Editing Hand-waving (section)

==In mathematics==
{{anchor|Mathematics|In mathematics}}
In mathematics, and disciplines in which mathematics plays a major role, ''hand-waving'' refers to either absence of formal proof or methods that do not meet [[mathematical rigor]]. In practice, it often involves the use of unrepresentative examples, unjustified assumptions, key omissions and faulty logic, and while these may be useful in expository papers and seminar presentations, they ultimately fall short of the standard of proof needed to establish a result. [[Proof by intimidation]] is one form of hand-waving.

The mathematical profession tends to be receptive to informed critiques from any listener, and a claimant to a new result is expected to be able to answer any such question with a logical argument, up to a full proof. Should a speaker apparently fail to give such an answer, anyone in the audience who can supply the needed demonstration may sometimes upstage the speaker. The objector in such a case might receive some measure credit for the theorem the hand-waver presented.{{citation needed|date=September 2015}} The opposite of hand-waving in mathematics (and related fields) is sometimes called '''nose-following''', which refers to the unimaginative development of a narrow line of reasoning that—while correct—can also end up making the subject dry and uninteresting.<ref>{{Cite journal|title=Polycategories via pseudo-distributive laws|journal=[[Advances in Mathematics]]|volume=218|issue=3|pages=781–827|arxiv=math/0606735|last1=Garner|first1=Richard|year=2006|doi=10.1016/j.aim.2008.02.001|bibcode=2006math......6735G|s2cid=17057235}}</ref>{{better source needed|date=May 2025}}

The rationale for this culture of hyper-critical scrutiny is suggested by a quote of [[G. H. Hardy]]: "[A mathematician's] subject is the most curious of all—there is none in which truth plays such odd pranks. It has the most elaborate and the most fascinating technique, and gives unrivalled openings for the display of sheer professional skill."<ref>G. H. Hardy (1940). "A Mathematician's Apology".</ref>