This fundamental "cheat" gave rise to some of the most important pure and applied mathematics known.

Can't solve the differential equation x^2 - a = 0? Why not just introduce a function sqrt(a) as its solution! Problem solved.

Can't solve the differential equation y'' = -y? Why not just introduce a function sin(x) as its solution! Problem solved.

A lot of 19th century mathematics was essentially this: discover which equations had solutions in terms of things we already knew about, and if they didn't and it seemed important or interesting enough, make a new name. This is the whole field of so-called "special functions". It's where we also get the elliptic functions, Bessel functions, etc.

The definition of "elementary function" comes exactly from this line in inquiry: define a set of functions we think are nice and algebraically tractable, and answer what we can express with them. The biggest classical question was:

    Do integrals of elementary functions give us elementary functions?

Risch gave us an algorithm to compute the answer, when it exists in elementary form.

:)

I still consider the article important, as it demonstrates techniques to conduct searches, and emphasizes the very early stage of the research (establishes non-uniqueness for example), openly wonders which other binary operators exist and which would have more desirable properties, etc.

Sometimes articles are important not for their immediate result, but for the tools and techniques developed to solve (often artificial or constrained) problems. The history of mathematics is filled with mathematicians studying at-the-time-rather-useless-constructions which centuries or millennia later become profound to human interaction. Think of the "value" of Euclid's greatest common divisor algorithm. What starts out as a curiosity with 0 immediate relevance for society, is now routinely used by everyone who enjoys the world wide web without their government or others MitM'ing a webpage.

If the result was the main claimed importance for the article, there would be more emphasis on it than on the methodology used to find and verify candidates, but the emphasis throughout the article is on the methodology.

It is far from obvious that the tricks used would have converged at all. Before this result, a lot of people would have been skeptical that it is even possible to do search candidates this way. While the gradual early-out tightening in verification could speed up the results, many might have argued that the approach to be used doesn't contain an assurance that the false positive rate wouldn't be excessively high (i.e. many would have said "verifying candidates does not ensure finding a solution, reality may turn out that 99.99999999999999999% of candidates turn out not to pass deeper inspection").

It is certainly noteworthy to publish these results as they establish the machinery for automated search of such operations.

This result itself is being described in those terms[1]:

> If this is true, then this blog post debunking EML is going to up-end all of mathematics for the next century.

This is very concerning for mathematics in general.

1: https://news.ycombinator.com/item?id=47775105

Well the author saysin that paragraph:

> In layman’s terms, I do not consider the “Exp-Minus-Log” function to be the continuous analog of the Boolean NAND gate or the universal quantum CCNOT/CSWAP gates.

But is there actually a combination of NANDs that find the roots of an arbitrary quintic? I always thought the answer was no but admittedly this is above my math level.