Computing the Arithmetic Derivative

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Jurij Kovič's paper "The Arithmetic Derivative and Antiderivative" contains a curious remark in Section 1.2. Having just stated the definition of the logarithmic arithmetic derivative (L(n) = n′/n = Σj aj/pj where the prime mark indicates the arithmetic derivative, and Πipiai is the prime factorization of n), Kovič writes:

The logarithmic derivative is an additive function L(xy) = L(x) + L(y) for any x, y ∈ ℚ. Consequently, using a table of values L(p) = 1/p (computed to sufficient decimal places!) and the formula D(x) = L(x)·x, it is easy to find D(n) for n ∈ ℕ having all its prime factors in the table.

... a table of values? Did I read that correctly? Surely there must be some mistake; surely a paper published in 2012 can't expect us to rely on a printed table, for all the world as if we were John Napier in the seventeenth century! But never fear, dear reader, for the situation is easily rectified—with just a few lines of Python, you can take all the arithmetic derivatives you like on your own personal computing device.

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Blood From a Stone

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Decision-theoretically speaking, there's no difference between punishment and lack-of-reward. (Von Neumann–Morgenstern utility functions are really only defined up to an affine transformation: if your behavior is described by u(x), then v(x) := au(x) + b does just as well.) Psychology isn't like that; punishment and lack-of-reward are very different things—although not quite so different as one might think. In an environment where behavior X is rewarded with praise and status, and behavior Y is ignored—not punished, not condemned, but ignored—what kind of mind would it take to persist in behavior Y? It would either have to be very stubborn, unshakeably convinced in the righteousness of Y, or very stupid, desperately willing to endlessly chase a satisfaction that will never, ever come.

Friendship Deficits

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I'm not sure, but I suspect that I'm running a friendship deficit—that I need my friends more than they need me.

In a naively romanticized world of "pure" and unconditional love, you would never have occasion to think of such things: friends are always there for each other, no matter what, and no one would dream of anything so monstrous as consciously evaluating whether it's worth maintaining a friendship given the costs and benefits of doing so (including the opportunity cost of forgoing something else that could be done with the same amount of time and attention).

And in a world of Bayesian expected-utility-maximizing decision agents, there would be no loyalty and no concept of friendship.

Part of the beauty of our world is that it lies somewhere between the two extremes, but that we don't know exactly where.

Telling Her There

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"And so," said Synthia, "if you feel too self-conscious to write an email, if you don't know what to say or are afraid of saying the Wrong Thing, it might help to lower your quality standard and just start typing as if it were realtime communication, and edit later. It's less tempting to procrastinate replying in a realtime medium like instant messaging, and hardly tempting at all in meatspace, so you can try to import that same mindset to start your email."

"I don't usually have that problem," said Quiana.

"But, arguably, it's still a good thing that I told you, because your analogue in a nearby alternate universe who does have that problem would have been grateful for the tip, and I had to tell you here to ensure that my analogue there tells her."

"Or you could have conditioned your telling me on whether or not your local Quiana had the problem."

"Well, the reason I had to tell you here in order to sure that my analogue told her is that neither I nor my analogue previously knew which of us was which," Synthia explained. She shrugged apologetically. "Now we know."

Bounded but Not Totally Bounded, Redux

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Theorem. An open set in real sequence space under the ℓ∞ norm is not totally bounded.

Proof. Consider an open set U containing a point p. Suppose by way of contradiction that U is totally bounded. Then for every ε > 0, there exists a finite ε-net for U. Fix ε, and let m be the number of points in our ε-net, which net we'll denote {Si}i∈{1, ..., m}. We're going to construct a very special point y, which does not live in U. For all i ∈ {1, ..., m}, we can choose the ith component yi such that the absolute value of its difference from the ith component of the ith point in the net is strictly greater than ε (that is, |yi – Si,i| > ε) but also so that the absolute value of its difference from the ith component of p is less than or equal to ε (that is, |yi – pi| ≤ ε). Then for j > m, set yj = pj. Then |y – p| ≤ ε, but that means there are points arbitrarily close to p which are not in U, which is an absurd thing to happen to a point in an open set! But that's what I've been trying to tell you this entire time.

The Quieted Scare Convention

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Everyone knows ("everyone knows") about "scare quotes," where you enclose a phrase in quotation marks to indicate that the literal interpretation of the words should be regarded with skepticism, but sometimes I do this thing where I'll use a phrase normally and then repeat it in scare quotes and parentheses, as if to say, "I do partially intend this sincerely, but also with some irony or skepticism, although not so much as to justify outright scare quotes."

Of course, this practice immediately suggests the "dual" (dual) practice of using a phrase with scare quotes and then repeating it unquoted in parentheses, as if to say, "I do partially intend this ironically or in a way that should be regarded with skepticism, but also with a some sincerity, although not so much as to justify not using any scare quotes at all."

Bounded but Not Totally Bounded

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The idea of total boundedness in metric space (for every ε, you can cover the set with a finite number of ε-balls; discussed previously on An Algorithmic Lucidity) is distinct from (and in fact, stronger than) the idea of mere boundedness (there's an upper bound for the distance between any two points in the set), but to an uneducated mind, it's not immediately clear why. What would be an example of a set that's bounded but not totally bounded? Wikipedia claims that the unit ball in infinite-dimensional Banach space will do. Eric Hayashi made this more explicit for me: consider sequence space under the ℓ∞ norm, and the "standard basis" set (1, 0, 0 ...), (0, 1, 0, 0, ...), (0, 0, 1, 0, 0, ...). The distance between any two points in this set is one, so it's bounded, but an open 1-ball around any point doesn't contain any of the other points, so no finite number of open 1-balls will do, so it's not totally bounded, which is what I've been trying to tell you this entire time.

Dreams

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Friend of the blog Alicorn tweets:

Why is the word "dreams" used to describe both pseudorandom nocturnal hallucinations and also heartfelt aspirations for real life?

A cynic might reply: because both the nocturnal hallucinations and the heartfelt aspirations are, for the most part, composed of lies. How many people, what proportion of the time, will actually lift a finger (or open a book, or make a telephone call) to work towards actually achieving what they believe to be heartfelt aspirations?

Don't Resent That No One Cares

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It's tempting to be resentful that other people don't value your time the way you do. You complain at every opportunity: "Why, why, why do I get socially rewarded for working on this-and-such random chore that doesn't even help anyone, when obviously my great masterpiece (in progress, in potentia, coming soon) on such-and-this is so much more valuable?!"

But I think it's better not to be resentful and not to complain, mostly because it doesn't work. Other people don't care about your great masterpiece on such-and-this. They really don't. Maybe someone, somewhere will care after it's done, but it's not reasonable to expect anyone's support in advance—or, alternatively and isomorphically, it is reasonable, but given that there's nothing you can do to force people to be reasonable, reasonableness is not the correct criterion to be paying attention to.

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The Sliding False Dichotomy of Idealism and Cynicism

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The Television Tropes & Idioms wiki has a page on the Sliding Scale of Idealism Versus Cynicism. Of course I understand why such a page exists, but part of me can't help but protest that it's not really a sliding scale. One of the most charming things about my native subculture is that we have heaps of both: cynicism in the style of "Humans are selfish, weak-willed hypocrites; the reasons people say they do things aren't always or even usually the real reasons, and even introspection itself is untrustworthy," and idealism in the style of "But knowing what we do now, we shall use the power of Reason to remake the world in accordance with our Values!"

Colon-Equals

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Sometimes I think it's sad that the most popular programming languages use "=" for assignment rather than ":=" (like Pascal). Equality is a symmetrical relationship: "a equals b" means that a and b are the same thing or have the same value, and this is clearly the same as saying that "b equals a". Assignment isn't like that: putting the value b in a box named a isn't the same as putting the value a in a box named b!—surely an asymmetrical operation deserves an asymmetrical notation? Okay, so it is an extra character, but any decent editor can be configured to save you the keystroke.

I'd like to see the colon-equals assignment symbol more often in math, too. For example, shouldn't we be writing lower indices of summation like this?—

\sum_{j:=0}^n f(j)

—the rationale being that the text under the sigma isn't asserting that j equals zero, but rather that j is assigned zero as the initial index value of what is, in fact, a for loop:

sum = 0;
for (int j=0; j<=n; j++)
{
    sum += f(j);
}
return sum;

[RETRACTED] Introducing the Fractional Arithmetic Derivative

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[NOTICE: The conclusion of this post is hereby retracted because it turns out that the proposed definition of a "fractional arithmetic derivative" doesn't actually make sense. It fails to meet the basic decideratum of corresponding with an iterated arithmetic derivative. E.g., consider that 225″ = (225′)′ = ((32·52)′)′ = (2·3·52 + 32·2·5)′ = (150 + 90)′ = 240′ = (24·3·5)′ = 4·23·3·5 + 24·5 + 24·3 = 480 + 80 + 48 = 608. Whereas, under the proposed definition we would allegedly equivalently have 225(2) = (2!·30·52 + 32·2!·50) = 50 + 18 = 68. I apologize to anyone who read the original post (??) who was thereby misled. The original post follows (with the erroneous section struck through).]

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Book Notes I

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Did you know that putting adorable foxes on the cover of your book will make it sell more copies??

Speaking of books with animals on the cover, is it wrong to mentally associate specific programming languages with specific colors based on the O'Reilly books?

Toni Morrison has a book titled What Moves at the Margin, and based on the title I keep hoping that it's a treatise on microeconomic theory, but that's probably not actually true.

Idiot or Alien? Incompetence or Evil?

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When you encounter someone who expresses a political or social opinion that you find absolutely abhorrent, it is instructive to consider the extent to which this person is making a mistake, and the extent to which they simply have different values from you. Is this opinion something that they would immediately relinquish, if only they knew they knew the true facts of which they are now ignorant?—or is it reflective of some quality essential to their agency, a basic motive far too sacred to be destroyed by the truth?

(Of course, it is also instructive to consider whether you're making a mistake. But that is not the subject of this post.)

Some would say that it is useless to consider such questions, that human cognition doesn't separate cleanly into beliefs and values, and that even if such a thing could be done, it is futile for any present-day human to consider the matter, given our ignorance of our own psychology. And yet, the question still seems to make sense to me. If I can't know, I can guess. And I don't guess the same thing every time.

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Facial Hair Is Gross

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I often go a couple days without bothering to shave, but never much longer, because the stubble quickly becomes intolerable: I end up compulsively touching my face out of what I want to describe as a mildly horrified perverse fascination, perhaps of the same kind that would motivate picking at a scab, or poking a tumor.