Last time we looked at the question of why we have to have postulates, which are not proved, rather than being able to prove everything. Often, this question is mixed together with a different question: Why do different texts give different lists of postulates, so that what one calls a postulate, another calls a theorem? How can we live in a mathematical world where the very foundations keep changing?
My book calls them postulates
We’ll first look at a simple version of the question, and then one that takes us deeply into geometry. First, from 2006:
Theorems and Postulates If SSS, SAS, and AAS are theorems, why do other books still use them as postulates? And can you show me the PROOFS that were used for these theorems? :) Sorry, I know you've answered questions about the topic many times but as I was reading the answers I realized that you were trying to say that SSS, SAS, and AAS are really theorems because they were proved (theorems need proofs). But, why do books and even teachers still teach students like us SSS, SAS, and AAS "POSTULATES". Do the words "theorem" or "postulate" really matter? I am a high school student studying geometry right now and your answers to my questions would be a great help for me.
Below, we’ll be looking at one of the earlier answers referred to here. But Alona is probably referring specifically to this page from 2001:
AAA, ASS, SSA Theorems Can you please tell me in detail why the ASS, SSA, and AAA postulates can't be used to determine triangle congruence?
Doctor Jubal started his answer by saying,
Just a side note: the SSS, SAS, and ASA triangle congruency theorems are theorems, not postulates. A postulate is something you just state and assume to be true. A theorem is something you can prove, based on your postulates.
Evidently Alona’s textbook calls these postulates, while Doctor Jubal’s calls them theorems. So what are they really?
Having already written about this, I gave Alona a succinct response, also referring to a page we looked at last time, and the page we’ll look at below:
These facts CAN be postulates, but they don't have to be. It's a matter of how an author chooses to present geometry to his audience. Different geometry texts choose different starting points. The best way to do geometry is to start with as few assumptions as possible, and prove everything from those. Many texts "cheat" a bit by using as postulates anything they don't want to bother proving (probably because the proofs are difficult and wouldn't really help their students understand the subject). Others use a good, small set of postulates, but state some theorems without proof, explaining that the proof is beyond the level of the text. I prefer the latter approach, but I can understand the "cheating". It is possible to take ONE of these congruence facts as a postulate and prove the others from it (so they become theorems). It is also possible to define congruence in such a way that all three can be proved from some more basic postulate about congruence. You can take them however your own text presents them; but be aware that they are all really equivalent facts, and which you take as postulates doesn't affect how you use them, which is what really matters. In other words, in answer to your question as to whether "theorem" or "postulate" really matters: it matters in presenting a specific systematic treatment of geometry, but not in USING the facts you learn, which are true one way or another regardless.
This is the key idea that any student should come away with, and is why I am covering this answer first: Whatever starting point an author chooses, once the theorems have been proved it doesn’t matter whether a given fact was stated as a postulate or as a theorem, it is still equally true.
Alona had a follow-up question:
Dear Dr. Math, thank you very much for answering my past question about "theorems" and "postulates". I now know that one of the SSS, SAS and ASA theorems can be considered as a postulate. It just depends on the starting point of the discussion. But they are really theorems (I hope I understood it the way you want me to understand it). My question is, is it really possible to prove theorems from theorems? What I mean is, is it possible to call all the three congruency theorems "theorems" and still prove each one using each theorem? In our geometry class, it is possible to prove theorems from previous theorems. Then why do we need to assume one of the congruency theorems as "postulate" when we could really prove it using "theorems". Thank you very much for your previous reply. It answered 70% of my questions. These questions are the remaining 30%. :)
I replied, pointing out the necessity for something to be a postulate, to avoid circularity:
Certainly you can prove a theorem from a theorem; you do it all the time, I would think. You can use any known fact, whether theorem or postulate, as the basis for a proof. What you CAN'T do is prove A from B, and B from C, and C from A! Such circular reasoning is not allowed, because you have to start with something that is known to be true. So if you call ALL THREE of these "theorems", then at least one of them has to be proved on the basis of something else (such as a definition of congruency that is more powerful than what elementary texts usually use). That's why the best approach is to take one of them as a postulate, and then prove the others as theorems. It doesn't matter which one you start with, but you have to start with one without assuming another is already true. If you do merely prove each from another of them, then what you have done is to show that they are all EQUIVALENT--that is, IF one is true, then they all are. But then either they are all true, OR they are all false. You don't know which! I think the pages I referred you to answer this question, by explaining the role of postulates as starting points. You may want to reread them with this new perspective in mind.
Euclid and others
With all this in mind, let’s take a look at that earlier discussion from 2002:
Theorem or Postulate? 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.
Maureen wants to get it right! In order to help, I dealt with both the general ideas discussed above, and some particulars about differences among textbooks:
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.
Mathematicians spend time rethinking known math, looking for new proofs and new ways to organize old information. Euclid’s Elements, the original geometry “textbook”, had its faults, and modern mathematicians have searched for better sets of postulates and definitions to start with (more consistent, more complete, and so on), so they typically have longer lists of postulates than Euclid (who sometimes didn’t recognize that he was using a fact he’d neglected to list). Meanwhile, textbook authors want to avoid making the subject too complex, so they try to simplify the list of postulates.
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.
Maureen responded with a more specific question about those congruence postulate/theorems:
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?
This is a very perceptive question; you will recall that I said in 2006 that some authors “cheat” by adding postulates just to avoid having to prove them, just as she suggests here. I responded with a short description of different options:
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 [postulates], 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.
Then I gave a bunch of references to discussions of various systems. Many of these references are now dead links (it’s been 16 years), so I’ll update them here.
First, for information about the benefits and deficiencies of Euclid’s postulates, you can start with the premiere Euclid site, David Joyce’s Euclid’s Elements . Commentaries on each proposition discuss the flow of thought and issues Euclid neglected. For an examination of the book, with a lengthy discussion of its limitations, see Euclid and the Elements.
To compare Euclid’s postulates with a few modern sets of axioms, you can look at Wikipedia on Hilbert’s axioms, Birkhoff’s axioms, and Tarski’s axioms. These are very different in their purpose and form. Many modern school texts, to the extent they use postulates or axioms at all, appear to take ideas from Birkhoff.
For an interesting look at how Euclid can be taught with the benefits of modern thought, see Teaching Geometry According to Euclid by Robin Hartshorne (who clearly disagrees with Birkhoff’s approach).
An excellent reference I gave was to Jim Loy’s site, which unfortunately does not exist now (except in the Wayback Machine). But the page I linked to, which shows how you can start with any of the congruence facts and derive the others (that is, they are all equivalent) was borrowed in this PDF, along with Joyce’s commentary on Euclid’s proof of SAS (which used an “obvious” fact that Euclid neglected to list as a postulate, making his proof invalid).
The next thing on this page is my response to a paragraph that was somehow omitted when this discussion was archived; I looked this up because what I wrote seemed like a non-sequitur. Here is the missing paragraph from Maureen:
As a math major and an algebra teacher I am finding this discrepancy among geometry texts to be a bit of a problem for any student who changes schools mid year. My friend's son has to prove that all right angles are equal. He seems to feel that that is so obvious that it should be a postulate. Euclid's 4th postulate does in fact state that all right angles are equal. However, the congruence postulates aren't nearly as obvious and yet they are postulates. As a math major I took a course in non-Euclidean geometry when I was in college. This course begins with the controversial parallel postulate which Euclid and many others tried to prove. It only became a postulate because Euclid couldn't prove it. I find that I am really surprised to see such an arbitrary treatment.
Here is my response:
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.
Let me add here another bit that was not included in the archive, namely an additional answer given by Doctor Fenton focusing specifically on the question of how to teach Euclid:
I'll also throw in my $.02. The American Mathematical Society also publishes a high-school text "Basic Geometry" by George D. Birkhoff, one of the top mathematicians of the Twentieth Century. It isn't a presentation of Euclid, but it is a logically rigorous modern development based on Euclid. There are logical problems in Euclid; he makes implicit assumptions (e.g. about properties of "between", and his first proof is invalid, because he assumes that two circles which appear to intersect do in fact intersect - but there is no postulate guaranteeing that). Birkhoff simplifies Euclid to some extent by introducing Ruler and Protractor Postulates, so that it is not a purely synthetic development. It's inexpensive ($26 for the text, and $9 each for the teacher's manual and answer book). I think it would be worth considering. Another text to consider would be Robin Hartshorne's "Companion to Euclid" (if you can find it - it appears to be out of print, but you might find it through online used book sellers) or one of his texts "Geometry" or "Euclid and Beyond". One of those is an updating of the "Companion to Euclid". It may be college level, but it does have problems and explains the shortcomings of Euclid in addition to presenting him.
Hartshorne is one of the authors I referred to above.
The same page contains a follow-up question from a reader in 2003, which I don’t have room for here. It deals specifically with the question of “superposition”, which is the unsupported “fact” Euclid used in “proving” SAS.