One common comparison made within philosophy of math circles is a comparison between the game of chess and mathematics. Variants of, in particular, formalism hold that mathematics is a sort of game with well defined rules that mathematicians play. I think the game of chess is actually quite apt for the comparison. Like all metaphors, it has limitations but I think it's still an interesting one.
The closest comparison, in my view, is between chess and geometry. Here's how it works.
Euclidean geometry (the geometry most are familiar with) has an infinite, continuous space. Chess, instead, has a finite, discrete space (in particular, 8x8 points).
Geometry has figures like lines and circles. Chess has figures like rooks, knights and pawns.
Geometry is governed by rules of logic and axioms. Chess is governed by the rules of the game.
Geometry has many relation properties such as line l is parallel to line m or point P is the midpoint of line n. Chess also has relational properties such as the rook has put the king in check.
There are many things one can do in geometry. One can construct an equilateral triangle but one cannot square the circle. One can checkmate with a queen and a king but one cannot with only a knight and a king.
The comparison attempts to question to what extent mathematics is invented or discovered. Many defend the idea that mathematics is mostly a discovery; we discover various properties and relations and truths. Contrast that with the game of chess and I think you'll find that most people believe that chess was invented. Chess had historical origins and the rules were set and determined by those who developed the game. (In fact, the rules evolved over time). There is no pretense that people playing chess or the inventors of chess were trying to discover something about the reality they lived in.
I have trouble showing how "discoveries" in mathematics differ from "discoveries" in chess. Consider an example from my own experience.
I remember approaching two individuals playing chess. They weren't actually playing a game (does that count as "playing chess"?) but one was showing the other a particular technique. The result of the technique is that one can checkmate with a bishop and knight. I had never seen any techniques like this before. Prior to encountering it, I didn't know the result was possible (though admittedly, I hadn't investigated the issue.) I certainly discovered that result on that day. Granted, it existed prior to my discovery. In fact, it may have already "existed" prior to the first human discovering it. That's because I contend that the result and the techniques already were present within the rules of chess itself. Given the rules, the result follows.
"Ah! But the rules of chess were invented by humans," one might say. The question is whether or not the "rules of mathematics" were also invented by humans.
I believe that this is actually a different sort of question. Where the rules of chess are more or less explicit, the rules of mathematics are less so. I think a large portion of the overarching norms that govern mathematical practice are ever-evolving. I think there are some basic norms that govern what mathematical practices are allowed but there's also some choice involved. (What told us to accept the law of excluded middle in spite of critics to the contrary?)
As a result I think the comparison - the metaphor - begins to break down. But I think the seed of doubt is sown so to speak. The question that remains is what makes the norms governing mathematical practice "discovered" but the rules of "chess" not.
The primary piece of evidence that is often cited in favor of the role of mathematics as being discovered is its overall usefulness in the sciences (particularly physics). So one way to distinguish chess and mathematics is by providing evidence for the following claims:
1) Mathematics is useful in understanding the physical universe.
2) Chess is not useful in understanding the physical universe.
3) Mathematics' usefulness is somehow related to it being "true" or "weaved into the fabric of the universe" or something of that sort.
Claim (1) I have little interest in disputing.
Claim (2) I find to be more complicated. I do not know of any present usage of "chess theory" in applications in the sciences. But there are plenty areas of mathematics which have no known uses. Couldn't it be possible that some areas of "chess theory" may turn out to aid understanding in the sciences? And furthermore, isn't possible that some areas of mathematics will never come to aid in any area of science?
Claim (3) I've come to doubt over the years. In my limited experience with how mathematics is used in modeling physical systems, I've come to believe that it's more of forcing "nature" to become mathematical then some feature of "nature". Granted, I think this claim is probably the most challenging (and one that I think has had a lot of devotion to within the literature.) I certainly don't think I can do this question justice in a simple blog.