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Isn’t “noise” in this context just another form of “nurture” (i.e., environmental factors at a molecular level)?
In these debates, “nurture” is typically taken to mean factors you as a parent or a society can control, whereas “nature” is heritable genetic information. “Noise“ is neither of those things, and so needs a new word.
> “nurture” is typically taken to mean factors you as a parent or a society can control, whereas “nature” is heritable genetic information.

??? If so, then the typical debate is very poor. Nature vs. nurture may be the common words used, but anyone with even a meager understanding of it realizes that it's actually a genes vs. environment issue, and that environment includes much, much more than "factors you as a parent or a society can control."

In fact, the linked article even acknowledges that: "Everything not chalked up to genetic control tends to get attributed to diverse environmental factors, ranging from nutrition to stress to idiosyncratic social interactions. It’s a line of thought that 'suggests that it must be something outside the organism,. . . "

The article seems to want to posit random noise as a third factor (beyond genes and environment), but I think the same point would be made by positing that recent studies suggest "developmental noise" may occupy an especially privileged role among environmental factors. Noise clearly falls into broad penumbra of "environment", which also includes many other random factors.

> Noise clearly falls into broad penumbra of "environment", which also includes many other random factors.

Why? Noise is an unpredictable development. There's no reason to expect it to be exogenous; it's very easy for the genotype to specify that an outcome should be more or less unpredictable.

Since what you wrote is true (you could have a gene where one allele increases observed phenotypic variance and another decreases it; in fact I'm pretty sure such genes exists), I suspect the people downvoted you did so because they'd rather not think clearly about these matters.
It's not quite at the level of a single allele, but e.g. I understand that the same mutation related to eye-color results in a lot more color variance in females than it does in males.
And male vs female variance in general would appear to be strong evidence that something is going on.
The terms do not need to be redefined or made more complex.

"Nature" is genotype.

"Nuture" is everything not encoded by genotype or that environmentally changes expression. It doesn't matter if it's parenting, culture, or random noise. It's the universe the genotype is exposed to.

When an egg and sperm fuse, the resulting cell contains more than just the genotype for the new organism. It has all of its inter-cellular contents and organization. I think a more reasonalbe usage would be that nature is this cell and nurture would be what influences this cell and decendents throughout its lifetime.
We benefit from having different levels of granularity in our terminology for different purposes. Noise and non-noise nurture (as you use the term) are sometimes, but only sometimes, worth differentiating between.

Those who see a benefit from making this distinction may also see a benefit in having different terms.

Both terms are meaningless in our current understanding of systems biology.
I think there are at least two discourse communities with opinions on the correct usage, and what is "typically" done depends on which you are interacting with.
I'd lean towards putting noise under 'nature' as it's uncontrollable.
A lot of people treat nature as "inherent". Noise has the implication that it is external.

Obviously, the case mentioned in the article is that there is an inherent property of the crawfish that modulates the effect of noise, which blurs this line.

Jushi: “Is nature or nurture the primary driver of human development?”

Sensei: “First tell me, what is the sound of one hand clapping?”

Jushi: “Huh? That doesn’t even make sense to ask! You can’t get a clap without two surfaces interacting with each other!”

Thus the jushi was enlightened

In addition to noise and small changes, the same genome can contain multitude of mutually exclusive anatomical and behavioral traits.

For example, high stress environment might trigger different genetic traits than low stress environment, it's neither nature or nurture but both. In some animals you can get so different phenotype by altering environment that it's hard to recognize that they are the same species.

The idea that the genes just run the same process for the same set of genes and the environment just linearly modulates some traits is not the whole picture.

Gene-environment interaction already falls into the 'heritability' variance component, so appealing to GxE can't explain what the nonshared-environment/error variance component is.

Is it mostly something trivially boring & unimportant like measurement error (which does account for a lot of it, especially behavioral stuff), or is it large meso-scale environmental factors which happen to not be exactly identical within a household such as which college one goes to (which are the bread and butter of sociology, fit nicely into theories, and can potentially be measured, controlled, and equalized), or is it low-level biological details like random cells mutating or random cells going haywire (possible but extremely difficult to measure, control, or derive any understanding from, which is depressing from both the scientific & social points of view)?

This might seem to be impossible to solve: behavioral geneticists have already used extensive checklists and inventories to measure all sorts of aspects of families to try to reduce the nonshared-environment down to clear constructs, and failed to explain more than trivial amounts of variance (indeed, it worked out a lot better at showing how the 'environment' is genetically-caused); but you could always argue that they were measuring the relevant environment badly or measuring the wrong things entirely (maybe it's not how the mother interacts with the child in a nurturing or authoritarian fashion, but the father, etc).

OP is about the clever research approaches which allow digging into it and distinguishing between high-level environmental variables, and just damn random noise and low-level problems, and quantifying how much of each.

Stephen Fry also insists on adding Nietsche to the list of necessary words with N important for the development.