Mathematics, particularly advanced mathematics, can be a very confusing subject.  So it is interesting that so many with at best a passing familiarity with it are vocal about its current state.  As with the natural sciences, there is first the topic itself: the research accumulated over centuries, the modern explosion of knowledge, and the influence of this knowledge on other academic disciplines.  Then there are the ancillary disciplines: the history of mathematics, the philosophy of mathematics, mathematics education, etc., that do not address the topic directly, but rather concern how it developed, how it is to be understood, and how it should be taught.  This entry point is much more accessible than the arcane symbolism of advanced mathematics, allowing amateurs to voice opinions on mathematical ideas.

There is nothing wrong with amateurs voicing opinions on areas of any discipline they understand, but this requires the self-discipline to limit those opinions to areas within the boundaries of their own knowledge and training.  This requires the amateur to not only know what they do not know but also to admit it.  However, just as with accepted theories in the natural sciences, some mathematical theorems produce results that the average person would not expect.  The net result is that some people set out to prove a scientific and/or mathematical consensus wrong without fully understanding the reasoning behind it. 

In the case of mathematics, an unsuspecting person might come across a magazine article, interview, television program, or YouTube video aimed at a general audience that  describes some theorem or idea in mathematics that is troubling – even shocking – to someone who has never heard anything like this before.  For most, the usual reaction is an immediate “this just can’t be true.”  This is no doubt exacerbated by the habit of many popular media treatments reveling in the scandalous assault upon the common sense of the rabble.  If, instead, they emphasized the restrictive conditions of some of these theorems and the possibility of alternate interpretations.

Faced with this jolt to their worldview, most shrug their shoulders, decide they will never understand mathematics, perhaps deride ivory tower nerds, and the matter will soon be forgotten.  However, some will not let go of this offense to common sense, decide they know mathematics better than those highly trained in the subject, and try to publicly challenge the upsetting theorems. 

This is most prevalent among those who understand just enough mathematics to be fully confident in their abilities without understanding their limitations.  They usually voice their dissent in blog posts and YouTube videos, which demonstrate their lack of familiarity with the material.  Aside from the lack of proper training in the subject matter, the most prevalent issue is their failure to grasp the context of the contested theorem.  As with many technical fields, mathematics relies on a specialized vocabulary with definitions agreed upon by those in the field and assumed without further explanation.  Furthermore, the defined terms often overlap with common vocabulary but are used in a very specific sense that does not correspond to common usage. 

Thus, when some of those challenging the results given by mathematicians present their arguments, it is clear they have not properly understood the context, as they are not using the word in the same manner as defined in the material and have committed the logical fallacy of equivocation.  This fallacy occurs when one challenges an argument by changing the meaning of the terms to something other than their intended meaning in the argument being challenged.

A similar error occurs when those challenging results do so without paying attention to the assumed preconditions.  For example, suppose a theorem begins “Let f(x) be on an open interval.”  If those contesting the theorem use a counter-example where f(x) is not defined somewhere on that interval, they clearly have not understood the necessary preconditions of the theorem.

The net outcome is that those challenging the results are wrong even when their arguments are otherwise sound, as they have ignored the context of the theorems being challenged.  It is a bit like someone playing chess but insisting that the pieces can move in a manner other than that agreed upon by everyone who regulates the game and by every professional chess player.  Even if your result is self-consistent, you are playing by rules other than those used in the original result.  When you are using rules of your own invention, you are playing an entirely different game.

Note that there is a difference between believing mathematicians should use different presuppositions and saying the results are wrong for the presuppositions they are using.  The latter would be like saying bishops do not move, as they obviously do by the rules everyone has agreed upon.  The former is arguing that the rules should be different.  Of course, this places the burden of the argument on those who desire a change to convince others to change.  Without a clear benefit to all involved, it is unlikely to sway the mathematical community, and so they instead convince themselves they have found some hitherto undiscovered error.

In order to see how such a desire for change could possibly happen, consider some recent work in the foundations of mathematics.  Throughout much of the twentieth century, the default foundation for pure mathematics has been some form of set theory derived from the work of Ernst Zermelo, with ZFC (Zermelo-Fraenkel set theory augmented with the Axiom of Choice) as the dominant system.  It has long been known that other systems could also be used, but there were both technical and pedagogical reasons for using the Zermelo set theoretical lineage.  In recent decades, however, newer challengers to the status quo have arisen, such as category theory and homotopy type theory, that have convinced a significant number of mathematicians that there should be a shift away from set theory to another system.  While this still remains a minority position, the default view could change within a few decades with a new generation of mathematicians.  Whether it does will be the decision of the community as a whole and not some amateur who fancies himself a mathematical Galileo suffering for the truth but is actually more of a Don Quixote figure jousting with windmills.

The important thing to remember when encountering such claims is that there is generally far less to them than meets the eye.  It might even be suggested that one use such videos to hone logical skills by attempting to discover the fallacy involved.  It certainly is preferable to take anything in such videos to heart.  They may, however, occasionally provide me with teaching moments.