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I often see arguments referred to as "technical" but I'm not sure what that really means. Does it mean "unintuitive"? Or does it mean "tedious"? Or something else entirely?

I have sometimes seen it used to contrast with an intuitive argument, but I find that even these "technical" arguments can be very intuitive once understood.

Is there an agreed (or mostly agreed) upon definition of "technical"?

I would also really appreciate some lucid examples of technical vs non-technical arguments.

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    $\begingroup$ I usually translate “technical” as “somewhat tedious, with a lot of detail you need to pay close attention to, and not very deep”. I would guess that an argument is technical when you struggle to understand it well at a glance after several times reading it in depth, and need some notes to explain it clearly at a blackboard (or a tablet), or when only an expert or a fully-focused attendant can understand someone explaining it during a seminar... but it’s not very clear-cut, more “I know it when I see it”. And I don’t have that much experience so perhaps the accepted meaning is different. $\endgroup$
    – Aphelli
    Feb 22, 2021 at 22:40
  • $\begingroup$ @peek-a-boo 's answer which was deleted seemed helpful to me; perhaps it will be re-posted. $\endgroup$
    – littleO
    Feb 22, 2021 at 22:50
  • $\begingroup$ This is not a matter of mathematical definition. So no, there is no agreed upon mathematical definition whatsoever. You're probably better served looking it up in the dictionary. As for how it's used, if a person reads a paper and finds it thick with many details and slow going, they may perhaps describe it to another person as "too technical". $\endgroup$
    – Lee Mosher
    Feb 22, 2021 at 22:57
  • $\begingroup$ @LeeMosher The closest definitions I see in Merriam-Webster are "of or relating to a particular subject" and "marked by or characteristic of specialization". I think your own description is quite good. $\endgroup$
    – littleO
    Feb 22, 2021 at 23:04

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I would think "technical" is a rather subjective term because what one person deems technical another might regard as fundamental. But in most cases it refers to a situation where a rather intricate argument is required to establish a certain statement, and usually one uses this term in the situation where this extra effort doesn't necessarily lead to any greater clarity of the big overall idea.

  • One example I can think of is the Fubini-Tonelli theorem in measure theory. I think intuitively this theorem makes a lot of sense "if you add non-negative things up in two different ways you should get the same result", but of course if you've seen the proof of this theorem it is no trivial task. For example the proof I know (the one in Folland's book) requires the use of the Monotone class lemma; I would count this as a rather technical lemma. It also requires $\sigma$-finite measure spaces, and so just by reading the proof one immediately sees that there's a lot of facts to be verified, and while all of them are plausible sounding, none of them is immediately obvious to prove.

  • Also as is common in measure theory, many results are usually easily proven in the case where the measure is finite, and then one extends the argument to $\sigma$-finite spaces. And in certain situations, one can then extend the argument to yet more general situations. For example, consider the duality theorem:

For any measure space $(X,\mathfrak{M},\mu)$ and any $1<p<\infty$ with $q$ the conjugate Holder exponent, the mapping $L^q(\mu)\to (L^p(\mu))^*$ defined by $g\mapsto \left(f\mapsto \int_X fg\, d\mu\right)$ is an isometric isomorphism of Banach spaces.

The real crux of this theorem lies in the case when $\mu$ is a finite measure. From here, extending to the case of $\sigma$-finiteness is rather simple (we're just adding up countably many results and patching everything together nicely). Then finally, one can extend the theorem to any arbitrary measure space by a rather intricate argument which essentially reduces is back down to the $\sigma$-finite case. Also, in trying to establish the more general cases, there are no "new ideas" involved in the proof.

So, one could say this theorem requires a rather technical argument to establish in full generality (and also, since most people work with $\sigma$-finite spaces anyways, this last bit of extra generality, while nice to know, will certainly be deemed as additional technicalities).

  • An example of a technical definition, rather than a technical argument is in the definition of integration on manifolds. For this, one commonly uses a partition of unity, and as you can see from the definition of a partition of unity itself, proving the existence (in the smooth case) is not a trivial task at all, and requires a rather intricate argument. Moreover, the definition of integration on manifolds using partitions of unity is hardly useful in actual computations (though very useful for theoretical proofs), and as such it is regarded as a very technical definition.

  • Also, take a look at the usual proofs of the multivariable change of variables theorem as presented in Spivak or Munkres... god that's an awful theorem to prove (sure, one can try to finesse the proof using some measure theory, using Radon-Nikodym derivatives etc, but this is, I think, one of the most basic yet technical theorems in integral calculus).

In fact, depending on how far along one is in their mathematical journey, even the proof that every continuous function on a compact interval $[a,b]$ is Riemann-integrable (a result which most people take for granted) is somewhat technical because it uses the notion of uniform continuity (well I'm sure you can avoid it as Spivak does in his calculus text, but still, this proof is not at all transparent). (I realize that so far all my examples relate to integration; this is unintentional... that's all that immediately springs to mind immediately).

Another example I can think of is the inverse function theorem from differential calculus:

If $f:\Bbb{R}^n\to\Bbb{R}^n$ is $C^r$ and $a\in\Bbb{R}^n$ is a point such that $Df_a$ is an isomorphism then there exist open neighborhoods $A$ of $a$ and $B$ of $f(a)$, such that the resitrction $f:A\to B$ has a $C^r$ inverse $f^{-1}:B\to A$.

Conceptually, this is simple enough to understand: a derivative by definition is a local linear approximation to the function. So, if the derivative at a point is invertible then it would be nice to expect that the function itself inherits this same nice property. In the linear case, what this is saying is that if $M$ is an invertible matrix then the equation $y=Mx$ can be solved for $x$ as $x=M^{-1}y$.

Now, in the course of proving the inverse function theorem, what one usually does is start by a sequence of approximations; then one has to invoke certain technical lemmas such as (Banach's I think) fixed point theorem, and then one has to show that the limit of the sequence of approximations actually converges to the right thing etc. These are all I think somewhat technical details. I think more important is realizing that this is one of many instances of "behavior of derivative implies something about behavior of function", and it is also important to recognize and know how to prove the equivalence between the inverse and implicit function theorems.

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  • $\begingroup$ This is great, thanks a lot (and extra thanks for the examples). You made an interesting point about technical arguments not adding any "new ideas". That definitely fits with how I've seen the term used. $\endgroup$
    – dmh
    Feb 22, 2021 at 23:39
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    $\begingroup$ @MONODA43 right, and in many situations, such technicalities arise when we're trying to prove a theorem in greater generality. Above I talked about the duality of L^p spaces; but we can also talk about similar stuff for the integral theorems like Stokes theorem. Usually if the domain is simple enough (a disk/ region bounded by two graphs etc) then the proof is a straightforward calculation; generalizing by weakening assumptions on the domain usually requires more involved arguments which are unnecessary for a first introduction. These may involve new ideas yet still be deemed "technical". $\endgroup$
    – peek-a-boo
    Feb 22, 2021 at 23:57

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