Popper Talk


Ellen Stuttle

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There keep being prognostications of a forthcoming discussion of Karl Popper's philosophy. In the frequent way of one subject leading to unanticipated side topics, the recently started thread "SCORECARD" detoured into epistemological issues. In post #92 of that thread, reponding to a question of mine about Popper's views on psychological induction, Bob Kolker provided a link to a site where Popper's "The Problem of Induction" (1953, 1974) is posted:

http://dieoff.org/page126.htm

On the same (long) webpage, somewhat more than about 1/3 of the way down, is the Popper essay "Knowledge without Authority" (1960). Popper concludes this essay with a listing of 9 numbered theses. Theses (8) and (9) struck me as so beautifully worded -- poetry of expression -- I felt an urge to copy and post them on the "Scorecard" thread. Then I thought...the whole numbered set seems a really good springboard to a discussion of Popper. A fair amount of what he says would likely be met with cries of aghastness from the convinced Objectivist. But...is he wrong? I think he's saying something profoundly right.

Knowledge without Authority (1960)

by Karl Popper

It is high time now, I think, to formulate the epistemological results of this discussion. I will put them in the form of nine theses.

(1) There are no ultimate sources of knowledge. Every source, every suggestion, is welcome; and every source, every suggestion, is open to critical examination. Except in history, we usually examine the facts themselves rather than the sources of our information.

(2) The proper epistemological question is not one about sources; rather, we ask whether the assertion made is true - that is to say, whether it agrees with the facts. (That we may operate, without getting involved in antinomies, with the idea of objective truth in the sense of correspondence to the facts, has been shown by the work of Alfred Tarski.) And we try to find this out, as well as we can, by examining or testing the assertion itself; either in a direct way, or by examining or testing its consequences.

(3) In connection with this examination, all kinds of arguments may be relevant. A typical procedure is to examine whether our theories are consistent with our observations. But we may also examine, for example, whether our historical sources are mutually and internally consistent.

(4) Quantitatively and qualitatively by far the most important source of our knowledge - apart from inborn knowledge - is tradition. Most things we know we have learnt by example, by being told, by reading books, by learning how to criticize, how to take and to accept criticism, how to respect truth.

(5) The fact that most of the sources of our knowledge are traditional condemns anti-traditionalism as futile. But this fact must not be held to support a traditionalist attitude: every bit of our traditional knowledge (and even our inborn knowledge) is open to critical examination and may be overthrown. Nevertheless, without tradition, knowledge would be impossible.

(6) Knowledge cannot start from nothing - from a tabula rasa - nor yet from observation. The advance of knowledge consists, mainly, in the modification of earlier knowledge. Although we may sometimes, for example in archaeology, advance through a chance observation, the significance of the discovery will usually depend upon its power to modify our earlier theories.

(7) Pessimistic and optimistic epistemologies are about equally mistaken. The pessimistic cave story of Plato is the true one, and not his optimistic story of anamnesis (even though we should admit that all men, like all other animals, and even all plants, possess inborn knowledge). But although the world of appearances is indeed a world of mere shadows on the walls of our cave, we all constantly reach out beyond it; and although, as Democritus said, the truth is hidden in the deep, we can probe into the deep. There is no criterion of truth at our disposal, and this fact supports pessimism. But we do possess criteria which, if we are lucky, may allow us to recognize error and falsity. Clarity and distinctness are not criteria of truth, but such things as obscurity or confusion may indicate error. Similarly coherence cannot establish truth, but incoherence and inconsistency do establish falsehood. And, when they are recognized, our own errors provide the dim red lights which help us in groping our way out of the darkness of our cave.

(8) Neither observation nor reason is an authority. Intellectual intuition and imagination are most important, but they are not reliable: they may show us things very clearly, and yet they may mislead us. They are indispensable as the main sources of our theories; but most of our theories are false anyway. The most important function of observation and reasoning, and even of intuition and imagination, is to help us in the critical examination of those bold conjectures which are the means by which we probe into the unknown.

(9) Every solution of a problem raises new unsolved problems; the more so the deeper the original problem and the bolder its solution. The more we learn about the world, and the deeper our learning, the more conscious, specific, and articulate will be our knowledge of what we do not know, our knowledge of our ignorance. For this, indeed, is the main source of our ignorance - the fact that our knowledge can only be finite, while our ignorance must necessarily be infinite.

We may get a glimpse of the vastness of our ignorance when we contemplate the vastness of the heavens: though the mere size of the universe is not the deepest cause of our ignorance, it is one of its causes. 'Where I seem to differ from some of my friends', F.P. Ramsey wrote in a charming passage, 'is in attaching little importance to physical size. I don't feel the least humble before the vastness of the heavens. The stars may be large but they cannot think or love; and these are qualities which impress me far more than size does. I take no credit for weighing nearly seventeen stone.' 5 I suspect that Ramsey's friends would have agreed with him about the insignificance of sheer physical size; and I suspect that if they felt humble before the vastness of the heavens, this was because they saw in it a symbol of their ignorance.

I believe that it would be worth trying to learn something about the world even if in trying to do so we should merely learn that we do not know much. This state of learned ignorance might be a help in many of our troubles. It might be well for all of us to remember that, while differing widely in the various little bits we know, in our infinite ignorance we are all equal.

There is a last question I wish to raise.

If only we look for it we can often find a true idea, worthy of being preserved in a philosophical theory which must be rejected as false. Can we find an idea like this in one of the theories of the ultimate sources of our knowledge?

I believe we can; and I suggest that it is one of the two main ideas which underlie the doctrine that the source of all our knowledge is super-natural. The first of these ideas is false, I believe, while the second is true.

The first, the false idea, is that we must justify our knowledge, or our theories, by positive reasons, that is, by reasons capable of establishing them, or at least of making them highly probable; at any rate, by better reasons than that they have so far withstood criticism. This idea implies, I suggest, that we must appeal to some ultimate or authoritative source of true knowledge; which still leaves open the character of that authority - whether it is human, like observation or reason, or super-human (and therefore super-natural).

The second idea - whose vital importance has been stressed by Russell - is that no man's authority can establish truth by decree; that we should submit to truth; that truth is above human authority.

Taken together these two ideas almost immediately yield the conclusion that the sources from which our knowledge derives must be super-human; a conclusion which tends to encourage self-righteousness and the use of force against those who refuse to see the divine truth.

Some who rightly reject this conclusion do not, unhappily, reject the first idea - the belief in the existence of ultimate sources of knowledge. Instead they reject the second idea - the thesis that truth is above human authority. They thereby endanger the idea of the objectivity of knowledge, and of common standards of criticism or rationality.

What we should do, I suggest, is to give up the idea of ultimate sources of knowledge, and admit that all human knowledge is human: that it is mixed with our errors, our prejudices, our dreams, and our hopes: that all we can do is to grope for truth even though it be beyond our reach. We may admit that our groping is often inspired, but we must be on our guard against the belief, however deeply felt, that our inspiration carries any authority, divine or otherwise. If we thus admit that there is no authority beyond the reach of criticism to be found within the whole province of our knowledge, however far it may have penetrated into the unknown, then we can retain, without danger, the idea that truth is beyond human authority. And we must retain it. For without this idea there can be no objective standards of inquiry; no criticism of our conjectures; no groping for the unknown; no quest for knowledge.

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This is a repeat of my post #93 from the "Scorecard" thread:

===

While you are waiting for you book, read this piece by Popper himself on the matter of Induction:

http://dieoff.org/page126.htm

The article pretty well states Popper's position on Induction.

Ba'al Chatzaf

Thanks for the link.

Here, succinctly expressed, is a paragraph saying just what I thought he thought:

The principle of empiricism (3) can be fully preserved, since the fate of a theory, its acceptance or rejection, is decided by observation and experiment - by the results of tests. So long as a theory stands up to the severest tests we can design, it is accepted; if it does not, it is rejected. But it is never inferred, in any sense, from the empirical evidence. There is neither a psychological nor a logical induction. [my emphasis] Only the falsity of the theory can be inferred from empirical evidence and this inference is a purely deductive one.

Thus he's disagreeing with Hume's belief (and as I understand some remarks of yours, with your belief) that a psychological induction is how we learn.

He goes on to spell out plainly both his agreement and disagreement with Hume:

I hold with Hume that there simply is no such logical entity as an inductive inference; or, that all so-called inductive inferences are logically invalid - and even inductively invalid, to put it more sharply [see the end of this selection]. We have many examples of deductively valid inferences, and even some partial criteria of deductive validity; but no example of an inductively valid inference exists. 2 And I hold, incidentally, that this result can be found in Hume, even though Hume, at the same time, and in sharp contrast to myself, believed in the psychological power of induction; not as a valid procedure, but as a procedure which animals and men successfully make use of, as a matter of fact and of biological necessity.

I take it as an important task to make clear, even at the cost of some repetition, where I agree and where I disagree with Hume.

I agree with Hume's opinion that induction is invalid and in no sense justified. Consequently neither Hume nor I can accept the traditional formulations which uncritically ask for the justification of induction; such a request is uncritical because it is blind to the possibility that induction is invalid in every sense, and therefore unjustifiable.

I disagree with Hume's opinion (the opinion incidentally of almost all philosophers) that induction is a fact and in any case needed. I hold that neither animals nor men use any procedure like induction, or any argument based on the repetition of instances. The belief that we use induction is simply a mistake. It is a kind of optical illusion.

What we do use is a method of trial and the examination of error; however misleadingly this method may look like induction, its logical structure, if we examine it closely, totally differs from that of induction. Moreover, it is a method which does not give rise to any of the difficulties connected with the problem of induction.

Ellen

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Ellen,

To pre-empt the arrival of your copy of "Objective Knowledge", and for those who aren't familiar with the issue, I will give a short rundown:

Popper breaks the problem of induction (what he called "Hume's problem") into two sub-problems:

H(L) The logical problem of induction

H(P) The psychological problem of induction

H(L) is formulated as follows: " Are we justified in reasoning from (repeated) instances of which we have experience to other instances (conclusions) of which we have no experience?"

Hume's answer to this is "No." Further Hume points out that this situation is the same even for probability of other instances. Popper agrees. Hume's logic is unshakeable.

H(P), however, is a different story.

We formulate H(P) as follows: "Why, nevertheless, do all reasonable people expect, and believe, that instances of which they have no experience will be conform to those of which they have had experience?"

Hume's answer is that humans must therefore be basically irrational: that human thought is merely mechanical "custom and habit", and that reason is in fact the "slave of the passions." It was this result that dealt a devastating blow to the hopes of empiricism, and that placed a "ticking time bomb" under Western philosophy, triggering the rise of various forms of ir-and-anti-rationalism in the past 200 years.

But it is this result that Popper criticises. If he can show that what humans use is not induction, then H(P) disappears, and we might successfully counter the irrationalist advance. Popper then argues that what looks like "induction" - that is, a mere irrational habit - is in fact a kind of optical illusion. What humans really do, Popper argues, is proceed by conjecture and refutation - not by totting up experiences in the illogical hope that they will one day congeal into truths, but by creating imaginative theories, which are then subjected to the tests of argument and experience. The fittest theories - the ones we hope are closest to the truth - survive.

This vision not only has the benefit of solving Hume's P, but like Alexander's fabled sword, solves a number of other philosophical conundrums at a stroke - leading to the theses you've outlined in your intro post above.

On the other hand, contrary to widespread belief, Rand never provided an answer to Hume's problem. In her own words, she never even began to think about it.

Why is this so important an omission on her part? What the Sam Hill has all this this got to do with Objectivism? Why do I keep harping on about it?

Because the whole of Kant's philosophy was created as a response to Hume...

Edited by Daniel Barnes
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But it is this result that Popper criticises. If he can show that what humans use is not induction, then H(P) disappears, and we might successfully counter the irrationalist advance. Popper then argues that what looks like "induction" - that is, a mere irrational habit - is in fact a kind of optical illusion. What humans really do, Popper argues, is proceed by conjecture and refutation - not by totting up experiences in the illogical hope that they will one day congeal into truths, but by creating imaginative theories, which are then subjected to the tests of argument and experience. The fittest theories - the ones we hope are closest to the truth - survive.

This vision not only has the benefit of solving Hume's P, but like Alexander's fabled sword, solves a number of other philosophical conundrums at a stroke - leading to the theses you've outlined in your intro post above.

On the other hand, contrary to widespread belief, Rand never provided an answer to Hume's problem. In her own words, she never even began to think about it.

Why is this so important an omission on her part? What the Sam Hill has all this this got to do with Objectivism? Why do I keep harping on about it?

Because the whole of Kant's philosophy was created as a response to Hume...

Your posting (in the entirety) is an excellent summary of Popper's position on induction. Popper takes a bad rap in some Objectivist circles and it really is not fair. His demolition of Logical Positivist verificationism is on point. In many of the lectures on physics and cosmology I have attended at Princeton, Popper's name comes up fairly often and he is probably the most favorably regarded philosopher among physicists.

Popper's solution was adumbrated in the philosophical analysis of scientific reasoning done by C.S.Peirce. Peirce considered a variant of induction, to wit, hypothesizing to likely or at least possible causes from observing phenomena as effects. He called this variant of induction by the name -abduction- which has a different logical structure from induction. I have often wondered why C.S. Peirce, one of the great American philosophers does not get more recognition than he does. If you want to hear abduction in action tune in on the PBS radio program -Car Talk-. The brothers Magliozzi, better known as Click and Clack figure out what is wrong with people's automobiles with scant clues and damned clever abductive reasoning.

Hume's contention that reason is subservient (in many cases) to passion is an accurate description of how people operate. Maybe they should not behave that way, but they often do. Man is the sometimes rational animal.

Ba'al Chatzaf

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(I don't know where this goes, so I have this posted here and at the other thread also.)

=================================================

But why why why?

If we look at the Popperian assertion regarding the single counterexample, why is this a falsifcation? It seems to ignore some basics that are not stated, but if they became the focus, might change some definitions completely.

1) Human logic must be defined in terms of human capabilities. A definition of logic that leaves out human capabilities is an oxymoron.

2) Every entity that exists, having a physical being, is finite. There is nothing that has been observed (that I know of) that is infinitely large. (Note: the "Universe" is a reified construct and cannot be adequately defined in relation to any other entity outside itself, except by theists.)

3) As finite human beings, we lack the capacity to perceive infinite entities using our finite sense data. We lack the capacity to observe something infinite in scope.

4) Our set of empirical observations (inferential knowledge) is by nature thus limited in scope and quantity, and the realm of these observations is defined and delimited by how we focus our attention. (This is the side issue of volitional consciousness.)

5) Every enitity exists in time and space, and when we focus on an entity, we focus on it in a specific defined place or realm (context).

6) Within every set there is a finite number of instances of finite entities within the defined realm.

7) If there is an instance of something (similar to the set members) outside of the defined realm, it is irrelevant. The rational mind cannot focus on two different defined realms simultaneously.

8) If there is an instance of something (within the defined realm) that is outside of the set, it is outside the set; it may be used as a delimiter of the set definition, but according to the Popperian definiton, it must invalidate or falsify the set. I don't see this as a valid inference.

9) Inferential knowledge is thus based in statistical probabilty, which I have referred to elsewhere as "the Confidence Factor". Never 100%, but it's what we've got.

What I am calling a "set" is what elsewhere is called a "concept", as formed within the basic S-I-R paradigm (with appropriate feedback loops). Sets of percepts are delimited empirically, and this is entirely a "psychological" process. The so-called "logical induction" of perfect concepts is an artist's dream, nothing more, and should be discussed under a separate subject (if not dismissed completely) of "Logic as an Art", instead of as a science.

Whereas inductive generalization is the process of locating in similar realms, duplicate or parallel sets of entities. But nothing goes on forever, additional realms are not necessarily identical, and so these derived generalizations tend to undergo their own peculiar "entropy" over time. (They "fall apart", usually through some ill-informed attempt at exegetical application.) But at least at first, in their generalities, they approximate reality well enough to be able to use them to succeed in some other goal-directed behaviour. This provides justification for continuing use of the "concept". ("Don't Argue with Success!") Until the defined realm is sufficiently different that the "concept" doesn't work any more.; it is then maintained as an irrational "falsehood" in the face of "reality"; or else it is abandoned for and replaced by a new organized set of percepts, derived from a new defined realm being brought into focus.

And no, AR did not address this topic in full detail.

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(I don't know where this goes, so I have this posted here and at the other thread also.)

=================================================

But why why why?

If we look at the Popperian assertion regarding the single counterexample, why is this a falsifcation?

Steve,

I'm not understanding what you're not understanding. Do you not understand that the existence of one black swan is logically contradictory to the proposition "All swans are white"? It can't both be and not be that all swans are white and that one swan is not white. The problem of induction is how would it ever be possible to justify making a statment of the form "All X are q."

In the rest of your remarks, which I admit to not entirely following, you seem to be talking about the limits of a concept. But the issue of induction pertains to propositions. I'm aware that Leonard Peikoff is apparently (if I'm understanding "the thrust") arguing that the process of conception formation solves the problem of induction, but I don't see how it could.

Ellen

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BTW Ellen, one of the most beautiful things I think Popper's stroke of genius restores is the status of imagination in science, and in rational thought in general. Scientists are no longer the accountants of the universe, boringly totting up experiences in the hope they will add up to a truth, but are a kind of artist. Creative invention becomes integral to the scientific process, thus holding up hope that the two battling "houses", the sciences and the humanities, may have been locked in a phoney war for the past few centuries.

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Steve,

Maybe it would help if I summarized Popper's message on induction, as I understand it. (If I'm misstating, I expect that Daniel, our Popper expert, will correct me.)

Popper is saying -- in agreement with Hume -- that the problem of induction is unsolvable. In disagreement with Hume, he's saying that there's no need to solve the problem of induction in order to have a rational method of knowledge. We have such a method in the process of conjectures and refutations. He's further claiming that this latter is the method humans and other animals actually use in learning, that the appearance of using induction is illusory.

Ellen

PS: I signed on to post this prior to noticing that Daniel had meanwhile posted a reply to Steve. Daniel is of course more comprehensive, but my little summary might still be of help.

Now you see it; now you don't! I was wondering where Daniel's post which I'd thought I'd seen had disappeared to. See below instead of above. ;-)

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(whoops...accidentally deleted my earlier reply to Steve. Here it is again, sorry for the confusion- DB)

Steve:

>If we look at the Popperian assertion regarding the single counterexample, why is this a falsifcation?

Steve, Popper's book that started it all was called "The Logic of Scientific Discovery." Emphasis "logic." Prior to that, the philosophy of science was considered to be all about induction. The dirty little secret of this belief, however, known since Hume's shocking result but not publicly confessed, was that induction was fundamentally illogical. Popper was attempting to restore a valid logical structure to the philosophy of science. (Falsification, unlike induction, uses a valid logical form: the modus tollens.)

>1) Human logic must be defined in terms of human capabilities. A definition of logic that leaves out human capabilities is an oxymoron.

This argument, which you've applied to logic here, is one of Rand's favourite ripostes and is reflexively asserted in Objectivism. Unfortuntately it is completely erroneous.

For example, you'd probably say I was appealing to a "Platonic" standard of logic - an abstract ideal that no human could achieve, so is therefore "invalid." And so forth.

But this whole line of argument is itself false. Abstract or theoretical standards, such as logic, are in fact of the greatest usefulness, and it matters not at all to their so-called "validity" whether we are entirely capable of achieving them or not. And not only is it false. Rand's argument, if taken seriously, leads directly to epistemological relativism!

For one clearcut example, I offer you the abstract standard of "absolute zero" in physics. This is the theoretical point (–273.15 °C) where there is zero energy in a system. Now, no human has ever been capable of achieving this standard. Further, if quantum physics is correct, then we will never achieve this standard, because doing so would violate the Uncertainty Principle. Does this therefore mean "absolute zero" is an "invalid, Platonic" standard because we are currently, and perhaps permanently, incapable of achieving it? Is –273.15 °C evil, "severing man's mind from reality"? Of course not. The idea is absurd. Rand's whole argument, while well-intentioned, is foolish, and on proper examination leads to any number of similarly embarrassing problems for those who adopt it.

Further, the unforseen consequences of Rand's argument is epistemological relativism. Using our same example, and applying the criteria you suggest - "human capabilities" - you will find that "absolute zero" becomes merely whatever humans are capable of achieving at the time. Yet this has changed radically as our technology has improved, so your "absolute" standard turns out to be, in fact, relative - a problem Rand unwittingly conceals by verbalism, by calling the situation "contextual."

So it turns out that it is not Popper that relies upon "oxymorons" to argue his case, but Rand. If you are sincerely worried about such inherent self-contradictions, I suggest you think long and hard about Objectivism's own central oxymorons, such as "contextual absolutes", "contextual certainty" etc.

>2) Every entity that exists, having a physical being, is finite. There is nothing that has been observed (that I know of) that is infinitely large.

In reply, I offer you the integer series. This is, of course, abstract rather than physical. But, as before, are you going to argue that this highly useful abstract system is somehow "Platonic", "invalid" or "severed from reality" as a result?

>3) As finite human beings, we lack the capacity to perceive infinite entities using our finite sense data. We lack the capacity to observe something infinite in scope.

Precisely Hume's point.

>4) Our set of empirical observations (inferential knowledge) is by nature thus limited in scope and quantity, and the realm of these observations is defined and delimited by how we focus our attention. (This is the side issue of volitional consciousness.)

This appears to be 3) restated.

>5) Every enitity exists in time and space, and when we focus on an entity, we focus on it in a specific defined place or realm (context).

This appears to be another restatement of your previous premises.

>6) Within every set there is a finite number of instances of finite entities within the defined realm.

This seems to be more or less a restatement of 1).

>7) If there is an instance of something (similar to the set members) outside of the defined realm, it is irrelevant. The rational mind cannot focus on two different defined realms simultaneously.

The point is not what the mind can "focus" on at any given time or place. The point is the search for the truth.

>8) If there is an instance of something (within the defined realm) that is outside of the set, it is outside the set; it may be used as a delimiter of the set definition, but according to the Popperian definiton, it must invalidate or falsify the set. I don't see this as a valid inference.

From the set of "white swans I have seen" we can validly infer that all the swans you have seen are white.

We can also validly infer Brand Blanshard's missing-the-point formulation that "some swans are white."

But if we consider science to be the search for universal laws - and I do - we are looking for "all swans are white." This cannot be validly inferred from the set of your swan-experiences. End of story.

If, however, you want to reject the search for the laws of the universe as the definition of science, and say instead that science is about statements like "some eggs are white" because you have observed a dozen eggs in your egg carton of course you will get no argument from me. I don't argue over the meaning of words.

BTW, you are right on this point: a black swan that you have not seen does not falsify the statement "all the swans that I have seen are white", nor Blanshard's "some swans are white." However, Popper never claimed that it did.

>9) Inferential knowledge is thus based in statistical probabilty, which I have referred to elsewhere as "the Confidence Factor". Never 100%, but it's what we've got.

Statistical probability, Steve, is deductive, not inductive. There is no valid inductive theory of probability - no way to infer the probability of future events from past ones. Check it out.

>What I am calling a "set" is what elsewhere is called a "concept", as formed within the basic S-I-R paradigm (with appropriate feedback loops). Sets of percepts are delimited empirically, and this is entirely a "psychological" process. The so-called "logical induction" of perfect concepts is an artist's dream, nothing more, and should be discussed under a separate subject (if not dismissed completely) of "Logic as an Art", instead of as a science.

Logic is a way of making sure our theories are internally consistent (with themselves)

Empiricism is a way of making sure our theories are externally consistent (with reality)

Get those two right and you're at least on the way to a true theory!

>Whereas inductive generalization is the process of locating in similar realms, duplicate or parallel sets of entities. But nothing goes on forever, additional realms are not necessarily identical, and so these derived generalizations tend to undergo their own peculiar "entropy" over time. (They "fall apart", usually through some ill-informed attempt at exegetical application.) But at least at first, in their generalities, they approximate reality well enough to be able to use them to succeed in some other goal-directed behaviour. This provides justification for continuing use of the "concept". ("Don't Argue with Success!") Until the defined realm is sufficiently different that the "concept" doesn't work any more.; it is then maintained as an irrational "falsehood" in the face of "reality"; or else it is abandoned for and replaced by a new organized set of percepts, derived from a new defined realm being brought into focus.

From what I can make out of the above, Popper's theory of conjecture and refutation, of the evolution of knowledge through selection via falsification, fits what you're saying - that "generalizations...fall apart...over time" - just as well as Rand's or the inductivists, without resort to oxymorons in the former and irrationalism in the latter. Hence it is the better theory.

>And no, AR did not address this topic in full detail.

Actually, Rand did not address Hume's problem at all. In her own words:"...the big question of induction...I haven't worked on that subject enough to even begin to formulate it." (ITOE p 304)

Edited by Daniel Barnes
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Mike:

>How about my statement from another thread?

>All black swans are birds.

So long as you define "swan" as a type of bird, then this is a priori true (which means true without the need for experience to justify it). So defined, I can say that "all black swans are birds" is true even if I never see a black swan in my life, or even if no-one in the history of humanity ever sees one. So it is not what is called an empirically testable proposition.

To contrast, a statement which is not a priori true, and is therefore empirically testable (ie requires experience), would be: "All swans are black" or "all black birds are swans."

(Of course, you can always play with the definition of "swans" to make these statements untestable and a priori true again - for example, define "swan" as "any black bird." But obviously we will not learn much by merely playing with words in this fashion!)

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Daniel,

How about my statement from another thread?

All black swans are birds.

How does that hold up?

Michael

Since swans are birds this is trivially and tautologically true..

To deny it would imply a bird is not a bird, a clear contradiction

Ba'al Chatzaf

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Ba'al:

>Your posting (in the entirety) is an excellent summary of Popper's position on induction.

Thanks Ba'al...;-)

>Popper...is probably the most favorably regarded philosopher among physicists.

Yes. Popper actually did physics and maths quite seriously - for example, his work on probability, and the objective interpretation of this in quantum physics. Despite their slow start, due to translation problems, Popper's own painstaking approach, and overt hostility from much of the philosophic profession, Popper's ideas have quietly spread until they seem now all-pervasive - as if they have always been there. Yet you only need to read some of the literature from even as recently as the '30s to see how different today's view of science is. This is why, I think, Frank Tipler called Popper's "TLScD" the most important book of the 20th century.

This is in complete contrast to the pomos who often cite allegedly physical concepts in their philosophic musings, and of course to the pomo's unwitting brother-under-the-skin, Leonard Peikoff.

>Popper's solution was adumbrated in the philosophical analysis of scientific reasoning done by C.S.Peirce.

Pierce anticipated Popper in a number of ways. Popper bemoaned discovering him so late, saying that it could have saved him a lot of work!

>Hume's contention that reason is subservient (in many cases) to passion is an accurate description of how people operate. Maybe they should not behave that way, but they often do. Man is the sometimes rational animal.

Yes, I agree. Popper's approach aims to improve matters, and move us away from the picture of man as inherently irrational towards something better. Popper and Rand's goals, at least, are the same in this, if not their conclusions. We can accept that much of what we think and do is irrational, but find a way to make this concession smaller rather than greater.

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In mathematics, there is what is called a proof by induction. A simple example is the mathematical formula:

1 + 2 + 3 ... + n = n * (n + 1) / 2

If this is true for a particular value of n, then add n + 1 to both sides of the equation:

1 + 2 + 3 + ... + n + (n +1) = (n^2 + n) / 2 + n + 1 = (n + 1) * (n + 2) / 2

Therefore, if the equation is true for some value of n, it is also true for (n + 1). Since the equation is true for n = 1, this implies it is true for n = 2, which implies it is true for n = 3, etc. Therefore, it has been proven inductively that the equation is true for all integers.

Is this a different meaning of the term "induction"?

Martin

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In mathematics, there is what is called a proof by induction...Is this a different meaning of the term "induction"?

Yes. Take a look here for the difference.

The key quote is:

"Mathematical induction should not be misconstrued as a form of inductive reasoning, which is considered non-rigorous in mathematics. (See Problem of induction.) In fact, mathematical induction is a form of deductive reasoning and is fully rigorous."

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In mathematics, there is what is called a proof by induction...Is this a different meaning of the term "induction"?

Yes. Take a look here for the difference.

The key quote is:

"Mathematical induction should not be misconstrued as a form of inductive reasoning, which is considered non-rigorous in mathematics. (See Problem of induction.) In fact, mathematical induction is a form of deductive reasoning and is fully rigorous."

It seems to me that science took off when mathematics did. If so, can we blame Aristotle for hundreds of years of lack of advancement in math, not just science? And if science needs math and we can't blame Aristotle can we continue to blame him for the non-advancement of science too? Chicken or egg?

--Brant

Edited by Brant Gaede
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So long as you define "swan" as a type of bird, then this is a priori true (which means true without the need for experience to justify it).

Daniel,

Two questions:

1. "A priori true" means concept formation?

2. How can one define "bird" "without the need for experience to justify it"?

Since swans are birds this is trivially and tautologically true..

To deny it would imply a bird is not a bird, a clear contradiction

Bob,

Two questions:

1. "Trivially and tautologically true" means concept formation?

2. Concept formation is a part of logic?

Michael

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Mike:

>1. "A priori true" means concept formation?

Probably not according to Rand, who argues all concept formation is derived from experience. She doesn't think there is such a thing as a priori knowledge.

>2. How can one define "bird" "without the need for experience to justify it"?

I don't understand your point?

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Daniel,

You consider calling a swan a bird "a priori knowledge." This means this is not derived from experience, if I have understood you correctly.

Yet the very concept of "bird" derives from experience. No experience, no mental category of bird.

Now look at the phrase from that lens: All black swans are birds.

How do you reconcile that?

Michael

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Is this a different meaning of the term "induction"?

Martin

Yes.

Mathematical induction is one of the Peano Axioms (or better postulates) for arithmetic and is part of a -deductive- system for arithmetic.

Peano Induction is so called because it is a way of inferring a universally quantified formula from the conjunction of an infinite set of particular (non-quantified) formulas. Empirical induction (the induction about which Francis Bacon wrote) is a way of inferring a universally quantified formula from a -finite- set of particular formulas and is not deductively valid.

Peano Induction is closely related to the idea of recursion in mathematical logic.

Ba'al Chatzaf

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Ellen, Daniel

I apologize for my obscurantism. While writing the post above, I was aware that some points would tend to be mistaken for each other, but hoped that my meaning would be clear. But it obviously wasn't.

The thing is, that to answer fairly, is akin to eating an elephant. You have to do it one bite at a time. So I will not address all points in this one message, but will break the explanations into pieces, to be read, and then go back and read the original post in that context.

(whoops...accidentally deleted my earlier reply to Steve. Here it is again, sorry for the confusion- DB)

Steve:

>If we look at the Popperian assertion regarding the single counterexample, why is this a falsifcation?

Steve, Popper's book that started it all was called "The Logic of Scientific Discovery." Emphasis "logic." Prior to that, the philosophy of science was considered to be all about induction. The dirty little secret of this belief, however, known since Hume's shocking result but not publicly confessed, was that induction was fundamentally illogical. Popper was attempting to restore a valid logical structure to the philosophy of science. (Falsification, unlike induction, uses a valid logical form: the modus tollens.)

Okay the thrust of my notes here is that "empirical induction" may not be "science" per se, but it IS the process of human concept formation, within the context of the human S-I-R learning paradigm (with feedback). As in "equals", "is the same as", "is just another name for", etc. And no, it isn't perfect, but it is quantifiable, and thus it is testable.

I understand the contrapositive argument. What I am discussing is falsifiability, or testability as an actual discipline.

As I have discussed elsewhere, the goal-directed practice of testability is an incremental, statistically-based, hands-on process, requiring multiple iterations, of empirically relating members of a set to the definition of the set itself, within a specific environment or context, rather than a simple a priori comparison between instance and definition. The purpose is to develop a reliable answer, i.e., a correspondence, between instance and generalization, of greater than N percent, and if necessary, to develop corrective measures (within a specific environment or context) to ensure that result. (N being selected according to the discipline involved: as I've stated elsewhere, "commonsense" logic usually requires at least 50% correspondence, and usually 67%-75%, in order to operate; "business" logic requires 78%-84% to operate at a minimal level or 84%-92% to operate successfully or 92%-96% to operate at a superlative level; scientific inquiry operates at the top of the scale.) It is thus akin to a technology of logic (like the scientific method), rather than (arbitrarily) a single rule or set of logic equations.

Please note that this does not necessarily imply the following syllogism:

1) All swans I had seen were white.

2) I just saw a black swan.

3) THEREFORE I MUST KILL ALL THE BLACK SWANS. {as a corrective measure}

What it does imply is that a single instance of a contrapositive may be a necessary condition, but it is not a sufficient one, to invalidate an entire definition or hypothesis a priori. It does, however, become part of the statistical mix, and if present in "enough" iterations of the hypothesis testing, indicates a need for at least a new "placeholder" concept, or "knowledge stub", if not a redesign of the initial concept.

Okay the wife is telling me to get off the computer. More later.

Steve

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What it does imply is that a single instance of a contrapositive may be a necessary condition, but it is not a sufficient one, to invalidate an entire definition or hypothesis a priori. It does, however, become part of the statistical mix, and if present in "enough" iterations of the hypothesis testing, indicates a need for at least a new "placeholder" concept, or "knowledge stub", if not a redesign of the initial concept.

Okay the wife is telling me to get off the computer. More later.

Steve

A single counterexample is -sufficient- to negate a universally quantified assertion. A single contradiction is sufficient to falsify a corpus of propositions or a theory. That does not mean ALL the assumptions of the theory are false. It means at least one of them is false.

Popper replaced verificationism in scientific theories with the doctrine of falsification which is the use of modus tollens to kill a hypothesis. No mater how many time a scientific theory is supported by experiment it does not prove the theory is true. Scientific theories, at best, are evidentially supported not yet falsified conjectures. As long as a theory has tons of evidence to support it, none to refute it, then you use it until it breaks.

Ba'al Chatzaf

Edited by BaalChatzaf
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BTW Ellen, one of the most beautiful things I think Popper's stroke of genius restores is the status of imagination in science, and in rational thought in general. Scientists are no longer the accountants of the universe, boringly totting up experiences in the hope they will add up to a truth, but are a kind of artist. Creative invention becomes integral to the scientific process, thus holding up hope that the two battling "houses", the sciences and the humanities, may have been locked in a phoney war for the past few centuries.

Einstein said that theories are NOT DETERMINED by facts, but are free creations of the human mind. Given a finite set of particular facts there are an infinite number of theories that will fit the given facts. Most of the theories are wrong in general. Popper was not revolutionary in his approach to science. In fact most scientists were Popperians before Popper. Popper's main opposition came from other philosophical camps.

Your remark about accountants of the universe is both charming and accurate.

I like to put it this way: Theories do not leap out of heaps of facts as Athena leaped out of the Brow of Zeus, fully armored and formed. Theories do not leap off of facts like frogs from lily-pads.

Nature provides the dots. Humans connect them.

Ba'al Chatzaf

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Daniel,

You consider calling a swan a bird "a priori knowledge." This means this is not derived from experience, if I have understood you correctly.

Yet the very concept of "bird" derives from experience. No experience, no mental category of bird.

Now look at the phrase from that lens: All black swans are birds.

How do you reconcile that?

Michael

Well, we can just make the formula completely abstract so you can see my point.

K is a kneezle. W is a weezle. No-one's ever had any sense-experience of kneezles or weezles, as they are entirely imaginary.

Substitute K for black swans, W for birds, and we have the following.

All Ks are Ws. This statement is true by definitiion., or a priori. That is, established via the rules of logic, thus not requiring experience.

Or we could make it even more abstract, and say All x's are y's, with x or y as mere placeholders.

Now of course you might protest and say that by merely manipulating definitions and playing with words you can "prove" almost anything you like with this rationalistic method.

And you'd be right. And then you'd be sounding just like me...;-)

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Ba'al:

>Theories do not leap off of facts like frogs from lily-pads.

Yes. This is charmingly put too.

And now, as a sidelight, we observe a moment of consilience - the unity of knowledge - as we note that, like theories, decisions (including ethical ones) do not leap from facts like frogs from lilypads either. Thus the is/ought distinction is consilient with human creativity, and human creativity becomes entwined with human responsibility.

Let's not derail into that subject however.

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