Mathematics

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Complex clocks, misleading DNA, and signs (not sines) of support

Well, it’s Wednesday, which follows Tuesday, which follows Monday, which follows Sunday, which follows Saturday, which follows Friday, which follows Thursday, which follows…

…that’s right, it brings us right back to Wednesday again.  It’s a little bit reminiscent of the Doe, a Deer song from The Sound of Music.  Or, maybe you could say it’s a little like Rock Around the Clock.  All these cyclical things are amusing in that each member of the closed, recursive group both precedes and follows itself.

That gives me an interesting idea.  At least, it’s interesting to me.  I’ve seen clocks (dial style clocks obviously, not digital ones) that designate the hours with radians, such that in place of the 12, for instance, there would be π/2, since the whole circumference of any circle is 2 π radians.

Six o’clock is then 3 π/2, nine o’clock is then π, and three o’clock is just 0 (or 2π).  The order of these numbers uses the tradition of beginning counting radians (or degrees) at the 3 o’clock position as it were.  The other numbers (1, 2, 4, 5, etc.) are various ratios of π, using the trig functions that give us the results for what we would think of (typically) as 30 degrees, 60 degrees, and so on.

Well, why not do a novelty clock with the hour markings represented by complex numbers?  In engineering and in physics, they are very frequently used because they are excellent ways of calculating cyclical things, using a unit circle (in this case, a circle with radius 1, centered on the origin of the complex plane).  Using such an arrangement, multiplying any complex number by i results in a 90 degree rotation (counterclockwise).

In the case of a complex number clock, the 3 o’clock position is just 1 + 0i, 6 o’clock is 0 – i, 9 o’clock is -1 + 0i, and 12 o’clock is 0 + i.  I’ll work out the specific complex numbers for the other hours, and I’ll put them in a footnote* below, but I don’t think I can do it off the top of my head while commuting.  For one thing, I don’t recall the sine and cosine of 30 degrees…

Wait!  The sine of 30 degrees (aka pi over 6 radians) is just ½.  And since we can take that as a leg of a right triangle with hypotenuse length 1 (since it’s a unit circle) then the Pythagorean theorem tells us 1^2  = ½^2  plus the cosine^2 of 30 degrees, so the cosine equals the square root of 1 – ¼ or the square root of ¾, in other words √3/2.  So 2 o’clock would be √3/2 + (½)i.

One o’clock (or 60 degrees) just reverses the ordering of the parts of the complex number:  ½ + (√3/2)i.  Then 4 o’clock is √3/2 – i, five o’clock is ½ – (√3/2)i, 7 o’clock is -½ – (√3/2)i, 8 o’clock is -√3/2 – (½)i, 10 o’clock is then -√3/2 + (½)i, and and 11 o’clock is -½ + (√3/2)i.  Neat and tidy!

I’ll leave it as an exercise for the reader to work out the complex numbers for the various minutes of the clock face**.  And, of course, if I’ve made any errors above, please let me know.

But anyway, I think you can see that this might make for a novelty clock every bit as exciting as the ones that just show you the hours as radians.  It might be slightly more rarefied than the other, but then again, engineers might prefer it (though they tend to use j rather than i to represent √-1, because i already stands for electrical current****, if memory serves.

Wow, how’s that been for some weird digressing?

Is “digressing” a word?  Google Docs seems to “think” it is.

Hopefully those quotation marks don’t offend Google Docs if it really does actually think.  If it’s any consolation (and even if it isn’t), I would be inclined to use such “scare” quotes when referring to the hypothetical thought processes of many a human.

On to another digression:  I had a weird thought just now, while my mind briefly wandered to imagining a potential crime thriller, maybe about a serial killer.  What if the killer, anticipating the potential for DNA evidence to be used to try to catch him*****, makes it a habit to pick up strands of hair, discarded tissues, used Band-aids, and various other things from various places such as public (pubic?) restrooms, barber shops, and so on.  He then leaves them randomly chosen and scattered at his various crime scenes to throw off investigators.

Of course, he’d want to be careful to get his DNA samples from widely chosen and semi-random places, such as maybe rest areas on major highways.  Otherwise the very pattern of those sources might give investigators something with which to work.  The FBI, for instance, can be impressively clever about such things.  At least, well, they were under previous administrations.

Jeez Louise, I’m really losing my mind, am I not?  Certainly it seems to have slipped its leash.  My apologies for this chaotic brain sneezing; I hope you don’t catch any mental viruses from me.

With that, I’ll bring this to a close.  I hope I end up feeling better today than I did yesterday, though so far it doesn’t seem promising.  More importantly, as always, I hope all of you have very good days.

*Well, there’s no point in doing this in a footnote now; I worked it out in the main body of the post.  See above.

**Just kidding.  Seriously, please don’t feel obligated to do this***.  I’m not going to do it (or might I say, “i is not going to do it”?)

***On the other hand, if one of you feels compelled to do the figuring, please share your results with me.

****That could almost be like someone’s protest sign, perhaps one that is held up and shaken outside a courthouse:

*****The vast majority of serial killers are men.  It’s a travesty, this wild gender/sex disparity; it’s the last bastion of unrepentant, unapologetic misogyny!  We need social programs to try to get more women out there serial killing!  Maybe we should provide subsidized child care services, so single moms won’t have as hard a time getting into the field…or wherever they might want to dump the bodies of their victims.

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“I said he’ll flip ya…flip ya for real.”

It’s Friday today, and though I am not a believer (and, more or less as I say in the meme below, I don’t want to be one, since I’m not impressed by “belief”) I will thank anyone and everyone who might be responsible for the fact that this is the end of my work week this week.  So, thank God it’s Friday, thank Cat it’s Friday, thank Frigg it’s Friday (thank Tony for his “Friggin’ Friday” comment), thank Cthulhu it’s Friday, thank Doom it’s Friday, thank Eru it’s Friday, thank Brahma it’s Friday…

I suspect that you get the idea.

Anyway, I’m very tired and need some rest.  It’s not that it’s been a particularly onerous week; it hasn’t really been.  If anything, it is I who has been more onerous than usual this week even than I usually am.  But I don’t think I’ve been much more onerous than usual.

I suppose that it’s up to those around me to say how much of a burden I’ve been, but I know their answers would be inconsistent and pretty confabulated.  This is not intended to insult them; it’s just the nature of human accounts of things that have happened to them when they are asked without having been paying attention to those things (in particular) in the first place.

Nature didn’t create a brain that keeps a running measure of its mood in relation to each individual around them, like some endless EEG plot.  What would be the use of such a function in the natural world?  Even if useful, would the use be worth the inevitable cost?  And could the drunken walk of the blind idiot god Evolution ever stumble upon such a process even if it were useful?

Well, given infinite time, it probably could, but we haven’t had infinite time to explore the space of possible biological innovations, and we probably will not have it.

Anyway, the point is, as Louis C K pointed out, it’s not up to me whether I’m being an asshole, at least not according to the impressions of others.  And I guess it’s not up to me whether others feel I am onerous or not.

Then again, it’s not up to them whether I consider myself to be a burden or not.  And I have the only front-row seats to me.  I am the one (and only) who is with me all the time*.

And consider something else:  all my meandering oddness (oddity?) that is on display here is me when I am tired, slowed down by literal decades of insufficient sleep.  Imagine how irritating I might be if I were fully rested and energetic!

Don’t make the mistake of imagining that it would necessarily make me calmer and more sedate.  I knew me back in the years when I used to get much more adequate sleep (though I have always slept less than most anyone else in my family).  I was not particularly sedate or calm then.

In closing, I’ll mention a weird thought that occurred to me when I was looking in the mirror (another daily burden) after my shower.  In principle, if one were picked up out of 3D space into 4D space, rotated in the proper way while there, then set down in 3D space again, one would be physically mirror-reversed***.

This would be a big problem, and not just because other people would be freaked out in subtle ways.  Every chiral molecule in one’s body would also be flipped from ortho to sinister, left to right (and vice versa).  That means you would need to eat similarly flipped food****, because your biological molecules only work on one version of most molecules.

Also, your DNA (which has chirality in spades) would be rotated in the other direction, so you certainly couldn’t reproduce.  But you couldn’t build new cells anyway, because many of the requisite molecules in that process are chiral.

Even your neutrinos/W and Z particles would behave differently, because this is one force/process in nature that is not parity-symmetric.  Meaning there is in a sense a handedness to the way the weak force acts.

Of course, that’s also the case with electromagnetism, come to think of it.  There is the famous “right hand rule” for how magnetic fields occur in response to changing electric fields/moving currents, and how electric fields respond to changing magnetic fields (see Maxwell’s equations).

So, heck, if you were so flipped, you might not even be able to interact with the electroweak sector of the quantum fields of our universe, and you might fall apart immediately at the subatomic level.  It would depend very much on whether such constraints are carried along with the stuff of one’s body or if they are imposed (and reimposed) by the fields in the universe around it.

I don’t even know what would happen with electric charge or hypercharge or spin or whatnot.  It almost makes me want to write Changeling in a Shadow World.  But I doubt that will happen.

With that weird stream of consciousness, I will let you go for the week.  I won’t be writing any new posts before Monday, exclusis improvisis.

*At least I’m the only one with a nervous system.  I suppose my skin and gut flora are with me all the time, but as far as I know, they don’t think in any reasonable sense.  Also, can any of them really be said to be with me always?  Their dividing/reproducing rates are very high compared to a human (or humanoid) lifetime, after all.  I guess it depends upon whether I consider a bacterium that undergoes mitosis to be two new organisms or just one increased organism that is continuous**.  Both options are reasonable and plausible, and nature doesn’t appear to care, outside the minds of humans, how such things are considered.

**This implies, in a sense, that all life on Earth could honestly be considered one organism, since there appears to have been one ultimate common ancestor to us all.  Thus, Life would really be a (so far) immortal organism.  I think that’s actually a pretty reasonable way to look at things, and it’s very much a sort of larger-scale, fractally self-similar pattern to how a human body is initiated with one cell−the “ensouled zygote” as I once mockingly made a meme about, after deciding it would be a good name for a band.

***If this is difficult to imagine, consider it in lower-dimensions.  Imagine an asymmetrical trapezoid in Abbot’s Flatland, with one of its two “legs” different than the other.  If it were picked up into three dimensions, flipped over, and then put back down, its asymmetry would now be in the other direction, and other Flatlanders might no longer even recognize it.

****This is part of why the possibility of being eaten by aliens, or of them eating us, makes little sense.  If you tried to eat food from an alien ecosystem, it would at best just go through you unabsorbed, and at worst would give you potentially fatal indigestion.  All your digestive processes are honed to work on other biological substances that occur within the ecosystem that evolved on Earth.  And this entails many lock-and-key enzymes and molecules that digest depending on chirality.  This was part of the motivation behind creating left-handed sugar and left-handed fat, which might physically behave similarly and perhaps even smell or taste somewhat the same, but would not be absorbed.  But such products almost inevitably lead to digestive troubles at best.

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Infinite distractions have limited effectiveness

It’s Wednesday now, and I’m writing this post on my lapcom.  There’s no particular reason I’m doing so, though as always, a cause or causes surely exist(s).  I just felt the impulse to take the lapcom back to the house with me yesterday afternoon, and I saw no reason not to indulge that impulse.  That’s about it.

It’s the 2nd day of September, which should come as no surprise, since yesterday was the 1st day of September.  Numbers don’t necessarily have to be put in any given order—and indeed, one cannot properly order the Real numbers, because there is an uncountable infinity of them between any two points on the number line, however arbitrarily close you might want to choose them—but the ordinal numbers have an inherent order.  Thus the name.  Duh.

I often marvel at the cool facts about uncountable versus countable infinities.  Think about it.  The “natural” numbers represent a “countable” infinity because you can make progress counting them and, if you had infinite time, you could in principle count them all, though it can be a bit mind-bending to try to imagine what that could really mean.  But it is true, mathematically speaking.

It’s also true that this infinity, the so-called countable infinity (ℵ₀, or “aleph nought”), is matchable one-to-one with all other countable infinities.  You can assign a natural number to every integer, and to every even or odd number, and to every rational number (i.e. every fraction) and so on.  Therefore, they are the same “size” or “cardinality”.

In case anyone wonders why that point might matter, all one has to do is to try to list the Real numbers (designated with ℝ).  These are all numbers including the integers and the rational numbers, but also all irrational numbers, i.e., numbers that can’t be expressed as a ratio of any whole numbers or integers.  The latter set includes pi (π) and e (e) and phi (φ, the so-called Golden Ratio) and so on, but these are simply a taste of what’s available.  All of these numbers have a potentially infinite number of digits* after the decimal point, and those digits do not have any repeating pattern, for if they were consistently repeating, they would be rational numbers.

Cantor’s “diagonal proof” demonstrates that there are more Real numbers than there are integers (or any other aleph-nought level infinities of numbers).  I won’t get into it here—it’s not that difficult, and it’s very clever and rather wonderful, but it would take a bit of time, so I just put a link to it for anyone who is curious.

What’s more, once we realize that fact and think about it a little, we realize not only that the Real numbers are uncountably infinite, vastly more infinite than the integers of the natural numbers, but something even more mind-boggling:  Between any two numbers you can choose, however arbitrarily close you want to make them, there is an uncountable infinity of Real numbers.

You could pick two numbers that have eighty-thousand zeroes after the decimal place before you start getting non-zero digits—such numbers would be far closer together than any separation we could possibly physically measure using any sensible unit of measurement—and the number of Real numbers between them is much, much larger than the number of integers, and is indeed equal in size to the Real numbers.

You can kind of understand why some of the people who worked with notions such as these, like Cantor and Gödel and so on, got pretty loopy at times**, though it is not possible to say definitively whether the mental atypia were caused by too much contemplation of infinities or were the reasons why such people got into studying them, or were pure coincidence.

Of course, the above raises for me the question whether the infinity of the Complex numbers is larger than that of the Real numbers.  I don’t believe it is***; I’m not looking it up to confirm right now, but I am just going to think it through out loud, as it were.

A complex number is made of two “parts”:  the so-called Real part and the so-called Imaginary part.  But the Imaginary part is just i, the square root of negative one, multiplied by some Real number.  So, each complex number is made up of two Real numbers, but two real numbers can be expressed as one Real number in various ways.  So, the Complex numbers have a higher dimensionality than the Real numbers (2 versus 1), but they have the same cardinality, as the terms go.

There are higher versions of countable infinities used occasionally in mathematics.  I find them a bit more contrived than the preceding, but perhaps that’s just because I’ve never had the call to use them nor to play around with them.  These are your ω and ω +1 and so on.  The idea behind it is that you can arbitrarily say that, once you’ve counted all the natural numbers (for instance), whatever the hell that might mean****, you can go on and count some more, calling the next number after infinity ω (that’s a small-letter omega, the counterpart to Ω, which is kind of a cooler-looking symbol).

Anyway, that’s enough of me nattering on about matters that don’t really ever seem to interest anyone around me other than me—I guess I’m not really around me, so I didn’t really need to add to caveat—but about which people sometimes smile indulgently for a moment or two, since I sometimes get rather excited when mentioning them.

This doesn’t happen often, though.  By which I refer to occasions on which I get excited.

This is all just distraction, anyway, and that’s getting more difficult for me over time.  I don’t really feel that I’m going anywhere or getting anywhere.  I’m just killing time while waiting for something to kill me.  I hate the world (by which I mean the universe, and maybe even the Universe), I hate my life, and I hate myself.  This is not a mood (though it entails a mood at times); this is a judgment, this is my (provisional) conclusion, this is my strongly held opinion.  I need to stop running from it; I suck, and so does everything and nearly everyone else.

Present company is excluded, of course (not counting me).  Indeed, I wish all of my readers the very best possible of every possible thing for which things can be the best.

*Though it is interesting to contemplate whether one can take any arbitrarily large run of digits of, say, π and always be able to find the matching sequence in any other irrational number.  I don’t know the answer to that question, so if anyone does, please let me know.

**Poor Gödel became convinced—or at least afraid—that someone was trying to poison him, or might try to poison him, and so he would only ever eat food prepared by his wife.  There is something rather charming, almost romantic, to the notion that he distrusted other people, in principle all other people, and yet he believed in his wife completely.  I wonder if she was flattered.  It was tragic, though, because when she became ill, he stopped eating, and he eventually more or less starved to death.

***It isn’t.  I eventually looked it up just to confirm (or refute) my judgement.

****You can say this, in a way, because (for instance) you can count the integers by first counting the positive integers, to positive infinity, and only then starting to count the negative integers toward negative infinity.  This means you can continue counting after you’ve counted an infinite number, at least in a sense.

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The title of this blog post is wrong

Well, to quote Emperor Palpatine (in Return of the Jedi), “Everything is proceeding as I have foreseen.”

Now, admittedly, I haven’t specifically foreseen very much, other than trivial things that are foreseen without any conscious thought.  But I did foresee (and did predict*) that I would go to the office today and that I would write a blog post today, which is what seems to be happening.  Hopefully, the Death Star will not be blown up around me.

Anyway, yeah, I’m going in to the office today.  Huzzah.

Okay, with reference to the preceding sentences:  Someone needs to tell all the word processor programs that just because we have a compound word “into” does not mean we cannot use the expression “in to” without it being a mistake!  The two things do not have precisely the same meaning, and the split words are in any case more fundamental than the compound word.

You’d think the editing software was worried that “in” and “to” were like neutrons or something, and that if you separate the compound words from each other they’ll just…decay with a half-life of about ten minutes, like free neutrons do.  Therefore, it’s got to keep them bound, because neutrons in a nucleus are really stable and can last until, I don’t know, the protons decay, if that even happens**.

Perhaps I’m overly impatient, but I get frustrated by the fact that a lot of the time, perhaps most of the time, when Google Docs gives me its grammatical suggestions, I have to choose the option “Suggestion is wrong”.

What bothers me about this‒one of the things‒is that I am pretty sure that I have more grammatical expertise than most people around me, and maybe more than most of the people who use Google Docs.  So, out there in the world where people are using it, mostly they are not correcting it, but are allowing it to correct them without even critically evaluating it.

And so, far from such grammar checkers improving people’s grammatical skill and knowledge, they weaken it.  One does not, simply by driving 26 miles every day, get in shape to run a marathon****.  If anything, one’s physical fitness tends to atrophy, and if one only gets around by motor vehicle, one tends to become far less fit, less healthy, weaker.

Of course, training too much can damage a person’s fitness as well.  Like the Laffer curve, we find our peak benefit somewhere in the middle of the arc.

Well, it looks like my footnotes today are even longer than the main body of this post.  I suspect that they are also somewhat more interesting than the main body of the post, but interestingness (interest?) is in the mind of the beholder, so only you can judge that.

I hope you all have a very good day today and a very good day tomorrow.  You should rest and prepare yourselves, because come Monday, I will be writing another blog post…as always, exclusis improvisis.

*It’s interesting that this word, which has a similar compound structure to “foresee”, is clearly of at least slightly different origin or evolution, linguistically.  Fore see:  to see before, to see ahead of time.  Pre dict:  to speak before, to speak the thing before it happens.  We pre dict but we don’t fore dict.  Likewise, we fore see, but we don’t pre see.  Curious.

**There are various theoretical reasons many physicists think protons should decay eventually, but that the half life is predicted to be something like 10 to the 35th or 10 to the 40th years or some such.  Now that’s a long half life, obviously.  But…there are a loooooot of protons.  There are so many protons (they’re everywhere, believe me, you just can’t seem to get away from them) that even if it will take 10 to the 40th years before half of them would decay, we should still be detecting some of them decaying now and then, if we look***.  But “we” have been looking for quite a while, and we haven’t been able to detect any instances of proton decay whatsoever so far.

***Think of it this way.  If there were a lottery in which to win you had to match 10 numbers, each of which could be between 1 and 100 (nonrepeating), the odds of matching one ticket would be roughly one in 6.28 times 10 to the 19th power, or one in 62,800,000,000,000,000,000 (sixty-two quintillion, 800 quadrillion).  Even so, if you had 10 to the 20th people (a hundred quintillion) playing it every week, you’d expect to have fairly frequent winners despite the long odds.  The chances of winning for one individual are so slight that they might almost as well be waiting to quantum tunnel through their front door, and yet the odds of there being no winner may be only about a coin flip in any given running of the lottery (I could work out the precise odds, but I don’t know that it’s worth the time and effort).  This is not a contradiction.  It’s a good reason to remind people not to be fooled by the fact that they may see on the news that someone often ends up winning the slightly more sane lotteries that we actually run.  Your own odds of winning are so close to zero that you’re almost certainly far more likely to die in a car accident on the way to buy your lottery ticket than you are to win the lottery, but when someone does win, it’s big news.  It’s somewhat analogous to the situation with airline crashes.

***Simply walking into Mordor, on the other hand, might be good for one’s fitness, if only there were not so many…environmental health risks.

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Migraines, after-effects, lesser evils, and greater goods

I’m a bit wiped out this morning, so I may only write briefly.  For those of you who may not know it, I developed a migraine headache yesterday.  It started with the surprisingly rapid onset of a “classic” migraine aura, which is to say a deficit in one’s visual field, marked by apparent, weirdly twinkling, almost rainbow-like lights.  It is not as lovely to see as one might suspect if one has not had it.  This first time it happened to me, way back in my early teens (while I was caddying), I worried maybe my retina had become detached.  Weirdly, this concern did not lead me to panic, nor to seek help, nor even to stop in the middle of that golf round‒work is work, after all.

The classic migraine, which is what this migraine with twinkling lights is called, seems to begin with a rapid, spreading depolarization of cells in the brain cortex, particularly (in this case) in the occipital lobe (thus the visual aura, because that is the region of the vision centers of the brain).  It spreads from there, and as part of its spread it seems to lead to vascular swelling and pressure in the meninges, which causes some of the pain.

The science has probably improved since the last time I studied it, so some of this description may be inaccurate.

In any case, as soon as I realized what was happening, I took steps to head things off.  Although I had taken a naproxen that morning, I popped three ibuprofen right away.  After a few minutes, as the aura progressed, I added two acetaminophen (extra strength) as well.  This was probably not ideal for my body‒kidneys, liver, and to a lesser extent, stomach come to mind‒but I didn’t really hesitate, because I have experience with migraines, and not just with treating them.  I also made sure to drink lots of water.

Anyway, I did get some dull pain (as well as just dull dullness) centering loosely on the left side of my head, but the Motrin™ and Tylenol® seemed to have done their work well, since I took them so quickly.  I did have pain, but it was not nearly as severe as it usually is, though I took one more ibuprofen a bit later when it seemed to be increasing.

With migraines, though, the pain is more an effect than a primary part of the process.  In a way‒and only in a way‒a migraine is a bit like a slow-moving seizure.  As such, it tends to cause quite a lot of fatigue in one’s central nervous system, and I certainly felt that yesterday and continue to feel the after-effects today.  For instance, I am making many more typos than I usually make right now.

I do seem to have slept slightly more than usual last night.  Tiring one’s nervous system by internally, directly causing the hyperactive discharge of whole populations of neurons seems to lead to slightly deeper sleep, no doubt because for the nervous system, it is something of an emergency and it must be cleaned up.  Trust me when I say, while the slightly longer sleep is nice as far as it goes, it is not worth it.

Even though I was able to prevent the worst of my migraine pain, the other effects were still quite palpable and unpleasant, and their after-effects remain.  They aren’t good.

This is not to say there aren’t many, many worse things in the world, and certainly when one has only two choices, one should choose the least bad of the two.  For instance, I would choose many migraines over a kidney stone.  The lesser of two evils is, mathematically, the greater of two goods.

If this fact is confusing‒I don’t know why it should be, but apparently it is, and that confusion has been consequential on large scales‒think of the old grade-school level number line, centered at zero.  If you have two negative numbers, such as negative two and negative twelve, you can certainly say that they are both negative.  But that’s a function of where you put the origin (where the zero is), which you can shift and put anywhere else, given the infinity of the number line and its inherent symmetry.

But even if you move the origin 100 places to the left, so that you now have 98 and 88 instead of -2 and -12, the distance between the two numbers, and their mutual relationship based on that, is unchanged.  One of the numbers is still less negative than the other.  And if that number measures the usefulness or the desirability of something, and if one has only the two choices, then one should pick the less negative one, which is by mathematical definition the more positive one.

A billion is less negative than a million.  And negative a million is less negative than negative a billion.

I’m not sure how I got from migraines to here, but that’s okay.  I’m sure it will make at least some sense during the edit.

Anyway, I hope that my mental fog will clear a bit today.  It would be nice to be able to sleep better without requiring general neurological, pathological exhaustion.  But then again, many things would be nice that are simply not so.  By which I mean they aren’t real, that they don’t exist, that they aren’t the case, not “many things would be nice that are simply not nice”, which wouldn’t make a lot of sense.

If wishes were horses, we’d all be drowning in horse shit.  And then we’d probably wish for it to go away, which would just produce more horse shit.  Alas.

I hope you all have a good day.

[PS:  I tried to do a brief audio recording on the way to the office from the train, while walking, because a thought had occurred to me and an occurrence had gotten me thinking, and I also wanted to “experiment” to see how well I’d be able to compensate for the noise of traffic and (to a lesser extent) wind.  It was unfortunately something of a disaster, and to correct it all would have required many hours of audio manipulation that may be beyond my skill.  So I just deleted it.  But I thought I would at least mention it.]

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Don’t regress if it’s just going to make you mean

Well, here we are again, starting another frikking work week.  It seems like only seven or eight days ago that we did this last time.  I don’t think there’s any more good that can be said about this occasion than on any of the previous occasions.  In some ways it is worse, for it merely continues the fruitless, pointless process it entails, without any clear increase in wisdom or knowledge or well-being.

There’s certainly been no gain in happiness‒for me, personally, or for the world at large.  Though probably there are some individuals who are better off this week than they were last week, by any criteria one might reasonably choose.  Gaussian curves tend to have outliers, after all; that’s the nature of the normal distribution.

Of course, there are negative outliers as well, down at the other end of the curve.  I feel for them.  Hopefully, their place in the lower percentiles is a fluke, a local, transient variation not produced by any particular process but merely by statistical fluctuations.  If so, they will almost certainly experience regression toward the mean next week:  when an exceptional occurrence in an iterated or longer term process is followed, almost always, by more average outcomes in whichever direction brings it toward the mean* of the curve.

That’s good.  It would be a shame if there were some process making it such that some individuals were persistently prone to be in the lower tail of the curve.

Of course, though I used the subjunctive just now, it’s not really a contrafactual situation; there really are plenty of people whose situation is such that each week for them tends to be worse in general than the weeks of the vast majority of other people.

This is almost certainly not by their own conscious choice.  It’s not easy to imagine a realistic situation in which someone deliberately chooses to be consistently worse off than most of their fellows.  Unconscious or subconscious urges, on the other hand, can lead to such things (presumably), as can bad habits that have taken hold.

Don’t look down on people who are stuck in bad habits that are clearly doing them harm.  The nervous system is built around habits, and does not change them lightly.  They are how people are able to do most of the things that they do.

If you had to decide de novo, without the grace of habits, what to do on any given occasion or in any given situation, odds are good that you would exhaust yourself rapidly, and you would screw things up at least as often as not.  There are good reasons why “System 1” takes care of most situations for most people**.

“System 2” requires much more time and effort and literal energy than “System 1” does.  And the brain already uses about 20% of a typical human’s calories.  So there, you see:  data centers have always been very energy-intensive, and they always will be.  There are good, sound thermodynamic and information theoretic reasons for this.

Okay, well, anyway, this has been a bizarre digression or series thereof.  It’s weird, and probably not terribly interesting to most people.  Such is the stuff of my thoughts; I can’t, in good conscience, recommend it to anyone.

Speaking of the stuff of my thoughts, I did a brief-ish audio recording yesterday afternoon, which I posted quite a bit later in the day.  I don’t know if it’s interesting or even coherent; I did not edit it for content, just sound quality and pause length.  You can listen to it if you want (obviously) and feel free to give me feedback on whether or not it is coherent.

Meanwhile, I hope that each and every one of you reading this feels better than I do right now.  If I could know that, I think it would make me feel slightly better.

*By mean, I mean (ha) the arithmetic mean, or what might be called the average in common parlance:  you add up the values of every “entry” and then divide by the number of entries.  This is not to be confused with the median, which is the value halfway from the worst to the best (or whatever) of a given particular listed and ordered data set.  It’s an important distinction, because confusing the two leads to some fairly common mental errors.  For instance, while by mathematical definition it is impossible for more than half of all comers to be above the median, it is entirely possible, if atypical, for more than half of people to be above average.  This just requires those who are below average to be significantly more below the average than the others are above it.  For instance, if you gave a test to 100 people, and 90 of them scored 50%, while 10 of them scored 0%, the mean would be 45, and so most people would be above the mean.

**There may be some evidence that people with ASD, for instance, tend to use System 2 far more often for far more things than neurotypicals do.  This may be part of why autistic people, even level 1 ones (which I am not, apparently, at least according to my official diagnosis), are so often just mentally exhausted, even by things that “ought” to be straightforward.

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Does the tread on a Mobius tire last twice as long?

“And it must follow as the night the day” that the second day of the work week (today) follows the first (yesterday).  This is what happens with ordinal numbers; they delineate (if that’s the word I want) the order of things.

Of course, as with numbers on the standard clock, or cycles of days of the week, or months of the year, the first day (or whatever) also follows the second day.  Indeed, in a small enough cycle, they can follow each other by exactly the same amount.  This is what happens in a binary cycle, like soldiers marching (or any biped walking).  The left foot starts off, and the right foot follows immediately after.  But then the left foot follows the right foot immediately after that.  And so on.

Of course, the weekdays have seven members, so that cycle may be more akin to the movement of the heptapods from Arrival.  And there are more months still, and more days in months than that, but the cycles still cycle.

Some cycles are surprising, of course.  For instance, if you go for one 360-degree turn around a Mobius strip, you’re not back where you started; you’re on the opposite side of the strip, “underneath” where you began.  To return to your starting point, you must take another 360 degree (or 2 pi radians, if you prefer) run around the strip.

This is a good analogy with (and a way of making some intuitive sense of) the spins of fermions, i.e., particles with half-integer spins, such as electrons and quarks.  When we say they have half-integer spins, we mean if you “rotate” one 360 degrees, it will end up in the opposite spin configuration to where it started; it requires another 360 degrees to return to its beginning.

That’s in contrast to the integer-spin bosons, which behave more “intuitively” in that they require 360 degrees to come back to where they started, like numbers on a clock.  Photons are probably the most obvious examples of spin-1 particles.

There is also a hypothetical spin-2 particle, the graviton (still a boson, because it has an integer spin).  This is easier to model in one’s head, because it “looks” the same after a 180 degree rotation.  It’s pretty easy to imagine 3-dimensional shapes to model that.  Think of football shapes, or even just rectangles.

Of course, it would be even easier to imagine “spin 3” or “spin 4” things:  think of equilateral triangles and of squares.  One can even quite easily imagine a spin-infinity object.  That’s just a circle; no matter how you spin it (maintaining its center point) it looks the same.

Of course, if you’re Archimedes (you’re not; I checked), you get closer and closer to finding out the value of pi (the ratio of the circumference of a circle to its diameter) by bounding a circle’s circumference between regular polygons that fit exactly within and exactly around the circle and taking their perimeters.  It’s hard to do that for a triangle, or at least it’s hard for me to do in my head, but a set of squares*, one nested in a unit circle**, the other in which is nested a unit circle, can be done.

The perimeter of a square in which is nested a unit circle is easy, because each side of the square is length 1, just like the diameter, so the perimeter is 4.  The interior square has diagonal length 1.  So any two sides make the legs of a right triangle, and a^2 + b^2 = c^2.  So side^2 plus side^2 equals 1^2.  Or, 2 times the side-length squared equals 1, or a single side squared is 1 over 2, and the length of a side is 1/√2 (we only use the positive root, since we’re talking about physically realizable lengths).  Thus, the perimeter of that square is 4 x (1/√2), or 4/√2, which is 2√2.  That ends up being around 2.8 ish.

So, it seems pi is between 2.8 and 4.  Which it is.  That doesn’t narrow it down much, but if you go from there to ever-increasing numbers of (equal) sides in nested/nesting figures, as Archimedes did, you can get arbitrarily close to the actual value of pi.

That’s all you can do, since pi is an irrational number.  There is an infinite amount of information encoded in the digits of pi, and indeed in any irrational number (making each irrational number its own version of the Library of Babel in a sense).  But pi is a fundamental geometrical number, so that fact hits home with it perhaps more than with any other specific number.

If the universe really is continuously divisible (this possibility is far from certain), then each spot in the universe contains a potential uncountable infinity of info.  If that is the case, nothing can truly, fully simulate the universe in detail.  But we can still make useful models.

I could get into the restrictions imposed by the Planck length and the Planck time and so on, but that’s a whole can of bees I don’t want to open just now.  Instead, I’ll call this to a close for today.  I hope you have a good one.

*A very crude starting point, but it’s good for demonstration.

**In this case I’m using that to mean a circle with diameter 1.  Sometimes one sees it used to refer to a radius-1 circle.  It doesn’t matter that much, but it’s very important to be clear which one you mean.  Just as it’s useful, even crucial, to define terms before beginning a discussion to avoid arguments over nonissues***.

***For instance, if you get two people to agree on what they mean by “sound” before you ask them, “If a tree falls in a forest and no one hears it does it make a sound?” then there is unlikely to be any disagreement, because the answer will be trivial (either way).  Thus, this is not actually a philosophical conundrum, despite its reputation; it is merely a demonstration of the folly of poor communication.

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He reads the post with just his fist and still believes he gets the gist

Well, I said yesterday that there would be roughly a 50/50 chance whether today I would write on the lapcom or on the smartphone, and guess what:  today I am writing this either on the lapcom or on the smartphone!  How’s that for an accurate prediction?

But wait.  Which one am I using?  Can you tell just by reading this post?  Are you sure?

Of course, I know which one I’m using.  It would be most ‘passing strange if I did not know whether I am writing this on my lapcom or on my smartphone.

Is there a way for you, the reader, to tell?  Probably.  Almost certainly.

But do you know what that way is and how to apply it?  I doubt it very much.

That’s not an insult, by the way; I don’t know what it is or how to apply it, either.  I’m just pretty sure there is such a way.

Of course, from my own point of view, the metaphorical wavefunction has already collapsed, and there is only one possible remaining outcome, whereas before there were (at least) two.

I say “metaphorical wavefunction”, invoking the quantum mechanical notion of the collapse of previously superposed quantum states into one final state, but there are good reasons for us to doubt that notion’s accuracy even within quantum mechanics.  After all, it would be the only known physical process in the universe that is not time-reversible and which destroys information about prior states of reality.  That oughtta be a pretty big red flag for scientists.  It’s almost as bad as finding a process that seems to violate the 2nd Law of Thermodynamics*.

I find the Everettian approach to quantum foundations much more intuitive, personally.  That doesn’t necessarily mean it’s more likely to be correct, but I think, I suspect, that it is.

Anyway, in the macroscopic world, the seemingly superposed possibilities that present themselves as we come to the point of a decision are not actual superpositions.  They are merely models we render in our minds of possible outcomes to try to improve our decisions.  In fact, in almost every case, it’s likely that the choice we make was “determined” ahead of time‒by the laws of physics, not by us.

I would guess that it was that way when Bohr’s and Heisenberg’s “Copenhagen Interpretation” of quantum mechanics became so dominant despite its failings.  The problem is, Bohr and/or Heisenberg (I don’t recall which one) was by reputation exceptionally charismatic, and he was well able to ensure that his/their notion(s) became predominant, not because the ideas were more convincing, but because the people were (or the person was).

That’s not a good reason.

This is part of why I dislike the practice of public “debates” about controversial topics at pretty much any level.  When it becomes a contest in and of the moment, the “winner” of the debate is not necessarily the one with the best evidence and the most consistent and clear reasoning.  It is, often, the one more skilled at mere rhetoric, the better sophist, the one with the better ability to manipulate human cognitive biases, the one with the better speaking voice, the better looking one, the one who makes the best jokes (especially at the other’s expense).

This is not a good or reliable or useful way to measure empirical reality‒except that part of reality that tells us who is more superficially persuasive to Naked House Apes.

That’s part of why the court system in general is so bad:  the one who wins in court is not necessarily (or even probably) the one who is right, but rather the one who has the better lawyer with more resources.  This usually translates to “the one who happens to have more money.”  That’s not a good basis for any kind of system that refers to itself with the term “justice”.

Oh, well, what are you gonna do?

Well, it would be nice if you could do your part toward at least improving these things in whatever way you might be able, especially if you are in any kind of influential position.  This here, this writing, is me doing at least some of my part, for whatever it’s worth.

In the meantime, I’d be interested to get your feedback:  do you think this post was written on the lapcom or on the smartphone?  Why do you think that?  Are those your real reasons?  Or are they the reasons you create‒some might say confabulate‒to justify a decision you made for reasons that are not clear to your conscious mind?

Please let me know in the comments.  And talk amongst yourselves there, too, if you like.

Also, please have a good day.

*This is not to say that it is impossible for net entropy to go down in a closed system.  It’s not only possible, but if you wait long enough, it’s going to happen somewhere, for the 2nd Law is statistical in character.  But for anything but the simplest situations, you’re going to have a wait for such an outcome.  Even if you’re just flipping 13 coins until you get all heads or all tails (or any other specific, ordered pattern you might want), then it’ll take a little while.  Getting all heads in a row (say) on 13 coins is a one in 8192 chance, if my mental arithmetic is right.  It would take some time, but you could pretty readily flip those 13 coins more than 8000 times, especially if you flip all 13 at once each time.  But anything much more involved than that (and just 2 more coins would require four times as many flips) becomes rapidly and astonishingly more unlikely.  If you’re waiting for any sensible region of, say, the Earth to experience spontaneously decreasing entropy, you’re going to be waiting such a long time that probably the current time (about 13.7 billion years) since our Big Bang would seem like an unnoticeably tiny fraction of the blink of an eye.  And, of course, the Earth is not going to be around that long‒not more than about another 4 or 5 billion years at most.  If that seems like a long time to you, you need to adjust your perspective.

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I almost forgot to put a title here again

I’m writing this blog post on my smartphone, as I did yesterday’s post, and in contrast to the posts from Monday through Wednesday.  I haven’t yet received any direct feedback on whether there’s a difference for the reader or what it might be, but the numbers seem to indicate that the phone-written posts are more popular than the lapcom-written ones.

This could, however, be mere statistical fluctuation, having no relation to whether readers find one or the other type of post better.  Quite possibly, most readers wouldn’t be able to tell one from the other without being told, even if the lives of their dearest loved ones were on the line.

Such is the difficulty with finding truly dispositive evidence in ordinary life.  But that’s not to say it can’t be done.  One just has to try very hard to be clear-headed and objective.  And I don’t mean “try to try” to be clear-headed, not just to be able to say “I tried to be clear-headed”, but actually to act with the true intent to be clear-headed.

Of course, the human senses and human brains gather a tremendous amount of information every waking moment, checking it against their hitherto-built model of reality, seeing if things meet expectations according to that model, and trying to improve that model, that map, of reality.  Mind you, there’s way more info in most places than anyone can take in.  That’s okay, for the most part.  Most of that detailed information is irrelevant to the life and reproduction of a far-flung African ape*.

Speaking of updating one’s models of reality, I just yesterday came across a video/written course on tensors (for physicists and would-be physicists).  The professor’s approach seems like it’s going to be a good one, so I’m planning on trying to go through the course.

I want to learn well about tensors (about which my understanding is not yet fully clear, though I get the gist of the basics) not just for my own curiosity‒which drives me, in principle, to want to try to understand everything in the universe‒but also because I will need skill in using them and manipulating them if I am to solve my longstanding point of curiosity in Special/General Relativity.

I have a specific question about what the theories predict would happen in specific circumstances, and I have not been able to find anyone who reliably answers it.  Really, no one has answered it at all, which is not too surprising, since it is fairly esoteric.

We’ll see whether I can commit to the bit.  I have a hard time maintaining focus on things for too long at once.  I dearly love to learn about new things and to develop new skills, especially in the sciences‒well, also in the arts‒but it seems that after (far too short) a time I get distracted by another interesting thing.  Either that or I just get mentally fatigued and need to distract myself, usually either with music or something funny.

I suppose that’s not really that unusual.  But lordy, it’s frustrating.  I wish I could actually want to do what I want to want to do.  Maybe I will be able to do so someday.  Maybe I will be able to devise or find more direct control of the regions of my brain which govern attention, focus, and drive.

Of course, we do have some somewhat direct ways to affect those brain regions.  The most widely used of these ways is caffeine.  The majority of people in the world use some form of caffeine on a regular basis.  At least, that was so the last time I looked.

Of course, there are other such tools, some more powerful in some ways than caffeine, but they come with their own sets of difficulties:  these include the amphetamines and related compounds and cocaine.  They can be useful in certain circumstances, but are difficult to use well, without significantly detrimental overall outcomes.

It would be easier if we could directly stimulate (and suppress) specific areas and processes of our brains at will.  Of course, the technology to do such things exists, more or less, in raw form.  One can stimulate the brain with implanted electrodes, or one can manipulate it more indirectly via externally applied electric and magnetic fields.

This has been done, of course, if only fairly crudely.  The technology I describe in Unanimity is (mostly) very real.  Is that what makes it scaaaary?

Probably not.

Okay, that’s enough for today, and for this week as well, since I am not working tomorrow (barring the truly very much unforeseen).  I hope you all have a good day and a good weekend.  Have a good meal or two while you’re at it.  Though, possibly, that’s implicit in most concepts of good days and good weekends, come to think of it.

Oh, well.  Have good ones nevertheless.

*That refers to humans, in case it’s not clear.

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Drop and give me twenty! (The prime factors of twenty, I mean)

It’s another lapcom blog post today, for the second day in a row.  Huzzah!

That doesn’t really count as any kind of streak—though no doubt WordPress will send me an automatically-generated “You’re on a streak!” message, since that’s apparently something that encourages people to keep using the site regularly.

I guess it would be nice if I were going for some kind of personal record—if I were trying to write as many days in a row as possible, for instance—to be aware of how many I had done without having to keep track of it myself.  And, I guess, if you’re recording streaks, you have to start somewhere, and the smallest possible streak is two.  You don’t really want to count one as a streak, because then everything is a streak and it loses much of its meaning.

This is somewhat analogous to the reason that 1 is not considered a prime number, though you cannot deny (correctly, anyway) that it is divisible without reminder only by one and itself (which is also one).  But for mathematical purposes dealing with primes, this would be a rather useless add-on to the set.

It would also make deciding the number of prime factors of a number pointless, instead of being specific and fixed for each number.  As it is, without one counted as a prime, it is specific and fixed.  For instance, the prime factors of 36 are 2, 2, 3, 3.  But if 1 were considered a prime, you could have 1, 2, 2, 3, 3 or 1, 1, 1, 2, 2, 3, 3, with any arbitrary number of 1s, because every 1 would not change the final product of those primes.  You could have an endless string of ones if you wanted.  And so, the number of prime factors of any given number would be up for grabs, whereas right now it’s fixed.

I know, I know, it’s not terribly important in day to day life for you to think about such things.  Some people are disdainful of even learning about such things unless they have a direct impact on their lives in a clear and obvious and simpleminded way (emphasis on the “simpleminded” there).

There are people who make jokes about the fact that they learned the Pythagorean theorem and don’t use it, or that they know how to do square roots but don’t really need them, or learned the quadratic equation and haven’t used it since that class in high school (or college, maybe).

Such people perhaps think that the point of doing push-ups or bench presses or squats or lat pull-downs is to get really good at doing push-ups or bench presses or squats or lat pull-downs.  They may imagine that the reason to do toe touches is to get really good at touching your toes, and if you’re not going to try to do that, then don’t do toe-touches.  These people probably think that everyone who jogs regularly is doing it so they can get better and better at jogging.

I think you probably see my point, but I’ll make it explicitly.  The purpose for all that physical exercise is not to get good at doing calisthenics (or whatever), not for the vast majority of people.  It’s to be comes stronger, fitter, more flexible, to develop better endurance, so that you will have those capacities to bring to bear on any other task in life.  In many, many matters, being physically fit will make a person more effective.  Sometimes it can even save someone’s life.

Likewise for doing things like math.  Some people will go on to be mathematicians, of course, or to become physicists, who use mathematics regularly in their work.  But everyone can strengthen their brains (far more than they can strengthen their muscles, which have a hard ceiling on improvement).  They can improve their ability to think logically and systematically, to recognize patterns and to be able to manipulate them in their heads, by practicing and understanding mathematics.

Understanding percentages will make it immediately clear, for instance, that something cannot sensibly be reduced in price by 600%.  Understanding probabilities can help one recognize why one should not invest in the lottery (unless you’re the one running it, in which case, by all means—if it doesn’t trouble your conscience).  And one should not blindly trust the representations of, say, managers of mutual funds and the like.  Even though what they happen to tell you may be real data, omission can be just as misleading as straightforward lies.  You also need to know what they have left out.

If they tell you their fund increased in value ten days in a row, implying thereby that you should want to invest in it, you need to think about (for instance) just how many funds they have.  How many funds are they picking from and for how long?  If they have a thousand funds, assuming equal chances of gains or losses, you should expect on average for one of them to go up for ten days in a row in any ten day period.  But the next ten days, it would be a different, random one that goes up*.  Or, if you can pick any ten days out of a given year, you can probably find somewhere where there’s a ten-day streak, or nearly so, even if you only have a few funds.

I have not done the math to figure that last bit out specifically, but it directly relates to the probability that you will have a string of ten heads in a row somewhere if you flip a coin 365 times.  That goes back to basic probability.

It can serve you well to study some micro- and macroeconomics.  It can also do you good to study a bit of basic chemistry and biology; you wouldn’t even have to have any more advanced medical knowledge than high-school level to know that while injecting enough bleach will kill COVID, it will also kill you, and so will not be a much better therapy than setting yourself on fire or detonating yourself with TNT.

I made a meme about a similar idea that I sent to a friend who was all too easily persuaded by conspiracy theories about various “natural” substance being able to kill cancer but that “Big Pharma” didn’t want you to know about them.  Actual knowledge of how cancer works and how lab tests in vitro work would have protected him from such claims, but I took a slight shortcut to hammer the point home.  Here it is:

Anyway, that’s already a lot for today.  I tend to write faster with the lapcom, and so I tend to write more.  I hope it’s not too irritating.

I hope that about myself and the things I do quite often, but I’m afraid I still end up being irritating more often than not.  My apologies.  I also hope you have a good day.

*Again, all this assumes about a 50/50 chance of going up or down.  Considering how many regulatory and other factors are in place to encourage markets to go up, it’s probably skewed slightly toward gains**, and this will increase the chance of ten-day streaks of gains.

**As evidence, if one had invested in a simple index fund without any significant churning (I think that’s the term) over the last several decades, one would have made around about 10% a year.  That’s roughly twice the rate of inflation, so there is still a real, adjusted net gain.  Given compounding, if you invested $1 in, say, 1990 with that return, you would now have, let’s see…$30.91.  If you invested a million, you’d now have $30.9 million.  Higher rates of return than this will tend to involve higher risk.