A friend of mine recently mentioned (do not read “endorsed” – I wouldn’t want anybody to get any false impressions about my anonymous friends) the idea that mathematical theorems are demonstrably, objectively true. It’s certainly a tempting conclusion to come to, especially if you’re the sort of person attracted to absolute truth, but in reality there isn’t anything absolutely true about math that’s of much interest. You can’t understand this, however, without looking deep into the bowels of mathematics to see what foundation it’s built on, so I’m going to try to do that for the uninitiated in an accessable way. To start with, I’m going to break mathematics up into three layers.
Top Layer: Popular Math
This is math as most people are familiar with it. The algebra and the geometry and the calculus and all that. It’s a bunch of properties about mathematical objects and rules for things you can do with them:
- Multiplying two negative numbers results in a positive number
- The sum of the interior angles in a triangle is always 180°
- The derivative of xn is n·xn-1
Most people find it incredibly dull and forget anything they’re forced to learn very quickly unless it happens to be helpful to them in some way.
Middle Layer: Mathematicians’ Math
Mathematicians’ math is the foundation for Popular Math. Some people would argue that this is the kind of math that the word “math” should refer to, and that Popular Math is something else entirely.
What mathematicians do is very much about discovery. They look for patterns, try to determine rules, and invent new ways of thinking about things. It’s a much more open and free activity than Popular Math, but it also takes a lot of genius and creativity to come up with anything new. A different summary of a mathematician’s activity is that they try to come up with interesting questions, and then try to find answers to those questions:
- Answered Questions
- Is there a relationship between the circumference of a circle and its diameter? If so, what is it?
- How many prime numbers are there?
- Does the square root of two have an infinite number of decimal digits?
- Unanswered Questions
- Are there an infinite number of twin primes? (a pair of primes whose difference is 2)
- Are there any odd perfect numbers?
Answers are called theorems, and they have to be formally proven before they will be accepted. Mathematicians try to make use of previously proven theorems mixed with insight and luck to prove something new and interesting, which itself becomes a theorem. All the results of mathematics boil down to a giant list of theorems.
This process of proving is (I suspect) what most people who argue for the absolute truth of mathematics are referring to. A mathematical proof is simply a list of statements, where each statement either follows logically from the previous one, or is itself some previously proven theorem, and the last statement is the answer to the original question. In the face of a correct proof, there’s not much that can be done except to acknowledge it. Answering the question of whether theorems reveal absolute truth, however, requires looking at the foundations of proof itself.
Bottom Layer: Logicians’ Math
A logician is concerned with the actual process of proving something. I said earlier that a mathematical proof is simply a list of statements where each statement follows logically from the previous one, or is itself some previously proven theorem. This sentence reveals the two legs that a mathematical proof stands on.
The first leg is the logical connection from one statement to another. What does it mean for one statement to follow logically from another? Most people have strong intuitions about this, but mathematicians aren’t generally comfortable with relying on human intuition if they can help it. Therefore, they rely on a formal system of logic, which is basically a list of rules for how one logical statement can be derived from another. These rules are stated using letters to represent abstract logical statements, and say things such as:
- If we know that A is true, and we know that A being true implies that B is true, then B must be true.
- If we know that A being true implies that B is true, but we know that B is false, then A must be false.
The rules of logic are simple statements that most people would intuitively agree with. Whether or not the rules themselves are true is not something a mathematician could comment on – that question is left to the philosophers. Mathematicians have, however, been able to show that the common set of logical rules are consistent – that is, you can’t use them to come to two contradictory conclusions.
Once logic has been formalized, we have a method for creating statements from other statements, in such a predictable fashion that a computer can do it. Theoretically, the work of a mathematician could be done by a computer (realistically, computers aren’t any good at deciding which theorems are interesting, so the amount of help they can provide is limited), so within a mathematical proof, there shouldn’t be any doubt about whether a connection is logically valid – either you can make the connection using one of the logical rules, or you can’t make the connection.
The second leg of a mathematical proof is the statements that don’t follow from previous statements, but are themselves previously established theorems. The first statement of any proof must necessarily be one of this sort, because there are no previous statements for it to follow from. Of course, if a proof uses a theorem that is actually faulty (i.e., wasn’t correctly proven), then the proof itself will be faulty. But a more glaring issue is a recursive problem: if this theorem relies on some other theorem, what does that theorem rely on? Probably a third theorem, but we could keep asking the question, like a child endlessly asking why. Where did it all start? This is where we get the idea of an axiom.
A system like this has to start somewhere, and every mathematical system has underneath it a set of axioms. Axioms are logical statements just like any statement in a proof, except that the axioms don’t have proofs. They are simply assumed. The idea is that if we start with a handful of well chosen axioms, we should be able to prove all sorts of interesting things. The famous ancient mathematician Euclid realized this, and came up with five axioms for geometry, which say things like “There is only one line between two points” and “circles can be drawn with any center and any radius”. These statements are the starting point for proving theorems within geometry.
Once again we can ask if the axioms are true, but once again the question is left to the philosophers. A natural question is if we can find a set of axioms that we’re intuitively comfortable with and that can serve as a foundation for all of mathematics. Maybe if we accomplished that, we could consider math to be “true enough”. Sure, it’s based on a few foundational concepts, but they’re concepts that nobody would argue with, and so at the very least math is the truest thing we can know. Unfortunately things get a little fuzzy from here.
A mathematical system (i.e., a set of axioms, and a set of rules for deriving theorems) has two fundamental properties: consistency and completeness. I mentioned consistency earlier when talking about the rules for logic. A system is consistent if it is impossible to prove two contradictory statements. If I can, using the axioms and the rules of logic, prove that something is true and then some other way prove that it is false, then the system is inconsistent. An inconsistent system is basically worthless, because one of the necessary rules of logic is that if a contradiction is possible, then anything is true. I had a professor once who wrote on the board “If I am the king of England and I am not the king of England, then I am the Pope” and said that it was a logically true statement. The idea is that if something is true and false, then we can use that to prove that any arbitrary unrelated thing is true. So in an inconsistent system, everything is true, which isn’t very interesting.
The second property is completeness. Completeness means that within a given mathematical system, any statement that can be made within that system is either true or false. This is obviously desirable, because it means that every question has an answer. It also says something else about our axioms besides consistency, something related to usefulness. If I decide to invent a mathematical system where the only axiom is “ten is a number,” then it wouldn’t be of much use. We could prove a few things, such as “If an object is ten, then it is a number” and “Something that is not ten is not a number,” but that’s about it. If somebody asked a question like “is seven a number?” then our system could not answer. It would depend on whether or not seven is ten, and we don’t have any axioms that address that issue. So for our ideal set of axioms, we’d like them to be both consistent and complete.
Unfortunately this turns out to be impossible. Within the last century it was proven that no system which is expressive enough to contain arithmetic can be both consistent and complete. One way of looking at this is that, given some consistent, complex mathematical system, there will always some statement within that system that is simultaneously true and unprovable.
So mathematicians play around with different sets of axioms, and use each for different things. The original geometric axioms of Euclid are just one way of doing geometry (it’s called Euclidean Geometry), and there are others (e.g., “what if lines weren’t really straight?”).
I’ve run through some pretty complex topics quickly, so if anything I said didn’t quite add up, leave a comment and I’ll try to clear things up.
Asking whether some statement is mathematically true is a bit like asking whether some action is legal: it depends on the context. In practice there are very good sets of axioms used in each field of mathematics which are either believed to be, or have been proven to be consistent, and result in all sorts of interesting theorems. For most of the history of mathematics, mathematicians didn’t question their axioms, and math has shown itself to be incredibly useful for all sorts of things. So we shouldn’t draw the conclusion that all of math is a lie. And usually we can get away with doing mathematical things without even thinking about the underlying system. But we can’t philosophize about the underlying truth without acknowledging that underneath everything else, math relies on human intuition, one way or another.
So if math isn’t necessarily absolutely true, why has it been so successful at describing the real world? Excellent question.