![]() 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. ![]() (Sometimes we later find an application, in which it actually is true but that doesn’t change the math itself.) The important thing either way is that we choose our postulates carefully so that we don’t have to assume more than necessary it should be as easy as possible to decide whether the math we’re doing applies to a given situation. Other math is just a “what if” exploration, so we start with a mere supposition and see what would result if it were true. So some math is intended to model the physical world, and we take our starting point there - for example, observations about how lines and points work on a flat surface. 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. 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. But on what grounds are they assumed, and why can’t they be proved? 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. Those facts we start with are the postulates (as they are traditionally called in geometry) or axioms (as used in much of the rest of math). So math is a process of reasoning from some basic assumptions to derive all that can be said about the subject of those assumptions. 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. I started out this way: The basic answer to your question is that we have to start somewhere. The answer will take us into the depths of what mathematics is! What it means to say that a postulate can’t be proved is a little subtle why that isn’t a problem is another question. Could you explain to me why it's okay that they're not proven? So the entire geometry is based on postulates that weren't and can't be proven. Postulates come first, and then theorems are formed from those postulates (right?). 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. I'm in Euclidean Geometry and the teacher said that theorems are proven postulates are not. We’ll start with this question asked by Julia in 2003: The Role of Postulates There’s a lot more to be said to expand that passing comment, which is technically correct but quite misleading. What?! All of geometry is built on statements that we just think are true, without proof? I thought math was all about proof and certainty! This is a postulate, not a theorem, meaning that it cannot be proved, but it appears to be true so everybody accepts it. What that means is, if you have two triangles, and you can show that the three pairs of corresponding sides are congruent, then the two triangles are congruent. Where Doctor Guy, in 1997, said in passing, SSS: the letters stand for "Side-Side-Side". They were never explained to me, just listed as ways to prove triangles congruent.” She then referred to our page Congruence and Triangles I was reminded of this topic by a recent questioner, who said about SSS, SAS, etc., “What I do not understand is why I should believe that these are true. This week, we look at how math can be built on statements that can’t be proved, and words that can’t be defined. I’m starting with questions about the structure of mathematics, particularly the postulates and theorems that are common in geometry classes. As I slow down the site for the summer, I plan to run a couple series of connected posts, one per week, on subjects that have seemed too large to cover in one post.
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