What do you call an object constructed as part of a proof?

Question

I find myself wanting to talk about parts of a proof, e.g. the role played by mathematical expressions within a proof.

When proving a theorem it is common to construct some kind of object and then prove this has certain properties. E.g. in the standard proof that there are infinitely many primes we assume the primes are finite $p_1,\cdots, p_k$ and the consider $n=p_1\times \cdots \times p_k +1$. We then establish properties of $n$ which lead to a contradiction. What do you call the act of constructing $n$, or $n$ itself?

I think that having names for things is important, especially when talking about them. For example, many simple proofs by contradiction could be a contrapositive instead. Having a word "contrapositive" is very helpful indeed in discussing such proofs, and explaining to students the difference between contradicting the hypothesis, and a general external contradiction such as $1=0$.

The word "ansatz" is widely used "is an educated guess or an additional assumption made to help solve a problem, and which is later verified to be part of the solution by its results" (Wikipedia). I find having a word for an ansatz is very helpful.

I don't have a word for "a particular object constructed as a device within a proof, built to establish certain conditions must hold".

I have lots of other examples of such objects, but no name for them.

My favourite proposal at the moment is "gadget", or "proof-gadget" for emphasis. This is a relatively positive, utilitarian word (not used elsewhere in mathematics as far as i can tell). Gadgets are something which a built for a particular purpose, and are often igneous.

I can, of course, use words as I see fit (The Humpty Dumpty defence) but I'd like to ask if anyone knows of words used routinely for this purpose?

Thanks, Chris

Answer 1

Two kinds of "objects" popping up in proofs come to my mind:

1. One defines object X and then proves that X has some required properties. In this case I'd call X a "candidate" when introducing it, and I think this is not uncommon.

2. One assumes something and then defines the object Y (as the number $n$ in your example) to disprove it. In this case I'd call Y just a "counterexample", no? In your case, $n$ is a counterexample to the claim that the prime divisors of any number are among $p_1,\dots,p_k$.

I'm not saying that those are the only cases of "objects in proofs", but perhaps they're the most important ones, and I can't immediately point at cases that are really so different from 1. or 2.

Answer 2

This question was asked four years ago, but never got an answer (though, it had some discussion in the comments). This question could have also been asked on academia.SE but it's not clear it's really on topic there either. Let me try to answer so that this doesn't linger forever on the unanswered queue.

With any question about writing, in addition to the guiding principle of not inventing new terminology for something that already has standard terminology, another good guiding principle is do what's best for your reader. Having prepared thousands of pages of writings in my research and teaching, I've found that what's best for the reader is usually whatever is clearest. Hence, when I make a construction in one of my papers and want to refer to it elsewhere in the paper, I give it a numbered environment, just like an equation, definition, lemma, proposition, theorem, conjecture, etc.

Construction 2.1: Under the assumption that there are only finitely many primes $p_1, \dots, p_k$, let $P = p_1p_2\cdots p_k + 1$.

Later in the paper, maybe I need to do the same trick again, and can say "Using the same technique as Construction 2.1, we now ..."

I've also had things like:

Standing Hypothesis 1.1: We assume all model categories are cofibrantly generated in this paper.

or

Agreement 2.1: Let us agree that by "operad" we mean "reduced operad" in what follows.

I think this is MUCH clearer than words like Scholium and Ansatz, that have the potential to throw off readers who have not seen words like that before. I teach so many students from Asia for whom English is already a second language, and words that draw from yet a third language tend to throw them off quite a lot.

Now suppose you're teaching a course on proof-writing, or writing a paper about how we write mathematics, and want a way to refer to the general idea of a construction done inside a proof. In that case, I think "gadget" is a good choice, even though it has a technical meaning elsewhere, because most students/readers know what a gadget is, especially if I mention the real-world meaning of the term and why I chose that word for this concept. Based on the comments, it sounds like there is no standard term. If you want to avoid "gadget" because of the connection to computational complexity, you can use "gizmo" instead.

Lastly, I want to point out that one way to avoid gizmos in proofs is to create lemmas, following Terry Tao's advice. You could imagine doing all your writing in such a way that, wherever you were constructing something inside a proof, you deliberately pulled that out into a lemma like:

Lemma 3.1: If $L = \{p_1, \dots, p_k\}$ is a finite list of prime numbers, then $P = p_1\cdots p_k + 1$ is divisible by a prime not in $L$.

This technique of writing is extremely clear and helps the reader focus on one proof at a time, instead of a proof within a proof. Incidentally, this gives me another idea. You could use "inception" to refer to the phenomenon of proofs/constructions within existing proofs of other statements.