We don’t learn numbers from set cardinality
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.
A few problems I have with this post:
Multisets are defined as a tuple of a set
ZFC was not built to be intuitive. It was built to unify and formalize all of mathematics. It is not supposed to be some kind of universal human way of seeing the world through a mathematical lens. It’s just a set of axioms we build on, and we can choose different sets of axioms at will.
Your example of a water molecule assumes we do not distinguish between the hydrogen atoms with indexing
In short:
You seem to be conflating the foundations of mathematics with the foundations of mathematical thinking. These are not the same thing. The former is how we justify doing math, the latter is how we map abstract concepts to intuitive patterns us meat-brained apes can understand.
Multisets can be defined in terms of sets, and conversely sets can be defined in terms of multisets. I don’t see either of them as “more fundamental” than the other necessarily, except in terms of which one we’ve chosen to give a more prominent place to.
If ZFC is not intuitive (and my opinion is that it is not), then we shouldn’t assume that any theorems it proves are actually true, and we should if possible rely on a weaker system which is intuitive, such as the Peano axioms. Peano arithmetic is in fact powerful enough to deal with almost all of modern mathematics (except for abstract set theory), it’s just that people are so used to using ZFC as the foundation that they rarely check to see whether their theorems can be proven in PA. See for example this book where the author develops a “strong undergraduate curriculum” using a system conservative over PA as the foundation.
On the atomic level, quantum effects prevent us from distinguishing
You’re missing my point, possibly intentionally. Lets go through your points in reverse order:
Yes I know hydrogen atoms are indistinguishable. The point was about objects in general, the atomic example was a holdover from the original post. If you need help abstracting think of two apples and an orange instead.
Truth
Yes of course. The point was that using multisets as a foundation isn’t significantly different from using sets as a foundation. It was a critique of the original post, not an assertion about the best foundation of numbers.
I’m getting the feeling you want to make a point but you’re not looking at the context of my comment.
Actually I was trying to make three separate unrelated points. I think we agree sufficiently on the first and third points that it would be unproductive to discuss them further. Regarding my second point:
Well I do like to plug for PA when I can (it’s a good foundation for math!), but I think I do have an actual point as well.
I mean, sure, there are a lot of unintuitive things that you can prove from true axioms. But with the axioms themselves, the only basis we could have for asserting that the axioms are true is that they are intuitive. I mean, why else would you think they are true? (In mathematics at least, of course in other fields you could have empirical evidence of something unintuitive.)
Ah okay, I think we have different understandings of the motivation for axioms. If I understand correctly you are saying axioms should be motivated by intuition. I disagree. Axioms should be motivated for their generative capacity, i.e. how much fun math we can squeeze out of them. Axioms don’t need to be “true”, rather they need to be interestingly generative.
When I encounter math I don’t immediately have an intuition for, I remind myself what Von Neumann said:
“In mathematics you don’t understand things. You just get used to them.”
For me at least, I have a hard time grasping the intuition behind a lot of math so I cling to this quote like dogma.
no… it contains { this H, O, that H }.
my lay understanding is that some particles cannot be distinguished. but it is unlikely that children are commonly coming across these while developing intuition for counting.
I’m not sure that the example justifies a move to multisets. H-O-H has three atoms: one is an oxygen atom and the other two are hydrogen atoms. The right way to read H-O-H is that the first atom in the molecule is of type H, the second is of type O, and the third is of type H. The H, O letters are really types/predicates. When conceived this way the problem disappears: {H, O} is the set of non-null predicates of the molecule (a second-order set, if you will), rather than the set of atoms in the molecule. Conversely, the set {2, 1+1} has only one element, not two: the two names are different, but they name the same element.
A better way to frame it is that the example treated the two hydrogen atoms in H-O-H as the same thing, when in fact they are not, in the same way that there are three fruits in a collection with 2 apples and 1 orange, not two, because the two apples aren’t the same thing. You can say that the set of atoms in H-O-H is {the first H, the second H, the O}
...except, you can’t? On a quantum mechanical level, two hydrogens don’t have XML tags labelling which one they are. This is easier to explain with electrons. You cannot say that the first electron is here, and the second there.
if i introspect while picturing a multiset, it feels like “a collection containing 12 pears and 6 apples”. or maybe “a collection containing pears and apples. the number of pears is 12. the number of apples is 6.”.
do you have a way of defining ‘multiset’ that is prior to ‘number’?