0

As you get serious with math the first order of business is to learn some set theory and get $\mathbb N$ going with addition and multiplication. Then you will usually learn Euclidean division. Next you will learn how to use exponents.

But hold on a second! You can get some abstract stuff going without even knowing how to represent integers using a base!

Suppose we have a binary operation $+$ on a set $M$ and have at our disposal the set of $\mathbb Z$ of integers with only the addition operation, and we only have symbols for 5 integers

$\quad \mathbb Z = \{\dots,-n,\dots-2,-1,0,1,2,\dots,n,\dots\}$

Now consider any bijective function $f: M \to M$. A great notation to adopt is $f^{-1}$ is the inverse, $f^{0}$ is the identity, $f^{1} = f$, $f^{2} = f \circ f$, etc. The following expression is well-defined: a $\tag 1 \sum_{k \in F} f^{k} \text{ with } F \text{ a finite subset of } \mathbb Z$

This is a pithy definition of a function that can be made early on.

Are there any introductory or advanced foundational books in math that develop the concept of exponentiating a function before introducing multiplication?

CopyPasteIt
  • 11,366
  • 1
    This question is unclear to me on multiple levels. Presumably $A$ and $F$ were supposed to be the same set? Is $f^k$ supposed to be the $k$-fold iterate? Also, the premise of your question seems false to me: a first 'get serious' math class does not start from axiomatic set theory/a formal definition of the naturals at all, that's more like a late undergraduate topic. – Mees de Vries Oct 25 '18 at 12:47
  • @MeesdeVries Thanks fixed it up. And since this is a 'soft question' I guess it is up to debate what a serious course is all about. – CopyPasteIt Oct 25 '18 at 12:51
  • So is your question about whether it makes sense to define iterated composition first in axiomatic set theory, or is it about whether there are pupils learning about it before they learn multiplication? It seems that your question is the latter, but the answer to that is as obviously "no" to me as to the question "are there pupils who study Steinbeck before they've learned to write", so then I do not understand why you'd even ask it. – Mees de Vries Oct 25 '18 at 12:54
  • I am not, and I don't appreciate the implication. The answer to your question is certainly no, but that is also impossible to prove without looking at everything that purports to be an introductory book in mathematics. – Mees de Vries Oct 25 '18 at 13:04
  • @MeesdeVries Sorry, I deleted the question, but it is certainly possible that someone has direct experience by reading some book. – CopyPasteIt Oct 25 '18 at 13:09
  • @MeesdeVries But my project here - https://math.stackexchange.com/q/2963140/432081 will make the case that exponentiating functions (when it makes sense) is not really 'crazy' and can be regarded as more primitive than multiplication. – CopyPasteIt Oct 25 '18 at 13:12

0 Answers0