Dufresne Posted October 17, 2005 Report Share Posted October 17, 2005 The problem with that type of reasoning is that it misses the point.The point being? Quote Link to comment Share on other sites More sharing options...

RSalar Posted October 17, 2005 Author Report Share Posted October 17, 2005 That isn't true. The Hubble Heritage project collects photographic images of distant bodies in the universe. Planets are among these (there's one really beautiful blue gaseous planet). It's beside the point, but inaccurate. Astronomers Confident: Planet Beyond Solar System Has Been Photographed By Robert Roy Britt Senior Science Writer posted: 10 January 2005 04:32 pm ET SAN DIEGO -- Astronomers are highly confident that they've taken the first photograph of a planet outside our solar system. Make that two photographs. A new image from the Hubble Space Telescope confirms with a high degree of confidence a picture made previously by astronomers at the European Southern Observatory (ESO) and reported by SPACE.com in September. The planet -- still just a candidate, actually -- is an odd duck in many respects. It does not orbit a normal star, and it is much more massive than the largest planets in our solar system. Still, if confirmed, it represents a landmark in astronomy along the road to the ultimate goal of finding and photographing Earth-like planets around other stars. The entire article can be found @ http://www.space.com/scienceastronomy/aas_...net_050110.html Quote Link to comment Share on other sites More sharing options...

RSalar Posted October 17, 2005 Author Report Share Posted October 17, 2005 This is a mistaken notion of induction. Have you read all the threads on induction on this forum? (You can search for threads whose titles contain "induction" with the search feature.) I highly suggest those if you are interested in understanding the error in your question. In particular, the argument that induction is method of causality (not counting) has been made several times. If there any specific parts of the argument that you don't understand or disagree with, I'd be happy to elaborate further. Thank you. I attempted to do what you suggested -- read through the threads about induction -- and WOW! How does one separate the wheat from the chaff? How about a good book on the subject? I thought that induction was basically the process of taking observed individual occurrences and making a broad true statement about them. Quote Link to comment Share on other sites More sharing options...

DavidOdden Posted October 17, 2005 Report Share Posted October 17, 2005 Are you saying that because it may be possible to discover another method, other than deductive inference, to describe the relationship between integers, that my statement is not universally true?Since your statement was a "would not" and not just a "did not", I would just say that your statement isn't true: specifically, it is arbitrary (universality wouldn't be meaningful given your question since the question itself includes the universal). I was asking you to move it out of the arbitrary category by giving some proof that inductive / experimental approaches could not also have created mathematical methods. Quote Link to comment Share on other sites More sharing options...

dougclayton Posted October 17, 2005 Report Share Posted October 17, 2005 Thank you. I attempted to do what you suggested -- read through the threads about induction -- and WOW! How does one separate the wheat from the chaff? The same way you separate the wheat from any chaff--lots of thinking and introspecting. I wasn't trying to claim you could read them once and understand everything. I remember reading the first thread on it over a year ago, and being shocked to read for the first time that induction was simply not counting swans or sunrises and generalizing into the future. It wasn't until I came back to this forum several months ago that the proper approach started to make some sense (the notion of "causality" is key). How about a good book on the subject?Sadly, I don't think there is a book (yet) that is better than the combined postings of the OO.net members (some more than others, of course). I thought that induction was basically the process of taking observed individual occurrences and making a broad true statement about them. Well, it is, if you stretch the word "basically" far enough. There are many different ways of "taking observed individual occurrences and making a broad true statement about them," and not all of them are valid logical methods. (The fallacies of "hasty generalization" and "post hoc ergo propter hoc" come to mind, for instance.) Quote Link to comment Share on other sites More sharing options...

AisA Posted October 17, 2005 Report Share Posted October 17, 2005 I thought about David's swan example in which someone inductively and properly arrives at the principle that all swans are white and then tries to apply that principle deductively to another swan not yet observed (which could be black). As Doug pointed out, induction is not achieved by enumeration but by causal identification so the swan example could be a little misleading unless the person has identified the cause of the color. But even then, the cause of the color could be absent in a yet unknown swan. Or there could be an additional cause superseding the cause for white. The other example was RSalar's astronomy example. From those two examples I induced that deduction is not 100% certain (regardless of context) because principles are created in a given context of knowledge but do not necessarily have to apply to facts that are only known in a bigger context of knowledge. Let me clarify my position. There is a distinction between a qualification and a contradiction. Consider a first-level generalization -- an induction -- that we have discussed in another thread: As a child, you induced by direct perception that an object called a "ball" will roll across a level, flat surface when pushed. (There are many such first-level generalizations learned by direct perception.) By a process of measurement-omission (just as it is employed in concept formation), you grasp that any size ball will have this property. Thus, when you see a new ball that is much larger than any you have previously seen, you may deduce that "This object is a ball, therefore it will roll when pushed on a flat surface." Is the knowledge gained by the induction "Pushed balls will roll on flat surfaces" certain? Yes, it is certain, and real, but it is also limited. "Limited" in this context means, "capable of being qualified by additional conditions." For instance, you may encounter a ball that is glued to a surface and does not roll when pushed. Does this contradict your previous induction? No, it merely qualifies that the induction depends on the previously unstated condition that the ball is not physically attached to the flat surface. For an excellent discussion of how new discoveries can qualify inductions -- and not contradict them --see pages 173 - 181 in OPAR. Quote Link to comment Share on other sites More sharing options...

dougclayton Posted October 17, 2005 Report Share Posted October 17, 2005 Mathematicians tend to eschew observation and induction (except for "mathematical induction" which is really a type of deductive inference), so they would be exceptions. You know, that's something I've been wondering about lately. I used to be bothered by the term "induction" used for the mathematical method of proof that consists of the following steps: 1. Show something is true for a base case (say, n = A). 2. Show that if something is true for n, then it is also true for n+1. 3. By extension, since it is true for A, A+1, A+1+1, etc, it is true for all n >= A. But the more I think about it, the more I think there is fair cause for calling that induction. Specifically, the essential aspect of the proof is step 2: showing that the truth for n+1 follows directly from the truth of n and the mathematical properties of numbers. And, in the end, we have a wide claim for all n that has only been directly observed for some n. This means that mathematical induction is ladder-like, in that there is a starting point directly observed and a "link" in a causal chain. Not all induction has this form, so mathematical induction is a subset of induction in general. Furthermore, some inductive conclusions seem to have this mathematical form. For instance, the (proper) claim that the sun will rise every 24 hours would take the "base case" that the Earth is observed to have a certain rotational velocity (by measuring the time elapsed from noon to noon). Then we show that, by the laws of mechanics, rotation will continue unabated since there is no external force acting to slow down the rotation. Thus, we know that the sun will appear to rise every day in the future. In a sense, the laws of mechanics fill in for #2, that is, if there is a celestial body with a given rotational velocity and orientation, then N hours later it will be in the same state again. Direct observation of the celestial positions today and tomorrow gives us #1, and it is then a deductive inference that #3 will hold. (Naturally, there is actually some slowing down that occurs, so you would have to speak of measurable deviation from the current length of "one day.") Although I am pretty convinced of the validity of this view, I definitely welcome comments or criticism. I don't want to commit the fallacy of "hasty generalization" myself. Quote Link to comment Share on other sites More sharing options...

dougclayton Posted October 17, 2005 Report Share Posted October 17, 2005 But the more I think about it, the more I think there is fair cause for calling that induction. Boy, there's nothing like posting something to a public forum for making you questions your assumptions. In particular, both my examples were the application of wider principles to a specific context (algebraic manipulation in one and Newtonian mechanics in the other). Thus I can see how they could be called deductive, rather than inductive. The truly inductive conclusions would be, say, the Newtonian mechanics in the first place. I still welcome feedback, but I will have to think more about this. Quote Link to comment Share on other sites More sharing options...

Felix Posted October 17, 2005 Report Share Posted October 17, 2005 Specifically, the essential aspect of the proof is step 2: showing that the truth for n+1 follows directly from the truth of n and the mathematical properties of numbers. And, in the end, we have a wide claim for all n that has only been directly observed for some n. I don't think this is correct. The reason this is called 'complete induction' is because it looks at every single n. That's why it is complete. Incomplete induction is what physicists do. They can only look at several parts of reality and find an essential similarity. Mathematicians can look at all n at once by means of a formula. The reason for this is the abstraction level of mathematics. I think that's why it's called induction. Step 2 lets you look at the next n, and then the next and then the next ... You show that the property you want proven is linked to the fact that n is a natural number. Complete induction derives the truth of the statement by testing it for every single n. It finds out the truth of the statement that says: If n is a natural number, then this statement is always true. Then it goes through all natural numbers. You start out with 1 as the first natural number and then the next and then the next. That's the beauty of mathematics. You can test it for all n till infinity at once. Inductively. The formula does all the work for you, practically. If you tried this in biology you would have to look at every single bird and look if it has the same characteristics as the last bird. Then the next, then the next ... In mathematics this hideous work is done by step 2. This is quite brilliant, now that I think of it. Quote Link to comment Share on other sites More sharing options...

dougclayton Posted October 17, 2005 Report Share Posted October 17, 2005 Complete induction derives the truth of the statement by testing it for every single n. But "testing it for every single n" is exactly what a proof by induction does not do. That would mean, for the claim that 1 + 3 + 5 + ... + (2n-1) = n^2, you would have to compute the sum of all odd integers from 1 to 2n-1 and see that the computation was equal to n^2, for every single n. In a proof by induction, you do this just once, for the base case. The equation is shown to be true for every n, but it is not tested for each n. If you tried this in biology you would have to look at every single bird and look if it has the same characteristics as the last bird. Then the next, then the next ... I hate to contradict you again, but this is what science does when it uses induction--only it no more looks at "each bird" than a mathematician evaluates the formula above for "each n." How do we know that man is mortal? Certainly not by looking at every single man to see if it has the same characteristics as the last man, then the next, then the next... Quote Link to comment Share on other sites More sharing options...

DavidOdden Posted October 17, 2005 Report Share Posted October 17, 2005 Doug, Well, I started to say stuff and wandered away, and I see there have been some changes in the interim. Now I gotta whack a few lines here and there. There is a point which I used to carp on incessantly, that in a formal system, induction and deduction are not different in an earth-shattering way: both are inferences. Whether you use the truth table method, named rules of inference like modus tollens, or symbol substitution and a list of axioms as in Kleene's textbook really isn't important. From a formal POV, induction refers to something like the rule inductive generalization, which introduces a universally quantified proposition. The main obstacle to integrating induction into formal inference has been the problem of deriving ^Ex(P(x)), which has been a problem given the acontextual nature of most logics. Here's my claim: from a formal point of view, "induction" is a rule of inference which introduces ^Ex(P(x)) when in a set of propositions that define a (knowledge) context P(y) is not present (for any value y). The rest is garden variety deductive logic. The problem from the classical POV is that you don't take into consideration such a thing as "the set of propositions admitted to be true" (a knowledge context), but once you have such a concept, it's obvious that introducing ^Ex(P(x)) when P(x) is not true does state the same kind of truth as AV^A. As far as real-world scientific induction is concerned, the only problem that exists (and it's not a small one) is that the relationship between the propositions that we recognise as true and the facts of existence is not automatically guaranteed. Most important is that we don't know everything (not all conceivable propositions that correctly describe reality are recognised by man -- any one man or all men), so for example Newton simply did not recognise certain other propositions which were later considered, which describe mutual attraction between objects. But we operate on the principle that if we have correctly identified the nature of reality, then when our knowledge context expands, our conclusions will remain valid. Sometimes that doesn't happen. As I understand the basis of mathematical induction and as you got (using 1+2+3..+n which was what I was thinking of), step 2, formally showing that "if true for N then true for N+1" is the hard part and that has to be proven deductively, by for example applying rules that do not require consideration of the entirety of a knowledge context. This similarity that you point to between mathematical induction, inferences about the sun rising, also the concepts of iteration and recursion, in fact science in general, all seem to be grounded in a fundamental fact of cognition and reality, which I supposed can be reduced to a slogan like "There is no magic". In the mathematical realm it means applying methodological principles consistently, and in the physical realm it means applying causal laws consistently. That is essentially what it means to be a law. Quote Link to comment Share on other sites More sharing options...

Dufresne Posted October 17, 2005 Report Share Posted October 17, 2005 Let me clarify my position. There is a distinction between a qualification and a contradiction. Consider a first-level generalization -- an induction -- that we have discussed in another thread: As a child, you induced by direct perception that an object called a "ball" will roll across a level, flat surface when pushed. (There are many such first-level generalizations learned by direct perception.) By a process of measurement-omission (just as it is employed in concept formation), you grasp that any size ball will have this property. Thus, when you see a new ball that is much larger than any you have previously seen, you may deduce that "This object is a ball, therefore it will roll when pushed on a flat surface." Is the knowledge gained by the induction "Pushed balls will roll on flat surfaces" certain? Yes, it is certain, and real, but it is also limited. "Limited" in this context means, "capable of being qualified by additional conditions." For instance, you may encounter a ball that is glued to a surface and does not roll when pushed. Does this contradict your previous induction? No, it merely qualifies that the induction depends on the previously unstated condition that the ball is not physically attached to the flat surface. For an excellent discussion of how new discoveries can qualify inductions -- and not contradict them --see pages 173 - 181 in OPAR. I've thought about what you wrote and read the recommended section from OPAR. Would you say that the following metaphor is adequate? Suppose the core of an onion represents the product of the first induction. It is a generalization that is based on a number of specific observations which constitute the "cognitive context" of the induction. Then new observations are made and a new layer is added to the onion. This new layer represents the qualification of the original induction. Thus it is the same onion, it just has become a little "richer". The inner core still refers to the first set of observations and thus is still true, the layer around the core refers to the second set of observations. But the whole onion still is one idea, just richer, bigger than before!? Quote Link to comment Share on other sites More sharing options...

Felix Posted October 18, 2005 Report Share Posted October 18, 2005 But "testing it for every single n" is exactly what a proof by induction does not do. That would mean, for the claim that 1 + 3 + 5 + ... + (2n-1) = n^2, you would have to compute the sum of all odd integers from 1 to 2n-1 and see that the computation was equal to n^2, for every single n. In a proof by induction, you do this just once, for the base case. The equation is shown to be true for every n, but it is not tested for each n. I thought a little again. You're right. Part 2 of induction bases the truth of the statement for n+1 on the truth of the statement for n. This is deduction. The truth of the statement for n+1 is based on the truth of the statement for n. The question now is: Why is it called induction when most of it is deduction? And I think I have an answer to this. It is called complete induction because doing the inductive step (step one) just once is enough. The rest can be proven via deduction. But you still need to perform one test. It doesn't derive directly from principles. (Well, in fact it does, but you have to test it for the case to find out.) I never thought much about the philosophical implications of mathematics (therefore my error, but also my post, I suppose), but this is fascinating. I hate to contradict you again, but this is what science does when it uses induction--only it no more looks at "each bird" than a mathematician evaluates the formula above for "each n." How do we know that man is mortal? Certainly not by looking at every single man to see if it has the same characteristics as the last man, then the next, then the next... Well, since I understood what you said above, this becomes obvious. But thanks again for pointing out. You're a clear thinker. Quote Link to comment Share on other sites More sharing options...

dougclayton Posted October 19, 2005 Report Share Posted October 19, 2005 (edited) Well, I have to say, David, that your familiarity with formal logic is both appealing and distracting: it's good to see someone who is clearly knowledgeable enough not to throw out the baby (formal notation and rigor) with the bathwater (the philosophy behind modern formal logic). On the other hand, it can be hard to read to someone whose last use of this was in college a decade ago. So let me try to translate: From a formal POV, induction refers to something like the rule inductive generalization, which introduces a universally quantified proposition. The main obstacle to integrating induction into formal inference has been the problem of deriving ^Ex(P(x)) Given that you talk about a universally quantified proposition, I would expect you to refer to deriving: for all x, P(x) is true Granted the upside-down A is hard to do in plain text, but typing in "& forall;" (without the space or quotes) and previewing twice does it: ∀ (at least in the Opera browser). But since you have what appears to be ∃ (& exist;) as a forwards E and ^ as a negative, I get: there is no x such that P(x) is true This is identical to for all x, ^P(x) But usually one does not express things in the negative: Newton's third law is not "there is no action that has an unequal or non-opposite reaction," even though it is logically identical. Thus I will assume you mean "for all x, P(x)." Here's my claim: from a formal point of view, "induction" is a rule of inference which introduces ^Ex(P(x)) when in a set of propositions that define a (knowledge) context P(y) is not present (for any value y). In plain English, I would expect this means that induction introduces a new universal proposition P that cannot be deduced from the existing known propositions. (If it can be so deduced, you get the flaw I referred to earlier, in that mathematical induction is not introducing new propositions, but extracting truths that derive from pre-existing propositions.) The rest is garden variety deductive logic. The problem from the classical POV is that you don't take into consideration such a thing as "the set of propositions admitted to be true" (a knowledge context), but once you have such a concept, it's obvious that introducing ^Ex(P(x)) when P(x) is not true does state the same kind of truth as AV^A.I could speculate on what this means, but several different interpretations I made all yielded contradictions, so I will just have to let you tell me. (I know AV^A is intended to be a tautology because it must be "true OR false" or "false OR true.") There is a point which I used to carp on incessantly, that in a formal system, induction and deduction are not different in an earth-shattering way: both are inferences. I am coming to believe this. I should probably get Dr. Peikoff's lectures on this to find out more on how "induction is measurement omission applied to causality." Addendum: I get blocks instead of the "forall" and "exists" symbols in Internet Explorer, which I expect many forum members use. Sorry. Edited October 19, 2005 by dougclayton Quote Link to comment Share on other sites More sharing options...

DavidOdden Posted October 19, 2005 Report Share Posted October 19, 2005 This is identical to for all x, ^P(x) But usually one does not express things in the negative: Newton's third law is not "there is no action that has an unequal or non-opposite reaction," even though it is logically identical. Thus I will assume you mean "for all x, P(x)." I guess I had forgotten that this board is basically an HTML repository, so I stuck with the old habit of using A and E. It is true that I want to get to something like "for all x, P(x) is true", since that is the ultimate conclusion that we would be interested in; and especially, from the perspective of seeking knowledge, I want to know what is, not what isn't. Whether or not this "P" thing is conceptually positive or negative wasn't crucial to me, although it is rather important to whether anyone understands what you're saying. There was a subtle point about doing this negatively having to do with concluding "There is no counterexample", but it was so subtle that I seem to have managed to make it go away over the past day, so I apologize for torturing you with too much negative existential verbiage. Later when I say that getting from a "there is no" to an "All X's are..." is by ordinary deduction, that simply means that "There is no x such that P(x) is false" is deductively equivalent to "All x's such that P(x) is true", but what the heck, why not go directly to the universal? Again doing what formal logic does, only looking at collections of symbols, and not considering actual cognition and knowledge, if I have an assemblage of propositions: P(y); P(j); P(l); Q(i); ^Q(k) then in this collection, for all individuals, P(x) is true. But (classically) I cannot introduce "∀xP(x)". OTOH, you can formally introduce "∃x(Q(x))", given Q(i). All's I'm saying here is that it is an arbitrary stipulation (one defining a particular logic) which declares that you can introduce "∃x(Q(x))" here but not "∀xP(x)". The reason for this difference is that the concept of "context" is alien to most versions of logic (this has to do with "monotonicity", which is a way of saying that if you can derive X from some premises, then you can still derive X no matter how much you add to your premises). I try not to go off on the formal tangents too often: it's usually just to make it clear that the notion of induction does have a solid position in formal logic which isn't just about "probability" (many people mistaken believe that induction is outside of the reach of formalization), and that the notion of "valid inference" is as applicable to induction as it is to deduction, even in formal logic. In plain English, I would expect this means that induction introduces a new universal proposition P that cannot be deduced from the existing known propositions. (If it can be so deduced, you get the flaw I referred to earlier, in that mathematical induction is not introducing new propositions, but extracting truths that derive from pre-existing propositions.)I suppose I'd agree: my reservation is in the word "deduced". You can introduce that new proposition, the integration of specific instances, and that is a formalizable operation but it is not called a "deduction".The formal models stuff is sort of interesting to me, but much less interesting than the question of how knowledge is actually created. Quote Link to comment Share on other sites More sharing options...

dondigitalia Posted October 19, 2005 Report Share Posted October 19, 2005 As I understand the basis of mathematical induction and as you got (using 1+2+3..+n which was what I was thinking of), step 2, formally showing that "if true for N then true for N+1" is the hard part and that has to be proven deductively, by for example applying rules that do not require consideration of the entirety of a knowledge context. This correlates to something I was saying to my roommate yesterday. My observation was that in mathematics, we deduce our premises, but apply them by induction, while, in everyday thinking, we induce our premises, but apply them by deduction. Quote Link to comment Share on other sites More sharing options...

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