A Criticism of the Thesis in David Harriman's
The Logical Leap: Induction in Physics
by Reginald Firehammer
I have been looking forward with some anticipation to finally reading the long-promised book on induction from ARI (Ayn Rand Institute). The book has been in the works for over 10 years and has finally been published as The Logical Leap: Induction in Physics. David Harriman is named as the author of the book, but it is clearly a collaboration between Harriman and Leonard Peikoff.
[Note: The is not a review of the book. If it were I would have good things to say about it, especially about the well presented and very interesting examples from the history of science. In this article I only address the essential arguments for the thesis of the book.]
There is a widely held view that the validity of science, in some way, depends on the validity of induction, and this book presents arguments which are supposed to be a defense of the inductive method in science. It is those arguments I am primarily interested in.
The entire argument is based on three concepts, "cause," "generalization," (by induction), and "mathematics." The book proceeds by means of illustrations of its premises with many good examples from the history of science. While those examples are excellent illustrations of the validity of scientific methods, they totally fail to demonstrate that induction is the basis of those methods.
My intention here is to explain why the concepts the book defends are either invalid or incorrectly understood. I'll first explain what Harriman seems to mean by the three main concepts. I will follow that by my criticism which includes the correct basis for the validity and objectivity of science.
[All quotes and page numbers refer to Harriman's The Logical Leap: Induction in Physics.]
I intend to present these concepts without bias or criticism as closely as I can to the meaning intended by Harriman.
Concerning cause, he writes, "The only justification for inferring the future from the actions of the past is the fact that the past actions occurred not arbitrarily or miraculously, but for a reason, a reason inherent in the nature of the acting entities themselves: i.e., the justification is that the past actions were effects of causes—and thus if the same cause is operative tomorrow, it will result in the same effect." [Page 21.]
This meaning of cause seems very much like the meaning Hume intended, when he described cause as a "... necessary connexion ... which binds the effect to the cause, and renders the one an infallible consequence of the other." [An Enquiry Concerning Human Understanding]
I point this out to make it clear that though Harriman admires Aristotle and credits his philosophy as the most important philosophical influence on the development of science, his meaning of cause is obviously not that of Aristotle.
Aristotle identified four "types" or "aspects" of cause:
1. Material cause: The substance of which a thing consists.
2. Formal cause: The "design" or "form" of a thing.
3. Efficient cause: The agent that brings a thing into existence.
4. Final cause: Reason or purpose of a thing.
Cause, as Harriman uses the term would only pertain to number 3, the "efficient cause," and the "agent" for Harriman would be whatever thing, event or attribute was responsible for that which is caused, that is, the effect. But Aristotle was thinking on much broader terms, and by "cause" he meant what Harriman meant when he wrote: "... actions occurred not arbitrarily or miraculously, but for a reason ...." [Page 21.] Aristotle is addressing "cause" as the "reason" for things, not just their "physical" cause.
Harriman's meaning of cause separates a "cause" from an "effect" as though they were two independent metaphysical existents with "cause" as the only connection between them.
"In seeking cause and effect, we are relating objects/attributes that are subsumed under different concepts. We are attempting to discover the effects of one type of existent on another, for example, to identify the effect of temperature on the pressure of a gas, or the effect of length on the period of a pendulum, or the effect of distance on the gravitational force between bodies." [Page 229.]
But, though he confuses them, there is another meaning for cause suggested by Harriman. It is included in the quote above, "actions occurred not arbitrarily or miraculously, but for a reason, a reason inherent in the nature of the acting entities themselves," and stated explicitly here:
"Let us start by noting that all generalizations—first-level and higher—are statements of causal connection. All assert (or imply) that an entity of a certain kind necessarily acts in a certain way under a given set of circumstances, which is the essence of the law of causality." [Page 21.]
Let me make the difference clear: the first meaning of cause is a description of the relationship between two different existents, the one being the cause of the behavior of the other; the second meaning of cause states that which "causes" an existent's behavior as its own nature. Harriman obviously believes these two descriptions of cause agree, perhaps even that they reinforce each other.
Harriman regards causality as a "corollary" of the axiom of identity implicit in the perception of action in the same way identity is implicit in the perception of entities.
"Knowledge of the law of causality is first gained by a child in implicit form, in the early, preconceptual stage of cognition; it is grasped as a corollary—a self-evident implication—of the law of identity, one of the fundamental axioms of philosophy. The law of identity states that to be is to be something in particular, i.e., to have a nature; causality is the application of identity to the realm of action, i.e. it states that an entity must act in accordance with its nature." [Page 22.]
One "possible" way a child gains knowledge of "cause," Harriman explains, is through the experience of causing things himself. He provides an example and explanation:
"A toddler, say, pushes a ball and it rolls away. ... the content of that concept [cause] is already present in the ... 'rolling' an object. To roll an object is to cause it to roll by a certain means. The experience of rolling a ball, therefore, is the experience of causing something to happen. It is a pure experience of causation, without which the concept of 'cause' could never be reached. The experience is directly perceptual. ... And if such rolling is an object of direct experience, as it clearly is, then causing, too, is an object of direct experience."
He then explains that this direct perception of cause is the basis of the child's first-level generalization (induction).
"He [the child] experiences the connection between what he does and what it makes happen. This is the basis of a child's first-level generalizations—and it gives him the explicit knowledge of 'cause' necessary for further progress." [Page 22.]
According to Harriman, our knowledge of "cause" begins with direct perception.
"Armed with an explicit concept of 'cause' (of one thing 'making' another happen,) he [the child] is ready to perceive, all around him further instances of it. ... 'The wind makes the leaves flutter,' 'The fire makes the paper turn into ashes,' 'The rain makes the ground wet.' In all such cases, the causal connection is grasped from a single instance, because we directly perceive the causation as it is occurring." [Page 23.]
"In regard to first-level generalizations, however, direct perception of cause and effect is essential—and sufficient." [Page 24.]
Harriman's own definition of induction is the following:
"Induction is the process of inferring generalizations from particular instances." [Page 6.]
His meaning of generalization is this:
"A generalization is a proposition that ascribes a characteristic to every member of an unlimited class, wherever it is positioned in space or time. In formal terms, it states: All S is P. This kind of claim, on any subject, goes beyond all possible observations." [Page 7.]
The following are examples of what he means by generalization:
"But all of this requires that men first have the concept of 'shadow'—which depends on our ability to distinguish the dark areas behind lighted objects from the objects themselves. And how did we learn this distinction? From a wealth of earlier data, such as 'The dark areas in contrast to the objects they abut, have no tactile properties' (a generalization) and 'The dark areas appear or vanish with changes in the light source, while the objects remain constant' (a generalization). From these (along with other such generalizations), we conclude that the dark areas are not objects, but rather an effect produce when an object blocks light (a generalization)—which gives us the concept 'shadow.'" [Pages 17&18.]
Harriman bases his justification for this kind of generalization on what he calls, "first-level," inductions, which I suppose means the same as a child's "first-level generalizations," described above under the, "Origin of the Concept Cause."
"Similarly, a toddler sees a particular ball, but his identification of it is simply 'ball.' At this early stage, the child does not and cannot know any wider integration or narrower subtypes .... The same applies to the child's experience of himself as the particular pushing agent. His identification must be of 'pushing' as such .... Inherent in forming and applying a concept is the understanding that what counts cognitively is only the identity of its referents ... because the concept of an existent subsumes all instances everywhere, past, present, and future.
"Because of his simple, first-level conceptual structure, our inducer, in the very act of naming what he perceives, automatically drops the measurement of the perceived cause and effect and thereby gains knowledge transcending the given concrete. This is how he is able to grasp that the cause pertains to pushing as such, and the effect to balls as such, no matter where or when the ball is pushed." [Page 27.]
Remember, according to Harriman, all generalizations are concepts of cause:
"Let us start by noting that all generalizations—first-level and higher—are statements of causal connection."[Page 21.]
Harriman nevertheless asserts that first-level inductions (generalizations) become concepts the moment a "word" is used to identify a "cause and effect," and this is automatic and self-evident.
"When the first-level inducer identifies his concrete experience of cause and effect in terms of words, his perceptual grasp of the causal relationship becomes thereby a conceptual grasp of it, i.e., a generalization. And since the application of first-level concepts is automatic and self-evident, the two aspects of a first-level generalization—the perceptual and the conceptual—are each, to a human mind, self-evident." [Page 28.]
Harriman makes this automatic self-evident concept of cause the basis for all knowledge of cause.
"How do you know that pushing a ball makes it roll? There is no answer, not even by Newton or Einstein, except this: Look and see. One cannot 'prove' such a generalization by deriving it from any abstract laws of motion. On the contrary, without a fund of such generalizations established at the outset, one could not discover or prove any laws of motion. The laws are valid only if their first-level antecedents are valid, not the other way around." [Page 18.]
[Note: On page 28, Harriman explains that a child's experience of "cause" becomes a concept when the experience is assigned a "word." On page 27, the only word Harriman gives as an example is "ball," and possibly by implication, thought not explicitly, the word "push." Harriman never gives an example of the child assigning a word to the supposed concept of cause.]
Harriman reduces all universal knowledge to mathematics, basing both the nature of the metaphysical and the epistemological on it, including the nature of the human mind. It is mathematics he says, that is the means by which cause is understood.
Mathematics is, "the science of relating quantities to one another, quantities that are ultimately related to perceivable objects. ... it is by means of relating quantities that scientists grasp and express causal relationships." [Page 84.]
For Harriman, "mathematics is the language of physical science." [Page 225.]
"What knowledge of astronomy is possible without mathematics?" he asks. [Page 109.]
Harriman insists that mathematics is not a mere product of the mind without reference to the perceivable world. He comments on the wrong view of mathematics as "detached from the world, ... its source ... placed entirely within consciousness ...:"
"Such views about the nature of mathematical concepts led Einstein to pose the unanswerable question: 'How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?' An answer to this question is possible only when we reject the premise that mathematics is independent of experience. Like every other science, mathematics applies to reality because it is derived from our observations of reality. It is a conceptualization of facts, which are ultimately reducible to observed similarities and differences." [Page 226.]
Continuing to emphasize that objectivity of mathematics, and after providing a somewhat fantastic explanation of how that method, which we call counting, was developed (as if anyone actually knows), he goes on to assert that "number concepts are integrations of similar concretes." and, "They refer to facts, as processed by our conceptual faculty; i.e., they are objective." [Page 227.]
He further states in this regard:
"Reduction to perceptual data is more complex for higher-level mathematical concepts.... If the concepts of higher mathematics are not derived from experience, however, then ... they are invalid." [Page 227.]
[NOTE: Because I shall strongly disagree with Harriman's view of mathematics relative to both metaphysics and epistemology in my criticism below, I will state here that I strongly agree with his view of the objectivity of mathematics expressed in the previous five paragraphs.]
Having asserted the objectivity of mathematics, he then asks the question:
"Taking for granted the objectivity of mathematics, our question is: Why is it only by means of mathematics that we can gain scientific knowledge of the physical world?" [Page 228.]
The answer, according to, Harriman is because all our knowledge is based on the principles of mathematics, and that answer as based on Ayn Rand's "measurement omission" theory of concepts.
"Rand identified that the similar concretes united by a concept differed from one another only quantitatively. We form a concept by noticing that two or more existent have the same characteristic(s), but that these characteristics vary along a quantitative continuum of more or less. By omitting the implicit, approximate measurements of the characteristics, we can integrate the existent and treat them as interchangeable instances of a single concept." [Page 228.]
Since it is by means of measurements (or rather, ignoring them) all concepts are formed, Harriman can say "Concepts are the means by which we identify the nature of existents, and they are based on our grasp of quantitative relations among their referents. In performing such an integration, our minds grasp that the various instances we perceive are commensurable, i.e., reducible to the same unit&mdahsh;and therefore that the instances are the same except for their varying measurements. ... Thus when we say 'I know what something is,' we mean 'I know what it is through a quantitative operation my mind performs,' i.e., through grasping the quantitative connection of instances to some concrete taken as the unit—and then dropping the measurements." [Page 228.]
Thus, for Harriman, all knowledge, at least all scientific knowledge, is reduced to quantity, even to the working of the human mind.
"Human consciousness is inherently a quantitative mechanism. It grasps reality—i.e., the attributes of entities and their causal relationships to one another—only through grasping quantitative data. In this sense, quantity has epistemological primacy over quality." [Page 231.]
Harriman does emphasize that the arithmetic nature of consciousness is epistemological.
"It is crucial to recognize that this point is epistemological, not metaphysical. Pythagoras was wrong to claim that quantity is the substance of reality; it is not true that "all things are numbers." Quantity is always quantity of something, i.e. of some entity or attribute. But quantity is the key to the nature of human knowledge. We can grasp and identify the qualities of things only through grasping quantity—and we can grasp causal relations between entities and actions only through grasping quantitative relations between them." [Page 231.]
Harriman considers induction the "compliment" of deduction, a kind of "converse" operation—induction is reasoning from the particular to the general; deduction is reasoning from the general to the particular.
"Induction is the process of inferring generalizations from particular instances. The complementary process of applying generalizations to new instances is deduction." [Page 6.]
The basis for the validity of induction, for Harriman, is the concept of cause, which is a fundamental concept of action in the same way identity is a fundamental concept of entities. Just as identity is implied by the perception of entities, cause is implied by the perception of action. Every action happens for a reason which is its cause.
The formation of the concept cause, "is automatic and self-evident," in the child, and is the basis of all future generalizations from particular instances, since "all generalizations ... are statements of causal connection."
These causal connections can be and are expressed in mathematical terms, because "it is by means of relating quantities that scientists grasp and express causal relationships."
Ultimately, mathematics, which is our means of comprehending quantity and quantitative relationships is the basis of all knowledge, because, "quantity is the key to the nature of human knowledge. We can grasp and identify the qualities of things only through grasping quantity—and we can grasp causal relations between entities and actions only through grasping quantitative relations between them."
[Note: This summary only addresses the concepts I am critical of. I am in total agreement or at least approving of many of the concepts in this book, specially the emphasis placed on the hierarchical nature of knowledge, the fact that new knowledge is always acquired in the context of current knowledge (which is frequently a prerequisite for that new knowledge), and the non-contradictory nature of true knowledge.]
Critique and True Nature of Science
Though the stated purpose of The Logical Leap is to defend the validity of induction, I begin my criticism with Harriman's concept of cause, because he says, "all generalizations ... are statements of causal connection." We are not going to know what he believes induction induces if we do not know what he means by cause.
Harriman states, "the same cause ... will result in the same effect." He also states, "In seeking cause and effect, we are relating objects/attributes that are subsumed under different concepts. We are attempting to discover the effects of one type of existent on another, for example, to identify the effect of temperature on the pressure of a gas, or the effect of length on the period of a pendulum, or the effect of distance on the gravitational force between bodies."
The key word in the second explanation of cause is "relating." It is obvious what Harriman means by "cause" here, is a particular kind of relationship, one in which an entity, attribute, state, or behavior corresponds in some identifiable way to some other entity, attribute, state, or behavior.
These kinds of relationships certainly exist, can be identified, and are identified; but why should they be called "cause?" The fact that a gas will have a higher pressure if its temperature is higher is simply a description of the relationship between two properties of a gas, not a description of a "cause." This is a good example because temperature and pressure in gasses are mutually determined. An increase in pressure produces an increase in temperature, and vice versa (so long as the volume remains constant). Which is the cause, and which is the effect?
The fact is, none of these are examples of one thing "causing" another. The correct explanation is given by Harriman, himself, "an entity of a certain kind necessarily acts in a certain way under a given set of circumstances ..." It is not, however, as he says, the essence of the law of causality," because an existent behaves the way it behaves because it is what it is and has the nature it has. Nothing "causes" it to behave the way it does.
The principle ought to be written, "the same entity in the same context always behaves the same way." An entity's context is its state and its relationships to all other things.
An entity is whatever all its qualities are. (By qualities I mean all of an entity's attributes, properties, characteristics, and states.) It is its own qualities that determine how it will behave in any context.
Now consider the examples of "cause" given by Harriman:
The temperature and pressure of a gas are attributes of the gas, an entity, and its behavior is determined by its own nature. It is not "caused" by something else. The fact that the attributes of pressure and temperature in a confined gas have a specific relationship is itself an attribute of gas. It does not exist in liquids, for example.
The length of a pendulum is a property of the pendulum. It behaves the way it does (has a specific period) because of its own attribute, length. It is not "caused" by something else.
Two bodies, relative to each other may be considered a system, an entity, and its behavior will be determined by its attributes, which of course includes what it consists of and the properties of those components. Otherwise each body might be considered part of the context of the other, but how each behaves in that case is determined by each body's own nature, that is, its own attributes.
There is nothing wrong with the concept "cause" meaning, as it does in common usage, the reason or explanation for a thing, usually for events, like the cause of an automobile accident or the cause of the power outage. In that sense it only means that things do not just happen, willy-nilly, but that everything happens as a consequence of some preceding conditions or events. Even though it is applied in some technical situations (looking for the cause of a circuit board failure, for example) it is not a technical term. The basis of the validity of the concept, however, is philosophical. It is based on the understanding that the physical world has an objective nature, that it is what it is independent of anyone individual's knowledge or consciousness of it, and that it's nature and behavior are determined by inviolable principles.
It is the purpose of the sciences to discover what those principles are. While those principles are the basis for the common concept of cause, the word "cause" is inappropriate as an explanation of those principles.
This is much more than a semantic issue. In both philosophy and science, the word cause usually has a much narrower meaning than its common one, and that narrower meaning is both incorrect and misleading.
Consider again Harriman statement that, "the same cause ... will result in the same effect."
Since events are only the behavior of entities, and since an entity's behavior is determined by its own response to its entire context, including all its relationships, identical "causes" would require identical entities in identical contexts, which is probably impossible. In the entire history of the world, there have probably never been two identical causes.
It is obvious that Harriman is convinced the validity of science, and perhaps all knowledge, rests on the validity of the concept, cause.
The concept "cause," in this case, is the philosophical one, which historically has always been mistaken. Aristotle's version was really about how things came into being and are what they are, and was quite unlike what Harriman thinks cause means. I've already mentioned that Harriman's explanation of cause most resembles Hume's.
In fact, I think philosophers like Harriman have been fooled by Hume. Hume defined cause in a way that is easily refuted implying that since our knowledge of scientific principles rested on the concept of cause, there is no certain basis for science. The big lie that Hume subtly put over was that the validity of science rested on some philosophical notion of cause, particularly the absurdity he himself put forth.
Harriman is fully convinced by that lie. For example he writes, concerning Kepler's laws:
"He thought of his laws in the following way: First, the sun exerts a force on each planet that causes it to move in an elliptical orbit (with the sun located at a focus); second, the solar force causes each planet to move so that the line from the sun to the planet sweeps out equal areas in equal time; third, the solar force diminishes with distance in a way that causes the cube of the mean distance from the sun divided by the square of the orbital period to be constant for all planets. Clearly, these are causal statements—as they must be in order to qualify as laws." [Page 104.] [Emphasis mine.]
Each of the statements in Harriman's description of Kepler's laws containing the word "causes" is incorrect. For example, the force the sun exerts on a planet does not "cause" it to move in an elliptical orbit. In fact, the sun's force does not "cause" it to move at all. The reason the planets move is their own inertia—they were already in motion and if there is a "cause" for that it would have to be their own entire history. In response to the force the sun exerts on a planet, it accelerates toward the sun and the resulting change in the direction of its own motion results in that motion conforming to an elliptical path.
In attempting to illustrate that Kepler's laws are examples of, "causation," Harriman misses the true basis for scientific laws, which is the metaphysical fact that every existent has a specific nature that determines how it behaves in every context. The behavior of the planets in the context of the suns gravitational field is not "caused" by the sun or the force it exerts, it is determined by the planets own nature (it accelerates toward other masses) and state (it is in motion at a certain velocity).
The validity of science does not rest on the notion of cause. The concept of cause, even if it could be made "scientific", is too simple. The validity of science rests on the fact physical existence consists only of physical existents, that every existent has a specific nature that determines its behavior, and its relationship to all other existents. The whole objective of science is to discover the nature of all existents and their behavior and relationships. The nature of existents, their behavior and there relationships are absolute, the discovery and identification of those existents, their behavior and their relationships constitute the inviolable "laws" of science.
Science, then, consists of to two aspects: 1. the process and methods by which the laws of science are discovered and 2. the body of established laws expressed as principles by which the nature of existents, their behavior, and relationships are understood—that is, the body of established scientific knowledge.
The reason Harriman emphasizes cause as a first-level concept, derived by direct perception (ostensively), is because his whole argument for the validity of induction as generalizations from even single incidents, depends on regarding all generalizations as notions of cause, and cause as an axiomatic (or at least corollary) concept inherent in all concepts of physical relationships.
"Knowledge of the law of causality is first gained by a child in implicit form, in the early, preconceptual stage of cognition; it is grasped as a corollary—a self-evident implication—of the law of identity, one of the fundamental axioms of philosophy. ... causality is the application of identity to the realm of action, i.e. it states that an entity must act in accordance with its nature," he writes.
There is a mistake here that is identical in nature to the mistake the idea of cause always introduces. It is a kind of false dichotomy. Just as the idea of "cause and effect" separate attributes of an entity, as thought they were independent existences, Harriman separates an entity's nature from its behavior.
An entity does no act "in accordance with its nature," as though its nature were one thing and its action another; an entity's action is an aspect of its nature. But Harriman must maintain this separation to insist that what a child perceives when perceiving events is "cause".
When he says a toddler, "pushes a ball and it rolls away," is "a pure experience of causation," he assumes the pushing the ball and the ball's rolling are somehow separated in the toddler's consciousness, else one thing being the cause of another could not be experienced. Of course no one can say exactly what the child's conscious experience is, but from everything we know about how we perceive things, it is much more likely a child would perceive his pushing the ball and the ball rolling as one single contiguous event, not two events, one being the cause, the other the effect.
The same false dichotomy shows up in all of Harriman's examples. It is unlikely a child perceives 'the wind makes the leaves flutter,' and if he makes any connection at all between wind and leaves, the fluttering leaves would probably be perceived as the wind; and I'm certain the child does not perceive 'the fire makes the paper turn into ashes,' but, which is much nearer to the truth, that fire is the paper turning into ashes.
None of these are examples of cause in any case, but are good examples of the principle that a thing's behavior or even state, are aspects of its own nature—leaves flutter in the wind, but rocks do not because of the difference in their nature; paper turns to ashes when it burns but alcohol does not because of the differences in their natures; and the ground becomes wet when it rains but the lake does not because of the differences in their natures.
There is, in fact, no perceivable axiomatic concept of cause as described by Harriman. What is perceivable is things as they are, and the fact that they are different, which implies differences in their attributes, though the nature of those differences may not be directly perceived. It is the job of science to identify those differences, conceptually, which will explain the differences in their behavior and their relationships to each other.
There is a widely held misconception clearly stated by Harriman:
"Induction is the process of inferring generalizations from particular instances. The complementary process of applying generalizations to new instances is deduction." [Page 6.]
The principles of correct reasoning first identified by Aristotle, is called logic. The process of using those principles, that is, thinking logically, is called deduction. There is no other kind of correct reasoning.
The principles of deductive reasoning have been formalized and are called "formal logic." Formal logic describes the principles of correct reasoning by means of strict rules and structure. It does not describe how we usually do our thinking, even when our thinking is correct, but all correct thinking can be put into the structure of formal logic, else it is not correct thinking.
All reasoning in formal logic is in the form of syllogisms, which all consist of three statements, call propositions. Those three propositions are called a major premise, a minor premise, and a conclusion. A proposition is a sentence that asserts something about something else. The following is an example of a syllogism:
All birds have feathers. (Major premise)
Penguins are birds. (Minor premise)
Therefore, penguins have feathers. (Conclusion)
The conclusion of a syllogistic argument contains the "inference." In this case the inference is that penguins have feathers based on the two premises, that is, since penguins are birds (minor premise) and all birds have feathers, (major premise) penguins therefore have feathers (being one of those things which all have feathers). If either of the premises were not true, the inference or conclusions could not be known from the argument to be true.
According to Harriman, our knowledge that "all birds have feathers" would be a generalization from particular instances; of birds, I presume. In reality our knowledge that "all birds have feathers," is based solely on our concept of birds, which like all other universal concepts, identifies a specific category of existents in terms of their attributes—which in the case of birds includes the attribute "feathered."
We do not first form our concept of birds from the observation of birds, and afterwards discover they have feathers— we form our concept of birds because we discover creatures that have feathers (among other attributes) and choose to name our identification of those creatures, "birds."
Harriman's description of generalization, in spite of invoking it, actually ignores the nature of concept formation:
"A generalization is a proposition that ascribes a characteristic to every member of an unlimited class, wherever it is positioned in space or time. In formal terms, it states: All S is P. This kind of claim, on any subject, goes beyond all possible observations," Harriman wrote.
Well, of course, if I observe a feathered animal, even if only one, and name it a bird, because it is feathered, every animal I discover after that which is feathered is a bird. The process is not generalization, it is identification, the identification of a thing's attributes, and the identification of entities in terms of those attributes. This is exactly how concept formation of universal concepts works. It is the identification of existents in terms of their attributes that make them the kind of existent they are. [Please see "The True Nature Of Concepts" below.]
So to say "all birds have feathers" is not based on a generalization, but a definition—"a bird is a feathered animal." If after seeing a bird fly, I say "all birds fly," that is a generalization, and of course it is wrong. (The penguin and ostrich do not fly.) An equally bad generalization would be that all flying animal's have feathers from observing flying birds, which do have feathers. (Bats and flying insects do not have feathers.)
Now consider Harriman's description of the concept of "shadow."
"And how did we learn this distinction? From a wealth of earlier data, such as 'The dark areas in contrast to the objects they abut, have no tactile properties' (a generalization) and 'The dark areas appear or vanish with changes in the light source, while the objects remain constant' (a generalization). From these (along with other such generalizations), we conclude that the dark areas are not objects, but rather an effect produce when an object blocks light (a generalization)—which gives us the concept 'shadow.'"
In each case where Harriman writes, "a generalization," it is actually, "an identification," specifically of an attribute, and what we mean by a "shadow" is something with all those attributes.
There is an oddity in this example of his idea of generalization as well. Remember, he wrote, "... all generalizations ... are statements of causal connection." Perhaps he would consider "the dark areas appear or vanish with changes in the light source," an example of a, "causal connection," but how can, "the dark areas in contrast to the objects they abut, have no tactile properties," be construed as causal?
No such thing as induction, in the sense of generalizing from some limited number of instances, can possibly be valid. The source for universal concepts is observation and identification of existents in terms of the qualities (attributes and properties) that are their nature. Conceptualization of an existent, even if only one has ever been observed, is as valid as it would be if an indefinitely large number had been observed.
This process is not inductive, not a complimentary process of deduction, but in fact a deductive process. (This will be more fully explained in terms of "The True Nature Of Concepts.")
While I believe Harriman's view of mathematics is badly mistaken, almost mystical, I do agree that mathematics is firmly rooted in objective reality, derived from observation and not some kind of disconnected abstraction of the mind. Beyond that, there is not much agreement between Harriman's view of mathematics and my own.
Before I discuss my view, however, I want to emphasize the fact that I fully agree that the use of mathematics in science is an invaluable and totally valid tool that has made it possible to discover the true nature of many things and relationships between them that would have been impossible without mathematics.
I've already provided what I think is a fair description of Harriman's understanding of mathematics. I do not intend to burden the reader with a description of my entire view of mathematics, but to understand my criticism of Harriman's view and use of mathematics, an understanding of what I regard as the limits of mathematics will make it easier to understand. The following is adapted from another work in progress.
There is in many people, maybe even most, an overwhelming desire for some one single ultimate answer or explanation for everything. That desire is born of an irrational fear of the unknown. For some, the ultimate answer for everything is God, for others it is the illusive Grand Unified Theory (GUT) of physics. All such ultimate answers, however, are really a kind of mysticism, and those who accept some supposed ultimate answer, whether God, or GUT, or something else, embrace it with religious fervor.|
Perhaps the most fervent of "true believers," are those who embrace what I call the Pythagorean fallacy or superstition. It is the belief that number or mathematics is, in some profound way, the ultimate answer or explanation for everything.
Pythagoras said, "all things are numbers." Modern Pythagoreans do not say all things are numbers, but do believe everything can ultimately be understood in terms of numbers or explained by mathematics. When the ancient Pythagoreans discovered incommensurables, some of them committed suicide, because that discover showed that all they believed, the very basis of meaning in their lives, was wrong. I hope the modern Pythagoreans will not react to what I have to say with similar despair.
Objective Base of a Method
Mathematics, like conceptualization and language, is a method, a human invention with the purpose of dealing with certain specific attributes of the perceived physical world.
All of what is called mathematics begins with the concept of numbers. At some level, the field of mathematics merges with geometry and some aspects of logic as well, but the strictly mathematical part of even the advance mathematical fields of trigonometry and the calculus, for example, depend on the concept of numbers.
The objective attribute of the world which numbers pertain to is multiplicity—that is, the fact existence consists of multiple discrete entities. That is the objective foundation of all mathematics.
Numbers are the conceptual tool of counting. Before men learned how to count, or even today where primitive tribes do not have that skill, there is no certain way to determine the quantity of things, such as how many cattle one has, or how many people there are in the village.
We do not know who the genius was who discovered that using a set of different symbols, always recited or recorded in the same order, assigning a different symbol to each item in a collection, the last symbol used would indicated the total number of items in that collection. The symbols that were assigned to each item are what we now call numbers or numerals. The process of assigning the names to items, we call counting. That discovery, wherever it has been passed on, has transformed the world, and nothing in the civilized world would be possible without it.
All of mathematics is an extension of that basic method of determining the number of things by counting. Addition and subtraction are just shortcuts for counting and "counting backwards." Multiplication and division are shortcuts of addition and subtraction. Fractions and decimals are shortcuts of division and methods of notation.
Another unknown genius of ancient history discovered that numbers could also be used to identify other characteristics of things, such as length, weight, or speed, as well as relationships between things, such as distance. The technique is called measurement.
Obviously this discovery has been just as important to the development of the civilized world as counting itself. Unfortunately, it also led to one of the first great mistakes in philosophy of which philosophy has never thoroughly rid itself.
Measurement uses the method of counting to determine a "measurable" attribute. All measurement requires "a unit of measure" commensurate with the attribute to be measured. If the attribute to be measured is length, for example, the unit of measure must be some length that is chosen as a "standard" length. If the attribute to be measured is weight, the unit of measure must be some standard weight.
The method of measurement is counting, and what is counted is the number of "units of measure" that equal the measure of whatever characteristic is to be determined. If we use length as an example, one way to measure it would be to take a small stick, as the unit of measure. The stick could be laid out on the length to be measured, starting at one end, then placing it next where it last ended, repeating this process, counting each time the stick is laid down until the end of the length being measured is reached. If the stick is laid down 10 times, the measured length is "10 'sticks' long."
While a stick is a metaphysical existent, and has length as an attribute, and its own length is a metaphysical fact, and it's length is being used as the "unit of measure" to measure something else, it is an arbitrary unit. As a concept for a "unit of measure" it is only a concept, there is no metaphysical existent, "stick-length."
When counting entities, counting is absolute. If there are thirty seven entities, counting will tell you exactly how many there are, that is, 37—and there are 37, absolutely.
When measuring something, the number of "units of measure" that are "counted" may, or may not be the exact measure of a thing, and in fact will almost never be perfectly exact.
The main reason for this is because units of measure are discrete; they are concepts and all concepts are discrete. But concepts have no physical existence, only mental or conceptual existence. There are no metaphysical inches, pounds, or minutes, there are only length, weight, and time, and they are all analog.
For any discrete unit of length conceived, there may be existents it can exactly measure, but there are, potentially, an infinite number of existents it cannot exactly measure. This is true of all units of measure. To suppose that everything can ultimately be known in terms of mathematics forgets that mathematics is only a method, a method of determining the numbers of things adapted as a method for dealing with measurable attributes of existents and their relationships. Measurement is only the application of the method of counting to that which cannot truly be counted, but with the invention of, "units of measure," some of the attributes of the physical can be treated as though they had "parts" that can be counted, which metaphysically, they do not.
Pythagoras' Devastating Discovery
The disillusionment of the ancient Pythagoreans followed directly from Pythagoras' greatest discovery that, where a and b are the legs (sides next to the right angle) of a right triangle, and c is the hypotenuse, (side opposite the right angle), a2 + b2 = c2. This led immediately to the discovery that in an isosceles right triangle, where a = b, there is no commensurate unit of measure that can measure both a leg of an isosceles right triangle and the hypotenuse.
Today many such "irrational" (no ratio) relationships are known, and very close approximations, such a pi, are used in calculations where such relationship need to be measured. It is difficult not to have the impression that irrationals, like pi, actually do have a value if one could just carry it out far enough. The ancient way of describing these irrational relationships is much clearer in demonstrating there is no such value.
Suppose the sides of an isosceles right triangle are one inch long. Let the length of the hypotenuse be represented by m/n. Since a2 + b2 = c2, substituting 1 for both a and b, and m/n for c, yields m2/n2 = 2. Divide out any common factor in m/n, now either m or n must be odd (because if both are even there is still the common factor 2).
Multiply both sides of the equation, m2/n2 = 2, by n2 to get m2 = 2n2. Therefore, m2 is even; therefore m is even. Suppose m = 2p (if m is even it must be 2 times something). Substituting 2p for m in the equation, m2 = 2n2, yields 4p2 = 2n2. Dividing both sides by 2 yields n2 = 2p2, therefore n is even.
If there were a unit of measure that could measure both a leg and hypotenuse of an isosceles right triangle, the length of the hypotenuse could be represented as m/n units, and either m or n, reduced to lowest terms, would have to be odd. Since both m and n can be demonstrated logically (or mathematically) to be even, there can be no unit of measure that can measure both a leg and hypotenuse of an isosceles right triangle.
This discovery was enough to demonstrate to the ancient Pythagoreans that not only is, "all things are numbers," not true, all things cannot even be described by numbers. It is even worse than that, however, for the modern Pythagoreans.
There is a class of physical events described by a set of concepts called "chaos" or "fractals" or "Lorenz attractors." The peculiar thing about such events is that they are determined, not randomly as chaos might imply, but strictly in terms mathematical functions, though the actual mathematical function for any real chaotic event or process can never be discovered, and the actual behavior of chaotic events and processes are impossible to predict (which is the real reason they are called "chaos").
True natural "chaos" events and phenomena are analog, not discrete, in nature, but scientists can simulate such events with digital computers using what are called iterative functions. The technology is too complex to describe here.
There are many aspects of the real world, however, that are examples of "chaos" theory. The human heart beat, for example, is never absolutely even, because the electronic nature of the heart behaves, apparently, like a Lorenz attractor. It is a feedback mechanism (like taking the output of one equation and using it as the input of the next). In fact, if the heartbeat were perfectly symmetrical, it would race uncontrollably, a condition which does happen called fibrillation.
The almost endless patterns of snow flakes are examples of fractals. Each is completely different, because the physics that forms them, though identical, begins with a different value for each snow flake (because the particles of dust all snow flakes form on are slightly different).
Ferns are another example. While ferns all look very similar, they are never identical. Broccoli exhibits the same fractal characteristics. There are, in fact, chaotic characteristics in all life. The venous and arterial systems in a human kidney, flowers, and trees are all examples, and DNA clusters form shapes that resemble Julia sets. Non-living examples include clouds, frost and ice formations, lightning, galaxies, and ocean currents.
Perhaps the most interesting example of chaos is the weather. It was while studying weather that the famous Lorenz attractor was discovered. Edward Lorenz, an MIT meteorologist was attempting to create a program that all meteorologists dream of. It was believed if one could map all the meteorological states of the world, one could predict all the weather, indefinitely. What they discovered was, because weather "feeds itself" it behaves chaotically, and therefore was unpredictable, which came as no surprise to anyone except meteorologists.
None of these limitations of mathematics in any way repudiates the value and utility of mathematics, especially in the sciences. All that men have achieved and accomplished in the fields of science and technology rests heavily on mathematics and without the knowledge of that method, all the benefits that contribute to the quality of life enjoyed by modern man in western civilization would not be possible. But mathematics is only a tool, a method invented by man. It is when mathematics, or any other single aspect of intellectual achievement, is raised to the level of mystic insight that will provide the answers and explanation for all things that it becomes a superstition.
Now I want to consider Harriman's question, "Why is it only by means of mathematics that we can gain scientific knowledge of the physical world?."
It is certainly true that a great deal of the physical world, especially those aspects of it studied by the physical sciences, are understood and expressed in terms of mathematics. Virtually all of the basic physics, "laws," from mechanics to quantum physics, are expressed in terms of mathematical formulas. But even for physics, those "laws" are not all of scientific knowledge.
Mathematics is only able to deal with two aspects of the physical world: that which can be counted, and that which can be measured. Before there can be counting and measurement you must first identify things to count and measure.
Certainly the identification of the heavenly bodies is part of that knowledge and precedes any possible knowledge even of the relative positions of the planets in the solar system. A great deal of scientific knowledge is descriptive identification involving no math concepts at all. What is a planet, a star, a galaxy, a comet, or an asteroid?
All of the examples in The Logical Leap are good examples of science dealing with those aspects of the physical world that are amenable to the methods of mathematics, but those are only a part of science.
The book is directed toward the science of physics, but draws broad conclusions about science in general. Outside of physics, there is even less in science that is strictly mathematical in nature, and becomes less so as science moves from the merely physical to chemistry (particularly biochemistry), and to biology, and medicine. (How much of taxonomy is "mathematical?" Does Harriman reject it as part of science?)
There are many attributes of the physical that are not mathematical in nature. There is no measurement of attributes like right-handed and left-handed, or mirror imaged. If something has "handedness, it is either right or left, there are no degrees. Such attributes are not mathematical, but different kinds of qualities altogether, Clockwise, counter-clockwise, and polarity, are others.
Are physical states, like solid, liquid, and gas, measurable? There might be measurable attributes associated with particular substances and their states, like temperature, but the states themselves, and the concepts for them, have nothing measurable about them. What is the measurable attribute of the property "sublimes." (Chemical substances that have only two states, solid and gas, like carbon dioxide and iodine.) What is the commensurate unit of measure for the attributes defining a plasma? A plasma is a plasma, period. One understands what a plasma is by description, not in terms of any measurable attributes. (Of course plasma's have measurable attributes, like temperature and charge, but they do not make a plasma what it is.)
All of the physical attributes science uses mathematics to measure are not themselves mathematical and cannot be described or defined in mathematical terms. Before anything can be measured, it must first be identified. We first have to grasp the identity of entities, attributes (qualities), relationships, and events before we can even notice differences or similarities between them. We first must observe that things have attributes like length, weight, temperature, volume, and speed, before we can discover differences in those attributes and before we discover there is any way to measure them.
The attributes of things are what they are and are absolute, but they are not measurements and, as attributes, cannot be described in mathematical terms. Consider two entities, A and B, with the attribute scringe. If B is much greater than A in its scringe attribute, five times say, the scringe of B may be described as 5 As. If A is a convenient quantity to use as a unit for scringe measurement, we can give the scringe of A a name, like kak and measure all entities with scringe in terms of kaks.
Notice that even though I can now identify the exact measure of scringe of any entity with that attribute, and can determine the exact difference in kaks of the scringe of any two entities, there is no way to explain what the attribute scringe is in terms of kaks, or scringe, or any mathematical terms, because scringe itself has no mathematical properties, and mathematics can only be used to compare differences or similarities in scringe.
The term scringe may be replaced with any of the measurable attributes of entities, like length, weight, temperature, volume, and speed, and the term kak replaced with any corresponding unit of measure like inch, pound, degrees C, gallon, or MPH and it will be obvious (or ought be) that the attributes length, weight, temperature, volume, and speed are no more mathematical in nature than scringe, and cannot be described in mathematical terms.
Mathematics can describe relationships between such attributes, but beyond that, all other aspects of the physical existence which the sciences study and we have knowledge of is in non-mathematical terms.
To say that "quantity is the key to the nature of human knowledge. We can grasp and identify the qualities of things only through grasping quantity," as Harriman does, is a kind of rationalism. He might write, "Pythagoras was wrong to claim that quantity is the substance of reality; it is not true that 'all things are numbers,'" but he still commits the Pythagorean fallacy, he just moves it from the metaphysical to the epistemological.
"Human consciousness is inherently a quantitative mechanism. It grasps reality—i.e., the attributes of entities and their causal relationships to one another—only through grasping quantitative data. In this sense, quantity has epistemological primacy over quality."
First I must emphasize that quantity is a quality and only one of the many qualities of reality.
Now there is a problem with the word "grasp" which Harriman uses throughout the book, but never makes clear whether he means "grasp" by direct perception, or "grasp" in the sense of conceptual understanding. I suspect he means "by direct perception" since that is the way he uses the word with respect to what he called "first-level concepts."
If by quantity is meant, "countable," it only pertains to collections of things like cows or pennies. If by quantity he means, "measurable," that only pertains to attributes of relationship. In either case "grasping" quantity at the perceptual level can only mean grasping that one group is larger or smaller than another (if counting is meant) or that one thing is larger, or heavier, or further away than another (if measurement is meant), but no count or measurements can be grasped by direct perception.
[Note: "Grasping" that one groups is larger than another, or one thing is larger than another perceptually, means only that a difference is perceived; that the difference is a countable or measurable difference, can only be "grasped" conceptually.]
Harriman's arguments are based on Rand's mistaken view of concepts, particularly her measurement omission theory. [The next section, "The True Nature Of Concepts" explains what that mistake is and what is wrong with it.]
"Concepts" Harriman wrote, "are the means by which we identify the nature of existents, and they are based on our grasp of quantitative relations among their referents. In performing such an integration, our minds grasp that the various instances we perceive are commensurable, i.e., reducible to the same unit&mdahsl;and therefore that the instances are the same except for their varying measurements. ... Thus when we say 'I know what something is,' we mean 'I know what it is through a quantitative operation my mind performs,' i.e., through grasping the quantitative connection of instances to some concrete taken as the unit—and then dropping the measurements." [Page 228.]
First, concepts do not identify the nature of existents, they are only the identify existents themselves—their nature is identified by means of propositions about them, such as a definition.
Beyond that, this is pure rationalism, and as fantastic as Kant's or Plato's. What in the world is a "quantitative operation my mind performs?" Since when does the mind simply "perform" anything. Since when is the identification of "quantity" some mystic operation of the mind. Do we not have to identify what quantity means, and what quantities entities are capable of having?
To his credit, Harriman does assert measurement is epistemological, and he says, "Quantity is always quantity of something, i.e., of some entity or attribute." but in case you missed it, he wrote, "Human consciousness is inherently a quantitative mechanism."
This is fantastic. Human consciousness is inherently qualitative. As I've already pointed out, quantity is only one of many qualities. The attempt to make everything comprehensible in terms of only one kind of quality (measurement) is the worst kind of rationalism.
To be sure the relationships in the physical realm mathematics is used to describe are absolute and exist independently of anyone's knowledge or understanding of them, but those relationships are not, themselves, "mathematical." To imply, however, that, because some attributes of physical existents and their relationships can be identified by the man-made methods of counting and measurement, (mathematics), they are actual attributes of the physical, (or of the mind), is reism and hypostacization of the worst kind.
[This is a very brief look at the true nature of concepts, adapted from a pending book on Philosophy. It is included here because the mistake of "measurement omission" as the basis of concept formation can only be corrected with a correct view of concepts.]
Two Kinds of Concepts|
A universal concept identifies things, not as particular individual things, but as members of a class or category of things. Most of our concepts are universal ones, and the ones necessary for all knowledge beyond that of immediate concretes.
The concept, "dog," for example, means any dog there has ever been, is, or will be, real or imaginary. There is a difference, however, in a concept like "dog" and a concept like, "Athenian," meaning a citizen of Athens. The difference is something that has been a bane to philosophy throughout its history called "essence."
"Athenian," like, "dog," refers to any Athenian there has ever been, is, or will be, real or imaginary, but a dog is a dog because that is it's nature; an Athenian is not an Athenian because it is his nature, but because Athens is where he lives. He would have the same nature (human) if he lived elsewhere.
The reason a dog is a dog, rather than something else, is inherent in the dog itself; it is the kind of being it is; the reason an Athenian is an Athenian is not inherent in Athenians, they would be the same kind of being if Londoners or Parisians. Philosophers call what is inherent in dogs that makes them dogs their essence. The problem for philosophy has been to explain exactly what essence is, and what it means in the case of concepts like Athenian. Most of the explanations have been disasters that have plagued philosophy with errors that make a rational epistemology impossible.
Intrinsic and Extrinsic Concepts
The only function of a concept is to identify the existents for which it is the concept or "identification."
[Note: The meaning of a concept, is the actual existents it identifies which are also called a concept's referents, units, or particulars.]
Except for concepts derived ostensively (pointing at an apple, saying, "that is an apple"), for "particular" concepts, the identification is accomplished by any definition that succeeds in isolating the particular existent from all others. The technical term for that isolation is differentiation; that is, it identifies what makes the existent different from all others. If the concept is a "universal" concept, it must not only identify what makes those existents different from all other existents, but must also identify what makes those existents members of the same class, category, or kind of existents, that is, what is the same about them.
For universal concepts like Athenian, cook, or trinket, the sameness is something external to the existents identified by the concepts and is determined by things such as where one lives, what one does for work, or what something is used for. I identify those kinds of universal concepts as extrinsic concepts.
For universal concepts of things that are members of the same category because of something inherent in the existents themselves, like dog, planet, or cloud, it is their very nature that is the same. I identify those kinds of concepts as intrinsic concepts.
Kinds of Things and Essence
What makes things the same kind of things? More importantly, what makes anything the kind of thing it is? The answer to the first question is their essence, everything with the same essence is the same kind of thing. It is the answer to the second question that is the explanation of what essence is. The answer to that question, is epistemological for extrinsic concepts, but metaphysical for intrinsic concepts. That difference is one of the most important concepts of epistemology.
For intrinsic concepts, all of metaphysics, especially ontology, bears on the explanation of what essence is.
The identity of an existent is determined by its qualities. By "determined" is not meant "caused" but "described." Listing all of a thing's qualities (attributes, characteristics and properties) is one way to identify a thing. A thing is whatever all its qualities are.
Essence and Qualities
Every existent has two kinds of qualities: necessary and possible.
Necessary qualities are all those qualities of an existent that it must have to be the kind of existent it is, and without which it would not be that kind of existent.
Possible qualities are all the qualities an existent may have, but will be the same kind of existent whether it has any of those qualities or not.
Necessary qualities also exclude any qualities that would make an existent a different kind of thing and determines what qualities are "possible" qualities for that kind of existent.
A dog is a physical entity, a living organism, an animal, a mammal, and a canine. Physical, living, animal, mammal, and canine are all necessary qualities of a dog. If something had all these qualities except one, it would not be a dog. A dog can have short hair, long hair, a tail, no tail, be very big, or very small. Hair length, tails, and any particular size are all possible qualities of a dog, but none are necessary; a dog will be a dog with or without any particular one of them.
From ontology we learn the identity of every existent is determined by three necessary conditions: 1. it must have some qualities, 2. it must be different in some way from all other existents; therefore, it must have some quality or qualities that are different from those of all other existents, and 3. every existent has some relationship to all other existents; therefore, it must have some quality or qualities it shares with all other existents.
It is, therefore, an existent's necessary qualities that determine the kind of existent it is and every existent with the same necessary qualities is the same kind of thing. The necessary qualities of entities of the same kind, are different from the necessary qualities of all other kinds of things—they may share some qualities, but cannot share all the same qualities, else they would not be different kinds of things.
The essence of a thing, therefore, consists of all its necessary qualities. A dog is a dog because it shares with all other dogs the same necessary qualities. It is not necessary that all of a thing's qualities be known or identified for the concept of a thing to be valid, but a valid concept always implies all of the necessary qualities of an existent or kind of existent, known or unknown.
For there to be more than one of any kind of thing, since everything that exists must have some quality or qualities that are different from all other existents, the difference in existents of the same kind must be differences in possible qualities.
Every dog has some combination of possible qualities that are different from the possible qualities of every other dog. No two dogs can have exactly all the same possible qualities. (The differences can be pronounced or very slight.)
The essence of any existent is all the necessary qualities it shares with all existents of the same kind, that is, all those identified by the same concept. Every unit or referent of that concept will have some different possible quality or qualities from all other units of that concept.
For all extrinsic concepts, the essence is epistemological (based on what is known about their function, use, relationships, or purpose). If the units of an extrinsic concept are physical, the units themselves have intrinsic essences, but as units of an extrinsic concept, the units intrinsic essences, that is, their necessary ontological qualities, are only possible qualities of the extrinsic concept, not necessary ones.
For example, I own many pairs of chop sticks. They are all units or referents of the concept, "chop sticks." Their essence as chop sticks is, "two straight sticks used as implements for eating." The necessary qualities are, "sticks" that are "straight" with the exact quantity "two" with the function of being used "to eat with." I'll call this essence, "chop-stickness."
The essence,"chop-stickness," is strictly epistemological. While sticks are physical entities, and straight is a physical attribute and those qualities are necessary to chop sticks, "chop-stickness" is not inherent in chop sticks, because straight sticks can have many other functions (cue sticks or drum sticks, for example) and those qualities are attributes of many other kinds of existents. Notice it is the function, "to eat with," combined with the physical attributes (straight stick) that differentiates chop-sticks from all other things, including eating implements of other shapes.
I have chop sticks that are made of wood, others made of bamboo, and still others made of plastic. Wood, bamboo, and plastic are all intrinsic concepts; the essences of wood, bamboo, and plastic are metaphysical, the necessary attributes of these substances are ontological. As any of these essences is instantiated in chop-sticks, they are only possible qualities, however, because chop sticks can be made from many different substances and still be chop sticks.
A Philosophical Mistake
There are a number of philosophical theories of concepts that are nearly correct. One is the so-called measurement-omission theory which describes a concept as the identification of things of the same kind, having the same distinguishing attributes or qualities with the particular measurements of those attributes left out.
The theory assumes that all physical existents of the same kind have a quality, or combination of qualities, common to them all that makes them the same kind of thing and the common quality or combination of qualities is measurable.
Though not called that, the common quality or combination of qualities is in fact the "essence" of the units subsumed by the concept. Therefore, this theory assumes the essence of the units of a concept is some measurable attribute that can be measured in terms of some commensurate unit of measure, common to all possible referents of the concept.
This theory differs from the correct one at two points. First, it makes no distinction between intrinsic and extrinsic concepts. The second difference is that it accounts for only one kind of possible quality, magnitude.
The first difference is important because most of our concepts are extrinsic. The only intrinsic concepts are those for physical substances, naturally occurring physical entities, organisms, naturally occurring physical events, relationships, and attributes, and all concepts abstracted directly from these, such as attributes of life, values, and scientific principles for example.
All other concepts are extrinsic, including all man-made physical existents, activities, services, and relationships between these things, as well as physical and intellectual methods of doing things, like cooking (physical) and mathematics (intellectual). Almost all extrinsic concepts are defined by their relationships, purpose, function, or use of the units subsumed by the concept, none of which have magnitude as an attribute, or at least a major defining one.
The first difference is interesting, because, for a small subset of intrinsic concepts, the theory works. The "measurement-omission" explanation does work for that subset of existents which are strictly defined by their geometric nature or quantitative attributes. A triangle is a triangle because it has three sides, and every triangle has those same qualities (attributes) and the difference between triangles is strictly the difference in the lengths of their sides, and of course the resulting difference in their angles. Other such existents would be all those whose physical shape is their essential attribute, including all other geometric figures, and naturally occurring balls and crystals as well as most substances which would be differentiated only by size or weight. It does not work for any other kind of existent, however.
Consider the example provided by that theory which is intended to describe the formation of the concept "table."
According to the, "measurement-omission" theory of essence, the formation of the concept table begins with the observation that tables are differentiated from other things, such as other furniture, by their distinctive shape: a flat surface with supports, for example. The concept table leaves out the measurement of the geometric characteristic of a table, but retains those geometric characteristics; for example: the concept leaves out the shape of the surface (round, square, octagonal), the number of legs, height, and weight, but retains the fact it must be some shape, have a leg or legs, be some height, and weigh something.
What actually distinguishes a table from all other things is not its shape, but its function. The possible number of different shapes a table can be is endless, and for almost all those shapes, there are other kinds of furniture and objects with exactly the same shape. In some parts of the world there are tables that look more like beds, and beds that look more like tables—if shape alone were what distinguished tables from beds—they could not be distinguished at all. A table and a bed might have exactly the same shape (and measurements); what distinguishes them is what they are used for.
This mistake is partly the result of the failure to distinguish between intrinsic and extrinsic concepts. Since "table" is an extrinsic concept, as are all concepts for man-made things, its essence will not be comprised of measurable necessary qualities, though it may have measurable possible qualities.
There is an essence of every concept, or more exactly, of the units of every concept that describes what they are, and that essence is all their necessary qualities, but only the units of intrinsic concepts have a metaphysical essence; the essence of all other things is epistemological, that is, identified conceptually, not from observation.
Entities are ontological existents—the physical, living, conscious, and rational/volition existents that are all of the ontological or natural existence, that of which we are directly conscious. But entities are not all that exists. Attributes, relationships, and events all exist; history, philosophy, and the sciences (as disciplines) exist; memories, concepts, values, and principles all exist; even fictions exist. They do not have ontological existence, but they are existents, and as existents can only be identified by means of concepts.
Very few concepts are of things that have any measurable attributes, which means, the difference between units of those concepts cannot be a differences in measurement since there is nothing to measure.|
In none of the following examples is there a measurable attribute of the units of the concepts with a commensurate unit of measure: yesterday, beer, cheese, milk, pain, pleasure, brother, uncle, history, man-made, cough, blink, malaise, smallpox, cancer, fruit, potato.
There are whole categories of concepts without any measurable attribute with a commensurate unit of measure, such as, Antibiotic, with units: Penicillins, Cephalosporins, Macrolides, Fluoroquinolones, Sulfonamides, Tetracyclines, Aminoglycosides, each of which are also concepts; Hormones, with units: testosterone, estrogens, thyroxine, epinephrine (adrenalin), insulin, norepinephrine, cortisol, corticosterone, aldosterone, corticotropin (adrenocorticotropin, ACTH), growth hormone (GH or somatotropin), thyrotropin, and thymosin, each of which are also concepts; Mind or attributes and functions of the mind, such as reason, concept, percept, memory, imagination, nostalgia; Philosophy and its branches and concepts, such as metaphysics, ethics, politics, aesthetics, logic, value; Language or its parts or structures such as verbs, nouns, pronouns, prepositions, adjectives, adverbs, conjunctions, declension, conjugation, subject, predicate, participle, gerund, infinitive, idiom, and metaphor; Adjective or particular adjectives, such as mysterious, pregnant, alive, dead, true, false, satirical, mysterious, new, old, desired, or shunned.
I wrote earlier, "the whole objective of science is to discover the nature of all existents, their behavior and their relationships. The nature of existents, their behavior and there relationships are absolute, the discovery and identification of those existents, their behavior and their relationships constitute the inviolable 'laws' of science."
To a large extent, the identification process of science is conceptualization—forming the concepts that identify existents, qualities, and relationships. I think this is what Harriman might have had in mind when he wrote: "Induction is the conceptualizing process itself in action." [Page 35.]
If that is, indeed, what he meant, it would be correct, and if that view were held consistently, there would be no need for the word "induction." Conception, meaning concept formation, is enough, and using "induction" for that is misleading.
Concept formation, alone, however, is not all of scientific identification. Many of those identifications are in the form of propositions which are statements of the ruthlessly derived logical relationships science discovers.
Science is not the search for causes. In fact, most of science is much too complex for such a concept to have any meaning. Most scientifically identified entities, events, processes, and relationships have many different complex factors determining and influencing them.
Try explaining a tuned circuit in terms of cause and effect, or just the current in an AC circuit in terms of both resistance and impedance, for example. The simplest law of electronics, E=IR (voltage equals the current times the resistance) defies the concept, "same cause same effect," simply because there are three variables, and neither current or resistance cause a voltage, they only indicate what it will be if you know their values and already have a voltage.
Harriman insists science is essentially inductive, that what it induces is cause, and that cause is always in terms of mathematical relationships. I think I have demonstrated that all three of these are incorrect.
What I want to demonstrate is that science is essentially deductive, that what it deduces are identifications (of existents, their attributes, and their relationships) which identifications are logical in nature. Mathematics is only one aspect of that deductive method as a subset of logic.
Since chemistry is the science I've had the longest experience with, and the examples provided by Harriman are excellent I will use some of his examples to demonstrate that it was not induction that led to the real discoveries in chemistry, but identification and deduction.
When Joseph Louis Proust announced in 1799 his law of constant composition, that different samples of a compound always contain the same elements in the same proportions by mass, though it was called a law, it was really the identification of an observation. Harriman provides the example of Proust's observation that copper carbonate always contains copper, oxygen, and carbon in the same mass ratio.
This, along with any other compounds Proust made this observation for are, identifications. The compounds he studied always had this same property, and for those compounds, their fixed ratio of component elements were qualities of those compounds, part of their identity as the compounds they were.
Here's the deduction:
- Copper carbonate always contains copper, oxygen, and carbon in the same ratio. (Major premise)
- "Some compound" contains copper, oxygen, and carbon in a different ratio. (Minor premise)
- Therefore, "some compound" is not copper carbonate. (Inference)
[For the logician, this is a variation of the formal syllogism named BAROCO. The minor premise is usually negative, as, "Some compound" does not contain copper, oxygen, and carbon in the same ratio. The logical meaning is identical.]
It is unlikely that Proust reasoned in this way, but if his reasoning was correct, this is the logical basis for it. What the reasoning means is if another compound is discovered with similar properties to copper carbonate with different ratios of elements, it is a different compound, because it has a different identity, since the identity of copper carbonate includes the specified ratios.
I suspect that Proust did attempt to generalize from his observations that the principle of constant composition would hold for all compounds, which generalization was challenged by other experimenters who found other compounds that seemed to defy it. Though further careful analysis by Proust demonstrated those questioned compounds did conform to the constant composition principle, the attempt to generalize was still a mistake, because it would have been induction—attempting to infer a general principle from a limited number of incidents, which is invalid.
The correct approach should have been ruthlessly deductive. For example, when John Dalton studied the three gaseous compounds of nitrogen and oxygen, he discovered the difference in weight of the combining elements in the different compounds were whole multiples. He obtained the same results for compounds of carbon and hydrogen. His conclusion that when two elements combine to form more than one compound, the element that combines with identical weights of the other are in simple multiple proportions, was not accepted by other chemists as a law, nor should it have been.
The reason it shouldn't have been accepted is not, as Harriman suggests, because more accurate data was needed, or because more instances were required, or because there were apparent violations, but because that would have been the very kind of, "induction," drawing an inference from a limited number if incidents, which is still invalid, but which Harriman is defending.
For the limited number of observations John Dalton made, the relationships between the various compounds was identified. For example, the three nitrogen oxygen compounds Dalton studied were apparently nitrous oxide (N2O), nitric oxide (NO), and nitrogen dioxide (NO2), so nitric oxide has exactly twice as much oxygen as nitrous oxide, and nitrogen dioxide has twice as much oxygen as nitrous oxide. I have no idea what names for these compounds Dalton used, but whatever they were, they identified the same compounds, that is, they stood for the same concepts, and those concepts were the identification of the compounds, with all the attributes Dalton had observed for each, by which he recognized them, including the proportional relationships between them which he observed.
For those compounds those relationships are absolute, because they are part of the identification of those compounds. For example:
- Nitric oxide has exactly twice as much oxygen as nitrous oxide. (Major premise)
- "Some nitrogen compound" has exactly 1.5 times as much oxygen as nitrous oxide. (Minor premise)
- Therefore, "some compound" is not nitric oxide. (Inference)
Even if a nitrogen compound had all the attributes of nitric oxide, but the ratio of oxygen to nitrogen were different, it would not be nitric oxide because it would have a different attribute. This is a pure deduction, not an induction.
There is, in fact, a nitrogen compound, dinitrogen trioxide (N2O3), which has exactly 1.5 times as much oxygen as nitrous oxide. It would never be confused with nitric oxide, however, since it is only stable, as a dark blue liquid, at or below -21 °C (-6 °F).
The observation and identification of any attributes of a specific kind of existent will be true of all that kind of existent, because an existent is whatever all it qualities, attributes and characteristics are. No principle, however, can be generalized (induced) from the observation and identification of any number of attributes of some particulars (e.g. specific compounds) of a class or category of existents (chemical compounds) applicable to all referents of that universal concept.
It can, however, by hypothesized that identified attributes or qualities of some particulars of a universal concept could or might be attributes of other, or even all other, particulars of that universal. Dalton's discovery of the multiple proportions principle relating to compounds of oxygen and nitrogen, and of carbon and hydrogen certainly suggested that. But the principle could only be validly applied to other compounds, with the knowledge then available, by actually testing them and observing and identifying those attributes for those compounds.
To a large extent, that is exactly what happened, and by 1816, the Swedish chemist, Jons Jacob Berzelius had tested most known compounds which all exhibited the principle of multiple proportions.
Harriman correctly points out the importance of the constant composition and multiple proportions principles to the development of the atomic theory.
Though it is known that copper carbonate is composed of copper, oxygen, and carbon, that knowledge does not indicate anything about how copper, oxygen, and carbon combine to form the compound. Discovering, however, that they always combine in exactly the same ratio suggests that the compound consists of discrete units (which would become known as molecules). It is not conclusive of course, but the reasoning is this:
Suppose I have a batch of copper carbonate.
Suppose I take a sample from that batch and discover it has copper, oxygen, and carbon in a certain ratio.
Suppose I take another smaller sample from the batch and discover it also has copper, oxygen, and carbon in the same ratio.
Suppose I continue taking smaller and smaller samples and discover they all have copper, oxygen, and carbon in the same ratio, no matter how small the sample is.
That implies that throughout any batch of copper carbonate, copper, oxygen, and carbon are combined in the same ratio, which is possible only if copper carbonate consists of many tiny discrete units of combined copper, oxygen, and carbon.
Here's the logic:
- If a batch of copper carbonate is not the same proportion of copper, oxygen, and carbon throughout, a sample with a different proportion will be possible to extract.
- No sample of copper carbonate is ever of different proportion.
- Therefore copper carbonate is the same proportion of copper, oxygen, and carbon throughout.
- If a batch of copper carbonate is not composed of discrete units of copper, oxygen, and carbon in the same proportion throughout, successfully smaller samples would eventually not be copper carbonate.
- The smallest samples are copper, oxygen, and carbon in the same proportion.
- Therefore copper carbonate is composed of discrete units of copper, oxygen, and carbon in the same proportion.
It is not possible from this logic to determine the size of the units (molecules) or whether they are single compositions of copper, oxygen, and carbon, or some multiple, but the fact that they are discrete units strongly suggest an atomistic composition of matter. At this point, it is only a suggestion however.
The multiple proportions principle is an even stronger suggestion of the atomistic nature of matter.
"Multiple proportions," is a slightly misleading name of Dalton's principle—perhaps, "whole number proportions," would have been better, because it was the fact that different compounds of the same elements always differed in whole number differences of proportion of the changing element.
The Oxides of Nitrogen
Atomic Ratio (N:O)
For each different compound of nitrogen and oxygen, for any 2 units of nitrogen, the change in the amount of oxygen for any compound is a whole number difference: 2:1, 2:2, 2:3, 2:4, 2:5.
The constant composition principle logically implies compounds are composed of discrete units of constituent elements. If variations in the compounds of the same elements, such as nitrogen and oxygen, vary only in whole number amounts, the element of varying amounts must vary in discrete units as well which are "added" singly to each discrete unit of the compound.
Even after Jons Jacob Berzelius had nearly exhaustively tested most known compounds, practically proving the multiple proportions principle, it did not mean that no compounds of the same elements could ever violate the multiple proportions principle (which would be induction), but only no example of such a violation was known, so the principle was valid for all known compounds.
At that time the principle could be stated, "for all known compounds, the composition is of discrete units of one element combined with discrete units of one or more other elements which form discrete units of the compound."
Of course we now know the discrete units of the elements are atoms, and the discrete combined elements are molecules, but neither of these were known at that time. It was the logical analysis of what the structure must be from what was known that formed the concepts the atomic theory would ultimately complete. That logic was deduction, not induction, and it was valid because it was based on observed and identified facts, not generalities deduced from them.
I've concluded my criticism of Harriman's thesis. This section consists solely of a few comments I think are needed regarding the nature of science itself.
Harriman wrote, "Modern science began with the full development of its own distinctive method of investigation:experiment. 'Experimentation' is the method of establishing causal relationships by means of controlling variables." [Page 36.]
Experimentation did not begin with modern science, of course.
Thales (585 BC) and especially Archimedes (250 BC) performed many early scientific experiments. Certainly experimentation in modern science is more rigorous, and well defined, but experimentation is only one method of modern science.
The emphasis on experimentation is actually a little surprising since a good quarter of The Logical Leap deals with the history of astronomy, which involves little experimentation, and is mostly careful observation and rigorous calculation by men like Copernicus, Galileo, Kepler, and Newton.
I like Harriman's description of science as investigation and would carry the analogy even further to describe scientists as detectives. Before a detective begins his investigation there must first be evidence of a crime. A scientist's investigation must begin with the evidence, real evidence, not just someone's, "testimony," whether that someone is an authority (religious, for example), or anyone making a claim of some experience. Such testimony might or might not be sincere or correct, but without evidence, there is nothing for the scientist to investigate. It is not enough that someone says, "I saw a dead body in the garage," if, when the detective arrives, there is no dead body, or any sign of one.
The scientist's methods, too, are like a detectives. Once a "crime" (any natural phenomena) has been discovered, the investigation begins with observation, as precise and careful observation as possible. That observation means discovering every possible relationship to the phenomena under study, and examining it all in the light of what is the already known. The context of new knowledge is always knowledge already established.
Just as in a criminal investigation, scientific research requires careful examination of all the evidence gathered by observation. The examination requires clearly identifying each piece evidence, every existent, every attribute of those existents, and every relationship between them.
The real research is that examination which must be ruthlessly deductive, discovering and eliminating every possible contradiction. It may include painstaking measurements, using precision tools, and endless trials and tests (experiments) that the evidence suggests. Much of science is eliminating the irrelevant, the misleading, and the mistaken. Much is finding new ways to observe what seems unobservable.
There is no one method of science, and the correct one in any field is dictated by the nature of that which is being investigated, and any method which will bring the truth to light is a correct one, only it must be logical and objective.
The correct method in all science is dictated by the purpose of science, which is the identification of all existents, their behavior, and their relationships, which, when identified are the inviolable principles (or 'laws') of science.
I do not discourage criticisms of anything I've written, and in fact encourage them. But I might be able to save the possible critic some trouble by pointing out a couple of things it might seem I've missed or been mistaken about, which are not the case.
First, my purpose has been a very narrow one, addressing only those particular issues in The Logical Leap I disagree with. There are many peripheral and related issues I intentionally did not address, such is the importance of mathematics in modern physics. I am quite aware of the importance of Maxwell's equations, Plank's constant (and quantum theory), gauge theory, and Lagrangian symmetry, for example. My comments about mathematics is not meant in any way to diminish that importance. While I am convinced there are mistakes, of a philosophical nature, in the interpretations of the mathematical in modern physics, those mistakes are not a repudiation of the validity or importance of mathematics.
I have already noted that Harriman wrote: "Induction is the conceptualizing process itself in action." [Page 35.] I commented there, "If that is, indeed, what he meant, it would be correct, and if that view were held consistently, there would be no need for the word "induction." Conception, meaning concept formation, is enough, and using "induction" for that is misleading."
Ayn Rand herself wrote something quite similar: "The process of observing the facts of reality and of integrating them into concepts is, in essence, a process of induction. The process of subsuming new instances under a known concept is, in essence, a process of deduction." [Introduction to Objectivist Epistemology, "Abstraction from Abstractions"]
In spite of both Rand's and Harriman's understanding that universals are established by means of concept formation (the identification of existents in terms of their qualities, i.e. properties, attributes, and characteristics), both have a baseless, almost mystic need to establish some universals on a totally irrational view of "induction."
I offer the following examples:
"Prof. H: This is a common question relating to induction. Someone is boiling water, and he notices that every time the water gets to a certain temperature, it boils. Now he wants to know: does all water boil at that temperature, or is it only due to some accidental feature about this particular water? How does he determine whether it's accidental or essential?
"AR: By whether you can or cannot establish a causal connection between what you have determined to be the essential characteristic of water and the fact that it boils at a certain temperature." [Introduction to Objectivist Epistemology, "Appendix—Philosophy of Science, Induction"]
This is a mistake. The temperature at which something boils is an essential characteristic of that existent. If something boils at a certain temperature, and that is one of the characteristics that identify it, and that something is named water, all water will boil at the same temperature. If something sharing other characteristics of water is discovered, but it does not boil at the temperature water does, it is not water.
Thought Rand was mistaken about the nature of concepts in terms of her measurement-omission theory, she did understand that existents are what they are because their qualities are what they are, and that all existents with the same essential (necessary) qualities are the same kind of existents
Nevertheless, she, like Harriman, thought there was some method of deriving universal concepts (what Harriman refers to as "generalities") beyond identification of existents in terms of the qualities that make them what they are.
"Prof. M: Take the example of Newton's theory of universal gravitation. He said that if the theory is true, then the planets will exhibit elliptical orbits with the sun at one of the foci. Now it is found in astronomy that the planets to follow that path. So what can one say then about Newton's theory? Is it a possible explanation? Is it correct, or what?
"AR: After it has been verified by a great many other observations, not merely the verification of one prediction, then at a certain time one can accept it as a fact. But taking your example as an illustration of what you are asking, if the sole validation for Newton's principle was that it predicted that orbit's will be elliptical, and then we observed that they are elliptical—that wouldn't be sufficient proof. Epistemologically , it wouldn't be enough. You would have to have other observations, from different aspects of the same issue, which all support this hypothesis [Historically, Newton validated his theory by means of a great many observations of widely differing phenomena.]
"Prof. M: The question is: when does one stop? When does one decide that enough confirming evidence exists? Is that in the province of the issue of induction?
"AR: Yes. That's the big question of induction. Which I couldn't begin to discuss—because (a) I haven't worked on that subject enough to even begin to formulate it, and (b) it would take an accomplished scientist in a given field to illustrate the whole process in that field." [Introduction to Objectivist Epistemology, "Appendix—Scientific Methodology"]
It is obvious from this discussion that Rand believed that "induction" was some kind of process of deriving universal principles of some sort (other than the identification of existents, remembering that existents include entities, events, attributes, and relationships), by means of some mysterious, yet to be discovered, sufficient number of observations, which would be confirmation of the validity of the, "generalization from observation."
This is completely wrong, of course, but no doubt the basis of both Peikoff's and Harriman's own same mistaken view.
There are, in fact, few if any truly new ideas in The Logical Leap. As an example, the bizarre idea that human consciousness is essentially quantitative is originally Rand's, as well. Harriman states it this way: "Human consciousness is inherently a quantitative mechanism. It grasps reality&mdashi.e., the attributes of entities and their causal relationships to one another—only through grasping quantitative data. In this sense, quantity has epistemological primacy over quality."
The following is unpublished, but Peikoff was certainly familiar with it:
"My hypothesis is that all consciousness is a mathematical process (or, rather, the function of any consciousness is a mathematical process). To prove this I would have to identify the basic principles common to perception and mathematics. (By perception I mean here the total process of human awareness, from sensation to perception to conceptions.) I would have to identify the wider abstractions underlying the processes of concept-formation and of mathematics. And I would have to integrate them with neurology on the one hand (with the physiological part of the integration of sensations into perception)—and with metaphysics on the other.
If my hypothesis is true, then algebra might give me the clue to the objective rules of induction—to a kind of 'Organon of Induction." [Aristotle's works on logic are called the 'Organon,' Greek for 'instrument.']" [The Journals of Ayn Rand, "16 - Two Possible Books"]
What Rand hypothesizes, Harriman asserts as a fact. "Human consciousness is inherently a quantitative mechanism." Though neither of them seems to realize that what they are advocating is physicalism.
Ayn Rand was clearly not a physicalist. Objectivism rejects physicalism. By physicalism I mean the view that everything can be reduced to and explained in the terms of the physical. By the physical I mean the world we are directly conscious of, the world that the physical sciences study. Consciousness and that which we are conscious of cannot be the same thing, as Rand expressed it, though she frequently interchanged the terms "physical" and "material" (which she actually used more often), meaning by both the physical existence of which we are conscious.
"Man is an entity of mind and body, an indivisible union of two elements: of consciousness and matter. Matter is that which one perceives, consciousness is that which perceives it." [The Journals of Ayn Rand, "14 - Notes While Writing Galt's Speech"]
Notice the two elements are consciousness and matter, meant to identify two unique things. But she is more explicit here:
"Man's consciousness is not material——but neither is it an element opposed to matter." [The Journals of Ayn Rand, "13 - Notes While Writing: 1947-1952"] [Emphasis mine.]
"Man is a being endowed with consciousness —an attribute which matter does not possess. His consciousness is the free, nonmaterial element in him." [The Letters of Ayn Rand, "The Fountainhead and Atlas Shrugged Years (1945-1959)," To Nathan Blumenthal, January 13, 1950] [Emphasis mine.]
There is no hint of either dualism or mysticism in Rand's view. She regarded consciousness as part of nature, part of the objective metaphysical world, but not a physical or "material" part of it. This is, in fact, my own view, except that I regard three aspects of reality non-physical—life, consciousness, and the human mind (volition, rationality, and intellect), because none of these can be explained in terms of physical attributes, though none can exist independently of the physical organisms of which they are the life, consciousness, or minds.
Rand does not make explicit why she regards consciousness as non-physical, but there is a hint in the last quote, "His consciousness is the free, nonmaterial element in him." The physical world, if it is to be known by means of the sciences, must be determined by inviolable principles. Everything happens, "for a reason," and the reasons are the physical laws which describe its nature. If consciousness were an aspect of the mere physical, it would be determined by the same physical laws as all other physical aspects of existence—volition would be an impossibility. Thus, consciousness must be an aspect of existence not accounted for by the physical alone.
I know there is much more to it than that, but this is enough to make the point, that Rand's and Harriman's idea that consciousness itself, particularly human consciousness, is mathematical in nature reduces it to the physical—a flat-out contradiction of Objectivism, and reality.
That which functions in conformance with principles which can be defined in terms of mathematical formulas is determined—what it does it must always do because its nature is determined by those mathematical principles. This is exactly what the physical is. If consciousness were mathematical in nature, its behavior would be determined by the mathematical principles that describe that nature, in the same way that all aspects of the physical are determined by the mathematical principles that describe its nature—it would make consciousness indistinguishable from any other phenomena of the physical—it would turn human psychology into exactly that imagined by the behaviorists and all other physicalists, and it would invalidate volition.