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I remember my first exposure to the Escher-Staircase and Hallucination Relations. It took me five minutes just to get past the names.

I have to wonder how laymen would ever take mathematicians' seriously.

Pair of pants

Considering that there exists a branch of mathematics known as "abstract nonsense", as a counterpoint to handwaving,...

So in my developmental psychology class, we're learning about how although toddlers know how to go "one, two, three, four," etc, they are terrible at actually figuring out the sizes of sets of objects. And by terrible I mean you ask for two things, they give you seven, and then despite having them count the number, they don't realize that they haven't given you only two things, but rather seven things, and that adding more doesn't get them any closer to two.

But all the while, I'm wondering about what is actually happening here. Why, despite them being able to count out the seven objects, do they not realize that they've given not two objects? How are they able to count to seven but not count out seven things?
Then I start wondering about the psychologists, because the inability of the child actually somewhat makes sense to me. It's evolutionarily stupid, but there is a difference between cardinal and ordinal, and apparently ordinal is easier for the child. They can list out iterations of the successor function, but judging cardinality is a different task.

I would have thought it would be the "more is better" idea too, except they don't always add; sometimes they subtract even if they've given you too few. It's entirely probabilistic; if you do the trials many, many times and then average them, the averages are actually very close to the numbers asked for, but the standard deviations grow with the number of objects asked for.
The explanation given in class for that is that our innate sense of cardinality is based on approximations, and that only as the brain develops do we tightly associate cardinality with ordinals.

One would think that it might simply be an executive function problem, their calculator processes not doing what their goal-direction processes tell them to, but it's not that they don't know cardinals at all; they know one and two object sets almost from birth, and they learn three, then four, and then five over the course of the first few years of life; but it's not until they learn five as a cardinal that they come to associate larger cardinals with the corresponding ordinals and learn to correct in the right direction if they don't give the right number of objects the first time.

Yeah, one of my friends did that with a nine-volt and a quarter. That led to the phrase "Move over, your pants are too hot."
Also a lot of screaming, but he does that anyway.

I don't know about mathematics, but I saw a similar type of tree mapped out for the various fields in science. It was a great piece of art and I wish I could find it again.