Is growth linear, not exponential?
rootsofprogress.org
rootsofprogress.org
Moore's law is in the saggy part of the S-curve, it's been morally dead for fourteen years in some ways, eleven years in others, and for eight years transistors flat out don't shrink at all they just pack more of them in the same space. I saw this coming, I did the math with the size of the transistors and the size of the chips and realized 99.5% of the area on a chip was empty back when 14 nanometer was the smallest. That's what new lithographies go for, they fill up that empty space. It ain't over til it's over, but when it is in fact over, it's over. It's over.
[1] South Korea, China, which is the new one? Is there a new one? A lot of the time it's artificial and carefully presented to market the exponential curve shit, mainly by lying about the beginning of the segment to make it look smaller and the end of the segment to make it look bigger. Shoehorning it. Also defining the segment as unique, like giving it an identity so it's meaningful to isolate it from the bigger picture which is not exponential because it has a past and a future. Moore's Law talks about transistors, why not talk about valves instead and look at the history since the nineteenth century, or since Leibniz for that matter? Like you can make a switch out of wood, people did, they transmitted energy by moving staffs of wood back and forth. Cuckoo clocks. Antikythera mechanism. The abacus. Mental arithmetic.
The big thing there is for usury, usurers have to keep the dream of exponential growth alive to charge exponential compound interest without getting tarred and feathered like so many usurers before them.
Why would anyone ever pay back a loan if it had zero additional interest for extending the term?
I have excellent answers to that. Well first off I would, because I pay loans in order from most benevolent to least benevolent. Second I figure out the mathematical theory that generalizes interest, and includes all types of interest as special cases, and compound interest is the special case number infinity, the most morally wrong case.
By this standard, every function is “fundamentally exponential”.
You can plot literally any (strictly positive, differentiable) function on a log scale and find the slope at every point. BAM! Look, an exponent (changing over time)!
If you don't have a causal graph explaining why your exponent is changing, then your plot doesn't make sense.
Yeah so that's oncologists for one, suppose I go to Cuba to a real doctor the likes of which cannot be found in America and get treated for cancer. That's precisely what I want is for that doctor to shut the growth of the tumor down, be a corrupt elite to that tumor, tell it the show's over. It's part of my body like the runaway growth thing is part of the economy, but if either growth is harming its surrounding environment, taking up resources or pushing things out, forget it. Tumor repression, oppression and suppression, all three!
Exponential growth is too unstable, it can't go on forever. If it had taken place since before the Middle Ages, or any other dark age, like it would be too stupid everyone would either own or owe a galaxy. It's not just that's too much, it's also that would be a highly stupid premise for a life to be born into. So it's marketed like "hey we'd all be rich" but come on, for there to be credit there has to be debt, no creditors without debtors. In these fantasies they never talk about who'd be the debtor, but here's a hint, between the speaker talking about we are all going to be too rich for sins to matter, and the listeners, it's not the speaker.
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https://www.nber.org/papers/w29766
Second most influential economic policy after Brookings, according to Wikipedia. Just qualifying the source for folks that may not know.
The big issue is that you get MANY more curve-fitting parameters to play with if you use a piece-wise linear model vs. an exponential model. (You get to choose HOW MANY breaks to make, what the slope is for each section, and WHERE to make the breaks.)
So... Let's say you created some synthetic data using an underlying exponential plus a normally distributed random number. Obviously, the BEST predictive model is an exponential one. However, for any arbitrary number of observations, I guarantee you there's trivially at least one piece-wise linear model that will have less error than the exponential one. Consider the one that is simply a straight line between EVERY point. Obviously that has zero error compared to the exponential model. Yet, it has very little predictive power compared to the exponential model.
Now, that's not what was done here... but there's actually quite a few parameters in the form of where to make the breaks and how many to make. Doesn't seem like a fair comparison.
The paper does cross-validate the models, and I am told that cross-validation properly penalizes overfitting with too many parameters… but I don't understand the statistics well enough here.
Cosmic inflation should guarantee an exponentially growing observable universe.
But with the amount of noise in economical data, I don't think is evidence of anything.
Yeah like the article mentions, they are basically making an analogy to the idea of “punctuated equilibrium” from evolutionary biology. Here’s a good exploration of how punctuated equilibrium works, vs the alternative which is called gradualism.
Appears to apply to the two preceding linear-scale charts.
I wonder if there is a really long compound German word for "an achievement whose greatness is best measured by the degree to which it disgusts experts in the field."
Perhaps even related to a human generation, or even living memory (like physics progresses one funeral at a time). Though the inflection points found lack that duration-scale and periodicity.
Just on curve fitting: you need some penalty for each extra line (noisy data can always be "fitted" better to enough lines). I expect the paper has a large section on this issue of statistical significance, but I can't access it.
I do feel it's kind of hard to say if such a pattern is "really" there, where exactly the breaks are, or if it's just a noisy artifact.
Perhaps there's something I'm missing, but it's weird that he isn't getting any credit for it.
Why not just use minimum description length?
And, I can't say without more reading more, what might be hidden by that weighting? Zooming into an exponential enough will make it look linear.
So cells. Cells do not multiply exponentially. DNA does not move faster than the speed of light! Nothing in the cell does! It may look exponential but actually that's cubic, they're easily confused, along with a shitty excel library, and bad measurement, measuring it on the small side early on. And what else? A bad education, being told exponential growth is real.
I bet if you actually had bunnies you could say fuck it they double every season and deal with them on that basis. You'd be wrong...but by how much, like get real? Plus the bunnies have the INTENTION of doubling, they each WANT to have lots of bunnies per season. I think that's the crux of it, despite the impossibility nature's program is exponential.
These two does not match economy growth vs global warming.
Y = AK^beta L^(1-beta)
with K being the amount of capital and L the amount of labour, A is just the unexplained "scaling factor" and it gets called TFP. But what TFP actually is... is a bit of a whatever-you-like.
One answer might be "TFP reflects technical knowledge; the more knowledge there is, the easier it is to generate new knowledge". But you'd have to get into growth macro for the details. Indeed, Romer is the person.
Yeah, that graph with the largest time scale on the article strongly disagrees.
For example the time until the next technology is 1/n, then the technologies per unit time increase exponentially.
This is true within a constant which is https://wikipedia.org/wiki/Euler%27s_constant
A justification for maybe why this is the right model is that the previous n technologies all help make the next technology possible in time ~1/n because they (n existing technologies) all contribute to innovating the next technology.
As an aside, the Falcon 9 flew 30 times last year and goes significantly over Mach 3.
interestingly, extremely high speeds turn out not to be that useful for military aircraft either. the fastest production fighter jet of all time, the mig-25, was introduced in 1970. the f-22 has a considerably lower top speed, and it's not because they lacked the funding to make it go faster.
The F-15 was designed primarily as an air-superiority fighter (and in fact misinterpretation of the Mig-25 design was motivation for increasing its maneuverability; the Mig-25 was a dedicated interceptor).
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In general the Soviets seemed to really like interceptors, the MiG-31 is arguably faster than the Mig-25; altitude and willingness to destroy the engines affect the calculation.
Not even CPU cycles.
Exponential-growth occurs when each unit of the growing-thing grows at a continuous rate. For example, if Alice invests $100 in a continuously-compounding bond, then keep re-investing the yields into more of the same bonds, then that'ld tend to be an exponential-growth process.
Linear-growth occurs when the growing-thing is produced at a regular rate. For example, if Bob keep making widgets, then the growth-rate of Bob's widget-pile would tend to be linear.
Anyway, apparently [this paper (2020) [PDF]](https://web.stanford.edu/~chadj/IdeaPF.pdf ) had its Equation-(1) basically parse to:
> dA/dt / A = alpha * S
, where "A" would be "ideas" (which seems vaguely defined), "t" is time, "alpha" is a constant-proportionality-factor, and "S" is an amount-of-scientists (who presumably generate the "ideas").
This equation is for an exponential-growth model. For example, if we reduce it to "dA/dt = k * A" (where "k" is a constant for alpha*S, to make this easier on WolframAlpha), then [the solution is an exponential-function](https://www.wolframalpha.com/input?i=dA%2Fdt+%3D+k+*+A ).
By contrast, it'd have been a linear-function if the authors instead assumed
> dA/dt = alpha * S
... this is, no "/ A" on the left-hand-side.
Anyway, a lot of comments on this thread seem to claim that any (first-order continuously-differential) function is approximately linear if we zoom in enough. Which, yup! -- we can look at both the linear-function and exponential-function as linear-functions by zooming in. So let's do that!
Basically, we can compare:
1. dA/dt = alpha * S (the linear-case)
2. dA/dt = alpha * S * A (the exponential-case)
where "dA/dt" is basically the rate at which "ideas" are generated, and then the right-hand-side of both equations is the marginal-rate (or instantaneous-rate), which is basically the slope of the linear-function that we'd see if we zoomed in enough on both functions such that they both appear (at least approximately) linear.
Practically speaking, we can ignore "alpha". It's basically just a fit-constant to be solved for. Then both equations also have "S", which is basically the amount of scientists who're working.
The big difference is that the exponential-case (which the 2020-paper linked above assumed) also includes a factor of "A" -- this is, the ideas. So, does it follow that "ideas" multiply how fast scientists produce more "ideas"? For example, if a scientist is working in a society that has 100 times more "ideas", then would that scientist produce new "ideas" 100 times faster?
If YES, then the exponential-form would seem appropriate. But if NO, then the linear-form would seem appropriate.
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EDIT: Skimming a few more sources, it looks like various folks may be trying to use the same equations/data/terminology, possibly for different things?
In the above-comment, I was mostly trying to comment on the basic-model that seemed to be presented in [this paper (2020) [PDF]](https://web.stanford.edu/~chadj/IdeaPF.pdf ), which the linked-article seems to be in-response-to.
However, it's unclear if the definitions cited, including of the variable "A", were necessarily representative of their usage elsewhere.
That said, [the linked-article's paper [PDF]](https://pages.stern.nyu.edu/~tphilipp/papers/AddGrowth_macro... ) starts its Section-5, "Conclusion", with:
> TFP growth is not exponential. New ideas add to our stock of knowledge; they do not multiply it.
, which seems to be in-line with the above-comment's interpretation from the other-paper.