider Galileo’s investigation of motion.
In my book, I discussed Galileo’s major discoveries in the following order: his law of pendulums, then his law of free fall, then his law of descent down inclined planes, and finally his law of parabolic trajectories. In essence, this was the order of Galileo’s investigations. However, I did not choose to present the full complexity of how he shuttled back and forth between these topics. He started his investigation of pendulum motion, then did some initial free fall experiments, then went back to pendulums, then went on to inclined planes, then back to pendulums and free fall, and so on.
In a scientific biography of Galileo, it would be appropriate to cover his discovery process in full depth and try to specify exactly what he knew (and did not know) at each stage. But such an approach would have been disastrous for The Logical Leap; it would have led to a much longer book in which the main epistemological points were lost in the details of history.
I did cover some of the “turns and bumps” on the road to discovery, but only when I thought they added to the focus on method rather than detracting from it.
I agree we don’t have to choose. I don’t see a big distinction anyway.
However the word “spiral” doesn’t seem a great one, it is vague. Presumably the notion is that one circles around gaining knowledge and correcting errors while converging on the correct answer in the centre. But one could instead visualize a jagged path, with forks in it. (Out hiking in the hills, for example, one has to be wary of apparent paths as they may have been made by many people who interpreted information such that they erred in taking a route through the rocks and trees. Sometimes the most trodden path is the false one, because so many people made the same error.)
The “iterative” question is more meaningful, but the amount of iteration needed depends on how well the work was done in the first place. Start with solid ideas, use good methodology, do quality work, keep your wits about you, and there will be little iteration.
But iteration, or even spiraling, are just wiggles in the overall path which should lead in an overall consistent direction, though one does not know that path at the beginning.
But all these visualizations have limits to their usefulness, that must be recognized.