Theorem or Postulate?
Date: 11/03/2002 at 17:36:59 From: Maureen Hamilton Subject: Geometry textbook inconsistencies I am homeschooling my daughter in math. I want to teach her proofs, which have just been removed from the high school curriculum. My problem is that there seems to be a great discrepancy between definitions, postulates, and theorems, from textbook to textbook. What is a postulate in one book is a theorem in the next, and vice versa. Is there a textbook that faithfully follows Euclid's Elements? I have the translation, but the language is quite formal and of course there aren't any problems to work out. Your help would be most appreciated in this matter.
Date: 11/04/2002 at 08:59:07 From: Doctor Peterson Subject: Re: Geometry textbook inconsistencies Hi, Maureen. It's important to be aware that there is no one "correct" axiomatization of geometry; a number of different schemes have been developed by reputable mathematicians, not to mention by textbook authors who are trying to keep things simple for students. Certainly the variety of postulates used in texts makes it hard for me as a Math Doctor to answer questions about proofs, since I have to ask what facts are available to the student; but in a sense that helps in our mission, since our goal is to help in understanding, not to give specific answers. By making the student look for whatever postulate or theorem in their book corresponds to a fact I mention, I help them learn how to put ideas together. That's relevant to you, because your goal is not to teach a particular "correct" set of postulates, but to teach reasoning (using whatever postulates and theorems are available) and geometric facts (all of which are agreed upon, even if we don't agree on which to call postulates and which to call theorems, or on what the theorems are ultimately based on). The educational results don't really depend on the details of the system used. It is definitely not true that Euclid is the one right way to do geometry. Despite the centuries when the Elements were treated as the Bible of geometry, there are many flaws in his treatment, and many new ideas have been introduced since then. For example, Euclid does not use the concept of "congruence" as we understand it today. So although going through Euclid can be a very enlightening experience, it need not (and probably should not) be a student's first exposure to axiomatic geometry. So I would recommend not looking for the text closest to Euclid, but for the text that best presents the concepts and demonstrates how proofs work. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 11/11/2002 at 10:06:51 From: Maureen Hamilton Subject: Geometry textbook inconsistencies Thank you for your response. However, I have many more questions. In all the textbooks that I am perusing I notice that there are three triangle postulates - SSS, ASA, and SAS. Shouldn't these be theorems? Are they postulates because the proof is beyond the scope of the high school student, or are they postulates because they cannot be proved?
Date: 11/11/2002 at 12:08:32 From: Doctor Peterson Subject: Re: Geometry textbook inconsistencies Hi, Maureen. As I said before, there are different axiomatizations of geometry. Some of them prove SSS, SAS, and ASA as theorems, as Euclid does; to do so properly they need a better definition of congruence than Euclid's concept of "superposition," with a clear set of axioms governing the motions that are allowed when one shape is put on top of another. Hilbert's famous axiomatic system takes SAS as a postulate, and derives SSS and ASA from it as theorems. One textbook I have makes ASA and SAS postulates, but proves SSS. Many texts seem to make all three theorems, probably, as you suggest, in order to avoid confusing students with a rigorous proof when they are not ready for such details. (I would prefer to present a good set of postulates and just say honestly that we are skipping the proof of some of these theorems because they are too difficult to go into yet.) In each case, whatever is chosen as a postulate is such because it can't be proved _from the other postulates that have been chosen for that presentation_. Some of these systems are clearly deficient, but it can be hard to say which is best. Here are some discussions of Euclid's, Hilbert's, and other systems that I have found: Euclid's Postulates - Canadian Society for Biomechanics http://www.health.uottawa.ca/biomech/csb/laws/euclid.htm Lecture Notes: The Axioms of Euclid and Hilbert, and the Parallel Postulate (PDF) - Jeff Connor http://www.math.ohiou.edu/~connor/geometry/chap3/betweenlct.pdf Axiomatic Systems for Geometry - George Francis (PDF) http://www.math.uiuc.edu/~gfrancis/M302/handouts/postulates.pdf Teaching Geometry According to Euclid - Robin Hartshorne (PDF) http://www.ams.org/notices/200004/fea-hartshorne.pdf Euclid's Elements, Book I, Proposition 4 - David Joyce http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI4.html Congruence Of Triangles - Jim Loy http://www.jimloy.com/geometry/congruen.htm It is true that many of the first theorems students are asked to prove are so obvious that they seem not worth proving; but that is only because they are deliberately easy, and are little more than a concatenation of postulates and definitions. That may well leave some students wondering why we need proofs, when they are so trivial. They need to be introduced early to a surprising proof, even if they can't follow it all yet, just so they can see the value of proofs. And they should also be shown a false proof, so they can see the need for care in each step, and will not think that the "obvious" is always true. The goal of an axiomatic system is to reduce the number of assumptions we make to a minimum (as far as possible) so that all our reasoning is based on readily accepted 'facts'. So postulates should be as 'obvious' as possible; yet the fact that something seems obvious is not enough to make it a postulate, since it may be provable from existing postulates, so that it would be redundant. On the other hand, it is not required that we prove a set of postulates is minimal in order to use them. As the last link I gave above mentions, SSS, SAS, and ASA are all equivalent postulates, so that only one of them need be postulated; but it is common to make them all postulates, and that is not illegal, just unnecessary. The problem is that this produces a bloated set of postulates and gives a false sense that geometry has to make unnatural assumptions. You will note that although Euclid's fifth postulate is such because it could not be proved from his other postulates, there have been many alternative ways to phrase it in order to make it seem less arbitrary. Any of those versions is a valid postulate. Math is at root a somewhat arbitrary endeavor; there are many ways to choose starting points for the same field. Yet ultimately it makes no difference; who cares whether SAS is a postulate or a theorem, when you need to use it in a later proof? You just write SAS and know that it is true, one way or the other. So although the variations in postulates will make a difference in the details of a student's work, and might cause some confusion if he moves to a different text in the middle of a course, none of the important things is affected: all the same facts are true, and the importance of proof is still being demonstrated. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 03/19/2003 at 14:21:43 From: Rahul Subject: Are SSS, SAS and ASA Congruences THEOREMS or POSTULATES? Dear Dr. Math, I see in my textbook that SAS is a postulate and others (ASA, SSS...) are theorems. I know that using SAS you can prove the others; therefore they are theorems but SAS is a postulate. But in some books SAS is named as a theorem and proved using superposition. I read on the Internet that Euclid first proved these using superposition, but he was not happy with his proof as he tried to avoid superposition in other situations. And now superposition is not legal as it contains complicated assumptions. In the Dr. Math archives I find some doctors saying that SAS, SSS, ASA, etc. are postulates and cannot be proved, but some of the math doctors still refer to them as theorems. Which are they? Thanking you, Rahul Radhakrishnan.
Date: 03/19/2003 at 17:03:06 From: Doctor Peterson Subject: Re: Are SSS, SAS and ASA Congruences THEOREMS or POSTULATES? Hi, Rahul. One man's theorem can be another's postulate. This is because we can choose different starting points for our development of mathematics, and often we find that two different facts can be proved from one another, so that it makes no difference which we take to be more fundamental. In our own answers, since we don't know the particular approach being taken by a student's text, we often use the word "theorem" to refer to any fact that is known to mathematicians, whether proven or taken as an axiom. In the best systems, no more than one of the congruence "theorems" needs to be a postulate; but it doesn't really matter which one that is! In some system, any of them can be a theorem. You will be especially interested in the last of the links I gave above: Congruence Of Triangles - Jim Loy http://www.jimloy.com/geometry/congruen.htm If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 03/23/2003 at 08:02:53 From: Rahul Subject: Re: Are SSS, SAS and ASA Congruences THEOREMS or POSTULATES? Dear Dr. Peterson, Thank you very much for your prompt and lengthy reply. I understand that all three congruence theorems (or postulates or whatever we call them) can be considered postulates though it is unnecessary. By considering one as a postulate we can prove others. So far I am clear. But if I say all three are theorems, am I right? I mean, is there any way of proving them without considering one a postulate (except superposition)? Second, is superposition valid now? Is it legal? Thanking you, Rahul Radhakrishnan
Date: 03/23/2003 at 23:15:52 From: Doctor Peterson Subject: Re: Are SSS, SAS and ASA Congruences THEOREMS or POSTULATES? Hi, Rahul. In order to turn all three rules into theorems, in the sense of being proved from postulates, you need something equivalent to superposition on which to base them. I would not say this is "illegal" in general; it simply is not used in the most popular current presentations of Euclidean geometry. As I said before, there is no one "correct" system of postulates, but many alternatives that can be chosen; and with a careful definition of the motions allowed when you move one figure onto another, as in transformational geometry, I think there is no reason not to turn superposition into a clearly defined concept on which congruence can be based. Here are some references to the validity of superposition in transformational geometry: Summary of Axioms of Congruence - Draga Vidakovic, Georgia State University http://www.cs.gsu.edu/~matdnv/Math6371/ReadingNotes/ch3congr/congruence.html Euclid did not assume SAS (Axiom 6) as an axiom but tried to prove it as a theorem. His proof involved taking one triangle and theoretically placing it on top of another. Because he had never stated an axiom that allows figures to be moved without changing their size and shape (a concept called superposition), the proof was invalid. Later on, transformational geometries would be created that would allow this proof to be valid. However, based on the given axioms, it is not considered a valid proof. Geometric Congruence - Eric Weisstein, World of Mathematics http://mathworld.wolfram.com/GeometricCongruence.html Two geometric figures are said to exhibit geometric congruence (or "be geometrically congruent") iff one can be transformed into the other by an isometry (Coxeter and Greitzer 1967, p. 80). This idea of transformation by an isometry (movement that preserves distances) is essentially "superposition". Analytic Foundations of Geometry - Robert S. Wilson http://www.sonoma.edu/users/w/wilsonst/Papers/Geometry/default.html The above is an example of a development of geometry from this perspective; look in section 6 for the definition of congruence, and a proof of SAS, SSS, and ASA based on carefully defined superposition. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 03/28/2003 at 06:47:43 From: Rahul Subject: Thank you (Are SSS, SAS and ASA Congruences THEOREMS or POSTULATES?) Dear Dr. Peterson, Thank you very very much for your prompt replies to my questions. It has helped me so much. The Math Forum is a very useful site. I thank Drexel University for creating such a marvelous site. Rahul Radhakrishnan
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