This doesn't sound like what you have in mind, but here is my favorite axiomatic result.
Start with an example, called Condorcet's Paradox. There are three people, 1, 2, and 3. They have preferences over alternatives A, B, and C. Their preference orderings are:
\begin{eqnarray}
1: A \succ B \succ C\\
2: B \succ C \succ A \\
3: C \succ A \succ B
\end{eqnarray}
Notice that each individual has a complete, transitive preorder over the set $\{A,B,c\}$. They have to pick one of the options as a group, like a political candidate or a tax code or whatever. They decide, ``Hey, let's use pairwise majority voting to establish a social preference ordering $\succ^*$ over the alternatives." They run A against B, and A wins, so $A \succ^* B$. They run B against C, and B wins, so $B \succ^* C$. They run A against C and C wins, so $C \succ^* A$. But then we have
$$
A \succ^* B \succ^* C \succ^* A \succ^*...
$$
and even though the individuals have complete, transitive preorders over the alternatives, society does not, using this voting rule. In practice, the outcome here would depend on agenda control: what sequence of alternatives is run off against which, to determine the winner?
Then you say, "Well, pairwise majority voting is maybe no good, or somehow pathological. Perhaps we can fix this." Well, what do you even mean by that? Presumably you mean that for any set of individuals and any preference orderings they have over alternatives, there exists some mapping from their orderings to a single ordering for society that satisfies some desirable criteria. But how do you make any of the preceding words in this paragraph make sense?
Suppose there are greater than two alternatives. Is there a complete, transitive preorder $\succ^*$ that aggregates the preferences of the people in society in an attractive way? The axioms Arrow picked are:
- Unresticted domain: over the $N$ alternatives, the voting rule must select an alternative for any possible set of preferences the people have
- Independence of irrelevant alternative: the social preference relation should depend only on the two alternatives under consideration; i.e., society has a complete, transitive relation over alternatives, where $x \succ^* y$ if society prefers $x$ to $y$, and this does not depend on whether some other alternative $z$ was available
- Positive responsiveness: if any individual switches their ordering to favor $x$ over $y$, $x$ cannot become less likely to be selected than $y$
- Pareto optimality: if every individual prefers $x$ to $y$, $x$ must be ranked by the social preference relation over $y$
- No Dictator: the social preference ordering does not simply equal the preferences of a single individual in society (because of unrestricted domain, this is more meaningful than it seems)
Each of these properties seems to capture some attractive feature of a decision making process for society, except perhaps for Independence, about which there has been a ton of additional research. There are literally thousands of papers that relax or substitute axioms in the above set.
Arrow showed that there does not exist a way of aggregating the complete, transitive preorders of the members of society into a complete, transitive preorder $\succ^*$ with the above properties. In fact, Arrow's proof is basically to show that if the other four properties hold, there must be a dictator, and any complete, transitive social preorder is equivalent to just endowing a single member of society with all decision-making power.
So you might have looked at Condorcet's Paradox and said, "well, let's give people points to spend on each alternative, and use those to aggregate." No, that's the Borda Count, it fails. Many clever people have played this game. Anything you come up with must violate one or more of the axioms.
Is that not shocking? Does that not both make you skeptical about our institutions, but also rationalize so much confusion and frustration you have experienced with society? You might have thought Condorcet's Paradox was some kind of anomaly, but it points at a much bigger and more serious problem. Groups of people are fundamentally not rational, unless their actions correspond to the preferences of a single person or they are all already fundamentally in agreement (so that everyone is a dictator).
Now, without the axiomatic framework, how could you have ever come to such a conclusion? You can't evaluate every possible voting rule for every possible profile of preferences. By using axioms that capture essential features of objects we are familiar with, we can study huge classes of them all at once, coming to conclusions by deductive reasoning that would never have been possible with inductive reasoning. This isn't asking what the essential properties of $x \cdot y$ are and extrapolating to inner product spaces or $\sqrt{\sum_{i=1}^n x_i^2}$ and abstracting to the concept of a metric.
The reason I like this so much is that before I saw the analysis, I didn't realize that thinking about such a thing was even possible. Being able to abstract way from examples to determine the essence of a thing in axiomatic form and then prove results is a super powerful way to access ideas that would never be available just by sticking with what we know, or by incrementally expanding definitions of known mathematical systems.
So I guess what I am saying is that nothing exists and the only truth is actually the math.
The other is like with the definition of a metric space. The is no notion of what a metric space is beyond its axioms. " I disagree with both this statements. Especially the second! A metric space is based on measuring distances. It's far more concrete and real then the definition of numbers. A map of your county is a metric space! The geometric plane is a metric space!
– fleablood Jun 08 '20 at 03:47