No human person has, so far as I am aware, managed to eradicate all irrationalities from their thinking.  They are unavoidable, and this is particularly distressing when the irrationalities are lurking in your brain like rats in the walls and you don't know what they are.  Of course you don't know what they are - they are irrationalities, and you are a rationalist, so if you had identified them, they would be dying (quickly or slowly, but dying).  It's only natural for someone committed to rationality to want to indiscriminately exterminate the threats to the unattainable goal.

But are they all worth getting rid of?

It is my opinion that they are not: some irrationalities are small and cute and neutered, and can be confined and kept where you can see them, like pet gerbils instead of rats in the walls.

I'll give you an example: I use iTunes for my music organization and listening.  iTunes automatically records the number of times I have listened to each song and displays it.  Within a given playlist, I irrationally believe that all of these numbers have to match: if I have listened to the theme from The Phantom of the Opera exactly fifty-two times, I have to also have listened to "The Music of the Night" exactly fifty-two times, no matter how much I want to listen to the theme on repeat all afternoon.

Does this make any sense?  No, of course not, but it isn't worth my time to get rid of it.  It is small - it affects only a tiny corner of my life, and if it starts to get in the way of my musical preferences, I can cheat it by resetting play counts or fast-forwarding through songs (like I could get around the chore of feeding a gerbil with an automatic food dispenser).  It is "cute" - I can use it as a conversation starter and people generally find it a mildly entertaining quirk, not evidence that I need psychiatric help.  I have it metaphorically neutered - since I make no effort to suppress it, I'm able to recognize the various emotional reactions that satsifying or frustrating this irrational preference creates, and I would notice them if they cropped up anywhere else, where I would deal with them appropriately.  I also don't encourage it to memetically spread to others.  I keep it where I can see it - I make note of when I take actions to satisfy my irrational preference, and acknowledge in so doing that it's my reason and my reason doesn't make much sense.

In short, I treat it like a pet.  If it started being more trouble than it would be to root it out of my brain, I'd go through the necessary desensitization, just as I would get rid of a pet gerbil that bit me or kept me up at night even if this meant two hours each way on the bus to the Humane Society.

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What you describe is not a factual mistake, nor does it strike me as a moral error. It is merely an aesthetic judgment. Though it might be a mistake if you are wasting too much attention and thus deprive yourself of superior experience.

Agreed. I have trouble accepting this as a true irrationality. It strikes me as merely a preference. You lose time you could be listening to song A because of your desire to have the same play count for song B, but this is because you prefer the world where playcounts are equal to the world where they are unequal but you hear a specific song more. Is that really an irrational preference?

I also agree with VN's disclaimer: this time spent [wasted?] on equalizing playcounts could probably be used for something else. But at what point does the preference for a certain aesthetic outcome become irrational? What about someone who prefers a blue shirt to a red one? What about someone who can't enjoy a television show because the living room is messy? Someone who can't enjoy a party because there's an odd number of people attending? Someone who insists on eating the same lunch every day? Some of these are probably indicators of OCD, but it's really just an extreme point on a spectrum of aesthetic and similar preferences. At what point do preferences become irrational?

The problem with irrationalities is that they hide. The leverage a rationalist has against them is that the hiding is active, not passive - it's a moment of choosing to look away, to dismiss the qualm, to reassure yourself instead of probing into the discomfort. I'm thinking it might be possible to catch those moments and break through the concealment - but it would take habit. A trained reaction to lash out mentally. And I think your pets would blunt that edge.

Why is an automatic lashing out more effective than an automatic noticing-and-evaluating? That's what I mean by "keeping them where I can see them". I don't let my pets fall back into the ether; I observe their activities carefully. An automatic lashing out could catch even some non-irrational things, that could be filtered and stowed safely if the automatic reaction was just to attend carefully.

The stuff you're noticing and evaluating, you've already discovered. I'm talking about an effort to expand the range of your discoveries. Subconscious ideas have to interact with the conscious mind to steer it, but they don't stay there for long.

I don't mean "lash out and destroy", so much as "lash out and grab".

The potential problem is that our cute and trivial irrationalities are seldom the sole instances of that class of irrationality, and indulging in a cute irrationality that you're aware of may be strengthening a more harmful irrationality of the same class that you're not aware of.

For example, if "spending time making the song counts equal" shares 'neural circuits' with other OCD-like tendencies, or with a need to invent arbitrary aesthetic judgments about trivial things that you'd rather not waste time on if you could consider things from a third-person perspective, then even though the irrationality is trivial and harmless in and of itself, indulging it is harmful because of the side effects of doing so.

I acknowledge that indulging it has a cost (it takes time and I don't always listen to exactly the song I would most enjoy). But eradicating it would also have a cost (it would also take time, and it would be emotionally uncomfortable at least for a while). I make this tradeoff rationally, even though my rational decision concludes that I should allow myself to be mildly irrational where iTunes is concerned.

I find the example here highly amusing because I think that while you have taken this quirk to an extreme I do a more moderate version of this balancing using the same feature of the same program. I don't have to have playlists line up exactly but I do keep things in balance. I find that I have a natural bias towards listening to the same tunes too much, with the result being that I don't do enough exploration and overplay the songs I like the most. A rule like this counteracts this bias, so this could also be an example of not letting the perfect become the enemy of the good.