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I need clarity on some definitions and mathematical "skepticism". In a recent video by Matt Parker, he says "(...) although the existence of the sign function, which says if a value is positive or negative, upsets some people (...)".

What does he mean by this, and can this "upsetting" be extended to functions like the absolute value or the indicator function of a set? Any suggestions for further reading would be highly appreciated!

Edit: Before this question is closed, I would like to clarify that by no means is this question intended to incentivize discord or negative emotions regarding certain groups of people. I do not wish to offend, nor suggest I am taking part in any ideology regarding the nature of mathematics and its definitions. I am merely curious about what Matt meant, and what is the deeper motivation behind his brief comment, hopefully generating a healthy discussion. Thank you.

sam wolfe
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    The persons who would be upset by such a thing are not worth catering explanations to in my opinion. Yes, some lay persons get confused and upset when being shown things in math which do not "look mathy" according to their sense of aesthetics... thinking that piecewise defined functions or descriptions using words or such do not belong. They simply lack mathematical maturity. – JMoravitz May 24 '23 at 16:23
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    Some closed-minded people believe that a function is something that needs to be expressed by an explicit "formula" and have a very limited definition of formula. They think that a function needs to be expressible as a combination of basic operations and elementary functions, but won't accept, for example, piecewise defined functions as a "formula" (except for the absolute value function. That one, for some arbitrary reason, is a valid function) – jjagmath May 24 '23 at 16:30
  • Just a guess, but maybe it's because the "sign" function is not an elementary function (https://en.wikipedia.org/wiki/Elementary_function)? I'm not even sure if this statement is true nor how to prove it (maybe something like this: https://math.stackexchange.com/questions/265780/how-to-determine-with-certainty-that-a-function-has-no-elementary-antiderivative). In contrast, abs(x)=sqrt(x^2) is definitely an elementary function – Scott Hahn May 24 '23 at 16:39
  • Even if it can't be represented as a nice formula and it isn't continuous, there are still widely-used theories where such functions can be reasoned with and computations can be done. Indeed, one can even take the derivative of such a function! It requires a lot of technical machinery to define though, so it is no surprise that it is not widely known by general audiences – whpowell96 May 24 '23 at 16:51
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    IMO, it's not worth spending energy on wondering why some people get upset at how a function behaves. I'm surprised some people are upset. If they don't like how the sign function behaves the way it does, then they might as well get upset at any other functions behaving they way they do. – Accelerator May 24 '23 at 17:40
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    We are not here to discuss videos and unknown origins of unmotivated emotions. – mick May 24 '23 at 17:44
  • Such videos are pure clickbait. Nonsense always attracted people , unfortunately , and some people try to make profit of this fact. – Peter May 24 '23 at 18:21
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    @Peter the video is not solely about the nature of the sign function. It is hardly even mentioned beyond a brief bit saying essentially "we talked about some ways we could describe 'equations of triangles' before, but they weren't aesthetically pleasing enough (partly because of the use of the sign function)... here are more ways that people came up with." MattParker's videos are typically very high quality and enjoyable, his skill speaks for itself and he doesn't need to rely on such tactics. – JMoravitz May 24 '23 at 18:28
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    @Peter I agree with JMoravitz. Matt Parker's videos have in general well presented and interesting mathematical content. – jjagmath May 24 '23 at 19:57

1 Answers1

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According to the most classical definitions, the sign function is a function and does exist.

However, in some alternative contexts such as intuitionistics / constructive mathematics, the sign function does not exist because it is not a continuous function, and in these mathematics theories, only continous functions do exist.

See for example:

(The question might actually be closed as duplicate, as a more general question exists in MSE, cf. above. But some people might not find it if they have only heard about "the sign function does not exist", which is the usual example, and not "only continuous functions do exist".)

Jean-Armand Moroni
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  • Indeed nobody would question that $f(x) = tanh(x/C)$ is a function, because it is smooth and continuous. Yet we can take the limit of $C$ to zero, and then the function becomes equal to the sign function. So at the very least, the sign function has a useful role as the limit of the $tanh(x/C)$ function. – M. Wind May 24 '23 at 16:56
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    To emphasize, in the posts you cite the meanings and definitions of words are different than how the vast majority of people use the words. Claiming that the sign function is not a function because these other people choose to define the word "function" in a nonstandard way is like saying that the number $1$ is not actually "a number" because some other people choose to define it as their preferred flavor of breakfast cereal and not as a mathematical object. Just because different people use words in different ways is not a good enough response in my opinion and deserves heavy disclaimers. – JMoravitz May 24 '23 at 17:01
  • @JMoravitz Brouwer was one of the first proponents of intuitionistic mathematics, as you know, and he was highly reputable. There has been a lot of undue turmoil about constructive mathematics. Now we have many variants of maths foundations, some of which are used in non-maths fields (cf. various logic flavours, e.g. linear logic, in computer science). So a more restrictive definition of "functions" - which has some use in computer science - is acceptable, as long as everybody clearly states what one uses. – Jean-Armand Moroni May 24 '23 at 17:17
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    @JMoravitz (?) Please do not downvote because that does not fit the mainstream theory. I am actually not a proponent of intuitionistics / constructive mathematics. I just recognize that these are alternate ways to build mathematics foundations, and they actually solve some problems that more mainstream (and permissive) theories have. Anyway, this has already been discussed at length during decades. – Jean-Armand Moroni May 24 '23 at 17:19
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    Recommend rewriting the first sentence of your post to something more like "According to the more classical definitions, the sign function is a function and does exist, however in certain alternative contexts such as those used by ..." If someone were to skim and not fully read your post it would sound as though you are suggesting that in every context (and importantly, even in the most common contexts which are taught in classrooms) that the sign function does not qualify as a function which is blatantly incorrect. – JMoravitz May 24 '23 at 17:23
  • @JMoravitz Yes, good idea. – Jean-Armand Moroni May 24 '23 at 17:25
  • Thank you for the links! I never thought much about his, but I am curious about what I will find. – sam wolfe May 24 '23 at 17:59
  • No serious mathematician would deny the existence of a function just because it is not continuous. Not even a constructive mathematician. The sign function is not at all pathological. What a constructive mathemtician might deny are nondescribable functions or uncomputable functions. – Peter May 24 '23 at 18:17