Proof

[Michael Stuart Kelly]

Since we have been discussing Objectivism and science for a while, here are a few Peikoff quotes on proof to chew on. Ayn Rand did not write very much on the process of proof itself. She did mention that the fundamental axioms cannot be proven, but instead must already be used in any process of proving them. She usually implied that identification of facts and logic were to be used. Peikoff has built on the implications of this to try to form an Objectivist approach to proof.

Since Peikoff stated so much about science in the DIM Hypothesis lectures, including the phrase, if I remember it correctly, that all scientific truths were ultimately based on induction, I think it is pertinent to look at his written statements about proof. The following quotes are all from Objectivism: The Philosophy of Ayn Rand.

("Chapter 1—Reality," p. 8)

Proof is the derivation of a conclusion from antecedent knowledge, and nothing is antecedent to axioms. Axioms are the starting points of cognition, on which all proofs depend.

("Chapter 2—Sense Perception And Volition," p. 39)

Proof consists in reducing an idea back to the data provided by the senses.

("Chapter 4—Objectivity," p. 120)

"Proof" is the process of establishing truth by reducing a proposition to axioms, i.e., ultimately, to sensory evidence.

("Chapter 4—Objectivity," p. 138)

Proof is a form of reduction. The conclusion to be proved is a higher-level cognition, whose link to reality lies in the premises; these in turn eventually lead back to the perceptual level. Proof is thus a form of retracing the hierarchical steps of the learning process. (As with conceptual reduction, so with proof: the process identifies not the optional variants, but the essential links in the chain, the necessary logical structure relating a mental content to observational data.)

Proof is not a process of deriving a conclusion from arbitrary premises or even from arbitrarily selected true premises. Proof is the process of establishing a conclusion by identifying the proper hierarchy of premises. In proving a conclusion, one traces backward the order of logical dependence, terminating with the perceptually given. It is only because of this requirement that logic is the means of validating a conclusion objectively.

("Chapter 5—Reason," p. 183)

A process of proof commits a man to its presuppositions and implications. It thus commits him to an entire philosophic approach—to the validity of sense perception, the validity of reason, the need of objectivity, the method of logic, the processes of conceptual knowledge, the law of identity, the absolutism of reality.

We have a number of scientifically oriented people on OL. I am curious to hear what they have to say.

Michael

[Brant Gaede]

Since we have been discussing Objectivism and science for a while, here are a few Peikoff quotes on proof to chew on. Ayn Rand did not write very much on the process of proof itself. She did mention that the fundamental axioms cannot be proven, but instead must already be used in any process of proving them. She usually implied that identification of facts and logic were to be used. Peikoff has built on the implications of this to try to form an Objectivist approach to proof.

Since Peikoff stated so much about science in the DIM Hypothesis lectures, including the phrase, if I remember it correctly, that all scientific truths were ultimately based on induction, I think it is pertinent to look at his written statements about proof. The following quotes are all from Objectivism: The Philosophy of Ayn Rand.

("Chapter 1—Reality," p. 8)

Proof is the derivation of a conclusion from antecedent knowledge, and nothing is antecedent to axioms. Axioms are the starting points of cognition, on which all proofs depend.

("Chapter 2—Sense Perception And Volition," p. 39)

Proof consists in reducing an idea back to the data provided by the senses.

("Chapter 4—Objectivity," p. 120)

"Proof" is the process of establishing truth by reducing a proposition to axioms, i.e., ultimately, to sensory evidence.

("Chapter 4—Objectivity," p. 138)

Proof is a form of reduction. The conclusion to be proved is a higher-level cognition, whose link to reality lies in the premises; these in turn eventually lead back to the perceptual level. Proof is thus a form of retracing the hierarchical steps of the learning process. (As with conceptual reduction, so with proof: the process identifies not the optional variants, but the essential links in the chain, the necessary logical structure relating a mental content to observational data.)

Proof is not a process of deriving a conclusion from arbitrary premises or even from arbitrarily selected true premises. Proof is the process of establishing a conclusion by identifying the proper hierarchy of premises. In proving a conclusion, one traces backward the order of logical dependence, terminating with the perceptually given. It is only because of this requirement that logic is the means of validating a conclusion objectively.

("Chapter 5—Reason," p. 183)

A process of proof commits a man to its presuppositions and implications. It thus commits him to an entire philosophic approach—to the validity of sense perception, the validity of reason, the need of objectivity, the method of logic, the processes of conceptual knowledge, the law of identity, the absolutism of reality.

We have a number of scientifically oriented people on OL. I am curious to hear what they have to say.

Michael

This is philosophy, not science. The question is whether this is a foundation for science? Nothing here about the scientific method, hypothesis and theory, double-blind studies, statistics, falsification, etc. "Proof" works until you crash and burn. When the first atom bomb was detonated it was conclusive proof--that the atom bomb had detonated. Some of the scientists wondered if the atmosphere would catch on fire, perhaps forgetting that oxygen supports combustion, but is not in itself combustible. Does philosophy work? Can we put it to the test the way we can an airplane? Probably not, but boy can we tell each other what is what!

--Brant

[BaalChatzaf]

Since we have been discussing Objectivism and science for a while, here are a few Peikoff quotes on proof to chew on. Ayn Rand did not write very much on the process of proof itself. She did mention that the fundamental axioms cannot be proven, but instead must already be used in any process of proving them. She usually implied that identification of facts and logic were to be used. Peikoff has built on the implications of this to try to form an Objectivist approach to proof.

Since Peikoff stated so much about science in the DIM Hypothesis lectures, including the phrase, if I remember it correctly, that all scientific truths were ultimately based on induction, I think it is pertinent to look at his written statements about proof. The following quotes are all from Objectivism: The Philosophy of Ayn Rand.

("Chapter 1—Reality," p. 8)

Proof is the derivation of a conclusion from antecedent knowledge, and nothing is antecedent to axioms. Axioms are the starting points of cognition, on which all proofs depend.

("Chapter 2—Sense Perception And Volition," p. 39)

Proof consists in reducing an idea back to the data provided by the senses.

("Chapter 4—Objectivity," p. 120)

"Proof" is the process of establishing truth by reducing a proposition to axioms, i.e., ultimately, to sensory evidence.

("Chapter 4—Objectivity," p. 138)

Proof is a form of reduction. The conclusion to be proved is a higher-level cognition, whose link to reality lies in the premises; these in turn eventually lead back to the perceptual level. Proof is thus a form of retracing the hierarchical steps of the learning process. (As with conceptual reduction, so with proof: the process identifies not the optional variants, but the essential links in the chain, the necessary logical structure relating a mental content to observational data.)

Proof is not a process of deriving a conclusion from arbitrary premises or even from arbitrarily selected true premises. Proof is the process of establishing a conclusion by identifying the proper hierarchy of premises. In proving a conclusion, one traces backward the order of logical dependence, terminating with the perceptually given. It is only because of this requirement that logic is the means of validating a conclusion objectively.

("Chapter 5—Reason," p. 183)

A process of proof commits a man to its presuppositions and implications. It thus commits him to an entire philosophic approach—to the validity of sense perception, the validity of reason, the need of objectivity, the method of logic, the processes of conceptual knowledge, the law of identity, the absolutism of reality.

We have a number of scientifically oriented people on OL. I am curious to hear what they have to say.

Michael

Newton provided "proof" that gravitation is a force. Einstein provided "proof" that gravitation is not a force.

Bohr, Heisenberg, Born, Dirac provided "proof" that both position and momentum cannot both be properties of a body. Bohm, DeBroigle provided "proof" that a body has both position and momentum.

Einstein provided "proof" that reality is local. Aspect et all provided experimental evidence that reality is not local by showing Bell's inequalities fail and the quantum predictions are correct.

There is "proof" that heat is a fluid. There is "proof" that heat is not a fluid.

There is "proof" that aether exists and there is "proof" that aether does not exist.

All of which goes to show that L.P. notion of "proof" is flawed. No surprise there!

Go figure.

Ba'al Chatzaf

[Dragonfly]

There are no proofs in science, only in mathematics. Another manifestation of the ASD.

[tjohnson]

My 2 cents;

'Proof' is a notion that belongs in mathematics only. "Proof' implies a sort of finality that is only possible in a field where we are speaking about imaginary things. 'Proof' is possible in mathematics because all particulars are in included in our definitions and so our deductions work absolutely, if properly made. As Baal and Dragonfly pointed out above, in physics we only have theories with a certain amount of 'similarity of structure' (a GS term) and at any given time they do not account for all phenomena.

[Robert Campbell]

Michael,

Interesting that you should raise this question just now.

In preparation for an article I'm writing, I'm trying to understand just what Peikoff means by "proof" in OPAR, and how it relates to the "reduction" (his term) of higher-level concepts to perceptual data.

Whatever the problems might be with the latter idea, Rand wrote pretty extensively about it.

Where has Peikoff ever explicated his notion of proof in detail? Not in OPAR, that's for sure. Not in his early 1970s course on logic, either.

Robert Campbell

PS. If I understand Peikoff correctly, one can "prove" that something is possible or probable. A "proof" might also turn out to be defeasible; i.e., new knowledge might lead one to revise or reject it. All the same, I tend to think of proofs as something that mathematicians do, not something we do in psychology--or that physicists and biologists and chemists do.

[BaalChatzaf]

PS. If I understand Peikoff correctly, one can "prove" that something is possible or probable. A "proof" might also turn out to be defeasible; i.e., new knowledge might lead one to revise or reject it. All the same, I tend to think of proofs as something that mathematicians do, not something we do in psychology--or that physicists and biologists and chemists do.

That is quite correct. Scientific theories are never -proven- (in the mathematical sense). They are either supported or refuted by experimental tests of their predictions. The closest thing to a -proof- in physics(say) is the derivation of a prediction from the core theory and the auxiliary assumptions specifying the conditions under which the experiment is to be made.

Ba'al Chatzaf

[Wolf DeVoon]

Does philosophy work? Can we put it to the test the way we can an airplane? Probably not

Well, if we had to rely on Peikoff for guidance, nothing in philosophy could be proved or disproved. Where I depart from the previous ASD thread and this new one on Proof is the presumed sanctity of empiricism. No one seems interested in predicate logic, so why bother?

For the record only, not expecting reply:

The Square of Opposition

All A is B .......... No A is B

Some A is B ..... Not every A is B

Propositions that are diagonal are contradictory; one and only one is necessarily true, the other necessarily false. Contradictions are unreal and cannot exist (a metaphysical principle).

Edited June 14, 2007 by Wolf DeVoon

[Michael Stuart Kelly]

In preparation for an article I'm writing, I'm trying to understand just what Peikoff means by "proof" in OPAR, and how it relates to the "reduction" (his term) of higher-level concepts to perceptual data.

Whatever the problems might be with the latter idea, Rand wrote pretty extensively about it.

Where has Peikoff ever explicated his notion of proof in detail? Not in OPAR, that's for sure. Not in his early 1970s course on logic, either.

Robert Campbell

PS. If I understand Peikoff correctly, one can "prove" that something is possible or probable. A "proof" might also turn out to be defeasible; i.e., new knowledge might lead one to revise or reject it. All the same, I tend to think of proofs as something that mathematicians do, not something we do in psychology--or that physicists and biologists and chemists do.

Robert,

Peikoff also makes a distinction about validation and proof. He considers proof to be one type of validation. Here are some excerpts on validation from OPAR.

("Chapter 1—Reality," p. 8)

The above is the validation of the Objectivist axioms. "Validation" I take to be a broader term than "proof," one that subsumes any process of establishing an idea's relationship to reality, whether deductive reasoning, inductive reasoning, or perceptual self-evidence. In this sense, one can and must validate every item of knowledge, including axioms. The validation of axioms, however, is the simplest of all: sense perception.

("Chapter 1—Reality," p. 14)

When a child has reached the stage of (implicitly) grasping "entity," "identity," and "action," he has the knowledge required to reach (implicitly) the law of causality. To take this step, he needs to observe an omnipresent fact: that an entity of a certain kind acts in a certain way. The child shakes his rattle and it makes a sound; he shakes his pillow and it does not. He pushes a ball and it rolls along the floor; he pushes a book and it sits there, unmoving. He lets a block out of his hands and it falls; he lets a balloon go and it rises. The child may wish the pillow to rattle, the book to roll, the block to float, but he cannot make these events occur. Things, he soon discovers, act in definite ways and only in these ways. This represents the implicit knowledge of causality; it is the child's form of grasping the relationship between the nature of an entity and its mode of action.

The adult validation of the law of causality consists in stating this relationship explicitly. The validation rests on two points: the fact that action is action of an entity; and the law of identity, A is A. Every entity has a nature; it is specific, noncontradictory, limited; it has certain attributes and no others. Such an entity must act in accordance with its nature.

("Chapter 2—Sense Perception and Volition," p. 54)

The proper order of philosophy, therefore, is not the chronological order of our actual development.

Chronologically, the sensation stage comes first, then the perceptual, and then the conceptual. Epistemologically, however, the perceptual stage comes first. If one seeks to prove any item of human knowledge, on any subject, he must begin with the facts of perception. These facts constitute the base of cognition. They are the self-evident and the incontestable, by reference to which we validate all later knowledge, including the knowledge that, decades earlier, when we first emerged from the womb, we experienced a brief sensation stage.

There are philosophers (David Hume is the most famous) who deny the perceptual level. Such men give the sensation stage epistemological primacy, then seek to determine whether the fact of entities (and causality) can be established by inference from it. This is a dead end; from disintegrated sensations, nothing can be inferred. A consciousness that experienced only sensations would be like the mind of an infant; it could neither perceive objects nor form concepts (which is one reason Hume ended as a paralyzed skeptic). Hume's dead end, however, is self-imposed. Entities do not require inferential validation. The given is the perceptual level.

("Chapter 2—Sense Perception and Volition," p. 70)

The concept of "volition" is one of the roots of the concept of "validation" (and of its subdivisions, such as "proof"). A validation of ideas is necessary and possible only because man's consciousness is volitional. This applies to any idea, including the advocacy of free will: to ask for its proof is to presuppose the reality of free will.

Once again, we have reached a principle at the foundation of human knowledge, a principle that antecedes all argument and proof.

. . .

The principle of volition is a philosophic axiom, with all the features this involves. It is a primary—a starting point of conceptual cognition and of the subject of epistemology; to direct one's consciousness, one must be free and one must know, at least implicitly, that one is. It is a fundamental: every item of conceptual knowledge requires some form of validation, the need of which rests on the fact of volition. It is self-evident. And it is inescapable.

("Chapter 5—Reason," p. 153)

One cannot seek a proof that reason is reliable, because reason is the faculty of proof; one must accept and use reason in any attempt to prove anything. But, using reason, one can identify its relationship to the facts of reality and thereby validate the faculty. The past four chapters are the gist of the Objectivist validation.

("Chapter 5—Reason," pp. 175-176)

So far, I have considered only two mental states, knowledge and ignorance, and two corresponding verdicts to define an idea's status: "validated" or "unknown." Inherent in the mind's need of logic, however, is a third, intermediate status, which applies for a while to certain complex higher-level conclusions. In these cases, the validation of an idea is gradual; one accumulates evidence step by step, moving from ignorance to knowledge through a continuum of transitional states. The main divisions of this continuum (including its terminus) are identified by three concepts: "possible," "probable," and "certain."

("Chapter 5—Reason," pp. 176-177)

"Evidence," according to the Oxford English Dictionary, is "testimony or facts tending to prove or disprove any conclusion." To determine whether a fact is "evidence," therefore, one must first define what proof of a given claim would consist of. Then one must demonstrate that the fact, although inconclusive, contributes to such proof, i.e., strengthens the claim logically and thus moves the matter closer to a cognitive resolution. If one has no idea what the proof of a conclusion would consist of—or if one holds that a proof of it is impossible—one has no means of deciding whether a given piece of information "tends to prove" it. If the terminus of a journey is undefined or unknowable, there is no way to judge whether one is moving toward it.

This is why there can be no such thing as "some evidence" in favor of an entity transcending nature and logic.

The term "evidence" in this context would be a stolen concept. Since nothing can ever qualify as a "proof" of such an entity, there is no way to identify any data as being a "part proof" of it, either. There is no way to validate such a notion as: "that which brings men closer to knowing the unknowable or proving the unprovable."

("Chapter 5—Reason," pp. 179-180)

Once, when I was a college student, an instructor defending skepticism declared to the class: "You think I am Professor X. But how do you know I'm not an impostor, a consummate actor taking the professor's place?" Transpose this question to your own situation. How can you be certain when you attend a lecture that it is Y, a man you know well, who is speaking, rather than an impostor?

In this case, the standard of validation is the direct testimony of your eyes and ears, as identified conceptually and then integrated to whatever other knowledge of yours is relevant. Judged by this standard, the proper conclusion to draw is outside the realm of doubt. All the available information—everything you observe and everything you know—leads to your identification of the speaker: the occasion, his appearance, his tone of voice, his facial expressions, his posture and gestures, the content of his prepared remarks, the quality of his extemporaneous jokes, his knowledge of your name and face, and so on. If a skeptic were to say: "But man does have the ability to impersonate others; so isn't it at least possible that the speaker is an actor?" the reply would have to be: "That is a non sequitur. On what basis do you claim someone's exercise of this human ability here and now? Is there an iota of evidence to support such a hypothesis in this context?" Of course, there is not.

("Chapter 5—Reason," p. 184)

A doubt susceptible of objective validation would also have to be finite, contextual, and bound by the rules of evidence.

As to reduction, Peikoff's meaning of reduction should not be confused with reductionism, which (as I understand it) means breaking entities down into their parts and treating those parts as new entities, which can then be broken down into their parts, which... and so on. Peikoff uses logical reduction to mean going backwards through a conceptual chain until arriving at the perceptual level. Here are a few more quotes from OPAR.

("Chapter 1—Reality," p. 34)

All definitions reduce ultimately to certain primary concepts, which can be specified only ostensively; axiomatic concepts necessarily belong to this category.

("Chapter 4—Objectivity," p. 133)

Context-keeping, as we know, is required if men's ideas are to be connected to reality. When the context is itself hierarchical, the successive levels of its structure are the connecting links. To keep the context in such a case is to identify and retain these links. This is where the process of reduction is necessary.

Reduction is the means of connecting an advanced knowledge to reality by traveling backward through the hierarchical structure involved, i.e., in the reverse order of that required to reach the knowledge. "Reduction" is the process of identifying in logical sequence the intermediate steps that relate a cognitive item to perceptual data. Since there are options in the detail of a learning process, one need not always retrace the steps one initially happened to take. What one must retrace is the essential logical structure.

Such retracing is a requirement of objectivity. Man's only direct contact with reality is the data of sense. These, therefore, are the standard of objectivity, to which all other cognitive material must be brought back.

("Chapter 4—Objectivity," p. 135)

The final steps backward, therefore, which I will not rehearse, do bring us eventually to first-level concepts, such as "table," "bed," "man." At this point, the reduction has been completed. It ends when we say: "And by this term—e.g., 'man'—I mean this," as we directly point to the entity.

("Chapter 4—Objectivity," p. 136-137)

In regard to higher-level concepts, reduction completes the job of definition. The purpose of a definition is to keep a concept connected to a specific group of concretes. The definition of a higher-level concept, however, counts on the relevant lower-level concepts, which must themselves be connected to concretes; otherwise, the definition is useless. Reduction is what takes a person from the initial definition through the definitions of the next lower level and then of the next, until he reaches the direct perception of reality. This is the only means by which the initial definition can be made fully clear.

Sometimes Peikoff uses "reduce" to mean to condense with a symbol, such as reduce a large number of existents to one mental unit.

This leads me to an essay highly critical of OPAR, but one that was fairly objective about Peikoff's view of proof (although some of the arguments are more semantics than meaning). David Gordon reviewed OPAR in The Journal of Libertarian Studies (the date is not given in the link and you have to scroll halfway through to get to the review). Here is a quote:

“At the culmination of concept acquisition stands definition, which "identifies a concept's units by specifying their essential characteristics" (p. 97). But even though the definition cannot list all the characteristics of the concept's units, the concept nevertheless refers to all of these. "A concept is not interchangeable with its definition-not even if the definition .. . happens to be correct. . . .[A] concept designates existents, including all their characteristics, whether definitional or not" (p. 102)

Thus for Peikoff meaning is pointing.

Compare this with the quote from OPAR above on reduction above (p. 135).

"And by this term—e.g., 'man'—I mean this," as we directly point to the entity.

Actually, this has to be broader because this system would not work for a blind person. I think Peikoff means "indicate a source of sensory stimuli."

Michael

[Robert Campbell]

Michael,

I'm familiar with the Peikovian notion of reduction as it is supposed to apply to concepts. All of that comes straight out of the Introduction to Objectivist Epistemology, whatever spin Peikoff may have ended up applying to it.

Also, Gordon's critique of reduction (I read his review a few months ago, even had a brief email exchange with him) applies to concepts. Which makes sense: that's the notion that Rand actually developed and it's the notion of reduction that Peikoff devotes most of his discussion to in OPAR.

Here are some problems I'm having with Peikovian proof:

(1) Peikoff assumes that reduction applies to propositions, just as it does to concepts. But Rand offered no theory of propositions. Peikoff's logic course assumed that propositions roughly have the subject-predicate structure required by ancient logic, but that is not a complete theory and he didn't carry any of it forward to his 1976 course, or to OPAR. So Peikoff has no theory of propositions either. He uses vague phrases like "advanced knowledge" or "cognitive items" to make it appear that the notion of reduction applies more widely.

(2) Proof is a species of validation. But what you have successfully validated you may conclude is true. So what you have successfully proven you may also conclude is true. What's happened to false propositions? What is Peikoff's theory of error?

(3) Rand stated (in the appendix to ITOE) that she needed a philosophy of science, and didn't have one. Peikoff himself says in OPAR that Rand wanted to develop her epistemology by incorporating further studies in mathematics and neuropsychology but never got around to it. Peikoff had definitely not developed a philosophy of science prior to writing OPAR, and there is none in the book. If Peikoff's notion of proof for propositions is sound, he ought to be able to show how the proof of a variety of important scientific generalizations proceeds. (For that matter, how it has proceeded, since if scientists have never actually carrried out a Peikovian proof, Peikoff's own ability to do so is mainly of academic interest.) He has not produced any such proofs, nor shown how others produced them, and appears to lack the theoretical resources necessary to carry them out.

(4) Peikoff's failure to specify in detail what the proof of a proposition consists of comes back to bite him when he makes the statement about evidence that you quoted:

To determine whether a fact is "evidence," therefore, one must first define what proof of a given claim would consist of. Then one must demonstrate that the fact, although inconclusive, contributes to such proof, i.e., strengthens the claim logically and thus moves the matter closer to a cognitive resolution. If one has no idea what the proof of a conclusion would consist of—or if one holds that a proof of it is impossible—one has no means of deciding whether a given piece of information "tends to prove" it. If the terminus of a journey is undefined or unknowable, there is no way to judge whether one is moving toward it. (pp. 176-177, my bolding)

So if you don't know what it would take to prove some proposition, you also don't know what would constitute evidence that the claimed state of affairs is possible, or probable. And if you don't know what evidence is required, you don't know when genuine evidence has not been provided; i.e., you don't know when a proposition has been asserted arbitrarily.

Yet Peikoff appears supremely confident in his ability to identify an arbitary assertion whenever he comes across one.

What does he know that he isn't telling us?

Robert Campbell

[Michael Stuart Kelly]

What is Peikoff's theory of error?

Robert,

Whatever disagrees with what Ayn Rand said?

On a more serious note, I do not mean to imply any lack of information or familiarity on your part by providing quotes. I put them together in the sense of hoping something might be helpful to the piece you are writing.

To my thinking so far, in terms of judging false propositions for science, I like very much the Popper stuff I have read. If you find one case that is false, the proposition is false.

I don't like that method for judging the overall value or for extracting the good parts from an incomplete or flawed body of ideas, or even a broad statement, since so many other factors are involved and the tendency is to throw out the baby with the bath water. But for for recognizing what the bath water is, that system works. Also, from what I have read so far, Popper is vague on initial concept formation and why logic applies to reality, but I have more reading to do before I can issue any real opinions or evaluations about that.

I think Peikoff would make the same mistake about throwing out the baby with the bath water , though. I think he would call Popper's theory "subjectivist" and ignore the part that works. He seems to do that with an awful lot of philosophy, except where parts align with selected parts of Objectivism.

Here is another affirmation about proof from OPAR that is very interesting (p. 109), taking your lead from above.

Ayn Rand regarded her theory of concepts as proved, but not as completed. There are, she thought, important similarities between concepts and mathematics still to be identified; and there is much to be learned about man's mind by a proper study of man's brain and nervous system. In her last years, Miss Rand was interested in following up these ideas—in relating the field of conceptualization to two others: higher mathematics and neurology. Her ultimate goal was to integrate in one theory the branch of philosophy that studies man's cognitive faculty with the science that reveals its essential method and the science that studies its physical organs. Unfortunately, she did not live long enough to pursue this goal systematically. All she could do was to leave us some tantalizing but fragmentary leads indicating the direction in which epistemology should be developed in the future.(38)

. . .

Footnote 38: Ayn Rand's philosophical notes will be published in due course.

That first statement jumped out at me. Proved? By what standard or method? I am going through OPAR slowly and making notes as I go along and, like you, I have not found a method of proof that would apply to something like that statement (or proposition, if you will).

Also, I do not know if the "philosophical notes" Peikoff referred to are the selected journal entries that were published, or if there is more coming. I did read some things in Rand's published journal on the beginning of her theory of epistemology, but nothing constituting an affirmation of proof.

Michael