The Logical Leap
introduction by Leonard Peikoff
New American Library: New York, 2010
The natural sciences have been one of humanity's great success stories over the past half millennium. In the purely cognitive realm, they have produced a series of great unifying theories, each of which has brought many seemingly unrelated facts together as referents of high-level concepts. In the realm of useful knowledge, science has gone from explaining how already invented technological devices worked, as with steam engines and thermodynamics, to creating entirely new technologies, as with electromagnetic theory and radio. Five hundred years ago, most human beings lived out their lives without ever being affected by science; now, even those who know nothing of it, or actively reject it, live in a world that it has transformed.
But science itself isn't nearly so well understood as the natural world that it studies. The historical facts about science are better established than ever before, through close study of scientific documents. But there is no one universally accepted theory of how scientific knowledge is achieved; philosophers debate radically different views of science, as different as geocentric and heliocentric astronomy, and interpret the same documents in their different lights. Some philosophers even deny that there can be a valid concept of scientific knowledge at all (see for example Paul K. Feyerabend, Against Method, rev. ed. (New York: Verso, 1988), which Harriman discusses on pp. 246–247). As a result, scientific researchers develop their methods without gaining much benefit from the philosophy of science.
In The Logical Leap: Induction in Physics, David Harriman proposes a theory of science as a process of induction, and illustrates it through applications to key achievements in the historical development of physics (and the associated fields of solar system astronomy and physical chemistry) up through the end of the 19th Century. By doing so, he seeks to create a philosophy of science that can guide the future development of science, and help scientists avoid being trapped in blind alleys.
As a secondary aim, Harriman seeks to complete one of the great unsolved projects of Ayn Rand's philosophy of Objectivism. In a workshop on epistemology (incorporated into Ayn Rand, Introduction to Objectivist Epistemology, expanded 2nd ed., edited by Harry Binswanger and Leonard Peikoff (New York: Meridian, 1990) as an Appendix, pp. 123–307.), Rand had the following exchange with a participant:
Harriman seeks to use his own knowledge of physics to provide that illustration. His work takes Rand's ideas on metaphysics and especially epistemology as its starting point, relying on her ideas on the validity of the senses and the axiomatic status of causality, and especially on her theory of concepts, in developing its account of scientific knowledge as the result of induction.
At the outset, Harriman frames his goals exactly in Rand's style, when he writes that "The problem is to identify the method of induction, not to seek its 'justification'" (8). Just as Rand rejected skepticism about logic, causality, and the validity of the senses, Harriman rejects skepticism about induction as such. His question is, "Given the validity of induction, how should one perform it so as to reach a knowledge of facts?" (8) That is, he starts out by rejecting Hume's doubts about induction. In effect, he is saying that induction is what Rand calls an axiom.
A widely accepted understanding of induction takes this form:
The second "therefore" is a conclusion from deductive reasoning, starting from a general premise. But the first "therefore" is a conclusion from inductive reasoning, arriving at a general premise from specific facts.
Harriman rightly dismisses this approach, "numerical induction", as unable to prove anything (8–9). On one hand, a generalization based in a long series of observed facts, such as "all swans are white", can be disproven by a single observation, such as "this is a black swan". On the other hand, as Harriman points out, science includes many examples of general theoretical conclusions based on a single key experiment, such as Benjamin Franklin's showing that lightning is a form of electricity (31–34). How is this done, in Harriman's view?
At the outset, induction requires immersion of the mind in what is known about a subject. Harriman describes, for example, Franklin's surveying everything known about lightning in his time (33–34), and Isaac Newton's extensive notes on "a wide range of topics in physical science, including the topic of light and colors" (59). Johannes Kepler's analysis of the orbit of Mars was founded on the observational astronomy of Tycho Brahe, which was both more accurate and more comprehensive than any previous astronomical data (89).
The next step is to look for patterns and regularities. For example, Harriman discusses Kepler's recognition that the orbital planes of all the planets intersect only at a single point: the position of the sun (90). This led him to infer that the sun must be the cause of orbital motion. At this point, Kepler turned to Brahe's observations of the relative positions of the sun and a single planet, Mars, selecting those that were not confused by other issues such as the position of the earth. His painstaking analysis of Mars's orbit (91–101) eventually showed that the orbit is an ellipse with the sun at one focus. And because he had formulated the general concept of a planet, he went on to generalize the same conclusion to all the planetary orbits (101–102).
This is Harriman's central idea: A concept is a "green light" to induction, which makes it not merely possible but obligatory to move forward from one case by generalizing to all similar cases (77). And conversely, he says, an invalid concept is a "red light" (78), which makes such progress impossible. "Valid concepts direct one on the path to true generalizations, and invalid concepts bring progress to a halt" (240). (Harriman credits this basic idea to Leonard Peikoff; his contribution is to apply it to the historical development of physics (2).) Harriman ties this view to Rand's theory of concepts, which emphasizes the open-endedness of concepts: "The formation of a concept provides man with the means of identifying, not only the concretes he has observed, but all the concretes of that kind which he may encounter in the future" (Rand, Introduction to Objectivist Epistemology, p. 27).
What distinguishes valid, "green light" from invalid, "red light" concepts? Harriman offers three criteria (186): that a valid concept must be derived from observations, that it must integrate concretes that are genuinely similar, and that it must have a proper definition that is neither too broad or two narrow. These standards, he suggests, are parallel to those that distinguish a proven from an unproven theory, and the achievement of a valid concept is a key step in the creation of a proven theory. As one example, Harriman traces the emergence of the atomic theory of matter, which gave rise to the concept of valence, or the ability of atoms of a particular element to combine with specific numbers of other atoms; to Mendeleyev's "periodic law" for the relationship between atomic weight and valence, and his prediction that new elements would be found to fill the gaps in his Periodic Table; and, in 1875, to the discovery of the new element gallium, which filled one of these gaps and had almost precisely the properties Mendeleyev predicted, confirming not only the periodic law but the entire atomic theory that it was based on.
One of David Harriman's crucial claims is that the process by which a theory is discovered is exactly the same as the process by which it is proven. "If, at the end, Newton had been asked, 'Now that you have this theory, how are you going to prove it?' he could answer simply by pointing to the discovery process itself" (145). Valid concepts are not products of pure abstract thought, nor arbitrary suppositions (the two approaches Rand rejected in developing her theory of concepts — Rand, Introduction to Objectivist Epistemology, p. 2). They are tools of human cognition, which must be grounded in reality — and a theory made up of valid concepts, such as Newton's concept of "mass" or Edward Frankland's of "valence", is also grounded in reality.
How does the scientist know his concepts are valid and not arbitrary? This, Harriman says, depends on the context of knowledge in which they are formulated: "the bridge from observation to generalization is not one premise, or even a hundred premises, but the total of one's knowledge properly integrated" (34). This includes not only the scientific concepts previously arrived at, but perhaps even more importantly, the "vast prescientific context of knowledge" (41) that a scientist can draw on to suggest directions for investigation, as when Kepler took the sun's great size as suggesting that it controlled the orbits of the planets (90), and to reject certain hypotheses as unfounded in reality, as when Lavoisier took early findings on the weights of chemical substances, which showed that the supposed element "phlogiston" would have to have negative weight, as ground for regarding the phlogiston theory as absurd (178–179).
In presenting his views, Harriman not only takes induction as his subject; he also, as Peikoff points out in his "Introduction" (xi–xii), relies on it as a method. In his own words, "The theory of induction presented here has itself been induced by observing the scientific discovery process in action" (236). This is, in his view, one application of a more general principle: "All knowledge of reality must be gained on the basis of observation, including the knowledge of how to gain knowledge" (233).
In developing a theory of scientific induction, Harriman says, "We have treated the history of science as our laboratory" (236). The point of his case studies is to derive a theory of how science works from an examination of how science has worked. His conclusion links each of his key ideas about science to historical examples that substantiate it (236–239). This is at once a book on the history of science and on the philosophy of science: it uses the history to support the philosophy.
However, Harriman makes an important exception to this approach: "Philosophy is and has to be an inductive subject in every branch except metaphysics" (233). He refers to Peikoff's discussion of metaphysical axioms (Leonard Peikoff, Objectivism: The Philosophy of Ayn Rand (New York: Penguin, 1991), Chap. 1, pp. 1–30), which is founded on Rand's discussion of axiomatic concepts (Rand, Introduction to Objectivist Epistemology, pp. 55–61). Harriman states that "I take for granted the law of causality (which states that the action of an entity follows from its nature) and the validity of sense perception" (9). Reliance on these principles is the basis for his rejection of rationalist and empiricist (including positivist) philosophies of science (211–223). His discussion of the late 19th-Century controversy over whether molecules really existed or were, as the positivists supposed, simply convenient fictions (216–222) is of particular interest.
Among the strengths and weaknesses of The Logical Leap, this focus on causality must count as one of its great strengths. Harriman's work parallels in important ways the realist movement in contemporary philosophy of science, which likewise rejects positivism and asserts that the entities described by scientific theories are not merely convenient fictions or mental models but actually exist. Harriman writes, for example, that "Kepler did know the cause of planetary orbits: He correctly identified the cause as the sun and some property of planets that responds to the sun" (104) — that is, to identify the entity that makes something happen is to identify the cause. This focus on entities independent of human consciousness is fundamental to science; any attempt to understand how science works needs to include the concept of such entities.
Harriman's decision to ground his theory in the actual history of scientific research is also promising. The practice of scientific research has been amazingly successful; careful investigation of how that success was achieved is needed to understand the scientific method theoretically and to devise better methods for future research. But Harriman's actual treatment of the history of physics raises some questions. In discussing Galileo, Harriman presents his dropping objects of different weights off the Leaning Tower of Pisa as an actual historical event. This is not recorded in Galileo's writings, but only in a biography by his pupil Vincenzo Viviani, whose accuracy many historians have questioned. Galileo's own statements about light and heavy bodies point out a logical contradiction that follows from assuming they fall at different speeds — a deductive rather than an inductive argument. The facts remain in dispute, and it wouldn't be reasonable to expect Harriman to resolve them; but in not mentioning that many scholars support a different account, he undercuts the informed reader's confidence in his other historical accounts. It also seems striking that the one great achievement of classical physics that he entirely omits, James Clerk Maxwell's electromagnetic theory, is a case where deductive reasoning and concern for theoretical elegance led to true conclusions of profound theoretical and practical significance. Such bold hypotheses were a big part of science in the Nineteenth Century, and indeed Harriman acknowledges one such, in Mendeleyev's work in chemistry; but where Mendeleyev predicted three specific new elements, Maxwell predicted the entire electromagnetic spectrum, as a result of adding a term to one of his equations that was not based on observation or experiment, but that enhanced the equations' mathematical symmetry. If Harriman is to take actual science as the inductive basis for his theory, that theory needs to account for this aspect of science, and to show how it differs from the speculative rationalism that he properly decries.
Harriman discusses the prescientific knowledge that is the context of scientific discovery in terms of plausible suppositions about how children come to understand causality, or about how tribal societies gained knowledge of the world. Both have been investigated systematically: by Jean Piaget's studies of children's reasoning (in particular, Jean Piaget, Biology and Knowledge, translated by Beatrix Walsh (Chicago: University of Chicago Press, 1971), offers an account of the broader philosophical aspects of this subject) and the work of later developmental psychologists, and by the work of anthropologists on subjects such as folk taxonomy (see, for example, Scott Atran, Cognitive Foundations of Natural History (New York: Cambridge University Press, 1990)). There is in fact an emerging field of "folk physics" or "naïve physics" that investigates this prescientific context in the physical sciences. Harriman briefly discusses one concept from this field, the medieval idea of "impetus", which regarded motion as the product of internal propulsive force imparted to the moving object — like a cartoon character who runs off a cliff, comes to a stop, and then falls straight down. His discussion of basic physical knowledge of shadows (16–18) and of pushing objects to make them move (16, 19, 21–22) could also have profited from findings on folk physics.
Concerning impetus, Harriman states, "this concept was based on a false premise but ... was nevertheless a first attempt at a valid and important integration" (78). How is this to be reconciled with his statement that valid concepts are "green lights" and invalid concepts are "red lights"? If the concept of impetus was not valid, but helped physicists to integrate the facts regarding motion more fully, and to arrive at the valid concept "momentum", it cannot be classed as a red light. And though it's the only such case Harriman identifies, his own history of chemistry provides several other examples of partial integrations (153–188). It took a century to go from Lavoisier's demonstration that combustion took place through oxidation (in 1774) to the discovery of gallium (in 1875). It could be argued that the concept of phlogiston, which Harriman dismisses as simply wrong, enabled the systematization of facts about combustion that led to the discovery that phlogiston would have to have negative mass (178–179). Similarly, the caloric theory of heat as a substance that flowed through other substances, though eventually proven wrong, helped created a systematic theory of heat conduction (164–165). All these cases fit Harriman's conclusion that "The inductive method is self-corrective" (210). If the history of science is the best basis for understanding how science works, it's a legitimate question whether early reliance on invalid concepts may be a necessary and useful step toward fully developed scientific knowledge.
Conversely, Harriman envisions "green lights" as guarantees that it's safe to go forward. For example, he writes, "When Newton discovered that the sun, the moon, Earth, apples, and comets all exert a specific type of attractive force ('gravity'), he was compelled to ascribe this force to all bodies" [italics mine] (77). Yet his own examples often show concepts in a different role: conceptual integration makes possible an observational or experimental test, as in Franklin's saying "Let the experiment be made" (quoted on 34) or Mendeleyev's predicting the properties of gallium and germanium (173–174). This is precisely the kind of single crucial observation from which, Harriman says, induction draws universal conclusions. On his historical evidence, the "green light" is sometimes not the concept itself, but the confirmation that it applies to perceived reality.
Harriman states that the cases where Newton's laws are said to fail are all "cases where his laws have been torn from the context in which they were discovered and applied to a realm far removed from anything ever considered by him" (147). Is this accurate? Consider a key principle of Newtonian mechanics and classical physics generally: the law of conservation of mass. As Harriman states (125, 144), Newton defined mass as quantity of matter. In classical physics, conservation of mass meant that before and after any physical change, the quantity of matter would be the same. However, the discovery at the end of the 19th Century of radioactive elements that could emit energy, seemingly without limit, whether or not energy had been put into them from outside, was explained by applying Einstein's theory of the interconversion of matter and energy to radioactive materials. Matter could be destroyed: conservation of mass, in Newton's sense, was not always true.
Is this far from the proper context of Newton's discoveries, as Harriman says? What was that context? Conservation of mass was not a hasty leap to conclusions. It was a law of Newtonian mechanics. Harriman says of gravity that "[Newton] was compelled to ascribe this force to all bodies." Was he not equally compelled to ascribe the indestructibility of matter to the substance of which all bodies were made? And later physicists and chemists had tested conservation of mass in every domain of physical science known to them.
Harriman objects, with reason, to Ernst Mach's claim that the only function of scientific theories is to help scientists remember previously discovered facts: "Of course, successful researchers do not regard theories in this way — they do not use them to remember past experiences, but to explain those past experiences and to predict never-before-seen phenomena" (221). But he also warns that "Further evidence is required if the laws are to be extended into previously unstudied realms" (147). For both these statements to be true, it has to be possible to identify realms that are not "previously unstudied" but that do include "never-before-seen phenomena." These realms are what Harriman calls "cognitive context[s]" (147). But how can they be identified? Harriman, in the Twenty-First Century, knows that radioactive decay is a context in which matter is not indestructible. But no physicist in 1895 could have identified it as such a context. Uranium had been identified as an element in the previous century, but all of chemistry showed that the quantity of an element was not changed by any physical or chemical process; a chemist who proposed that uranium was an exception would have been making an arbitrary assertion. The limits of the cognitive context for predictions of an unchanged quantity of matter could not have been known. Identifying that context could not be part of the method of scientific investigation; it could only be its end result.
One of the pitfalls of history is "20:20 hindsight": projecting present knowledge onto people in the past in interpreting and evaluating their actions. The historian of science cannot simply look backward, seeing the past through his own eyes. He must be able to adopt the perspective of a past researcher looking forward into what was, then, not yet known. This is in fact a corollary of the idea of cognitive context. But Harriman seems not to have identified it explicitly. At any rate, he does not discuss, for example, the difference between the question of whether we can identify the specific context within which Newton's theories are valid, and the question of whether Newton himself could identify that context.
At the end, the question Rand declined to answer comes back in full force: "When does one decide that enough confirming evidence exists?" How to identify the proper context within which a scientific theory can be treated as certain is the problem of induction itself, in more specific language. It's a problem Harriman has not solved. He views a concept as a "green light" that both permits and requires generalizing to all similar cases; but what counts as "similarity"?
The difficulty may be that Harriman is trying to answer more than one question. His discussion makes three key points about inductive reasoning:
Suppose a scientist is told, "Your conclusions are open to doubt." He can ask for the evidence on which that doubt is founded, following the example of Newton (66). If the critic supplies such evidence, then far from attacking induction, he has contributed to it by making possible further self-correction. On the other hand, if the critic's doubt is founded in general skepticism, based for example on the mere possibility of human error, without any specific evidence for any particular error, then it can be dismissed as arbitrary, as Leonard Peikoff has discussed at some length (Objectivism, pp. 163–181). And Harriman does dismiss any attack on induction as such, and any demand for a justification of it. So these two points complement each other.
How does the idea that a theory validated by induction is never overthrown fit together with the other points?
On one hand, if a theory had been validated, it would never need to be corrected, if this claim is true. The self-corrective process of induction would be complete and could stop. But that would have physicists decide for some theories that it was no longer necessary to make and test predictions, or to check whether they were consistent with other theories. Any theory regarded in that way would be more religious dogma than science.
On the other hand, saying that a theory has been validated and will never be overthrown is not necessary to defend it against arbitrary doubt, skepticism, or agnosticism. It is sufficient for the physicist to say, "You have not shown reason to doubt my findings"; he need not say, "You cannot show reason to doubt my findings." In treating a validated theory as one that justifies cannot, Harriman seems to propose a defense of physicists' theories against any possible doubt, including arbitrary doubt, and in doing so inadvertently cedes too much legitimacy to arbitrary doubt.
In the cognitive domains of narrow theories, Harriman's second point may be defensible. Such theories apply to delimited ranges of phenomena that can be studied exhaustively, as with Newton's theory of color or Kepler's theory of planetary orbits. No one after Newton was going to discover a new, previously unseen color! But fundamental theories, such as Newtonian mechanics and the atomic theory of matter, apply not merely to new phenomena of known kinds, but to new kinds of phenomena, to "never-before-seen phenomena.". Such applications are unavoidably risky. Yet these fundamental theories are science's great achievements, the ones that bring entire aspects of nature within human comprehension. To limit generalization to the scope of narrow theories, for the sake of avoiding all cognitive risk, would amount to stopping the motor of science. This would be the wrong lesson to take from Harriman's theory. The right one is that induction as a method is self-corrective: scientists must not attempt to avoid all possibility of error, but pay constant attention to the facts, and trust that reality itself will guide them to the truth. The scientist is not, in effect, an administrator following a set routine of verification and generalization, but an entrepreneur on the watch for any opportunity to gain new understanding — and willing to take risks in the process.
David Harriman's own theory in The Logical Leap: Induction in Physics is founded on the historical facts about scientific research. This review has argued that his selection of facts is flawed: he works with selected cases rather than first surveying the entire subject, he omits key cases such as Maxwell's electromagnetic theory, and at least a few of the facts he relies on, concerning Galileo's experiment at the Leaning Tower, may be more dubious than he acknowledges. But if some of his conclusions are premature or unjustified, this need not be a "red light" stopping further progress, but a first step toward a better theory. And if the study of induction itself is to be self-corrective in this way, the path forward lies through deeper and wider study of the historical facts about science.
© 2013 William H. Stoddard
For an early science-fictional mention,