Some readers may wonder how my theory of causality and induction differs from Objectivist philosopher Leonard Peikoff’s as expressed in his 2003 course, "Induction in Physics and Philosophy."

In my judgment, Dr. Peikoff’s course consists of essentially incorrect though intelligent answers to some good questions, which I had already answered essentially correctly in my own writings and lectures on causality, induction, propositions, and mathematics.

The closest Dr. Peikoff comes to being on the right track are in the following two ideas of his:

1.      Dr. Peikoff is correct that the basic evidence for causality is in causal relations between oneself and external objects. However, Dr. Peikoff—like many other thinkers—thinks the basic evidence is in instances in which the self is the cause and the action of other objects is the effect. E.g., my pushing the ball makes the ball roll. The reverse is true. In keeping with Ayn Rand’s principle of “the primacy of existence,” the basic evidence of causality is in instances in which external objects are the cause, and one's awareness of the objects is the effect. E.g., the ball makes me feel it. 

Dr. Peikoff's "first-level generalizations" (e.g., my pushing the ball makes it roll) are not, as he maintains, self-evident; they must be induced from many instances and relying on the more basic evidence I cite above. 

2.       (On this second issue, Dr. Peikoff is closer to a correct track.)  Dr. Peikoff is correct that concepts are what he calls a “green light to induction,” and that something is needed to fill the conceptual “file folders” (Ayn Rand’s metaphor) created by the forming of concepts. However, much of what Dr. Peikoff attributes to mathematics in this process is actually more generally attributable to propositions. Specifically, Dr. Peikoff claims that mathematical formulas (e.g., d=16t²) relate qualities to each other by “put[ting] back in”—“in numerical terms” —some of the measurements that were omitted during concept-formation. But it is in virtue of mathematical formulas being propositions that they relate quantities to each other and “put back in” some measurements that were omitted during concept-formation. In Dr. Peikoff’s claim above, it is only the phrase “in numerical terms” that applies specifically to mathematics. Non-mathematical propositions, as in biology and the humanities, also relate qualities and “put back in” omitted measurements, along with performing other important functions of propositions. (For my theory of the nature and cognitive role of propositions, see my essay, A Theory of Propositions.) If one understands the function of propositions, one can reduce Dr. Peikoff’s long argument to: “Mathematical formulas are propositions that use numbers.” His argument reveals unawareness of the cognitive role of propositions, and sheds little light on the cognitive role of mathematics per se.

The essential cognitive role of mathematics is to achieve unit-economy and open-endedness regarding measurement.

Mathematics achieves unit economy by conceiving and integrating groups of units that are uniform in the attribute being measured. For example, all inches have the same length, all pints have the same volume, all dollars have the same value. “The uniform unit is the key to the unit-economical power of mathematical measurement. If you have 25 pouches of water of different sizes, to recall how much water you have in total requires that you recall the size of each pouch. But if each pouch is the same size, you need recall only two mental units: the number 25 and the size of your standard pouch.”

Mathematics reaches conclusions regarding open-ended classes of measurement. For example, “2 x 3 = 6” means that two groups of three units each equals six of those units in total, whether those units are inches, pounds, or some other kind of unit encountered for the first time. Mathematics achieves this open-endedness by starting from the single mathematical premise—the axiom of mathematics—that all the units being considered are uniform in the attribute being measured, i.e. that 1 = 1.

For an in-depth treatment of the nature and cognitive role of mathematics, see the articles by Glenn Marcus and me originally published in The Intellectual Activist; the latest revised online versions are available for sale here. In particular, see our first article, “The Foundation of Mathematics,” originally published in 1994 in two parts, and from which the above quotation is taken.

Regarding induction beyond the field of mathematics proper, one essential application of mathematics is “the identification of underlying similarity through the lawful quantification of differences.” This role is described in my 2002 draft of A Theory of Knowledge.

In his course, Dr. Peikoff spends a good deal of time on the philosophical foundation of mathematics. In my judgment, the treatment of this subject by Glenn Marcus and me in our mathematics articles mentioned above is far more precise, fundamental, and thorough. Nevertheless, Dr. Peikoff’s course did remind me of something that induced me to add two words to the latest revision of our first article. 

Dr. Peikoff also spends much time arguing for and applying his claim that mathematics cannot be used in measuring consciousness, and therefore cannot be used in philosophy. A glaring counterexample to Dr. Peikoff’s claim is probability theory, which measures states of knowledge; probability theory quantifies the continuum from possibility to probability to certainty. Indeed, a correct theory of probability is an essential element of a correct theory of induction, but Dr. Peikoff dismisses probability theory out of hand.

In short, my theory of causality and induction addresses much of the same subject matter that Dr. Peikoff’s course addresses, but my conclusions are essentially contrary to Dr. Peikoff’s.

Ron Pisaturo