infinity

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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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There’s an infinity that shapes our ends, despite having no end itself

It’s Friday now.  It will in fact be Friday now until midnight tonight, local time.  Indeed, one could argue it will be Friday now until finally midnight strikes at the international date line, when this Friday will finally be gone from the entire Earth, forever.  So, though as a matter of physics there is no universal “now”, and even for individuals, the “now” is an evanescent thing, a constantly moving and infinitesimal single frame of the movie of one’s existence, nevertheless that “now”, for me and for most others on Earth, will still be Friday for some time.

How many such “nows” are there, even for one individual?  Well, that depends a bit.  If the Planck time (5.39 x 10-44) is just an artifact of our lack of complete knowledge or ability to calculate, and time is truly continuous, then there is an uncountable infinity of such “nows” in any given day, or indeed in any given hour, or in any given second, or in any given picosecond, or indeed, in any given Planck time*.

Such is the nature of the uncountable infinity, as in the case of the real numbers:  between any two numbers, no matter how arbitrarily close you want to make them (as long as they are not identical) there is an uncountable infinity of numbers, larger than the number of possible quantum states in the visible universe, larger than the “countably” infinite number of integers.  In fact, that uncountable infinity between any two such real numbers is as large as the uncountable infinity of the set of real numbers itself, of which it is a subset.

Infinities are weird.  You need to be careful with them.  I doubt that contemplating them has actually driven anyone to madness‒though it’s easy enough to imagine that it might exacerbate depression‒but maybe minds somewhat prone to madness are more likely than others to contemplate infinities in the first place.  In any case, contemplating them can put other things into perspective.  For instance, no matter how arbitrarily large a number you might pick, it is just as far from infinity‒even the boring old “aleph nought” infinity‒as is the number one.

An interesting thing to contemplate is that, if you could pick a truly random number from, say, all positive integers, you would almost certainly get some number far huger than any number ever named or contemplated by humans, larger than a googolplex, larger than Graham’s number, larger than TREE(3), larger than the time required for a Poincare recurrence of the cosmos.  Graham’s number (for example) is big; the information required to state it precisely, if contained within the space equivalent to a human brain, would cause that space to collapse into a black hole!  But Graham’s number is nevertheless finite, and so there is a finite number of positive integers lower than Graham’s number but an infinite number of them larger than it.

It’s interesting to note the related fact that the chance of you randomly picking any particular integer is mathematically equivalent to zero‒so I’m told‒and yet you will pick some number.  Let that bake your noodle for a bit.

By the way, when I earlier compared the moments between two points in a continuous time stream to the number of possible quantum states in the visible universe, I was being a bit contradictory.  After all, our designation of the maximum number of possible states in a given enclosed region of spacetime‒which is “equivalent” to the number of square Planck lengths (each such square being 1.6 x 10-35 meters, squared, or 2.6 x 10-70 square meters) in the surface area of a sphere surrounding such a region**‒is based on quantum mechanics, and thus implicitly entails time being only sensibly divisible down to the scale of the Planck time.  So comparing that to a continuous time is comparing two fundamentally incompatible realities.

Oh, incidentally, I’m writing this post on my smartphone today.  I just didn’t feel up to bringing the lapcom with me yesterday, and I didn’t expect to write any on The Dark Fairy and the Desperado today.  I did, however, have a bit of a thought, as I’m prone to do when conscious, whether I want to do it or not.

That thought was that, perhaps, I can try to write my blog posts in the evenings‒on the way back from work, say‒but set them up still to be published the following morning and work on fiction in the morning.  Writing fiction seems to give me a boost, mental health-wise, when I do it in the morning.  It’s quite ego syntonic, as they say, or at least it seems to be.  But I don’t really want to stop writing this blog.  Then I’d just be floating in the void all alone, writing fiction that I like but that almost no one else will ever read.  That is a discouraging thought.

In any case, I don’t think I’ll be writing a post (for) tomorrow, since I don’t think I’m going to be working tomorrow.  If I am, and if I cannot get out of it, I guess I will write a post, and it will likely be a grumpy one if it happens.  But I may start next week writing the following day’s blog post on the evening before and doing fiction in the morning.  One good aspect to writing fiction in the morning is that the initial writing and the editing process are separate.  I don’t have to edit what I write each day on that day, which I have to do with this blog.

We shall see what happens.

In closing, I leave you with this juxtaposition of two notions:

*If time is not sensibly divisible even in principle below the Planck time, then the maximum number of “nows” in a given day is just 24 hours divided by the Planck time, or about 1.6 x 1048 “nows”.

**See Bekenstein-Hawking black hole entropy calculations and the Holographic Principle.

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It’s a prime day for a (slightly) shorter blog post

Morning has broken!

Does anyone out there know a good, reasonably priced morning repair service or person?  I really don’t have the money or time to go pick out and buy a new morning.

Ha ha.

Sorry.  I know that’s quite a stupid joke.  Still, it should give you some idea just how tired I am.

I should have been able to get a decent sleep last night, but I did not.  That shouldn’t be anything new to me, but some days the effects of the insomnia seem to pile up far more than on some other days.  And today is one of those “pile up” days, it seems.

Thus, I intend to make this a relatively short blog post today if I can.  To that end, I’m setting my “target” number of words to be 701 instead of the usual 800*.  701 has the added advantage of being a prime number, which always makes things at least a little bit better from my point of view.

It’s funny how, as numbers get bigger, there seem to be fewer primes (they fall off as something like the natural log of the number range at which you’re looking, if memory serves).  And yet, there are an infinite number of them‒the same “countable infinity” as the natural numbers, “aleph nought”:

Also, we have the twin primes conjecture‒which apparently most mathematicians think is correct‒that there are an infinite number of primes that are with 2 of each other.

It has apparently been proven that there are an infinite number of primes separated by no more than about 500 or some such (it’s probably a lower separation now, but I haven’t looked into it in a while).  That may not seem like much of an accomplishment, but remember, we’re talking infinity here.  No matter how big the numbers get** you will never stop finding new sets of prime numbers that are no more than about 500 apart.

That’s not particularly useful to anyone but number theorists, I suppose, but it seems very interesting to me.

Incidentally, 701 is not part of a pair of twin primes, since 703 is not prime (it has four factors) and 699 is obviously divisible by 3.  And of course there is only one even prime (the number 2) because all other even numbers are divisible by 2.

Sorry, I know many people find these things boring, but I’m a fan of prime numbers.  In any case I’m trying not to be so negative in my blog posts, since I think it bothers a lot of people and may even be contagious at times.

It certainly doesn’t appear to do me any more good than does being a voice crying out in the wilderness, so to speak.  And despite the excellent biblical reference***, voices crying out in the wilderness usually are not heard or received by anyone or anything that can even understand them, let alone offer them any help.

I guess I can still talk about “imaginary” and complex numbers, because they just involve the square roots of negatives, but are not necessarily negative themselves****.  As long as I avoid multiplying them together, I should be able to steer relatively clear of negativity.

Ha ha, again.

I’m trying to try to avoid making other people miserable by expressing my own dark thoughts, so instead it seems I will make others miserable with my bad jokes.  You’re welcome.

Thinking about complex numbers makes me start feeling like I want to learn more about quaternions and spinors and so on, which seem truly fascinating, but about which I have only highly superficial knowledge.

It would be nice to learn more about them.  I probably will not, knowing me and my fatigued and distractible mind, but at least I can maybe be on the lookout for Numberphile videos about related subjects.  There’s at least one YouTube channel with a series on spinors.  Unfortunately, PBS Infinite Series stopped making new videos some time ago, but at least PBS Spacetime is still going strong, as are all of Brady Haran’s excellent channels.

And now, I’m over 701 (in the first draft), and so I’m done (for today at least).  I hope you all have a good day, and that your subsequent good days scale as the number of days, not as any logarithm thereof.

*I almost always go over my target, but at least it gives me a noteworthy place to decide it’s “time to wrap this up”.

**And there are described numbers so large that if you could memorize every digit of them, the information would be so concentrated as to turn your head into a black hole.

***It’s from the book of Isaiah.  John the Baptist supposedly quoted it and referred to himself as that voice, but then again, a lot of the writers of the “New Testament” shoehorned in references to supposed prophecy in the “Old Testament” to make the whole thing seem more sexy-cool (I guess).

****I just have to try to keep to the right upper quadrant of the complex plane.