Some of my academic critics have crossed the line by belittling the achievements of Galileo and Newton.

They claim that Galileo did not understand the concept of “friction,” despite the indisputable fact that Galileo abstracted from the effects of friction in order to discover his laws of motion. Furthermore, Galileo made an original discovery about friction. He was the first to realize that bodies falling through a resistive medium reach a terminal speed, and this happens because the frictional force is proportional to speed and therefore this force increases until it equals the weight of the falling body. How could he reach such a generalization without the idea of friction?

The attack on Newton claims that he didn’t understand the concepts of “inertia,” “acceleration,” and “momentum.” Allegedly, he was still in the grip of the medieval concept of “impetus.” In contrast to the medieval thinkers, however, Newton was able to calculate — in one case after another — motions and the effects of such motions. In some of his calculations, he symbolized “acceleration” by a “v” with a dot over it, and the dot indicates a time derivative. If he could symbolize and calculate acceleration, in what sense does he not have the idea?

Such people cannot possibly have studied the mathematics of Galileo and Newton.

As to my personal reaction, I’ll paraphrase Jack Nicholson in the movie A Few Good Men:

“I don’t want money and I don’t want fame. What I do want is for academics to stand there in their girlie graduate gowns and extend to Galileo and Newton some freakin’ respect.”

Every generalization is induced within a specific context of knowledge, and to claim that it is true is to claim that it applies within that context. The available context of knowledge determines the referents of the concepts that are related in a generalization, i.e., it determines the meaning of the generalization. So, in order to know what we mean, we must be able to identify the context. But what exactly does it mean to “identify the context”?

Rationalists will claim that “identifying the context” means specifying the boundaries of the generalization’s domain, i.e., specifying exactly where and how the generalization breaks down. Only then, they argue, do we really know the context within which the generalization is true. So, according to this view, we cannot identify the context for Kepler’s laws until we know Newton’s laws, and we cannot identify the context for Newton’s laws until we know Einstein’s laws, and so on. The implication is that omniscience is the standard of knowledge; we cannot know anything until we know everything. This view is a widespread error that Ayn Rand rejected.

So how do we properly identify the context? In essence, we do it by specifying the evidence that led to the generalization. Kepler, for example, cited evidence that the sun exerts a force on the planets, and evidence that the resulting orbits obey his laws. He recognized that the observational data are always limited in both precision and range. If Kepler was asked: “Will your laws predict the angular positions of planets to an accuracy of one arc-second?” His proper response would be: “I don’t know; no such data exist. But I have correctly identified causal laws that explain and integrate the available data, which are accurate within about two arc-minutes.”

This answer won’t satisfy those who demand omniscience, but it will satisfy those who demand objectivity.

The first chapter of my book presents Leonard Peikoff’s analysis of “first-level” generalizations. These are generalizations that a toddler can grasp without antecedent generalizations; for example, “pushing a ball makes it roll.” On page 19, I quote from Dr. Peikoff’s lectures:

“A ‘first-level generalization’ is one derived directly from perceptual observation, without the need of any antecedent generalizations. As such, it is composed only of first-level concepts; any form of knowledge that requires the understanding of higher-level concepts cannot be gained directly from perceptual data.”

Some readers have raised the issue of whether concepts of attributes and actions are properly regarded as “first-level” concepts. They point out that we first grasp concepts of perceived entities, and conclude that the first-level should be restricted to these concepts. According to this view, concepts such as “red,” “round,” “pushing,” and “rolling” are higher-level.

I think this view derives from a failure to grasp what is meant by “level” in Objectivist epistemology. The concept “level” pertains to the hierarchical nature of knowledge. In Objectivism: The Philosophy of Ayn Rand, Dr. Peikoff writes:

“A hierarchy of knowledge means a body of concepts and conclusions ranked in order of logical dependence, one upon another, according to each item’s distance from the base of the structure. The base is the perceptual data with which cognition begins.”

This base obviously includes the data integrated by concepts of attributes and actions that are directly perceived. If we classify such concepts as “first-level,” then the idea serves an important function: we can talk about the reduction of higher-level knowledge to the first-level. But if we made the error of restricting the first-level to concepts of perceived entities, what function would this idea serve? The first-level would be so impoverished that nothing could be reduced to it. This restricted idea is a definition by inessentials.

This spring, The Logical Leap will be a required text in a course on scientific method at Western Carolina University. And the book has already been used in a critical thinking course at Augsburg College in Minnesota.

At Claremont Graduate University, I remember taking a philosophy of science course taught by a Kuhnian (a professor who believed that scientific knowledge is merely a subjective, social construction). He made a strange confession to me one day. In the past, he said, he enjoyed shocking and upsetting the students with skeptical arguments. But now, he explained, the students are already skeptics, so they just shrug and yawn at his arguments. It wasn’t fun for him anymore.

I’m hoping that my book will put the fun back into such courses. Let the students read both The Logical Leap and Kuhn’s The Structure of Scientific Revolutions. The debates should be lively.

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.

Imagine going on a “free fall” ride with Galileo at an amusement park. Despite the fact that Galileo packed a few extra pounds (he enjoyed Italian food and wine), you and he would fall at the same rate. Galileo, of course, would not be surprised by this result. Very early in his career, he knew that bodies of the same material fall together, regardless of differences in weight.

Galileo made enormous contributions to understanding the nature of free fall, but the above generalization is not one of them. Several of Galileo’s predecessors (including Giambattista Benedetti, Giuseppe Moletti, and Simon Stevin) had already understood this point and published it. From high places, they dropped bodies of the same dense material but different size and weight, and saw them hit the ground at the same time. They found this to be a convincing demonstration.

Of course, people also knew that a cannonball falls faster than a straw hat. One must appreciate the effects of air resistance in order to reach Galileo’s broad principle that all bodies accelerate at the same rate in free fall (“free” means “in the absence of friction”). I emphasized this point in my book, but I did so without citing the specific observations that contributed to Galileo’s appreciation of friction (for example, the experiments in which he dropped bodies through fluids). Strangely, this has led some of my critics to charge me with “historical inaccuracy.” In fact, it is simply one example of how I condensed and essentialized the material. I wanted to keep the emphasis on what is revolutionary about Galileo’s physics—and that is his brilliant quantitative experiments that led to mathematical laws.

My presentation of Galileo’s investigation of free fall is consistent with that of Stillman Drake, who gives little emphasis to the experiments on bodies falling through liquids. I suspect that Drake made this choice for the same reason I did.

Now, I will begin to face my critics.

I have been accused of presenting an “unconventional” history of science. Apparently, to depart from convention is a serious crime. How do I plead? Am I guilty or innocent?

In a very important sense, I confess to being innocent. My book does not contain any new discoveries in the history of science. What I say about the discoveries of great scientists can be found in the writings of the scientists themselves or in the writings of the best historians of science (see the references at the end of the book). For example, my account of Galileo’s discoveries relies heavily on the work of Stillman Drake, whom I regard as the best of the Galileo scholars. There was a time when Drake’s view of Galileo was very “unconventional”; most historians regarded Galileo as a Platonist who arrived at conclusions by thought experiments and mathematical deduction, whereas Drake showed that Galileo was a brilliant experimentalist. Scholars such as Drake conduct painstaking investigations to discover previously unknown facts about what scientists actually did. This is not the type of work I do, so I rely on those who do it.

In another sense, however, I am proud to be guilty as charged. My account of the history is unconventional because I have condensed an enormous amount of material in order to reveal the essential logic of the discovery process. For example, starting from more than 2,000 pages of material written by Galileo himself, or by Drake and other historians, I boiled it down to a 20 page section of my book that focuses on the crucial discoveries and the method that led to them. By today’s standards, this is unconventional; any essentialized account is typically dismissed as simplistic. Contemporary academics suffer from a disease that can be characterized as “complexity worship.” As a result, they bury the important points under mountains of trivia. I have tried to sift through the details and clear most of them away—in order to reveal the buried treasures.

At times, this ruthless process of essentializing forced me to make painful decisions. For example, I omitted discussion of Newton’s famous “rings” experiment, in which he associated a wavelength with each color of light. This is one of the most brilliant experiments of the 17^th century, and a personal favorite of mine. But it was not absolutely necessary for what I needed to say about Newton’s theory of colors, so I cut it out.

On the other hand, I sometimes included experiments that are omitted in most other historical accounts. For example, I included an experiment that Newton himself left out of his Optics book—the one where he looked through a prism at a thread painted half blue and half red. In this case, I judged the experiment to be essential to the logic of his discovery process. Another case can be found in Chapter 5 on atomic theory: most historians omit the discovery of ozone, but I think it is crucial because it refuted a major objection to Avogadro’s law.

In every case, my selection criterion was the same: Is a particular point or experiment absolutely necessary to arrive at the generalization I am trying to reach? My selection criterion was not: What do most other historians usually say?

To date, The Logical Leap has received more than a dozen positive reviews, and it is gratifying to me that these reviewers have found the book so valuable and said so publicly.

Perhaps the most significant of these reviews appears in the November issue of Physics Today (a well-known and respected science magazine). The reviewer is Ulrich Gerlach, a mathematics professor at Ohio State University. Here is Prof. Gerlach’s concluding paragraph:

“The Logical Leap is the most satisfying resolution of the ‘problem of induction’ that I’ve come across. It not only shows how inductive reasoning comes about but also demonstrates that it is the sine qua non of progress and success in physics and, more generally, in science. Harriman’s brilliant work is destined to be the fountainhead of future studies in the philosophy of science.”

Thanks to this wonderful review, sales of the book have increased in the past couple of weeks. The Logical Leap is beginning to reach physicists, who are a key part of my intended audience.

Another significant review has been written by Steve Canipe, who directs the science and mathematics E-learning programs at Laureate Education and Walden University. In recommending The Logical Leap to members of the National Science Teachers Association, Dr. Canipe wrote:

“Deductive reasoning in science has been much more the norm for students of science at all levels rather than the inductive reasoning covered in this book. This book pushes that envelope and will definitely take the average reader out of his/her comfort zone. But those who tackle it will find this book to be a valuable resource.”

This review should help The Logical Leap reach science teachers, who represent another crucial part of my intended audience.

In The Logical Leap, I have tried to shed light on the nature of scientific method by examining the discoveries that led to Newtonian mechanics and the atomic theory of matter (Chapters 2-5). In the rest of the book, Chapter 1 deals with the foundation of such advanced knowledge, and Chapters 6 and 7 discuss the various ways that scientists can go off-track, the role of mathematics and philosophy, and the implications of my thesis for evaluating contemporary theories in physics.

Before the 19^th century, physical science used to be called “natural philosophy.” Those who investigated nature recognized that philosophy played a crucial role in their work. They engaged in explicit discussions (and often heated debates) about proper method. The greatest of these natural philosophers—men like Galileo, Newton, and Lavoisier—were innovative in epistemology as well as science. And they insisted that their understanding of method was at the root of their success.

Today, however, most scientists dismiss philosophy as irrelevant to their work. This attitude is a disastrous error, and scientists are paying the price for it. But it is the philosophers who are primarily to blame. Scientists look at the hash of avant-garde skepticism and silly word games that pass for philosophy today—and they ask: Of what use could this nonsense be to me (or to anyone)?

But I urge scientists not to equate the field of philosophy with the ramblings of its worst practitioners; in other words, don’t throw out the baby with the dirty bathwater.

The Logical Leap offers a different kind of philosophy.

The Logical Leap is written for anyone who wants a deeper understanding of how we discover generalizations. When we generalize from observed cases, how do we know that we’re right? My book tries to answer this question for the field of physical science, but the basic points of method can be applied to any field. It’s a “self-help” book (albeit on a more philosophic level).

I will use this blog to elaborate on interesting topics in epistemology and in history of science. Also, I may give some background on how the book developed and what I learned while writing it. And, occasionally, I’ll comment on reactions to the book (positive and negative).

So, if you found The Logical Leap interesting and valuable, I hope you will visit here from time to time. I plan to post something every week.