The "Right" Answer in Mathematics

Question

I apologize if this question seems too vague for this site. I have a general question regarding how we determine "correct" answers in math.

I am currently a college student studying physics/engineering. When given a math problem, my approach is to conceptualize the problem to figure out what ideas might apply. Then, once I start to see a path to the solution, I try to take logical steps to reach that solution. Up to this point, everything I am doing seems logical to me. After doing all this, how do I know if my answer is right? I check with my professor's answer. If the answers match, great (if the professor used a different approach, I will typically take the time to understand it). If not, I go back to my work and I am usually able to find the flaw in my logic.

I was wondering what the process of solving a problem for a professional mathematician is. I imagine that a mathematician goes through a similar series of logical steps (at least after he/she has conceptualized the problem). Once they get their solution, though, how do they know if it is right? Just because the solution seems logical, it does not necessarily mean it is right (like I said, my solutions seem logical until I see the "right" answer). Math is not like physics where you can run an experiment to test your answer. So how do mathematicians know if their solutions are right?

Answer 1

Despite what many a philosopher, theologian, and mathematician will argue, truth is fundamentally one of two things:

1) An opinion or agreed upon belief

In this case, an answer is correct when it is agreed upon by the consensus (if not majority) of people considering it. Alternatively the person or people in question may agree upon certain rules (formally called axioms in mathematics, also definitions) which are considered to be incontrovertible truths.

In the latter case, you can determine whether a statement is true (i.e. - whether or not your answer is correct) by verifying that it either naturally follows from or is confirmed by these axioms. A statement is false (i.e. your answer is wrong) if it contradicts itself or these axioms. This is the means by which mathematicians verify proofs.

2) The agreement of a statement with a physical referrent

In this case an answer is correct when it accurately describes something which can be observed. Such observations may take the form of personal experiences ("objects raised above the ground fall back towards it") or experimental evidence ("objects with mass accelerate towards each other according to a force proportional to their masses.). This is the means by which scientists verify hypotheses.

Generally, an "experimental" approach to mathematics is acceptable at least in the case of numerical problems (e.g. $x^2=2$), where the answers can be computed exactly via arithmetic. In a more abstract setting (e.g. Given ... prove that $S$ is a subgroup of $W$), mathematicians rely more heavily on axioms.

Answer 2

I think this is a deep question that will resist a simple, easy answer. Here are some thoughts.

Math is not like physics where you can run an experiment to test your answer.

It can be! Mathematical statements make predictions about the behavior of mathematical objects, and you can test those predictions. Nobody tells you this or encourages you to do it, but it turns out that you can do it anyway.

A simple example is if you're studying some sequence of interest to you and you think you've figured out a closed form for it; say, Binet's formula $F_n = \frac{\phi^n - \varphi^n}{\phi - \varphi}$ for the Fibonacci numbers. That closed form makes predictions: most simply, it predicts values of the sequence, and you can test those predictions by computing them. It also makes more interesting predictions: for example, in this case, Binet's formula predicts that $\frac{F_{2n}}{F_n} = \phi^n + \varphi^n$ is an integer (it is in fact a Lucas number), and you can also test this prediction.

This is one available tool mathematicians have for deciding if they're right, but there are others. One curious feature of mathematics is that mathematicians often give incorrect proofs of correct statements, which means the process by which mathematicians arrive at correct statements is not by giving correct proofs of them in general.

Rather, it's a more complicated process of deeply understanding the mathematical territory, building and refining intuitions about the behavior of mathematical objects, making predictions and checking them via proof, experiment, and/or consulting other mathematicians, etc. When you have many overlapping intuitions relevant to a mathematical situation they serve as a check on each other: if one of them says one thing but all the rest say another, then something's funny about the first one and the other ones probably have a point. See also Terence Tao's There's more to mathematics than rigour and proofs.

There's a related process of learning how to trust your ability to write correct proofs; for example, I know several proofs of Binet's formula and I consider them all airtight, because I understand all of the mathematical tools I'm invoking in those proofs and have a lot of experience using them to prove other results in a way which hasn't caused me to arrive at a contradiction so far, has resulted in me proving things that other people agree are true, etc. Proving things is a skill I was less good at before, practiced a lot, and am now better at, like every other skill.

Answer 3

In mathematics, "right" (or possibly "true") just means "consistent with the given system", where the given system usually starts with basic logical inference and some rules that set up arithmetic and set operations, and then extends into whatever else is relevant (e.g. some geometric axioms, definitions of various constructs, etc.).

As far as we can tell, the system of mathematics is self-consistent (i.e. it does not contradict itself), but it is not complete (in the sense that there are some mathematical statements you can express, but cannot prove to be true or false). As it turns out, this is about as good as you can get. There are actually quite a few different systems we could be using, but the main ones we use are chosen because (a) they have a certain richness in what you can prove, and (b) they are somewhat consistent with observations of the real world and intuition on how things work (although neither of these are guaranteed, and both can potentially fall over in certain situations).

So, when a mathematician "proves" something, they are applying a set of rules (which tell you how you are allowed to manipulate the various objects you are working with) such that they operate in a consistent fashion. For example, if you combine true statements in certain ways then you should only ever wind up with other true statements. Hence, anything that is proven using such statements is correct, within the given system.

Additionally, for some results it is possible to test whether the answer is right regardless of the process that led to it - for example, if I've been told to solve $3x + 4 = 13$ for $x$, then when I wind up with the answer $x = 3$ I can go back and check - does $3 \times 3 + 4 = 13$? Yes, then my answer is right (even if I accidentally made a bunch of rookie errors along the way that happened to cancel out).

Answer 4

What makes a result in mathematics true is different from the manner in which an answer in physics is true. In mathematics, a lot of the time, we often sit here just messing with abstract notions that almost certainly have no analogous representation in reality, or at least an experimentally-verifiable one.

How, then, do we know if a result in mathematics is true?

It follows from the axioms of the mathematical framework in which we are working. (I mean, yes, if the context is appropriate, there are other ways in which to check your answer/logic/whatever it is may be. But I'm talking about the more "fundamental" notion of correctness, since this also has to apply to those methods of checking your answer.)

This is a powerful thought, because of two things. For one, it means that anything either is or is not true - there is not some notion of "well, close enough" or "kinda true" or whatever. The laws of physics have to be revised to account for new observations and is dynamic in that sense - some observations might be true, but not for the reasons we think - whereas mathematics is comparatively static: given a proper set of axioms, something either does or does not follow from it. The only thing is to determine if, indeed, it does follow.

(Of course, depending on the axioms involved, something might be literally unprovable. Godel's incompleteness theorem might be worth looking at. I believe one example of something that's unprovable would be the continuum hypothesis, at least in the framework of ZFC set theory.)

What I find more interesting is that it doesn't specify the axioms or mathematical framework in itself.

This is powerful in that we can take "obviously false" results from our current understanding, plop them in a different framework or axiomatic system, and see what comes of it. You could make the argument that, in a sense, this is where complex numbers came from. We saw square roots of negatives, decided they didn't exist, until we decided to investigate what happened if they did exist. And hell, who would've thought that from a simple notion that the field of complex analysis would be born - that it would even have applications outside of mathematics? These "obviously false" results could always be worked into a system in some way: if nothing else, we can discuss what comes of them. Who knows, they might even be useful!

So in this sense, almost everything in mathematics is actually true, if placed in the right context. It might not be true in this framework, but it might be true in this framework. Physics is sometimes concerned with similar notions - alternate universes, alternate laws of physics, etc. - but their ideas can be at best difficult if not impossible to verify experimentally (and thus be "right" in the physics sense). In mathematics, there's an unparalleled freedom since we can pick and choose our "universe" by playing with the axioms as we so choose.

Of course, this almost begs another question - why do we often favor the axiomatizations we do? After all, we have a literal sense of freedom in this respect, so why favor them?

That's a broader question I'm even less qualified to answer, but I believe it comes from two things:

• They just make sense and are thus somewhat useful. I feel like there's a sort of underlying notion of intuition in mathematics, or that we at least seek this intuition to what degree we can. If we deliberately make systems that are hard to wrap our minds around in even an intuitive sense, not much is going to come from it, and certainly few applications. It's sort of like coding in that Brainfuck computer language: sure, you can, but why would you want to do that to yourself, and what could you realistically do with it that wouldn't be way easier with Python, Java, C++, or any number of other languages?

• There's a "richness," as another answerer, ConMan, put it. There's a lot of interesting results that ultimately follow from the axiomatizations that we favor. An axiomatization that results in a lot of results bears further examination in its own right, particularly if said results are interesting or useful.

This also limits us somewhat. Okay, we can axiomatize however we want, yet we want axiomatizations with these certain, nice properties - intuition (well, maybe, I for one at least highly value it), richness, and the ability to be applied usefully are probably the big main properties we want, ignoring the obvious "we don't want the axiomatization to be self-contradicting."

So, in short:

• A result in mathematics is true if it logically follows from the axiomatization of your framework.
• What that axiomatization or framework is: that's effectively up to you, within a certain extent. There are results to be had from tweaking the axioms or framework a little to allow results that aren't normally worth considering.
• That said we favor certain axiomatizations for the reason that they're useful, that they work, and they're relatively easy to grasp. (And not self-contradicting hopefully.)
• Physics, as opposed to this, tends to rely more heavily on experimentally-verifiable results, which can be limiting in certain contexts, and also tends to have a dynamic set of laws and theorems that change as new observations come to light. This results in a fundamental groundedness in reality and experimentation that mathematics need not always have.

Answer 5

There is a whole "battery of tests" that allows you to check if your argument is "logical enough to be correct". It is somewhat field-specific and none of the tests is completely fool-proof, but it still helps. Here are some most common general techniques to check yourself before you show your argument to someone else.

1) Pinpoint exactly where you used every condition in your setup. If you haven't used something, ask yourself if the conclusion really has any chance to hold without this assumption.

2) If you have a long and convoluted argument, see if you can split it into short steps with a clear understanding of what you gain at each step and of how these small gains combine into the final derivation. In general, the shorter and the simpler some particular passage is, the easier it is to verify or refute. Also, quite often, when doing so, you discover that your initial proof went in circles in places and shortcuts are possible.

3) Check that neither of your intermediate claims contradicts something already known (to you).

4) If some of your claims are about reasonably simple and generic objects, consider a few examples of such objects and see if what you claim makes sense in these cases.

5) If your proof depends on some miraculous cancellation rather than on the fact that a linear combination of several terms out of which one is much bigger than any other is approximately the dominating term, check your algebra several times preferably deriving the same formula in several alternative ways.

and so on, and so forth.

The ultimate trick of self-checking is to become your own enemy and read the argument you wrote as if your single aim were to show that it is a piece of gibberish. If anything is not so crystal clear that you would have to grudgingly accept it, criticize it as much as you can. Then switch sides and try to respond to the criticism by rewriting the text. The main mistake people make when proofreading their own work is taking a relaxed and benevolent attitude towards the text, while one should really try to exhibit as much negativism as one is capable of. Acquiring the error detection skills also requires time and practice, and your own writings (at least in the beginning) are excellent samples to try them on.

Of course, one seldom goes through the full routine of self-checking (mainly because the human time is finite and if you don't find your own mistake, you may be pretty sure that it will be discovered by others sooner or later) but it is always worth the effort to do at least a few steps.

As to "the truth is an opinion or agreed upon belief", if one explains carefully what is meant by that, perhaps not many people would disagree, but as far as these words stand by themselves in a single sentence, they certainly are way too provocative and an immediate response comes to mind: "If somebody gets enough power to burn on stake anybody who denies that 2 times 2 is 5 (never really tried) or that some god judges all our actions and rewards or punishes us after death, or, alternatively, that all religions are pure delusions (both tried many times), will the identity $2\times 2=5$ become true, or will the gods appear and disappear at human will?", though I prefer to answer such provocative words with equally provocative "We are not searching truth in mathematics, we are just figuring things out".