Can anyone please explain this further? It seems like he’s moving the goalposts.
The post's argument is different than the usual Galois theory result about the unsolvability of the quintic, in that it shows a property that must be true about all EML(x,y)-derived functions, and a hypothetical quintic-solver-function does not have that property, so no function we add to our repertoire via EML will solve it (or any other function, elementary or not, that lacks this property).
You can't solve an equation? Why not just introduce a function that is equal to the solution of the equation! Problem solved.
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.
Bring radicals don't. They're just defined as a solution to this particular quintic.
Kinda the similar story with the Lambert function.
Like sine or exp, it also has a nice series representation:
sum(k = 0 to inf) binom(5k,k) (-1)^(k+1) a^(4k+1) / (4k+1)
x <- x - (x^5 + x + a)/(5x^4 + 1)
Why not invite it to the table of functions?
Ellipses are simple and beautiful figures known to every child, but why do we rarely invite the elliptic integrals to the table too?
I guess my point is that "nice geometric interpretation" is a little subjective and hasn't led to much consistency in our choice of which functions are popular or obscure.
Erm... No. It's not great.
> Why not invite it to the table of functions?
Because it's too arbitrary.
> Can't solve the differential equation y'' = -y? Why not just introduce a function sin(x) as its solution! Problem solved.
But that's not how sine was introduced. It's been around since classical geometry. It was always easy to solve the differential equation y'' = -y, because the sine had that property, and we knew that.
Heck, you can tell this just by looking at the names of the functions you mentioned. "Sine" is called "sine", which appears to have originated as an attempted calque of a Sanskrit term (referring to the same function) meaning "bowstring".
"Square root" is named after the squaring function that was used to define it.
Introducing an answer-by-definition gives us negative numbers, rational numbers, imaginary numbers, and nth roots... but not sines, come on. You can just measure sines.
All of these concepts, from sine to real numbers, Bring radicals to complex exponentials, can all be defined in different, equivalent ways. What is interesting are the properties invariant to these definitions.
It still doesn't seem to me that a square root should be any more or less contrived than a Bring radical. Maybe we should call it a ultraradical instead?
Eg you can get complex numbers from matrices.
But if you want to go in your direction: you can say we get fractions and negative numbers this way.
I am a professional mathematician, though nowhere near this kind of thing. The result seems amusing enough, but it doesn't really strike me as something that would be surprising. I confess that this thread is the first I've heard of it...
Some of my favorites:
DoctorOetker: "I'm still reading this, but if this checks out, this is one of the most significant discoveries in years."
cryptonektor: "Given this amazing work, an efficient EML operator HW implementation could revolutionize a bunch of things."
zephen: "This is about continuous math, not ones and zeroes. Assuming peer review proves it out, this is outstanding."
[1] https://news.ycombinator.com/item?id=47746610
[2] https://www.reddit.com/r/math/comments/1sk63n5/all_elementar...
> 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.
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.
I think it really comes down to what set of functions you are calling "elementary".
(I'm not a mathematician, so don't expect me to have an opinion as far as that goes. But the author also writes well in English, and that language we do share.)
> 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.
Compare https://arxiv.org/abs/1108.1791 and why computational complexity is often more interesting that computability.
However by the same token couldn't you use the same brute force approach with exp minus log?
What im really asking, are NAND gates really different here?
This can be done in polynomial time as well.
This is fairly obvious if you think about that your computer can do the same thing and it’s just a fancy circuit.
It wouldn't be a math discussion without people using at least two wildly different definitions.