Wikipedia entry Problem of Induction.

This is a youtube by Oxford's philosophy department on point https://www.youtube.com/watch?v=ET9oRKEw....

I think the key point you made above is that Hume's formulation ignores the law of identity, which is something I noticed in watching the video.

I was talking about so-called pure math and I stand by the idea that pure math is worried about proofs that are not checked by empirical evidence. It is purely concerned with logic but not evidence.

I think it would be a mistake to overly formalize the exact process of how knowledge is gained. For instance, serendipity in discovering the structure of benzene or inventing post-it notes does not invalidate reason. Both have to be checked by logic and empirical facts.

What in that is from Wikipedia and what did you write?

Without getting very far into it or with much detail, the notion that induction is based on generalizing with nothing but some number of repeated observations, such as the "All swans are white" example, is the fallacy of induction by simple enumeration. The emphasis on Hume further builds into the discussion an anti-conceptual epistemology that led to his infamous skepticism as the culmination of British Empiricism, including denial of causality and a quandry of how induction could ever be possible at all, rather than how to do it.

Causality is the principle of identity applied to action. To be is to be something. To be something is to be something in particular. It has a specific identity and behaves accordingly. That is the basis of the "regularity" that makes science possible.

Science, not philosophy, discovers the relevant identities, actions, and causes, in terms of correct concepts classified and defined by the relevant essentials. The problem of induction is the epistemology of how this is done in general, not whether it can be done, which it has been throughout science for centuries. It is analogous to the epistemology of how, in general, we form concepts.

The last paragraph in the post above follows the common false division of knowledge into two classes, knowledge that is about reality which is said to be uncertain, and certain knowledge which is not about reality. See the first appendix in IOE on the Analytic Synthetic Dichotomy in its various forms.

The concept of a line is not just "inspired" by the real world, it is a concept _of_ lines in the real world (geometry originated from surveying in the ancient world). Knowledge about which someone doesn't care if it's consistent with the real world is pathological, not knowledge. Knowlege _is_ knowledge about the world.

Mathematical knowledge is based on perception of instances of numbers of units, points, lines, etc., which gives rise to the most basic mathematical concepts of number and geometry. But those are already higher levels of abstractions. They are not about any particular entity or kinds of entities, but do pertain to measurement in the real world no matter what the entities are.

Mathematics in its fundamental, elementary form is directly about how to formulate and relate measurements. It is a science of _method_, not a science of physical objects. In that sense it is "about the world" in that it refers to and pertains to real measurements in the real world, but it is not "about the world" the way physical sciences like physics are.

When you form concepts you are always omitting measurements of essential characteristics in common between the units referred to, maintaining the specific measurements only implicitly (such as Ayn Rand's example of the concept of length, which refers to any length, with the particular measurements omitted in the abstraction of what they have in common).

That a line has "no width" means that the width is small enough to be neglected as irrelevant, not that it literally has no width. It has some width; what particular width is irrelevant, but it essential for being a line that whatever it is it has to be negligibly small in comparison with the length. The geometry of lines does not, therefore, need not and does not deal with their widths. Likewise for sizes of points, straightness of lines, curvature of a flat surface, etc.

You don't say for a small house that the earth is flat, that would be misapplying concepts out of context, but you do say that the terrain of the lot is flat, i.e., a plane, if it's not on a hill, etc. because the curvature of the earth and other local features are irrelevantly small.

If the thickness of a line matters, then you have to use several of them to define a shape with another dimension of measurement. When applying a concept like 'plane' you must ensure that in the particular context the curvature is in fact negligible or your use of the geometry won't work. Likewise for features of the local surface. Then you invoke more complex geometry and work with elevations, slopes, etc. to the extent you have to to get it all right and accurately depict the surface and its relation to the house, but it won't include the curvature of the earth.

So the idea of "omitted measurements" in the process of generalizing through concept formation has its complement in applying the concepts, especially in science and engineering where the quantifiable discovery of what measurements are essential and must be explicitly included, and which are irrelevant, and the accurate calculation or estimation of their sizes to some specific degree of precision, are crucial.

Much of applied mathematics is determining what measurements are relevant and assessing the accuracy of calculations to ensure that all relevant precision is included for whatever accuracy is required throughout chains of calculations.

"Error analysis" in this sense became a major aspect of numerical analysis particularly with the advent of computers doing elaborate calculations, but it had also already been important with electric calculating machines and even the slide rule before that.

So we do very much care about whether or not our concepts and methods in geometry are consistent with the real world, and what features and their measurements are necessary to ensure that we are using mathematical methods that are sufficient to apply to reality in any particular problem or class of problems.

The "pure" mathematics dealing only with relations between already established concepts, like proofs in Euclidean Geometry, always has essential measurements implicit, but they are not typically dealt with explicitly. That they are implicit does not mean that "mathematics doesn't care", what we care about depends on the context. Each facet of reality referred to by concepts is always part of the meaning of the concept, and we may or may not need to focus on it depending on the purpose.

When formulating and applying theories in the physical sciences, the context dictates what characteristics are relevant and essential, but now extended to kinds of entities, physical characteristics and phenomena, not just numerical magnitudes and relations. That is how conceptual theories are generalizations established and used in a certain context and is what is meant by the "contextual nature of knowledge", which makes generalization possible.

But you have to be careful that when formulating and validating generalizations you understand essential characteristics and causes, thinking with appropriate conceptual classifications in terms of essentials. You can't just make an assertion and then claim that the first contradictory fact observed the next day doesn't matter because it is out of the context. The contextual nature of knowledge is not a get out of jail free card for erroneous theories.

Much of science is devoted to discovering and better understanding the "limits" and extent of the proper context to increasing precision in terms of essentials. It is all related to the "problem of induction", which in turn depends on having, but is not the same as, proper principles of concept formation

Doing this properly results in the expansion of knowledge through new discoveries for different ranges of measurement, more precision, or new phenomena, in the way you cited. It makes science an ever expanding base of systematic knowledge in terms of principles of ever increasing levels of abstraction rather than a series or exploded fallacies in the way it is often falsely regarded -- Hume's anti-conceptual skepticism is part of the reason why.

According to wikipedia the problem of induction is:

The problem of induction is the philosophical question of whether inductive reasoning leads to knowledge understood in the classic philosophical sense,[1] since it focuses on the lack of justification for either:
Generalizing about the properties of a class of objects based on some number of observations of particular instances of that class (for example, the inference that "all swans we have seen are white, and therefore all swans are white," before the discovery of black swans) or
Presupposing that a sequence of events in the future will occur as it always has in the past (for example, that the laws of physics will hold as they have always been observed to hold). Hume called this the principle of uniformity of nature.[2]

First, all knowledge is inductive originally unless you are a mystic and principles are handed down from god or otherwise revealed. If the person rejects mystical revelation then they must argue against induction using language which implies that he believe he has knowledge and that the receiver has knowledge, which has to based on inductive reasoning.

The problem of induction is that since you do not know every fact you cannot know that the next fact will conform to the universal (All swans are white), which means that you have to know everything to know any universals according to this line of argument.

Humans deal with this in two ways. 1) In math we create a definition, such as for a line, and then we never care if this concept is consistent with the real world, although it was and has to be inspired by the real world, or 2) In science we deal with this by understanding domains. It is perfectly valid to say the world is flat if you are building a small house. BTW when we find that scientific knowledge does not extend to a new domain, this is a chance to expand our knowledge, not a failure as so many populist philosopher's suggest.

Understanding that knowledge is contextual and that knowledge through scientific general principles does not mean omniscience is crucial, but it doesn't solve the problem of the epistemological principles of the induction. The problem of induction is often misconstrued to mean showing that it induction is possible at all, but that isn't what is meant here.

Ayn Rand knew that the "problem of induction" had not been solved, and said so. She discussed it at the workshops and described what was involved and the kind of knowledge of both science and philosophy that would be required to solve it. You can find some of that in the appendix to IOE. (That is also where you will find discussions of the important ideas of first level and higher level concepts in a hierarchy, and their significance.)

Leonard Peikoff's lectures on induction in his 1970s course on logic is also very good on the nature of the problem, how to frame it correctly, and some classical insights on it such as Mill's methods.

Dave Harriman's book doesn't include any kind of personal disputes. The fact that some people are involved in personal squabbles over it is not a reason for you to not read it. Most of it is very good, interesting, and informative in an objective philosophical analysis of the history of science, even though it doesn't accomplish the more sweeping claims made for it.

We have to be able to question Rand and other Objectivists without being called heretics I think Peikoff and other o's are just dead wrong when they argue against self ownership for instance.

I don't think any of this formulation is in Rand's books. I have not read David Harriman's book and the fighting within the O community is part of why.

I have looked at the "problem of induction" and I think it is what I call the problem of "perfect knowledge." Knowledge is always contextual, sort of like the domain of an equation in math. The problem with the perfect knowledge argument against induction is that in order to have perfect knowledge you have to know everything and you can't know anything until you know everything under this argument. That is clearly not how knowledge works.

He claims to have solved the "problem of induction" in the philosophy of science (and more broadly). He released his theory in recorded lectures (which may no longer be available) and in condensed form as the first chapter in Dave Harriman's otherwise generally very good book, The Logical Leap.

He calls this his own "application" of Ayn Rand's epistemology, but he clearly regards it as much more, calling it "Objectivism's" solution to the problem, and when John McCaskey questioned some particular use of the history of science in Dave Harriman's book, Leonard Peikoff accused him of "attacking the philosophy".

The theory is based on two main ideas. This description, which I am trying to condense, presupposes that you know Ayn Rand's epistemology and what the general "problem of induction" is.

The first idea claims that there are "first order generalizations" which are known automatically to be true, analogous to perception preceding conception. First order generalizations are said to be general propositions containing only first order concepts. But there are no general propositions consisting only of first order concepts, as Ayn Rand describes the hierarchy of concepts (concepts of actions and attributes must be at least second order; you can't utter a statement at all with only first order concepts). And the idea of infallible first order generalizations doesn't work under the most cursory examination. A later attempt (by Harry Binswanger, who endorsed the theory) to patch this up with a new supposed "ïmplicit first order" after the misuse of "first order concept" was pointed out, did not help.

The second main idea is that generalization in propositions is no more than the generalization inherent in the process of forming concepts. Ayn Rand had called generalization in concept formation an inductive process, but contrasted it with induction in propositions, and explicitly rejected the idea that they are the same process (this was all at the workshops). This was in the context of a discussion of examples of the problem of induction in science, but it if you think about the difference between what is being generalized when you form a concept with contextual definitions versus a general assertion claiming to be contextually universal, you see that they are very different and require different kinds of knowledge and validation. General propositions are much harder. It has been done many times successfully of course, but there is no epistemological principle explaining fully what is necessary.

So what Leonard Peikoff calls an application of Ayn Rand's epistemology is not only embarrassingly inadequate, it misconstrued some of Ayn Rand's basic concepts in epistemology, misused the terminology, and contradicted her earlier explicit rejection of one of the main ideas decades before.

I don't write this to attack Leonard Peikoff. He is an honest, intelligent, and knowledgeable man with great integrity and dedication, but has shown some poor judgment in his later years, especially in areas he lacks first hand knowledge. It doesn't detract from his previous superb expositions of Ayn Rand's philosophy and its relation to the history of western philosophy, mostly based on work he did when she was alive. It's important to read and evaluate everything independently and not take anything for granted.

yes. Your assertions that LP made some logical errors. Interested to know about that and why

He doesn't know what concepts are, let alone Ayn Rand's formulation of the nature of universals, or how to apply it to the concept 'line', or how we would ever come up with the concept of a line if there were no lines from which to abstract the concept. That isn't uncommon and is understandable given the general state of philosophical understanding in math or anywhere else; the loutish belligerence and lack of desire or ability to discuss it is another matter.

You mean #2 on the lectures on logic? I'm not sure what you mean.

I saw the self-ownership thread but haven't had time to post about it.

It is not clever to deny concepts. And the fact that you are using words means that at the same time you are using concepts and denying that they can exist. Kinda of pathetic.

ewv,
To your second point, would you be willing to to expound on your comments in a separate post?
second, I don't believe I saw you contribute to the self ownership post. I would appreciate your feedback.

Leonard Peikoff's lectures in the 1970s were endorsed by Ayn Rand. They aren't his own interpretations differing from her. In fact he once said (on his radio show) that Ayn Rand was responsible for the Objectivist ideas in those lectures. His OPAR book was of course written after Ayn Rand died, but that book was based on the earlier lectures and articles, and was intended to represent her thoughts, not his own additions or interpretations. There are some points where his success in that can be questioned, but it's nothing like his later works expounding his own ideas and what he believes are applications of Ayn Rand's philosophy.

Ayn Rand also agreed that epistemology is fundamental, yet we have seen so little written about it since then.

Plusaf's post is non-responsive. It doesn't matter what he calls himself. He is promoting a-philosophical utilitarian Pragmatism often promoted by conservatives. It is also promoted by the small number of utilitarian "libertarians" and others. It has nothing to do with Ayn Rand, who was not "all about" "how people learn things and make decisions" and was not anything like his "law" of everything is a "tradeoff". If he doesn't understand what that is based on he should read Ayn Rand.

Savvy Street is not about presenting academic analysis. It is about getting readers to think more deeply about the world and to be an asset to your flourishing life. One would expect and encourage readers to delve into these areas of study when they can.

Hello ewv,
I have read Peikoff's book, Objectivism: The Philosophy of Ayn Rand. It was, without a doubt, an in depth examination and worthy read no matter one's opinion of every precept Piekoff has offered. There are, as you know, other notable "objectivist's" that hold disagreements on some points of his interpretations, as well as his own (sometimes questionable, as you have mentioned) extrapolations.

I quite agree, this article is nothing more than an interesting introduction... the basics.
I would hope it impetus for deeper investigation for those truly interested, particularly in the field of epistemology... a field of great import for any philosopher. It is after all what philosophy is all about at its most basic level. How do we know, what we know? is essential in knowing.

Respectfully,
O.A.

"The whole World is a Trade-off"
This is one-dimensional thinking. Knowledge is not a trade. If we look at it from an economic perspective alone (one dimension) we might say you take time from other actions to learn. However, that is not foundation. It's a metaphysical point? The only context for which this is essential would be in economics. This makes no sense in epistemology. Since you have not been clear how this statement applies within the context of this post, I assume you are saying there are no absolutes epistemologically or morally. so on an Ayn Rand site, you are going to run up against some heavy criticism, since in Objectivism Law of Non-Contradiction two contradictory statements cannot both be true simultaneously and in the same way. Your statement of trades suggest that nothing is wrong or right, just a series of trades. You no longer have to adhere to the premise that A is A or that Reason is no more powerful to gaining knowledge than mysticism.

This kind (Robbie and Plusaf) of a-philosophical utilitarian Pragmatism often promoted by conservatives has nothing to do with Ayn Rand's epistemology or any other part of her philosophy depicted in and which made possible Atlas Shrugged, and it explains nothing about the nature of knowledge, which is the topic of the thread.

The question and answer format is due to the fact that the extensive appendix in the 2nd edition is an edited and reorganized transcript of about 1/3 of the many hours of questions and answers in her Epistemology Workshop, held for a small group of professionals over several days around 1970.

But that was just the appendix. Without the systematic presentation in the first part of the book as a base (originally the entire 1st edition based on a sequence of articles in her journal The Objectivist), coherent questions and answers would not have been possible. The second edition also includes Leonard Peikoff's systematic article on the analytic synthetic dichotomy and its variants (also from The Objectivist), which is also necessary for a proper understanding.

But that is primarily about concepts, which are fundamental, but not all there is to the subject. Other complementary systematic elaborations are :

1. The first part of Leonard Peikoff's comprehensive book Objectivism: The Philosophy of Ayn Rand, which systematically covers the nature of knowledge in accordance with the axioms of existence, identity and consciousness; and the nature of perception plus more on propositions, logic, and the principle of the objective versus the intrinsic and subjective. This comprehensive treatment supersedes several recorded lecture series he gave on Ayn Rand's philosophy.

2. Leonard Peikoff's two recorded lecture course series from the 1970s on Logic and on Induction (not to be equated with his more recent lectures on induction that are his own theory claimed to be an application of Ayn Rand's philosophy -- with some interesting elements but not correct in some major ways).

3. Leonard Peikoff's 1970s lecture series on the history of western philosophy, which describes the major issues of philosophy and how the various philosophers from the Greeks through the 20th century addressed them, and Ayn Rand's answers in her own philosophy. Much of that is necessarily epistemology. It shows how ethics and political philosophy depend on epistemology and provides an important context for understanding Ayn Rand's philosophy and its significance.

Any one of these is better and much more than the article posted above, and all are necessary to understand the significance and meaning of what that article is referring to. It is good that someone is defending knowledge and induction in general terms, but much more is needed to explain it, and that can't be done in a single article for the general public.

He (Danno) doesn't understand conceptual thought and its relation to reality at all, let alone induction, and it would take a lot more than that article to fix it.

A good book to read on induction in mathematics is the very well written and highly readable George Polya's Mathematics and Plausible Reasoning: Induction and Analogy in Mathematics.and Patterns of Plausible Inference (2 vols).

Inductive reasoning in mathematics _is_ induction and there are many forms of it, typically employed to establish or discover patterns. Strict deduction is used within axiomatic systems, after fundamental concepts and principles have already been established inductively, to keep the hierarchy of the relationships straight and as an aid in avoiding subtle errors and unjustified leaps. Symbolic deductive proofs, in particular, become a form of 'calculation' themselves. But cognition precedes calculation. and cognition requires concepts, which are a form of generalization established in advance.

The "principle of mathematical induction" that you are referring to is one method of generalization, incorporated as a principle in Peano's axiomatic system for the positive integers. Poincaré described it as an infinite sequence of deductions, based on the infinite sequence of numbers in Peano's system, but there is much more to it than that. It employs deduction as a means to establish the _conceptual_ nature of the principle (relating _any_ 'n' and 'n+1') that leads to the generalization from the first particular instance (typically 0 or 1) in an open ended sequence.

It is not possible to perform a 'completed infinity' of deductions. The generalization is possible only with a conceptual understanding of a principle relating _any_ successive terms in the sequence. The open-ended nature of concepts makes it possible to understand the principle behind Poincaré's idea of an infinite sequence of deductions. (There is also a so-called "strong form" of "the principle of mathematical induction", but it does not change the nature of how it all works conceptually.)

Deduction is commonly used within complex inductive reasoning to establish particular elements within the analysis process and such use of deduction does not make the final generalization not a generalization, i.e., not inductive.

Danno is not going to understand that any more than anything else you explained.

Two points specify the direction of a line but do not "form" it: The concept of a line requires abstracting out the thickness as negligible and irrelevant, and that is what he seems to be missing. There is no such thing as a physical line with zero thickness. That does not mean there are no lines. Mandelbrot has nothing to do with it, and Mandelbrot's analysis and examples could not even be discussed without mathematical concepts of lines, angles, etc.

But that is typical of the way Danno tosses around bromides with no understanding as he jumps from one irrelevancy to another. (Look at how he ignored everything you said as he flitted off to one disconnected slogan after another.)

What is more interesting and important is how common it is for many of those educated in formal mathematics to become defiantly complete rationalists with no depth of understanding beyond the process of symbolic manipulations, all in the name of "logic" and "precision" in their floating abstractions. That is easily avoidable, but there is nothing else in their education to properly explain it either. This is why it is hopelessly impossible to discuss philosophy and mathematics with someone who does not already understand both the mathematics and Ayn Rand's epistemology.

It depends on what he thinks it's "cracked up to be". If it means that understanding with our conceptual faculty is not superior to bears sniffing around the edges of camp grounds searching for garbage (or people) to eat then he's dead wrong. But for those who "crack up" thinking to mean imagined mystic insights accepted on faith, and other fallacies and acts of destruction, then no, thinking isn't what it's "cracked up to be", but that's because they've cracked up thinking.

That the capacity to think includes the capacity to commit fallacies and the pursuit of irrationalities, because our thinking is not infallible and requires rules to reach correct conclusions, does not diminish the value and necessity of thinking for human beings or its overwhelming superiority to the more limited forms of consciousness of other creatures.

Thank you, Robbie! My "law" or 'observation' is even more fundamental than that.. it asserts that EVERYTHING involves 'tradeoffs'... choosing anything instead of something else means accepting the risks and benefits of the choice as well as forgoing the risks and potential benefits of ALL other alternative choices.

Choosing to mandate safety features has costs and benefits, as does allowing the market's choices to drive such availability.

What's usually not part of the decision on which choice to make or implement is any kind of really thorough analysis and publication OF the benefits AND costs of Either or Any of the Alternatives.

An almost trivial measurement might be the selling price adder for implementing the New Feature compared to any kind of forecast of the number of injuries or deaths possibly prevented by the implementation.

Unfortunately, that immediately becomes a 'moral discussion' about the 'value of a life or a limb,' which is pretty much logically impossible to factually determine without some layer of value judgment not quantifiable by rational logic.

As for heinous crimes and capital punishment, I maintain that defining the 'line' that, when crossed, describes 'heinous' is a moral and NOT a 'logical' demarcation, so the ice gets very thin under that decision point very quickly.

Or, if ending one person's life under XYZ circumstances 'differs' from ending a life under other circumstances, well... that's a moral decision, too... and not based on logic either.

But deciding that, if a person commits some kind of activity which is so repugnant to society or a culture that the punishment is clearly advertised to be "we're going to find you and when we do, we're going to put you somewhere where you can't do that again to members of our society," well... that's Truth in Advertising, and the Free Market in Heinous Deeds will determine whether or not anyone (or 'manyone') will CHOOSE to risk that tradeoff!

For me it fits together well.
If, for you or someone else, that doesn't make sense, 'agreeing to disagree' also does not advance the discussion to any agreement, either.

:)

Of course a line exists in nature. Any two points form a line. To say lines do not exists is to deny concepts exist. It is like say people do not exist because they are all different. But to make that statement you have to assume concepts.

If we look at Mandelbrot's work, it is clear a "line" does not exist in Nature. Molecules binded together oriented in one direction is all. Then decay. Assuming parallel lines caused much bad intellectual work. If you read the book I referenced, you will better understand my argument that senses including the brain are measurement instruments.