Are abstract definitions in math still subject to our understanding of concrete objects? To what extent?

Question

It seems like in mathematics there are two separate meanings of the word axiom:

One is like the real or natural numbers, whose axiomatization is based off on concrete objects.

The other is like with the definition of a metric space. The is no notion of what a metric space is beyond its axioms.

Here is what bugs me about this: when discussing axioms in the sense of concrete objects, we can add new axioms; it seems like we can't do the same with vector or metric spaces. I get the first type of axiom, but not the second. The second supposedly sounds like game formalism, which not very many people adopt. Yet, this is the apparent definition, which paints mathematicians, in my mind, as being game formalists not knowing what their symbols are for or how they apply even to pure math (obviously not in their specialty, but at least some of the stuff they fly by in undergraduate math). Clearly, I am missing something. Now to be specific about what I'm asking:

Does this distinction (between axioms based off of concrete things and... I don't know what the other one is based off of) really exist or am I imagining it? If I am not imagining it, then can someone explain what this type of axiom is based off of and why we study them so much?

If I am imagining it, then my question becomes why? Are things like metric spaces merely generalizations of what we know to exist, just like with real numbers? The second part of my question is then:

If we're were to find something that everything we know as metric space has in common, would we change it's definition to accommodate the axiom just like for the reals?

This question has two parts, so please comment on both if you are going to respond.

Answer 1

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.