# The Universe of Discourse

Mon, 20 Nov 2017

Mathematicians tend not to be the kind of people who shout and pound their fists on the table. This is because in mathematics, shouting and pounding your fist does not work. If you do this, other mathematicians will just laugh at you. Contrast this with law or politics, which do attract the kind of people who shout and pound their fists on the table.

However, mathematicians do tend to be the kind of people who quibble and pettifog over the tiniest details. This is because in mathematics, quibbling and pettifogging does work.

Mathematics has a whole subjargon for quibbling and pettifogging, and also for excluding certain kinds of quibbles. The word “nontrivial” is preeminent here. To a first approximation, it means “shut up and stop quibbling”. For example, you will often hear mathematicians having conversations like this one:

A: Mihăilescu proved that the only solution of Catalan's equation !!a^x - b^y = 1!! is !!3^2 - 2^3!!.

B: What about when !!a!! and !!b!! are consecutive and !!x=y=1!!?

A: The only nontrivial solution.

B: Okay.

Notice that A does not explain what “nontrivial” is supposed to mean here, and B does not ask. And if you were to ask either of them, they might not be able to tell you right away what they meant. For example, if you were to inquire specifically about !!2^1 - 1^y!!, they would both agree that that is also excluded, whether or not that solution had occurred to either of them before. In this example, “nontrivial” really does mean “stop quibbling”. Or perhaps more precisely “there is actually something here of interest, and if you stop quibbling you will learn what it is”.

In some contexts, “nontrivial” does have a precise and technical meaning, and needs to be supplemented with other terms to cover other types of quibbles. For example, when talking about subgroups, “nontrivial” is supplemented with “proper”:

If a nontrivial group has no proper nontrivial subgroup, then it is a cyclic group of prime order.

Here the “proper nontrivial” part is not merely to head off quibbling; it's the crux of the theorem. But the first “nontrivial” is there to shut off a certain type of quibble arising from the fact that 1 is not considered a prime number. By this I mean if you omit “proper”, or the second “nontrivial”, the statement is still true, but inane:

If a nontrivial group has no subgroup, then it is a cyclic group of prime order.

(It is true, but vacuously so.) In contrast, if you omit the first “nontrivial”, the theorem is substantively unchanged:

If a group has no proper nontrivial subgroup, then it is a cyclic group of prime order.

This is still true, except in the case of the trivial group that is no longer excluded from the premise. But if 1 were considered prime, it would be true either way.

Looking at this issue more thoroughly would be interesting and might lead to some interesting conclusions about mathematical methodology.

• Can these terms be taxonomized?
• How do mathematical pejoratives relate? (“Abnormal, irregular, improper, degenerate, inadmissible, and otherwise undesirable”) Kelley says we use these terms to refer to “a problem we cannot handle”; that seems to be a different aspect of the whole story.
• Where do they fit in Lakatos’ Proofs and Refutations theory? Sometimes inserting “improper” just heads off a quibble. In other cases, it points the way toward an expansion of understanding, as with the “improper” polyhedra that violate Euler's theorem and motivate the introduction of the Euler characteristic.
• Compare with the large and finely-wrought jargon that distinguishes between proofs that are “elementary”, “easy”, “trivial”, “straightforward”, or “obvious”.
• Is there a category-theoretic formulation of what it means when we say “without loss of generality, take !!x\lt y!!”?

[ Addendum: Kyle Littler reminds me that I should not forget “pathological”. ]