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I seem to remember reading once a story that some mathematician had written to justify the use of categories, or isomorphisms or equivalences, or something like that. The story goes something like this:

Once upon a time, people did not know what equality was. Instead, they only thought about things up to isomorphism. For example, they did not say that two sets had the same number of elements, but that they were in bijection. Today with category theory go back to these roots.

Does anyone have an idea of who told this story and what the full story is?

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    $\begingroup$ When we teach kids to count we create a bijection between the fingers on a hand and the objects that we are counting. $\endgroup$ – Thomas Rot Oct 6 '17 at 14:49
  • $\begingroup$ That's the idea yes. I've actually seen a public lecture for non-mathematicians where the speaker started from there to the notion of bijection between infinite sets, to the proof of the fact that N is not in bijection with R. People seemed to really enjoy it. $\endgroup$ – Maxime Lucas Oct 6 '17 at 14:54
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This sounds an awful lot like TWF week 121:

To understand this, the following parable may be useful. Long ago, when shepherds wanted to see if two herds of sheep were isomorphic, they would look for an explicit isomorphism. In other words, they would line up both herds and try to match each sheep in one herd with a sheep in the other. But one day, along came a shepherd who invented decategorification. She realized one could take each herd and "count" it, setting up an isomorphism between it and some set of "numbers", which were nonsense words like "one, two, three,..." specially designed for this purpose. By comparing the resulting numbers, she could show that two herds were isomorphic without explicitly establishing an isomorphism! In short, by decategorifying the category of finite sets, the set of natural numbers was invented.

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    $\begingroup$ See also the paper Categorification by Baez and Dolan. And don't overlook the same parable being told in Paul Taylor's Practical Foundations of Mathematics. $\endgroup$ – Todd Trimble Oct 6 '17 at 17:27
  • $\begingroup$ @Todd have you a more precise reference as to where in Paul Taylor's book? It's a beast! $\endgroup$ – David Roberts Oct 7 '17 at 5:02
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    $\begingroup$ @DavidRoberts "Bo Peep" makes her first appearance in Exercise 1.1, page 60, and later the assertion that any injective endofunction on a finite set is a bijection is called "Bo Peep's theorem". I've also heard Paul relate the parable at conference dinners. $\endgroup$ – Todd Trimble Oct 7 '17 at 8:02
  • $\begingroup$ I think the story is Georges Ifrah's, available in English as From 1 to 0. "Let us imagine a shepherd, unable to count, who has a flock of sheep that he keeps in a cave. There are 55 of them, but he has no understanding of what "the number 55" means. He would like to be sure that all of them come back every evening. One day he has an idea. He sits down at the entrance of his cave and has the sheep go in one by one. Each time one of them passes, he makes a notch in a bone. When all the sheep have passed, he has made exactly 55 notches, without knowing the arithmetic meaning of that number." $\endgroup$ – David Treumann Oct 15 '17 at 22:00
  • $\begingroup$ and a little later from the same book, now describing an archaeologist's discovery: "...without knowing it, the uneducated servant had repeated a procedure used by equally uneducated shepherds who lived in that region 3500 years earlier. The clay envelope had belonged to an accountant (who, unlike the shepherds, knew how to write). The shepherds had gone to see him before taking their master's flock to pasture. He made as many clay balls as there were animals in the flock and put them into the envelope, which was then closed." $\endgroup$ – David Treumann Oct 15 '17 at 22:08
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As Todd has already written an answer for me, maybe I can claim it as an Answer:

Exercise 1.1 in my book Practical Foundations of Mathematics (CUP 1999) reads,

When Bo Peep got too many sheep to see where each one was throughout the day, she found a stick or a pebble for each individual sheep and moved them from a pile outside the pen to another inside, or vice versa, as the corresponding sheep went in or out. Then one evening there was a storm, and the sheep came home too quickly for her to find the proper objects, so for each sheep coming in she just moved ANY one object. She moved all of the objects, but she was still worried about the wolf. By the next morning she had satisfied herself that the less careful method of reckoning was sufficient. Explain her reasoning without the aid of numbers.

My reason for putting it in the book was to try to get some anthropologist to say when and what the original "proof" was, ie the cognitive basis of the long-universal belief that this is valid, which provides the justification of counting with numbers.

What I am trying to imagine is how one of our distant ancestors with the cognitive abilities but not the education of a mathematician might approach this. Of course they would not have formulated Peano Induction or Euclidean Infinite Descent. They would have an argument (that we would more or less accept as rigorous) that the result is true for three sheep, then four and five. After that they would use Induction in the naive epistemological sense to convince themselves that it is true for arbitrarily large sets.

It seems plausible that anyone who is challenged to come up with a proof would give the following (albeit non-constructive) proof.

Suppose that some sheep $s_0$ is missing in the evening. Then the pebble $p_0$ that served as its "name" in the morning was used for some other sheep $s_1$ in the evening. But then $s_1$ must have been named by a different pebble $p_1$ in the morning, which named yet another sheep $s_2$ in the evening. And so on. All of the sheep $s_0$, $s_1$, $s_2$, ... are different individuals, Likewise all of the pebbles $p_0$, $p_1$, $p_2$, ... But, essentially as Euclid says in Book VII, Proposition 31, this is impossible for a set of sheep.

By chance, this issue came up recently following an internal seminar by Martin Escardo in Birmingham (where I am now an Honorary Research Fellow). He was developing the foundations of arithmetic (in the setting of Homotopy Type Theory, though this was not essential) in such a way that $3\times 5=5\times 3$ could be seen in a primary-school fashion as transposing a rectangle.

He based this on a function $F:{\mathbb{N}}\to{\mathsf{Set}}$ with $F0=\emptyset$ and $F(\mathsf{succ} n)=F(n)\coprod{\mathbf{1}}$. In his treatment the most difficult Proposition is $$ F(n)\cong F(m) \Longrightarrow n=m, $$ which he deduced from the Lemma $$ X\coprod{\mathbf{1}}\cong Y\coprod{\mathbf{1}} \Longrightarrow X \cong Y. $$

This Lemma holds in any lextensive category, i.e. one with finite limits and stable disjoint coproducts. The Proposition follows using Peano induction, since then $$ F(n+1)\cong F(m+1) \Longrightarrow F(n)\cong F(m) \Longrightarrow n=m \Longrightarrow n+1=m+1. $$

I think it is reasonable to suppose that Bo Peep could formulate this Lemma, but I feel it is more plausible that she would use the "infinite descent" argument than the Proposition.

I am not sure whether this answers the original question about justifying "the use of categories, or isomorphisms or equivalences", although maybe Martin's treatment of arithmetic does so.

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    $\begingroup$ It seems that one could develop much of a theory of permutation groups before inventing the concept of number to ease the development. Does Martin Escardo's development resemble such a development of permutation groups? Gerhard "Who Needs These 'Numbers' Anyway?" Paseman, 2017.10.15. $\endgroup$ – Gerhard Paseman Oct 15 '17 at 16:21
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    $\begingroup$ I see the temptation of permutation groups. Also, HoTT (Martin's setting) was partly motivated by (higher dimensional) groupoids. I think, however, Martin wanted to eliminate (as much as possible) the repeated use of induction in Giuseppe Peano's 1889 paper to prove things like commutativity, associativity and distributivity of $+$ and $\times$ in favour of (the more appropriate) primary-school set theory. (At least, Peano presumably used induction repeatedly: his treatment is extremely terse.) $\endgroup$ – Paul Taylor Oct 15 '17 at 17:53
  • $\begingroup$ This suggests to me using Boolean circuit (flowchart) analysis to restructure proofs to minimize certain component usage (for example, building a three inverter circuit using ands, ors, and just two inverters). Do you know of anyone who has diagrammed the proofs and done a similar kind of analysis on them? Gerhard "Computer-Aided Proof Optimization Through Clones!" Paseman, 2017.10.15. $\endgroup$ – Gerhard Paseman Oct 15 '17 at 18:58
  • $\begingroup$ Another simple argument (though I cannot speak as to its historical plausibility) which avoids explicit consideration of "infinite descent" might be like so: The sheep go out in some order, one pebble marked with corresponding sheep's name tossed onto the end of a row for each one, building up a row yea long. [cont] $\endgroup$ – Sridhar Ramesh Oct 15 '17 at 21:39
  • $\begingroup$ If the sheep were to return in precisely the opposite of the order in which they went out, then we could easily knock their corresponding pebble off the end of the row with each return; this would all play out in precise reverse of the build-up, and the row would return to its initial emptiness when, and precisely when, all the sheep had returned. BUT! [cont] $\endgroup$ – Sridhar Ramesh Oct 15 '17 at 21:39

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