The Role of Postulates
Date: 03/29/2003 at 16:53:02 From: Julia Subject: Geometry based on non-proven postulates I'm in Euclidean Geometry and the teacher said that theorems are proven; postulates are not. Why? Who decided what were postulates and what were theorems? I asked my teacher if postulates *could* be proven and simply weren't, and she said that they couldn't be proven. This is my current question. Postulates come first, and then theorems are formed from those postulates (right?). So the entire geometry is based on postulates that weren't and can't be proven. That just doesn't seem right to me. Could you explain to me why it's okay that they're not proven? Thanks a lot.
Date: 03/31/2003 at 09:48:12 From: Doctor Peterson Subject: Re: Geometry based on non-proven postulates Hi, Julia. The basic answer to your question is that we have to start somewhere. The essence of mathematics (in the sense the Greeks introduced to the world) is to take a small set of fundamental "facts," called postulates or axioms, and build up from them a full understanding of the objects you are dealing with (whether numbers, shapes, or something else entirely) using only logical reasoning such that if anyone accepts the postulates, then they must agree with you on the rest. Now, these postulates may be (and were, for the Greeks) basic assumptions or observations about the way things really are; or they may just be suppositions you make for the sake of imagining something with no necessary connection with the real world. In the first case, we want to choose as postulates facts that are so "obvious" that no one would question them; in the second case, we are free to assume whatever we want. In both cases, we want a minimal set of postulates, so that we are assuming as little as possible, and can't prove one from another. Euclid's problem was that one of the postulates (the fifth) didn't seem simple enough, so people over the centuries tried to prove it from the other postulates, rather than be forced to accept something that didn't seem immediately obvious. Eventually it was realized that there are in fact different kinds of geometry, some of which don't follow all of Euclid's postulates; and that you could replace his parallel postulate with a contradictory assumption and still have a workable system. In particular, spherical geometry - the way things work on a sphere, if you think of a "line" as a great circle - is a very real example of this, in which parallel lines just don't exist. Spherical geometry follows different rules, yet is just as valid as plane geometry. So we have to take as our starting point some postulates that simply define the particular mathematical system we are studying. If we take a different set of postulates, we get a different system, which may be just as useful as the original - and therefore just as "true" - yet different in its conclusions. The postulates we choose are the connection between the abstract concepts about which we are making proofs, and the "real world" ideas that they model (if any). Without postulates, we would not have such a connection, and would be reasoning about nothing! Here are a few Dr. Math archive discussions of the role of postulates in math: Unproven Fundamentals of Geometry http://mathforum.org/library/drmath/view/52290.html Understanding Mathematics http://mathforum.org/library/drmath/view/52289.html Properties and Postulates http://mathforum.org/library/drmath/view/52472.html What is Mathematics? http://mathforum.org/library/drmath/view/52350.html If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 03/31/2003 at 15:49:39 From: Julia Subject: Thank you (Geometry based on non-proven postulates) Thanks for your answer. "[...] or they may just be suppositions you make for the sake of imagining something with no necessary connection with the real world." I never really realized this - thanks. I appreciate it.
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