[lotaezenwa]

The author essentially says that the quintic has no closed form solution which is true regardless of the exp-minus-log function. The purpose of this blog post is lost on me.

Can anyone please explain this further? It seems like he’s moving the goalposts.

[reikonomusha]

"The quintic has no closed form solution" is a theorem that is more precisely stated (in the usual capstone Galois proof) as follows: The quintic has no closed form solution in terms of arbitrary compositions of rational numbers, arithmetic, and Nth roots. We can absolutely express closed form solutions to the quintic if we broaden our repertoire of functions, such as with the Bring radical.

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).

[lotaezenwa]

Cool explanation, thanks!

[cyberax]

Bring radicals are just cheating.

You can't solve an equation? Why not just introduce a function that is equal to the solution of the equation! Problem solved.

[reikonomusha]

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?

The answer is "no" and Liouville gave us a result which tells us what the answer does look like when the result is elementary.

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

[eru]

That's one way to get at complex numbers and the sine function. But it's not the only one.

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.

[cyberax]

Sure. But the square root and the sine function also have nice geometric interpretations.

Bring radicals don't. They're just defined as a solution to this particular quintic.

Kinda the similar story with the Lambert function.

[reikonomusha]

The Bring radical has a great geometric interpretation: BR(a) is where the curve x^5 + x + a crosses the x axis.

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)

We can compute its digits with the very rapidly convergent Newton iteration

    x <- x - (x^5 + x + a)/(5x^4 + 1)

and so on.

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.

[cyberax]

> The Bring radical has a great geometric interpretation

Erm... No. It's not great.

> Why not invite it to the table of functions?

Because it's too arbitrary.

[thaumasiotes]

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

> 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.

[reikonomusha]

You can calculate, measure, draw, construct, write a power series for, express as hypergeometric function, etc. the Bring radical too.

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?

[xigoi]

For me, what makes the square root more “natural” is that, although it’s usually introduced as an “answer by definition”, it can also be arrived at by wondering what happens if you take something to the halfth power.

[markgall]

Can anyone provide a link that "Some are going as far as to suggest that the entire foundations of computer engineering and machine learning should be re-built as a result of this", or anything similarly grandiose?

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...

[saithound]

It's a fun, but unsurprising undergrad-level result. It got picked up and overhyped on HN [1] and /r/math [2] earlier this week.

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...

[DoctorOetker]

:)

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.

[renewiltord]

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

[seanhunter]

Why on earth would it upend all of mathematics? Secondly, even if it did that, why would that be concerning for mathematics?

[renewiltord]

Yes or no, it's too early to tell.

[DevelopingElk]

His claim is that we exp-minus-log cannot compute the root of an arbitrary quintic. If you consider the root of an arbitrary quintic "elementary" the exp-minus-log can't represent all elementary functions.

I think it really comes down to what set of functions you are calling "elementary".

[throwanem]

The author discusses this in his third paragraph, and states explicitly in his fourth that he considers the result faulty for its unrealistically narrow definition of elementarity.

(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.)

[bawolff]

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.

[reikonomusha]

Combinations of the NAND gate can express any Boolean function. The Toffoli (CCNOT) or Fredkin (CSWAP) can express any reversible Boolean function, which is important in quantum computing where all gates must be unitary (and therefore reversible). The posited analog is that EML would be the "universal operator" for continuous functions.

[zeroonetwothree]

Yes, NAND gates can implement root finding algorithms for arbitrary polynomials. For example a variant of Newton’s method can be used (there are also better algorithms for circuits specifically).

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.

[floxy]

Surely you can use EML to do root finding approximations also.

[meindnoch]

Solving polynomials over finite fields is trivial. Just try all combinations.

[eru]

You probably want a fast algorithm.

Compare https://arxiv.org/abs/1108.1791 and why computational complexity is often more interesting that computability.

[bawolff]

Sure, i guess i should have said something like with a polynomial circuit size or something.

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?

[zeroonetwothree]

How can you brute force real numbers?

[AlotOfReading]

The argument is that a universal basis would be capable of solving arbitrary polynomial roots. The rest is an argument that the group constructed by eml is solveable, and hence not all the standard elementary functions.

It wouldn't be a math discussion without people using at least two wildly different definitions.

[petters]

Yes, that blog post could have been much shorter….