We don't learn numbers from set cardinality

azergante
Sun, 03 May 2026 11:33:37 GMT

As pointed out in Where Mathematics Comes From (WMCF), we are born with an innate sense for numbers, which gets fuzzier very fast as the numbers grow bigger. We can also subitize collections, that is to say instantly determine the cardinality, of collections of up to 3 objects.

It is likely that children learn about bigger integers by playing with collections of objects, adding or subtracting objects from them, merging collections, and linking the resulting cardinalities to their innate number sense.

The authors of WMCF also remark that there is a correspondence between operations on collections and basic arithmetic operations. For example, merging a collection of cardinality 1 with a collection of cardinality 2 results in a collection of cardinality 3, which maps cleanly to the addition 1 + 2 = 3.

Now to my point, conceiving of numbers as the cardinality of collections of objects is reminiscent of, though not the same as, the definition of numbers seen in Zermelo-Fraenkel set theory (ZFC), where a number is the set of all smaller numbers, with 0 the empty set.

However, the formal ZFC definition is not the most intuitive. A more approachable way to conceive of numbers is as the cardinality of sets. And indeed, if we take a set of 1 fruit, another set of 2 fruits, then take their union, we end up with a set of 3 fruits, mirroring the behavior of real-world collections. A neat correspondence between collections of objects, sets, and arithmetic, right?

But there is an issue. Consider a water molecule H-O-H. Naturally you can subitize its elements and tell that it contains 3 atoms. However if you take the set of atoms in H-O-H: it is {H, O}! not {H, O, H}, and it has a cardinality of 2, which doesn't map to the 3 atoms of our water molecule, because elements of sets must be distinct. This uniqueness constraint is the issue: it breaks the mapping from collections of objects to sets.

To map cleanly to real-world collections, we need a mathematical object that preserves their properties, that is to say an object that allows the same element to occur multiple times. That mathematical object is called a multiset.

So, to summarize, it is likely that we learn numbers from the cardinality of collections of objects, and these collections of objects correspond not to sets, but to multisets. In other words, we don't learn numbers from set cardinality, but it is likely that we learn numbers from multiset cardinality.

You may be wondering if any of that matters. I argue it does because maths should be built on intuitive concepts as much as is sensible. This requirement arises from a very pragmatic concern. We can conceive of maths as a program that runs on our brain. We look at notation, compute results, and understand meanings. All things equal, the faster the math program runs, the better.

Our brain comes pre-equipped with modules that understand collections of objects and their cardinalities, i.e. multisets, not sets. So it follows that multisets are a more brain-native representation of numbers, and that thinking of maths as built in terms of multisets should be faster and more intuitive than thinking of maths as built in terms of sets, because we can offload some of the reasoning to our specialized object collection modules, without requiring extra processing steps to remove duplicated elements.

Taking a step back, the more general question is:

What are the mathematical objects that allow our brain to run math programs fast?

I expect the answer to point to mathematical objects that cleanly preserve the properties of real-world objects, for which our brain has built intuition, or in other words for which our brain has dedicated neuronal wiring that makes processing more efficient.

Above, I added the caveat "all things equal", by which I mean that there are other considerations before deciding to use multisets, such as whether mathematical foundations can be conveniently built out of multisets.

What I gather from a cursory search is that it is convenient enough, as evidenced by the Wayne paper cited below that "develops a first-order two-sorted theory MST for multisets that "contains" classical set theory."

For an introduction to the mathematics of multisets, I highly recommend the extremely pedagogical video A multiset approach to arithmetic | Math Foundations 227 | N J Wildberger.

Or for a more rigorous approach defining multiset theory you can refer to Multiset theory or to The development of multiset theory for a survey of different multiset theories and their usage, both by Wayne Blizard.

Discuss