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. I’m starting with questions about the structure of mathematics, particularly the postulates and theorems that are common in geometry classes. This week, we look at how math can be built on statements that can’t be proved, and words that can’t be defined.
Why can’t postulates be proved?
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. They were never explained to me, just listed as ways to prove triangles congruent.” She then referred to our page
where Doctor Guy, in 1997, said in passing,
SSS: the letters stand for "Side-Side-Side". 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. This is a postulate, not a theorem, meaning that it cannot be proved, but it appears to be true so everybody accepts it.
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!
There’s a lot more to be said to expand that passing comment, which is technically correct but quite misleading.
We’ll start with this question asked by Julia in 2003:
The Role of 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?
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. The answer will take us into the depths of what mathematics is!
I started out this way:
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.
So math is a process of reasoning from some basic assumptions to derive all that can be said about the subject of those assumptions. 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). 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. 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.
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. Other math is just a “what if” exploration, so we start with a mere supposition and see what would result if it were true. (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.
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.
The fifth postulate (called the Parallel Postulate) is, as Euclid stated it, “If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” Others since have identified many similar facts that could be put in place of this, such as Playfair’s axiom, “In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.” What mathematicians in the 19th century discovered was that it couldn’t be dropped, because it was independent of the other postulates; and replacing it with an alternative (such as having no parallels, or many parallels) did not lead to contradictions.
This example illustrates the fact that postulates are true only in the particular “world” that the math deals with. A postulate can be replaced with another, and the resulting math is still just as valid — it just tells us about a different “world”, such as a spherical one rather than a flat one.
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!
Note that, since postulates form the “ground floor” of a mathematical system, there is nothing before them from which they could be proved! A proof would require going outside the system, and therefore would not be a mathematical proof. And that is why postulates can’t be proved.
Do we have to take math on faith?
Here is a similar question from 1999, considering not only postulates, but also undefined terms:
Unproven Fundamentals of Geometry I was inspired by some of the answers in your archives to further investigate why the fundamentals of geometry are necessarily unproven/undefined. It seems that in every human system of thought discoveries and inventions must be built upon faith. Less vaguely, in geometry, the most basic unit - the point - cannot be defined. What are some other important postulates or axioms that geometry cannot exist without, but cannot prove, either?
Euclid, in his Elements, started with “definitions” that really didn’t define anything in a formal sense; they just indicated what he had in mind. Modern versions of geometry replace these with “undefined terms”, from which other terms are defined, just as theorems are derived from postulates (axioms).
As discussed above, Euclid then listed five postulates, which are assumed to be true.
Is this all just blind faith? Doctor Rick pondered this:
Hi, Han, I like thought-provoking questions like this. I agree with you about the necessity of faith as it relates to our knowledge of and interaction with the real world. In math, though, I see things a little differently. Math in itself is not intrinsically connected to the real world. It is possible, and perfectly okay, to develop a mathematical system that doesn't relate to anything in the real world. It is, as you say, necessary to have "undefined terms" describing entities in the system, and "postulates" (unproven facts relating those entities). But these are not so much matters of faith as "rules of the game." They are rules that we must adhere to if we are going to prove theorems within the particular mathematical system.
Just as postulates define the subject we are analyzing (such as the way lines interact to form a plane), undefined terms are necessary in order to name what the postulates are talking about. The terms are like the objects used in a game (chessmen, for example), and the postulates are the rules for playing (how a knight can move, how a piece is captured, etc.). All of this may be merely hypothetical (“imagine a world where …”), or applied (“let’s assume the world …”). “Faith” would apply, if at all, only in the latter, not in the math itself.
We aren't allowed to introduce additional assumptions (undefined terms or postulates) or alter them without explicitly stating the new assumptions. When we do so, we are no longer working in the same mathematical system. It may be a perfectly valid system, but it isn't the same one once its rules have been changed, even the slightest bit. Many mathematical systems - probably all until the last two centuries or so - were motivated by attempts to describe and explain things in the real world. At this point, math overlaps with science, and faith becomes relevant. Do the undefined terms and postulates of our system correspond to elements of the real world and their interactions? We can't know. In all likelihood, they don't correspond exactly, but they may make a good approximation.
So “faith” is needed in applying the math, either in taking a teacher’s word for it that gravity follows certain laws, say, or in trusting that our experiments reveal truth about the underlying rules of the world. (Generally, what we discover is close enough to use, given the accuracy of our measurements.)
For instance, a "point" in geometry can be thought of as something with no length, width, or breadth. Everything in the real world has some length, width, and breadth; we can only approximate a point by making a dot with the sharpest pencil we can get. (Physicists now think that electrons may actually be points, but electrons obey the laws of quantum physics, which is rather more complicated than ordinary geometry.) Still, somehow, geometry is very useful in describing the real world, even though strictly speaking, it describes things that don't exist in the real world.
As we become able to measure smaller things, we discover that rules we thought we knew aren’t quite exact; but they are still good enough for rough use. When we apply math to the rules we choose, we are determining how things would work in that “world”, which is not quite the real one but is close.
I said that you can change the rules and come up with a new system. Euclid had 5 postulates in his system of geometry. You can see them here, along with his undefined terms (he called them "definitions", but not all of them are) and "common notions" (actually postulates that are more fundamental than geometry): Euclid's Elements, Book I (David Joyce) http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html The fifth postulate was a lot more complicated than the others. It wasn't very pretty, but it seemed to be needed in order to prove some basic facts about real-world geometry - for instance, that the angles in a triangle add up to 180 degrees. Over the years, people tried to prove the fifth postulate, thinking that something so complex must somehow follow from the simpler postulates. They failed. In the nineteenth century, mathematicians tried a different tack: try changing the postulate, and see what happens. They found that they ended up with several varieties of "non-Euclidean" geometry that were completely self-consistent, but different from Euclid's geometry. Changing the "rules" made a new but perfectly good game. So what do you think happened next? Einstein came along and discovered that these non-Euclidean geometries were just the thing to describe the real-world interactions of objects with mass - that is, to describe gravity. This is a case where the mathematical system was invented with no consideration of the real world (and therefore no faith element), but it turned out that this system does appear to describe the real world. The experiments to show that Einstein's theory of general relativity do describe the real world better than any other mathematical system are very tricky; it is still possible that another system would do better. We can be absolutely sure that the results of general relativity theory follow from its assumptions; the only question is whether or not those assumptions match the way the real world is.
Science, as we see, also makes assumptions; its assumptions are about the real world, and can therefore be wrong; the assumptions in math just define what the math is about, and can’t be wrong until we try to apply the math to something pre-existing.
Let me put it another way. There are two kinds of truth; I'll call them mathematical truth and real-world truth. Mathematical truth means that a statement is consistent with the assumptions of a particular mathematical system. In a sense, people created that system, and they can tell absolutely whether the statement is true within that system. ... Real-world truth is of a different order: it means that a statement is consistent with the particular system that is the real world. There is only one real world, and no human created it; no one knows exactly what the rules are. Scientists try to make rules that seem to describe the real world, but they can't possibly know whether these rules really describe everything in the universe. So yes, faith is necessary, because we did not create the real world, so we can't know absolutely what the rules of this system are ... unless someone from outside this system - the creator of the system - lets us know the rules.
Can’t I define the undefined terms?
To close out, let’s look at a student’s attempt to define “line”:
Undefined Geometry Terms I know that they call point, line, and plane the undefined terms of geometry, but is there a way to give those terms a definition? I've been thinking, and you may not be able to give all them a definition, but a line could a line be defined as the inconclusive conjunction (or joining) of two rays going in separate directions. I've never really thought that anything couldn't have a definition, so is it possible for any of these geometric terms to be defined?
Jake is wise to suppose only that some of the undefined terms could be defined, in terms of others (just as perhaps a claimed postulate might be found to be provable in terms of others, and thus demoted to a theorem). But his attempted definition, which I might refine as “A line is the union of two rays going in opposite directions from the same point, ” proves the point, as I explained:
Your "definition" would require us to first define "ray" and "direction." Can you do that without reference to "point," "line," and "plane"?
A definition has to be given in terms of previously known terms. In the usual presentation, “ray” is defined in terms of “line” and “point”, and “direction” isn’t really given a definition at all. So his attempted definition has to either be circular, or go outside the system.
Think of it this way: Math is a huge building, in which each part is built by a solid line of inference upon other parts below it. What is the foundation? What is everything else built on? There must be some lowest level that is not based on anything else; otherwise the whole thing is circular, and never really starts anywhere. The "undefined terms" are part of that foundation, along with other things like rules of inference that tell us that logic itself is true. The goal of mathematicians like Euclid has not been to make math entirely self-contained, with no undefined terms, but to minimize the number of them so that we have to accept only a few basics, and from there will find all of math to be absolutely certain. Also, the goal is to make those terms "obvious," so that we have no trouble accepting them, even though we can't formally prove their existence.
Interestingly, geometrical concepts have been applied to non-geometrical ideas, by assigning the undefined terms differently: if you take “point” to mean “person”, “line” to mean “family”, and so on, you might define a “geometry of people”, for example. This is another reason for undefined terms; it separates the logic from the application.
To put it another way, these terms do have a definition, in human terms; we can easily understand what they mean. They simply don't have a mathematical definition in the sense of depending only on other previously defined terms.
Next time, we’ll look into the fact that different geometry textbooks list different postulates.