The definition of exponentiation provided is that of cardinal exponentiation, in that it takes two cardinals and gives back a cardinal. So it must indeed work on the set-theoretic structure of the von Neumann representatives, though, as mentioned, it only uses the fact that the representatives in question have a given cardinality (compare with ordinal exponentiation, in which we also want to take into consideration the underlying ordering)---that is, it uses the fact that 2 has cardinality 2.

Thanks, I got it when I thought about it again. I got into a habit of thinking of numbers as abstract objects that just satisfy certain requirements, like Peano axioms, whose representation and internal structure need not concern us. I lost track of the obvious idea that a number can also be thought of as a collection with that many distinct objects - a cardinal*. When you mentioned von Neumann construction, I thought about Zermelo's construction with its Russian doll structure - that wouldn't work so well. I guess that's one of the advantages of the von Neumann model, right?

* Heh, now that I think about it, that was exactly how numbers were introduced to us in preschool, using little sacks with different amounts of marbles :)

But I see your point, I think. IF numbers work as they did when we all first learned math (rigidly), then normal mathematics is objectively true. — Gregory

Oh no, not at all. I make no claims of truth for math. Only logical coherence and interestingness, those are the criteria by which mathematicians judge math. If you have a logically coherent, interesting system in which 2 = 9 you have every right to pursue it. Math is not about truth; only about whether given assumptions entail particular conclusions.

As Bertrand Russell said:

“Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true. [...] Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate.”

https://www.goodreads.com/quotes/577891-pure-mathematics-consists-entirely-of-assertions-to-the-effect-that

You can say 9 is the same as 2 in my mind. — Gregory

It just occurred to me that 2 = 9 in the integers mod 7. This is perfectly ok in standard math. They study things like this in number theory and abstract algebra. Your idea's fine. Math accommodates a lot of different possibilitles.

Interesting. Imagine this: your father bequeaths to you a painting of a beautiful girl. Before he passes away, your father says to you, "the painting was the work of a famous painter but", he says, "I have no idea whether the girl in the painting was real person discovered by the painter or just someone invented".

Now imagine you're walking down a street in your idyllic town and something amazing happens; you see a girl, who's identical in appearance to the girl in the painting. Two possibilities immediately cross your mind: either 1) this is a coincidence and the girl is an invention of the painter or 2) the girl in the picture is actually the girl you see before you. How would you which of the two possibilities is true?

Note: I see no third option here. We have only one method available at our disposal, the method of elimination whereby we negate one of the disjuncts (options) and then accept/affirm as true the option that wasn't eliminated.

How does one eliminate a coincidence? One good method according to a book I read is to check for persistence over time and space and different situations. For instance, punching someone and the ensuing pain isn't a coincidence because despite the immense variety in people, the shape of their hands and whatnot, and despite differences in time and location, pain will always follow a punch. If one utilizes this principle then math, which seems almost universally applicable, must be embedded in nature and so is a discovery and not an invention.

Another method to eliminate coincidence is to look for some kind of plausible explanation for a state of affairs. In other words, why is math everywhere? Every domain of the physical world seems to be mathematical. Why? I have no satisfactory answer to this question but imagine a world where there's no math: the laws that operate in this world wouldn't, couldn't, appreciate what maybe called nuances of the forces involved. For instance a world could operate under a law such as: If x moves and hits y then y should move. As you can see, such a system of laws would fail to capture variations in the movement of x and would fail to translate that into differences in the movement of y as expressed in a mathematical law such as: if x hits y with amount of force A at an angle B then y should move at speed C in the direction D. Perhaps math is just the inevitable result of the variations inherent in the universe.

It seems we've satisfied two conditions that rule out the possibility that math's ability to describe the world is just a coincidence or, more to the point, that math is an just an invention and not a discovery. The girl you discovered in the street is too identical to the girl in the painting for the painting to be an invention of the painter.

Fascinating!

We as a human species tend to do things without question while under authoratative governing principles, similar to math but I'll get to that point in a little bit. A lot of us get up and work from 9-5 because it is universally accepted to be a part of society without question of it. We read words from the dictionary and almost never question the origins of such words. We engage in religious activities without second thought of whether this is actually true or not.

The only real question is,

Why don't most people want to question these things themselves or try to understand why they do these things without question? I hope it isn't out of fear of possibly thinking for yourselves and drawing your own logical conclusions. And the same can be said for mathematics to a varying extent. — flame2

I'm not sure that's the only real question.

I confess, I'm as boggled as you are about how it comes to pass that human beings tend by and large to take so much for granted, to neglect their own critical powers, and to behave like insensate and unwitting rule-followers. The way they flock to follow fashions! As if cultural trends or matters of convention were "laws of nature" or "objective matters of fact". The way they neglect to criticize their own beliefs, and to identify their own prejudices! As if driven by some monstrous power of vanity.

I suspect the answer has something to do with our animal nature and something to do with problems of culture.

Most people on any topic automatically want to think they are right because they are afraid of possibly being wrong.
I used to be that kind of a person but I am not anymore and it has allowed me to keep an open mind and question everything in the world in which we reside in. Of course it has also allowed me to be more accepting of people's differing opinions if you are thinking. That's just a way of life. — flame2

Congratulations on your transformation into an open-minded critical thinker!

How would you say this transformation came to pass?

But this all leads up to my main topic which is about math, — flame2

I'm not sure I follow all of your discussion on this topic.

If I follow well enough: I think you're right to say that mathematics involves abstract concepts -- chiefly concepts of number and of numerical operations. But I suspect you're wrong to infer from this that math "has no real physical purpose". Doesn't it seem rather plausible that the original purpose of our number concepts is to count physical things, and that increasingly sophisticated applications of number concepts become progressively available once the most basic operations are established?

Your example of the volcanoes suggests as much. It seems reasonable to suppose that our abstract number concepts were formed by virtue of our exercise of conceptual capacities aimed at objects of exteroception, like volcanoes. Would you agree such perceptual objects are paradigmatically physical things?

The story I like to tell about number concepts has been influenced by my encounter with Frege's Foundations of Arithmetic. You might find it an interesting read.


I'm not sure the distinction between invention and discovery is especially relevant or informative here. Clearly number concepts are products of mind and culture. Clearly we use them to make true or false assertions about objective matters of fact. Counting is a custom that emerges in history; but each instance of counting is a real thing that involves a set of physical processes. Once we learn to count, we have number concepts, and arithmetical operations seem to follow "of necessity", as the saying goes.

Likewise, abstract concepts like "dog" and "red" are products of mind and culture. And we use them to make true or false assertions about objective matters of fact. And our dispositions to affirm or deny statements purporting to describe our observations of dogs or red things seem to follow "of necessity".

Of course we refine our concepts of "number", "dog", and "red" over time. But it seems we tend to revise such concepts in light of objective -- formal or empirical -- matters of fact.