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Reducing induction

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 Lathanar Posted 8/12/06 1:06 AM Link to discussion Lathanar Dallas, TXPost #: 143 Well Tom and Ahmad, I was wondering what the two of you were so engrossed in out there at the last meeting. I have a question for both of you. Are geometric proofs reduction or induction, and if one or the other, why? I was taught three ways of figuring them out, starting with the postulates and working to the conclusion, starting with the conclusion and working backwards, and an approach where you start from both ends to get to the center. In the end though, all are written as a logical progression working your way from base concepts to the conclusion.- Travis A former member Posted 8/12/06 8:32 AM Link to reply Post #: 39 Well Tom and Ahmad, I was wondering what the two of you were so engrossed in out there at the last meeting. I have a question for both of you. Are geometric proofs reduction or induction, and if one or the other, why? I was taught three ways of figuring them out, starting with the postulates and working to the conclusion, starting with the conclusion and working backwards, and an approach where you start from both ends to get to the center. In the end though, all are written as a logical progression working your way from base concepts to the conclusion.- TravisGeometric proofs can be done inductively, and there are several websites that illustrate this. However, I don't have time to look them up right now.For your examples of how to do a geometric proof, it would really depend upon the nature of the postulate. If the postulate is at the perceptual level -- i.e. can easily be grasped via perception -- then starting with the postulate and working your way to the conclusion would be inductive. Starting at the conclusion and working your way back to the postulate would be reductive. Starting at either end and working your way to the middle would be a combination of induction and reduction.However, many of the geometric proofs I dealt with in classes were much more rationalistic -- i.e. they started with a premise and worked to another premise -- and never dealt with the perceptually given. For it to really be either induction or reduction [not deduction], one would have to have, at a minimum, a diagram, and not merely statements or mathematical equations.And I think that to have a full inductive proof one would have to have diagrams covering the range of possibilities -- i.e. for the Pythagorean Theorem dealing with right triangles one would have to have one triangle with angles (in degrees) 10, 80, 90; one triangle with angles 80, 10, 90; and one triangle having 45, 45, 90; and one triangle with angles not including a right angle (90) to act as a differentia. And then one could have squares on all the sides of the various triangles, and re-arrange them so that one could see that the square of the hypotenuse is equal to the sum of the squares on the other sides; and see that this doesn't work for a triangle that doesn't have a right angle in it.Even though this is usually done with merely one right triangle, I don't think that is sufficient for a proof because there wouldn't be anything to integrate or to differentiate. Following the general outline of how concepts work in Introduction to Objectivist Epistemology, one would need those three to four illustrations to really "get it."[edited to correct reduction versus deduction] Edited by User 2,429,252 on Aug 12, 2006 4:06 PM A former member Posted 8/12/06 4:39 PM Link to reply Post #: 40 One thing that occurs to me regarding the nature of induction is that Miss Rand gave inductive validations (if not inductive proofs) of her philosophic positions via the characters and stories of her novels.The primary conflict in The Fountainhead is the first-hander versus the second-hander. When Roark decides to figure out the principle behind the dean, he does it inductively, by observing the various people he associates with both professionally and personally; and comes to the conclusion that there are people who go by a first-hand assessment of the facts and those who go by second-hand evaluations that may or may not be based on the facts. Even though she doesn't present it in these terms in The Fountainhead, this is basically primacy of existence versus primacy of consciousness (other people's consciousness).The primary issue in Atlas Shrugged is the role of man's mind, and is it a primary in production? One can inductively see the differences between the characters and readily tell who is using their minds and who isn't. This makes it possible to integrate the good guys as those who use their minds productively, versus the bad guys who don't use their minds (except to manipulate others by force or fraud).Her ability to present issues inductively is the hallmark of Ayn Rand. And I would think it is one reason she decided to write novels rather than philosophic treatises, at least at the beginning of her career as a writer. After she had written the novels she could then point to the various characters to illustrate philosophic principles, which is still inductive in nature. A former member Posted 8/12/06 6:22 PM Link to reply Post #: 41 I found a website that has 67! various proofs of the Pythagorean Theorem, some are so complex that it boggles the mind.http://www.cut-the-kn...­Proof number 36 of this page comes the closest to what I think has to be done in order to understand the Pythagorean Theorem inductively, complete with variations in the range of sizes of the sides and variations in the angles.http://www.cut-the-kn...­It's an applet, so you have to let it load up before it will work. The points of the triangle are the controls, use your left mouse button pressed down to move the points. The upper point controls the length of the right-hand side of the triangle; the lower left point controls the length of the bottom side of the triangle; and the bottom right point controls the position of the drawing.Due to the color coding of the squares on the sides of the triangle, one can quite easily perceive (inductively) that the sum of the squares on the sides (right and bottom) equals the square on the hypotenuse.\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$­\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$Philosophic essays based on the philosophy of Ayn Randhttp://www.appliedphi...­Applied Philosophy Online .comWhere Ideas Are Brought Down to Earth!tmiovas@appliedphilosophyonline.comAll rights reserved 2006 by Thomas M. Miovas, Jr. Hammad Hussain Posted 8/12/06 8:12 PM Link to reply user 2469690 San Marcos, TXPost #: 35 To Travis,This is intended _only_ to answer to your question above, speaking for myself. It is not, in any way whatsoever, intended to address or reply to what Tom has written in this thread. (For the reasons, please read the OPAR Study Group thread on this message board.)My position (both in regard to the what the Objectivist view is, and what the truth of the matter is) is that it is a mistake to consider "reduction" to be a third method of inference, in addition to induction and deduction. If one treats it as that (as if Ayn Rand had discovered a third method of inference), I think it indicates a lack of understanding of what reduction is. Reduction, unlike induction and deduction, consists, not of making an inference, but of checking and tracing back, to the percepual level, the inferences (whether inductive or deductive) supporting a non-self-evident item (whether a concept or proposition). As such, reduction is the method of _validating_ a non-self-evident item, whether it is a concept or proposition. (The vadidation of a self-evident item is simply direct observation.) I agree with Dr. Peikoff's definition of "proof" on p. 120 of OPAR, which indicates that all proof is the reduction of a non-self-evident propositon to the self-evident (perceptual data and axioms). The two basic types of reduction, in other words, are conceptual reduction and proof. Observe that the argument "All men are pigs; Socrates is a man; so Socrates is a pig" does not prove the propostion that Socrates is a pig, because the argument does not fully reduce it. The conclusion that Socrates is a pig can be reduced _only_ to the premises that all man are pigs and that Socrates is a man. But one of the premises--the premise that all men are pigs--cannot _in turn_ be reduced. When we try, we find that it contradicts what we observe.But the argument "All men are mortal; Socrates is a man; so Socrates is mortal" does prove the proposition that Socrates is mortal (as Dr. Peikoff indicates on p.138 of OPAR), because the proposition is fully reducible. The conclusion that Socrates is mortal reduces to the premises that all men are mortal and that Socrates is a men. The premise that all men are mortal can be, in turn, reduced to countless observations of men dying. The premise that Socrates is a man (assuming that you live in Ancient Athens and can observe Socrates) can be validated by direct observation. Observe that in this reduction of the conclusion that Socrates is mortal, one is tracing back both deductive and inductive inferences. The conclusion that Socrates is mortal is deduced from the two premises. The major premise that all men are mortal is induced from countless observations. Hence, it is no cotradiction to say that this proof is reductive (as all proofs are) and as well as being partly deductive and partly inductive. If, on the other hand, we reduce the Objectivist proposition that reason is man's basic means of survival, the inferences we will be tracing back will only be inductive inferences. Hence, the proof for the principle that reason is man's basic means of survival will be reductive (as all proofs are) _and_ inductive. This is no contradiction, because the adjectives "reductive" and "inductive" refer to different aspects of the proof. "Reductive" refers to the fact that the proof logically grounds the conclusion in perceptual data; "inductive" refers to the fact that the logical inferences that _do_ the grounding are of a certain type (inductive). (This is similar to saying "The ball is both round and red." "Round" and "red" refer to different aspects of the ball, and it is no contradiction say the ball is both.) With regard to geometric proofs, you asked, "Are geometric proofs reduction or induction, and if one or the other, why?"As I should have made clear, that question rests on a false alternative. A proof does not need to be only one and not the other. Your believing that it does, I think, probably comes from taking reduction as a third method of inference, in addition to induction and deduction. But, as I have said, I think that this is the result of misunderstanding what reduction is. Further, you said, "I was taught three ways of figuring them out, starting with the postulates and working to the conclusion, starting with the conclusion and working backwards, and an approach where you start from both ends to get to the center. In the end though, all are written as a logical progression working your way from base concepts to the conclusion."Which ever way you figure out geometric "proofs" (they are actually not full proofs, but I will say more about that later) is fine, as long as it works for you. Using whatever way you choose, if it works, the conclusion will be _reduced_ to the geometric axioms and definitions. (It, however, is not a FULL reduction of the conclusion, and that is my next topic.)The problem with Euclidean geometry as it is presented in high school is that it doesn't, in fact, present full proofs. This is because it deduces various threorems (propositions) from geometric axioms and definitions, and hence, the theorems can be reduced to the relevant geometric axioms and definitions, but the geometric axioms and defintions are not, _in turn_ reduced to perceptual data. (Geometric axioms, unlike philosophic axioms, are not perceptually self-evident.) Because of this, Euclidean geometry is presented rationalistically. (Rationalism is the epistemological doctrine that knowledge can be derived or validated without reference to sense data.) But the smart, Objectivist student should be able to reduce the geometric axioms and defintions to perceptual data, thus reducing his geometric knowledge to perceptual data, and completing the proofs. Further, after listening to Dr. Peikoff's course _Objectivism Through Induction_, I thought that Euclidean geometry could be reorganized to present inductive rather than deductive proofs. (This, of course, would make it non-Euclidean geometry, but that doesn't matter.) If geometry were presented with inductive proofs in high school, the proofs, I think, would be FULL proofs (i.e. FULL reductions to perceptual data), because the teachers would also prove the geometric axioms and definitions inductively (and thus reduce them perceptual data), instead of erroneously taking geometric axioms and definitions as "self-evident," as teachers of Euclidean geometry do. In fact, I believe that a certain Objectivist intellectual gave a certain course at a conference on reorganizing geometry into a system that is inductive (instead of deductive). I cannot remember her name, but her recorded course should be available from the Ayn Rand Bookstore (www.aynrandbookstore.com). (You should be able to do a search on the website to find the course.)--Ahmad Hassan Lathanar Posted 8/13/06 2:45 PM Link to reply Lathanar Dallas, TXPost #: 145 Okay, so in conclusion, deduction and induction are how you figure out concepts and reduction is how you prove them. When trying to prove something already known or proposed, you use induction and deduction to figure out the logical steps to go from perceptual data to the conclusion, but when looking at the finished product, the proof itself, it's reduction. Is that it in a nut shell?- Travis Hammad Hussain Posted 8/13/06 5:59 PM Link to reply user 2469690 San Marcos, TXPost #: 37 To Travis,You wrote: "Okay, so in conclusion, deduction and induction are how you figure out concepts and reduction is how you prove them."Yes, but proof only applies to propositions, not single concepts. Single concepts, like "organism," "culture," and "justice" are not, strictly speaking, proved, but validated (by reduction to the self-evident). (Remember from OPAR Ch. 1: "Validation" is a concept that is wider than "proof." All proof is validation, but not all validation is proof.) Propositions are statements, and consist of more than one concept, like "The current culture is bad." Just as the validation of a non-self-evident concept is reduction of the concept to the self-evident, so the proof of a non-self-evident proposition is reduction of the propostion to the self-evident. A concept like "goblin" or "ESP" is invalid because it cannot be reduced to the perceptually self-evident. Simalarly, a proposition like "President Bush engineered the attacks of 9/11/01" is false (or arbitrary) because it cannot be reduced to the perceptually self-evident (i.e. it cannot be _proved_). (If _some_ evidence could be provided, then we could consider the proposition "possible" rather than "false" or "arbitary." But in this case, I don't think there is _any_ real evidence.) Further, you wrote:"When trying to prove something already known or proposed, you use induction and deduction to figure out the logical steps to go from perceptual data to the conclusion, but when looking at the finished product, the proof itself, it's reduction."I think you are close to correct here. But if you really understand "reduction," and its relationship to induction and deduction, remember that proof IS reduction (of a non-self-evident proposition to the self-evident). Thus, you should be able to replace "prove" in the first part of your sentence with "reduce." Hence, it would read, "When trying to _reduce_ something already known or proposed, you use induction and deduction to figure out the logical steps to go from perceptual data to the conclusion..." And that is a true statement. Now, all we have to do is replace your conjunction "but" with "and." The result is: "When trying to reduce [i.e. prove] something already known or proposed, you use induction and deduction to figure out the logical steps to go from perceptual data to the conclusion, _and_ when looking at the finished product, the proof itself, it's reduction." And that's a true sentence. Is that it in a nut shell?With my modifications, yes.I hope my comments help. --Ahmad Hassan Okay, so in conclusion, deduction and induction are how you figure out concepts and reduction is how you prove them. When trying to prove something already known or proposed, you use induction and deduction to figure out the logical steps to go from perceptual data to the conclusion, but when looking at the finished product, the proof itself, it's reduction. Is that it in a nut shell?- Travis A former member Posted 8/16/06 8:59 PM Link to reply Post #: 42 The topic is basically: What is induction?Taking a concept (or a conclusion) and working ones way back down the hierarchy and pointing to an item representative of the concept (or conclusion) is not inductive; it is reductive. So, reduction does not involve induction, even though both deal with the perceptually self evident; reduction at the end of the mental process, and induction at the beginning of the mental process. And it is important to realize that we are talking about mental processes here, and not the final product per se. The final product of both a reduction and an induction is that one has tied one's concepts (or conclusions) to the perceptually self-evident; but one goes from the conclusion to the evidence (reduction) while the other goes from the evidence to the conclusion (induction).Induction, in general, follows the outline of forming concepts in Ayn Rand's Introduction to Objectivist Epistemology: One has to have made enough observations within a contextual range to be able to make an integration of the items into a concept (or into a conclusion). One example that I have already given is observing that the sum of the squares of a right triangle are equal to the square of the hypotenuse; by noticing that this relationship holds true for all right triangles by varying the lengths of the sides and observing that the relationship holds true for all of them. In this type of mental operation, it's all observation and integration.Reduction, in general, means taking apart the integration into it's components, as Dr. Peikoff did with the concept "friendship" in OPAR (pg 133). Taking things apart is different than putting them together. And so long as one realizes that reason is an integration of sensory data, then taking a concept (or a conclusion) apart ultimately means being able to point to something in reality. For a complicated reduction, one may have to do this with the various concepts integrated into the concept one is reducing, each one being reduced to something that can be pointed to.\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$­\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$Philosophic essays based on the philosophy of Ayn Randhttp://www.appliedphi...­Applied Philosophy Online .comWhere Ideas Are Brought Down to Earth!tmiovas@appliedphilosophyonline.comAll rights reserved 2006 by Thomas M. Miovas, Jr. A former member Posted 8/16/06 11:35 PM Link to reply Post #: 43 I've drawn a chart to better illustrate my position regarding induction and reduction. Read induction from bottom to top; read reduction from top to bottom.\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$­\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$Philosophic essays based on the philosophy of Ayn Randhttp://www.appliedphi...­Applied Philosophy Online .comWhere Ideas Are Brought Down to Earth!tmiovas@appliedphilosophyonline.comAll rights reserved 2006 by Thomas M. Miovas, Jr. Edited by User 2,429,252 on Aug 16, 2006 11:38 PM A former member Posted 8/17/06 3:00 PM Link to reply Post #: 44 Since Ahmad has made the nature of proof part of this discussion, I decided to do a little research using The Ayn Rand Lexicon by Harry Binswanger and the Objectivism CD-ROM composed by Phil Oliver.In The Ayn Rand Lexicon, I found an entry under Proof:"'Proof,' in the full sense, is the process of deriving a conclusion step by step from the evidence of the senses, each step being taken in accordance with the laws of logic. [Leonard Peikoff, "Introduction to Logic" lecture series (1974), Lecture I.]"By my understanding of the nature of induction, this is proof by induction, not reduction; even though in OPAR (written much later, 1991) Dr. Peikoff says that all proof requires reduction, which, by my understanding of reduction means going from the conclusion to the evidence backwards down the hierarchy.Here are some entries from the Objectivism CD-ROM:[Note: in this part of the research, I did not find a formal definition of proof given by Ayn Rand. However, it is clear in her usage of the term that she means cognitive content must be logically tied to the evidence of the senses irrefutably, but she doesn't say anything about it must be presented going up the hierarchy or going down the hierarchy. For some issues, such as a discussion about music theory, she goes into a lot of details about what would be required for the theory to be proven; she takes the theory apart in some detail, and shows what facts would have to be presented for the theory to be proven.]Unfortunately, the Objectivism CD-ROM does not include quotations from Dr. Peikoff's lecture series; but here are some quotes from Dr. Peikoff, that are a more formal definition of "proof" from OPAR:"This kind of chain and nothing less is what Objectivism requires as "proof" of an idea."'Proof' is the process of establishing truth by reducing a proposition to axioms, i.e., ultimately, to sensory evidence. Such reduction is the only means man has of discovering the relationship between non-axiomatic propositions and the facts of reality." [OPAR, pg 120]"Proof is the process of establishing a conclusion by identifying the proper hierarchy of premises. In proving a conclusion, one traces backward the order of logical dependence, terminating with the perceptually given. It is only because of this requirement that logic is the means of validating a conclusion objectively."[OPAR pg 138]So, Dr. Peikoff, in my understanding of the issue, says that a proof can be established both inductively and reductively.Now, here's one way I can resolve the issue for myself. For most instances of proving something, one will start with the conclusion and then offer a proof of it. Or someone will state a conclusion and someone else will say: Prove it! Which means one is starting from the conclusion and working towards a proof i.e. one is trying to prove an assertion ultimately by referring to evidence.However, I think there are two ways of doing this: 1) jump immediately to the evidence and logically work one's way up to the conclusion (inductive proof), and 2) logically follow the reasoning backwards down to the evidence (reductive proof).Both methods require logically connecting the content of consciousness to the evidence, but there is a difference in how one does this: forwards or backwards.In my mind, this brings up a question: Does an inductive proof require reduction?Referring to my induction / reduction chart, the question becomes: If someone were to start with the concept "organism" and then points to a grasshopper, a flower, an ant, and a blade of grass, would that be sufficient in saying that he reduced the concept "organism"? Because if it is sufficient, then even an inductive proof would mean doing a reduction.I'll have to think about this some more.
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