28 What Axioms Can There Be for Human Mathematics?

When they were first developed in antiquity the axioms of Euclidean geometry were presumably intended basically as a kind of “tightening” of our everyday impressions of geometrythat would aid in being able to deduce what was true in geometry. But by the mid-1800sbetween non-Euclidean geometry, group theory, Boolean algebra and quaternionsit had become clear that there was a range of abstract axiom systems one could in principle consider. And by the time of Hilbert’s program around 1900 the pure process of deduction was in effect being viewed as an end in itselfand indeed the core of mathematicswith axiom systems being seen as “starter material” pretty much just “determined by convention”.

In practice even today very few different axiom systems are ever commonly usedand indeed in A New Kind of Science I was able to list essentially all of them comfortably on a couple of pages. But why these axiom systems and not others? Despite the idea that axiom systems could ultimately be arbitrary, the concept was still that in studying some particular area of mathematics one should basically have an axiom system that would provide a “tight specification” of whatever mathematical object or structure one was trying to talk about. And so, for example, the Peano axioms are what became used for talking about arithmetic-style operations on integers.

In 1931, however, Gödel’s theorem showed that actually these axioms weren’t strong enough to constrain one to be talking only about integers: there were also other possible models of the axiom system, involving all sorts of exotic “non-standard arithmetic”. (And moreover, there was no finite way to “patch” this issue.) In other words, even though the Peano axioms had been inventedlike Euclid’s axioms for geometryas a way to describe a definite “intuitive” mathematical thing (in this case, integers) their formal axiomatic structure “had a life of its own” that extended (in some sense, infinitely) beyond its original intended purpose.

Both geometry and arithmetic in a sense had foundations in everyday experience. But for set theory dealing with infinite sets there was never an obvious intuitive base rooted in everyday experience. Some extrapolations from finite sets were clear. But in covering infinite sets various axioms (like the Axiom of Choice) were gradually added to capture what seemed like “reasonable” mathematical assertions.

But one example whose status for a long time wasn’t clear was the Continuum Hypothesiswhich asserts that the “next distinct possible cardinality” 1 after the cardinality 0 of the integers is 20: the cardinality of real numbers (i.e. of “the continuum”). Was this something that followed from previously accepted axioms of set theory? And if it was added, would it even be consistent with them? In the early 1960s it was established that actually the Continuum Hypothesis is independent of the other axioms.

With the axiomatic view of the foundations of mathematics that’s been popular for the past century or so it seems as if one could, for example, just choose at will whether to include the Continuum Hypothesis (or its negation) as an axiom in set theory. But with the approach to the foundations of mathematics that we’ve developed here, this is no longer so clear.

Recall that in our approach, everything is ultimately rooted in the ruliadwith whatever mathematics observers like us “experience” just being the result of the particular sampling we do of the ruliad. And in this picture, axiom systems are a particular representation of fairly low-level features of the sampling we do of the raw ruliad.

If we could do any kind of sampling we want of the ruliad, then we’d presumably be able to get all possible axiom systemsas intermediate-level “waypoints” representing different kinds of slices of the ruliad. But in fact by our nature we are observers capable of only certain kinds of sampling of the ruliad.

We could imagine “alien observers” not like us who could for example make whatever choice they want about the Continuum Hypothesis. But given our general characteristics as observers, we may be forced into a particular choice. Operationally, as we’ve discussed above, the wrong choice could, for example, be incompatible with an observer who “maintains coherence” in metamathematical space.

Let’s say we have a particular axiom stated in standard symbolic form. “Underneath” this axiom there will typically be at the level of the raw ruliad a huge cloud of possible configurations of emes that can represent the axiom. But an “observer like us” can only deal with a coarse-grained version in which all these different configurations are somehow considered equivalent. And if the entailments from “nearby configurations” remain nearby, then everything will work out, and the observer can maintain a coherent view of what’s going, for example just in terms of symbolic statements about axioms.

But if instead different entailments of raw configurations of emes lead to very different places, the observer will in effect be “shredded”and instead of having definite coherent “single-minded” things to say about what happens, they’ll have to separate everything into all the different cases for different configurations of emes. Or, as we’ve said it before, the observer will inevitably end up getting “shredded”and not be able to come up with definite mathematical conclusions.

So what specifically can we say about the Continuum Hypothesis? It’s not clear. But conceivably we can start by thinking of 0 as characterizing the “base cardinality” of the ruliad, while 20 characterizes the base cardinality of a first-level hyperruliad that could for example be based on Turing machines with oracles for their halting problems. And it could be that for us to conclude that the Continuum Hypothesis is false, we’d have to somehow be straddling the ruliad and the hyperruliad, which would be inconsistent with us maintaining a coherent view of mathematics. In other words, the Continuum Hypothesis might somehow be equivalent to what we’ve argued before is in a sense the most fundamental “contingent fact”that just as we live in a particular location in physical spaceso also we live in the ruliad and not the hyperruliad.

We might have thought that whatever we might seeor constructin mathematics would in effect be “entirely abstract” and independent of anything about physics, or our experience in the physical world. But particularly insofar as we’re thinking about mathematics as done by humans we’re dealing with “mathematical observers” that are “made of the same stuff” as physical observers. And this means that whatever general constraints or features exist for physical observers we can expect these to carry over to mathematical observersso it’s no coincidence that both physical and mathematical observers have the same core characteristics, of computational boundedness and “assumption of coherence”.

And what this means is that there’ll be a fundamental correlation between things familiar from our experience in the physical world and what shows up in our mathematics. We might have thought that the fact that Euclid’s original axioms were based on our human perceptions of physical space would be a sign that in some “overall picture” of mathematics they should be considered arbitrary and not in any way central. But the point is that in fact our notions of space are central to our characteristics as observers. And so it’s inevitable that “physical-experience-informed” axioms like those for Euclidean geometry will be what appear in mathematics for “observers like us”.