Category Archives: Ideas

More generally.

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Remember σ(pn) = p * σ(n) + σ(n / pα)?

More generally, σ(pkn) = pk * σ(n) + (pk – 1) / (p – 1) * σ(n / pα).

Again, p is a prime and alpha is its multiplicity in n’s prime factorization. σ is the divisor function, of course.

All I need to do now is figure out how to extend it to a composite number and I’ll have a complete multiplicative recurrence on the divisor function, which I can use to obtain a closed-form rate of growth. I’ve empirically calculated it to grow at approximately 1.6449*n, but my goal is to obtain a tight worst-case bound. I could not find anything special about this number, except that is the 90% critical value of a normal distribution.

Here’s a messy Maple worksheet containing the derivation (among a whole bunch of stuff not related to the derivation that I was experimenting with today).

Conjectures

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Do highly composite numbers of the form n^2-1 occur infinitely often? What about when n is prime? Most values of n for which this holds seem to be. (Which makes sense, because if n is highly composite, n+1 is going to be deficient (because n’s prime factors will all go away), and squares of primes certainly fit the bill).

(Actually, can we always say that a highly composite number + 1 is either prime or the square of a prime? All of the values I checked were.

Things start to change around 5040…)

Persuasive Writing: Always Have a "Take Away" Idea

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When trying to present an idea or vision, one important aspect of the presentation is establishing a central idea that will stick in their minds long after the remainder of the prose has faded.

For instance, take this:

“Because curricula would be problem and goal-oriented, spanning diverse subjects, and chosen primarily by students in accordance with their own goals, and because performance would be measured by proficiency rather than completion of a set number of credits, graduates of such a university would emerge highly versatile, with the insight, confidence, and experience required to pursue not only the challenges that society will set before them, but their own private visions of how the world should be. In fact, because students tailor their own experiences and can easily change fields without losing time due to our proficiency-oriented measure of progress and our goal-directed curriculum, they will have already exercised a great deal of personal choice, ensuring that their values are fully developed and their ultimate visions are well-defined and inline with their training. In short, they would satisfy the criteria for personal advancement set forth in Maslow’s concept of self-actualization or Dabrowski’s positive disintegration. Polymath graduates would not only be well-versed, but further along in their human development and realization of their potential.”

Very few people will read through that whole paragraph, and very few of those who do will actually understand and remember it all. But the one message that the paragraph (and the text of the site as a whole) conveys is:

“We are going to train polymaths”.

That’s it. In an honest world, I could trim the text down to this sentence and be done with it.

Even though this is the primary thematic idea people take away from the text, it’s a very powerful and compelling one. It captures the very essence of the vision, and thus represents the project’s goals. Those who agree with it resonate with the goals of the project and very often become its most ardent supporters.

This is a natural consequence of idea-orientation as well – the prose grows around an idea because that is the way the writer is thinking. It’s a component of natural leadership.

Wow, TV/Movie Fantasy has become stupid.

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I was watching the premiere of a fantasy series based on one of my favorite books today. I’ll do my readers a favor and not bother naming it, because what I saw was a two hour train wreck. I honestly do not understand how the producers of this show managed to obtain the author’s approval of their screenplay.

The show took many liberties with the text. This is understandable to an extent, as the medium and audience are very different, but every deviation from the text was executed very poorly. Now that I think about it, what little TV fiction I’ve watched recently has also exhibited the same general characteristics as this show:

The world is portrayed in absolutes: there is Good and there is Evil. The job of Evil is to take over the world. No motive for this is given, and it’s never because the Evil person wants to make the world a better place. The job of Good is to stop Evil, and thus Save The World.

It’s always personal: This originally begins as a personal vendetta after Evil lashes out at the protagonist, but this is quickly subsumed into a sense of duty to Save The World by killing the minions of evil, usually in elaborate, drawn-out battle scenes. Nevertheless, as Good Triumphs Over Evil, a protagonist will invariably make some remark about having given meaning to those who have fallen or having achieved his revenge.

All motivations are exogenous, most caused by Evil: if someone on the side of Good is a traitor, it is because he was bribed or coerced by Evil. If a character is attacked by wild animals, it was somehow Evil’s fault. If it rains and a character gets wet, it must be the Wetness of An Evil Storm.

Morality Determines Causality: Just as most motivations are Evil, most of the plot consists of Evil’s machinations. Nothing can happen independently; it must all be the result of the actions of the protagonists or antagonists. There is literally no setting; it has become an extension of the characters.

No patience for unknowns: This is a bit more specific to the show I was watching. There was an aspect of the main character’s identity that the book kept the reader guessing at for at least 100 pages. I was shocked when the show merely blurted it out, as if it were known all along. And everyone picked it up and acted as if it were perfectly normal once it was revealed!

Violence solves everything: This book had several instances where the characters talked their way out of problems and used their wits. Part of the idea was to avoid unnecessary violence, which is always a smart thing to do. On TV, if one character so much as breathed too near another, out came the swords.

What Philosophy?: Finally, the motivation of Good is to Save the World simply because the Good Guys are Just Plain Nice. They don’t have those pesky attributes of real morality, like a set of personal values or decisions that require them to really think about these values. This makes the characters come off as completely inauthentic. It’s as if an average person were to suddenly become a hero, yet retained the morality of an average person rather than anything that could be construed as heroic. The deeds are heroic, but why are the characters performing them? Think Superman.

That’s my rant for today.

MBTI types of generalists?

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I’m a member of several groups on polymathy and on harnessing talent in multiple areas (it’s something I’m interested in myself, after all). One of the more interesting things I’ve noticed is that when asked about their MBTI types:

1. Everyone knows them already.

2. Almost everyone is an INTJ. Next common is INTP, then ENTP.

3. Most historical polymaths are thought to be INFJs. (What happened to cause the shift? Did thinking types suddenly become more in-tune with their artistic sides recently, or was there something cultural to it?)

How much faith one can put in MBTI types is questionable, but it does support my hypothesis that nonlinear intuitive thought, not straight logic, is required to see the connections between disciplines.

A Meta-Idea

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From penicillin to global warming, ideas require two people to flourish: one to generate the idea and another to popularize it. Very seldom are these functions performed by the same person. In fact, it’s common for a great deal of time to elapse between generation and popularization.

False negatives in animal tests.

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Lots of treatments work very well in mice but fail to show benefits in human trials. They’re false positives, and they get lots of people excited over treatments that never end up working in humans.

(Why do they work so well in mice, I wonder? Is it because so much of our research uses them? I wonder, if we were willing to completely throw morals out the window, could we get those sorts of results in humans by experimenting on them directly? Not that I’m advocating this.)

I just realized something blindingly obvious: there are false negatives too. But how are these handled? Treatments that don’t work in mice never make it to human trials, even though they may work in humans. Without doing human trials on treatments that failed to work in mice, we can’t evaluate a false negative rate, but it could potentially be high. Certainly it’s nonzero in any case.

This is another example of snap judgments shooting down ideas, but this is far less clear-cut than most criticism because failing to analyze the treatment prior to human trials can endanger people’s health.

I think that what we need are better computer models.

Maybe killing malignant cells isn't the answer.

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Unless the specificity of an anticancer drug approaches 100%, anything that kills off cancer cells is going to kill off some normal ones as well. This means side effects, often quite nasty.

But what if, instead of killing off cancerous cells, we just shut down their invasive potential?

I’ve been reading up on what differentiates noninvasive cancer cells – carcinoma in situ – against invasive cells. The literature on this has been surprisingly sparse, so either I’m not looking for the right things (quite possible), or this is a very understudied approach. The papers I’ve read have identified a few gene loci and a protein called Twist, but that is as far as I can take my search, lacking the resources to experimentally pursue such lines of study.

My point is this: carcinoma in situ is harmless except in its ability to become invasive cancer. Most of the proteins that seem to cause aberrant behavior in cancer cells seem those that are present during embryonic development (which makes sense in a way, since embryonic development is high-rate controlled, regulated division, whereas carcinogenesis is high-rate uncontrolled, unregulated division), but these proteins are all but absent in adults.

So rather than attempting to kill off the cancer cells, why not attempt to remove their ability to invade (and thus metastasize, destroy tissue, and cause other problems)? Even if the treatment were nonspecific, side effects should be far milder than the “killer” drugs, since normal cells are not known to depend on the function of the identified proteins. And unlike drugs that kill cells, there is little selective pressure against the treatment.

I see so many solutions to this problem. How I wish I could take part…