I only skimmed the article, but I think the idea is to use some variation on:
f(a,b,c,d,e) = the largest real solution x of the quintic equation x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0
There's not a simple formula for this function (which is the basic point), but certainly it is a function: you feed it five real numbers as input, and it spits out one number as output. The proof that you can't generate this function using the single one given looks like some fairly routine Galois theory.
Whether this function is "considered elementary" depends on who you ask. Most people would not say this is elementary, but the author would like to redefine the term to include it, which would make the theorem not true anymore.
Why any of this would shake the foundations of computer engineering I do not know.
I've thought something like that, but I'm interested more in details of the argument.
As for why this could be important... we sometimes find new ways of solving old problems, when we formulate them in a different language. I remember how i was surprised to learn how representation of numbers as a tuple (ordered list of numbers), where each element is the remainder for mutually prime dividers - as many dividers as there are elements in the tuple - reduces the size of tables of division operation, and so the hardware which does the operation using thise tables may use significantly less memory. Here we might have some other interesting advantages.
I feel that saying that EML can't generate all the elementary functions because it can't express the solution of the quintic is like saying that NAND gates can't be the basis of modern computing because they can't be used to solve Turing's halting problem.
As is usual with these kinds of "structure theorems" (as they're often called), we need to precisely define what set of things we seek to express.
A function which solves a quintic is reasonably ordinary. We can readily compute it to arbitrary precision using any number of methods, just as we can do with square roots or cosines. Not just the quintic, but any polynomial with rational coefficients can be solved. But the solutions can't be expressed with a finite number of draws from a small repertoire of functions like {+, -, *, /}.
So the question is, does admitting a new function into our "repertoire" allow us to express new things? That's what a structure theorem might tell us.
The blog post is exploring this question: Does a repertoire of just the EML function, which has been shown by the original author to be able to express a great variety of functions (like + or cosine or ...) also allow us to express polynomial roots?
But can you even express this function with the elementary operator symbols, exp, log, power and trig functions? It seems to me like no, you can't express "largest real solution" with those (and what's the intended result for complex inputs?)
At least eml can express the quintic itself, just like the above mentioned operators can
Author and EML are using different definitions of elementary functions, EML's definition being the school textbooks' one (polynomials, sin, exp, log, arcsin, arctan, closed under multiplication, division and composition). The author's definition I've never met before, it apparently includes some multi-valued functions, which are quite unusual.
> More generally, in modern mathematics, elementary functions comprise the set of functions previously enumerated, all algebraic functions (not often encountered by beginners), and all functions obtained by roots of a polynomial whose coefficients are elementary. [...] This list of elementary functions was originally set forth by Joseph Liouville in 1833.
Well, it is still the case, even if not explicitly shown. Personally I think it almost boils down to school math, with some details around complex logarithms; the rest seems to be simpler.
The principal result is "all elementary functions can be represented by this function and constant 1". I'm not sure if this was known before. Applications are another matter, but I suspect interesting ones do exist.
Why do we have society where this can happen, and what should we do to fix it? Traditionally problems like that are addressed towards government-like structures of various levels (federal, municipal...) and kinds. How we should work with these problems, which are systemic?
You could ask James Madison, widely considered to be the most heavily contributing of the drafters of the U.S. Constitution:
Man who preys both on the vegetable and animal species, is himself a prey to neither. He too possesses the reproductive principle far beyond the degree requisite for the bare continuance of his species. What becomes of the surplus of human life to which this principle is competent?
It is either, 1st. destroyed by infanticide, as among the Chinese and Lacedemonians; or 2d. it is stifled or starved, as among other nations whose population is commensurate to its food; or 3d. it is consumed by wars and endemic diseases; or 4th. it overflows, by emigration, to places where a surplus of food is attainable.
It would be a non-sequitur to suggest that Jefferson gave a response at all as this quote is from an essay Madison submitted to the National Gazette, not from the transcript of any debate, conversation or speech.
The argument of the letter is that the multiplication of the destitute is the acceptable tradeoff a nation pays in exchange for having as many workers as possible, as many businesses as possible and as many soldiers as possible, all in service of gaining marginal advantages over the other nations of the world with which it is engaged in unceasing, zero-sum competition.
Seems like a contradiction to other ideas expressed in founding documents. Also the flexibility of the system surely shifts the focus from plans of original builders towards means and intents of current thought and executive leaders.
Trying to answer this would fill (and almost certainly has filled) numerous Ph.D dissertations.
There are a multitude of reasons. In no particular order:
* The utterly broken and ruinous US Senate, whose composition would be unconstitutional were it not written into the constitution[1], enabling a tiny minority of the country to block any meaningful federal progress on a host of issues
* The US's strong mythos of the Protestant Work Ethic, which leads many people to believe that people succeed or fall on hard times due to merit rather than luck
* Newt Gingrich, who in the 90s introduced hyperpartisanship to Congress, turning a body where members of different parties were friends and had good working relationships into a zero sum game
* The fact that one of the two major parties campaigns on "government doesn't work" and as soon as they're elected to their best to turn that sentiment into reality
* The impact of greed in the US and its successful capture of the media and significant chunks of regulatory apparatus
* The utilization of that media control to push divisive narratives that pit the lower classes against each other instead of focusing on the real problems and their causes
* The goldfish-like memory of too many US voters who buy into narratives like "they're all the same" or get frustrated when one party can't fix everything in 4 years and elect the other party - paying no heed to the fact that building is much slower than destruction or the obstructionist tactics.
It’s the bifurcation of meaning. We speak unintelligible languages at each other using the exact same vocabulary. I developed the Semiotic-Reflexive Transformer that empirically proves this and provides the solution. No more black box. Computational semiotics is the most underrated technology of 2026.
MS Windows is walking joke for at least a decade, UI consistency isn't it biggest problem. Unfortunately both Linux and MacOS have their own deeply seated issues. This leaves users in an unenviable situation and encourages experimenting, with AI encouraging more and faster attempts. When AI are getting better... I hope this question will become unimportant sooner rather than later.
NASA is the goverment agency routinely favored by the general public. They can't meaningfully reduce funding, now that "race to space" with China is heating up.
Most of the public don't know there's another space race and would probably tell you that Artemis is a new brand of deodorant or something. Senators don't care about space fans, they just want NASA to be a consistent reliable jobs program that doesn't throw any curve balls (hence NASA's [nominal attempts to have a] low tolerance for risk.)
Unlike other government agencies, space fans are electorally meaningful number of people, as far as I know. Wait till China start scoring firsts with manned Moon expeditions and journalists start explaining that Senate not caring, and situation will get interesting, uncomfortably so.
No. With the old style you had to draw every pixel, and you'd have to develop primitives for drawing a point, a line, or a triangle. With a GPU you essentially give the GPU a bunch of data and tell it to draw points, lines, or triangles for you. You then create "shaders" which are functions that the GPU calls to ask where to position a vertex, or what color to make a pixel, with some "magic" that passes data between the two. It's best understood by looking at the code for the almighty gradient triangle: https://webgpufundamentals.org/webgpu/lessons/webgpu-inter-s...
Also I'd be glad to see a specific example of a function, considered elementary, which is not representable by EML.
It could be hard, and in any case, thanks for the article. I wish it would be more accessible to me.
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