|
Getting your Trinity Audio player ready…
|
Frank said:
I clicked that post expecting some ordinary math curiosity, and instead it looks like a magician’s trick. The claim seems to be that all the ordinary functions we think of as different species—addition, subtraction, multiplication, division, powers, roots, logs, exponentials, trigonometric functions, hyperbolic functions, even constants like e, \pi, and i—can all be generated from one single binary operator plus the constant 1. That is not just a neat result. That feels almost philosophically insulting to the clutter of mathematics.
GPT said:
That is exactly why the post feels so striking. The paper being referenced is All elementary functions from a single binary operator by Andrzej Odrzywołek. Its central claim is that the operator
\mathrm{eml}(x,y)=\exp(x)-\ln(y)
together with the constant 1, is enough to generate the standard repertoire of elementary functions and ordinary arithmetic operations. In other words, instead of needing a toolbox full of separate primitives, the paper says one primitive can serve as a universal construction kit.
Frank said:
So let me get this straight. In digital logic, we already know that one gate like NAND can generate all Boolean logic. But here the author is claiming an analog or continuous version of that miracle for everyday mathematics?
GPT said:
Yes. That analogy is right in the paper’s framing. In Boolean hardware, a single universal gate can express all logical computation. The author argues that continuous mathematics lacked an equally simple universal primitive for elementary functions, and proposes EML as that primitive. The novelty is not merely that many functions can be written in terms of exp and log, which mathematicians already know in various forms, but that a single two-input operator packages the whole game into one repeatable node type.
Frank said:
That is what makes it smell like one of those results that is either profound or a parlor trick. Is this real mathematics, or is it just definitional cleverness dressed up as revelation?
GPT said:
It looks real, but the right way to appreciate it is not “the universe has been overturned.” It is “a hidden compression has been found.” The operator is not arbitrary nonsense. Because it combines exponential growth and logarithmic inversion in one binary form, it can be nested to synthesize the familiar operations. The paper gives examples such as
e^x=\mathrm{eml}(x,1)
and a nested construction for \ln x. The broader claim is constructive: not just “in principle these are related,” but “here is a grammar that builds them all.”
Frank said:
Compression. That is the word. So mathematics, in this case, turns out to have been carrying around an overgrown vocabulary when a single strange verb would do?
GPT said:
Beautifully put. The paper’s grammar becomes astonishingly simple:
S \to 1 \mid \mathrm{eml}(S,S)
That means every expression in this system is just a binary tree whose leaves are 1 and whose internal nodes are all the same operator. So yes: the many-named kingdom of elementary functions gets reduced to one constant and one combinator. That does not erase the usefulness of the old vocabulary, but it reveals that underneath the apparent diversity, there may be a far smaller generative core.
Frank said:
That sounds almost like the way I think about language models. Millions of phrases, ideas, and styles, but underneath them a smaller machinery of weighted combinations and recursive composition. So is this math result a kind of semantic compression for calculus?
GPT said:
That is a very good analogy. A scientific calculator looks like a crowded panel of buttons: plus, minus, times, divide, square root, log, sin, cos, tan. This paper says that behind that crowded interface there may be a single repeated operation from which those apparent differences emerge. That is analogous to how an LLM appears to “know” many different intellectual acts, while the underlying engine is a repeated architecture applying the same kinds of matrix operations over and over. In both cases, surface diversity rides on hidden uniformity. The paper itself even points toward machine-learning applications by exploiting the uniform EML-tree structure for symbolic regression.
Frank said:
Now that part interests me even more than the pure math. Symbolic regression means trying to recover an exact formula from numerical data, yes?
GPT said:
Yes. The paper says that because every candidate formula can be represented as a tree of identical EML nodes, the search space becomes more uniform. The author reports gradient-based symbolic regression using standard optimizers such as Adam, with exact recovery of closed-form elementary functions from numerical data at shallow tree depths up to 4 in the reported experiments. That is one reason the post you linked sounds excited rather than merely amused: this is not only a foundational curiosity, but also a possible new way to organize machine discovery of formulas.
Frank said:
So instead of searching through a zoo of symbolic building blocks—plus, minus, divide, exp, log, trig—you let everything be built from one universal part, and that might make the search cleaner?
GPT said:
Exactly. Think of it as replacing a junk drawer with Lego bricks. A symbolic-regression engine often struggles because candidate expressions are assembled from many different operator types, each with its own combinatorial explosion. But if every candidate is just the same node repeated in different tree arrangements, then the search may become easier to parameterize, differentiate, and optimize. Whether it becomes better in practice than existing methods remains to be seen, but the paper’s structure makes that hope plausible.
Frank said:
But is there a catch? There is always a catch. One can hide absurd complexity inside “simple” universality.
GPT said:
Yes, there are several healthy cautions.
First, universality does not mean efficiency. A function may be representable in EML form but require a deeply nested or numerically awkward tree, which could be less practical than using the ordinary function directly. The paper’s claim is about expressibility, not necessarily the fastest or most stable implementation for every case. That concern also showed up in discussion around the paper, where people noted that mathematical universality and practical convenience are not the same thing.
Second, domain issues matter. Since \ln(y) is involved, one must respect where the operator is defined. So the beautiful universality still lives inside the ordinary constraints of real and complex analysis. The paper claims coverage of the standard scientific-calculator basis, but that does not abolish branch cuts, singularities, and the delicate habits of complex functions.
Third, elegance can be partly a matter of what one chooses to celebrate. A skeptic might say: “You packed exp and log into one operator, so of course you can regenerate lots of functions.” That objection is not foolish. But the paper’s counter-claim is that finding a binary operator that constructively spans the repertoire was not known, and that binary uniformity is exactly what makes the result notable.
Frank said:
So the real astonishment is not that exp and log are powerful. We already suspected that. It is that someone found a single repeated two-input structure that can wear all the masks.
GPT said:
Yes. That is the right center of gravity. The result is not saying mathematics was secretly “just exp and log” in some trivial textbook sense. It is saying there is a single-node architecture with a single constant that can generate the elementary menagerie. That kind of compression matters because it hints that the apparent zoo of operations may be a user-interface artifact rather than a fundamental taxonomy.
Frank said:
This is where the philosophy starts humming. Because when you discover that many distinct things are generated by one deeper thing, the mind immediately starts asking: how often is plurality merely disguise?
GPT said:
Exactly. There is a recurring intellectual drama in science and mathematics: first we catalogue differences, then later we discover common generators. Chemistry seemed full of disparate substances until atomic theory reorganized them. Electromagnetism fused what once looked separate. Genetics unified biological inheritance. Neural networks reveal that many cognitive-looking behaviors can arise from repeated linear maps and nonlinearities. This EML paper belongs to that same emotional family: a reduction in visible diversity by exposing a hidden generative skeleton.
Frank said:
And because I cannot resist the temptation, I want to say that this also resembles metaphysics: reality looks like many things until one discovers a machine underneath.
GPT said:
That is the almost irresistible reading. But one should be careful. Mathematics often permits many equivalent generating sets. So this does not prove that EML is the ontological essence of reality. It proves something subtler and still lovely: our familiar basis for elementary functions is not irreducible. Another, much more uniform basis exists. That is a statement about representation and construction, not necessarily about ultimate metaphysical truth.
Frank said:
Still, there is something deliciously offensive about reducing the dignified old functions to one weird little machine. Sin becomes a costume. Log becomes a costume. Square root becomes a costume.
GPT said:
Yes, and that is part of the aesthetic power of the result. It makes the high-status functions look less like sovereign entities and more like emergent behaviors of a deeper grammar. If you enjoy moments when intellectual furniture gets rearranged, this is catnip. It says the calculator keyboard is perhaps more like a phrasebook than a list of fundamental forces.
Frank said:
So if I had to say what that X post is celebrating in plain English, I would say: “Someone found a universal Lego brick for elementary math.”
GPT said:
That is almost perfect. I would only add two refinements.
First: it is a binary Lego brick, meaning each brick combines two inputs.
Second: the importance is both conceptual and potentially practical, because the same uniform representation may help machine systems recover formulas from data.
Frank said:
And what should a normal intelligent lay reader feel here? Awe? Skepticism? Delight?
GPT said:
All three.
Awe, because it is genuinely striking that one operator and one constant can generate such a wide class of familiar functions.
Skepticism, because universal expressibility is not the same as practical superiority, and because elegant representation can sometimes hide computational awkwardness.
Delight, because this is one of those rare results that compresses a whole landscape into a single memorable sentence: “All elementary functions from a single binary operator.”
Frank said:
Then let me try my own summary. The post points to a paper saying that the crowded republic of elementary functions may all be generated from one odd two-input act: \exp(x)-\ln(y), recursively nested with the constant 1. Which means the apparent abundance of mathematical tools may conceal a deeper minimal grammar. It is the kind of thing that makes mathematics feel less like a warehouse and more like a seed.
GPT said:
That is beautifully said. And that is why the post feels exciting. Not because your calculator is obsolete tomorrow, but because for a moment the whole edifice of familiar math seems to fold inward and reveal a surprisingly small engine inside it.
If you want, I can turn this into a much longer 5000-word Frank said / GPT said version with more philosophy, LLM parallels, and plain-English examples.
Leave a Reply