Problems

Cantor’s infinities are meaningless and stupid

Overall, this video provides a good intro to some important math ideas:

Here are some random thoughts on the video:

I don’t think principles like undecidability necessarily imply a “hole” or “flaw” in mathematics. It only appears to be a “hole” if you’re assuming something should be there, like decidability. But that’s based on your assumption that it should exist in the first place.

The video soon turns to Cantor, giving me the perfect opportunity to finally rant about how foolish he was! (I’ve been intending to do so for some time.)

Cantor’s infinities are a bit of a pet peeve of mine. His foolishness, and the foolishness of those who nod in bedazzled wonder and agreement with his nonsense, stem from a lack of understanding the implications of infinity. Infinite size means no size. That can be confusing, because in this instance “no size” does not mean a size of 0. So how can something have no size without the size being 0? By being infinity. One must stop thinking about infinity as a number, but as a concept parallel to that of number.

So when Cantor asks (at 4:30 in the video): “Are there more natural numbers or more real numbers between 0 and 1?”

Woah, back it up, back it up, beep, beep, beep!

That is a nonsense question. It’s like asking: “What is five divided by green?” By definition, there cannot be “more” or “less” of something that has no amount to begin with. There are infinite natural numbers. There is not an amount of natural numbers, because there are an infinite amount, which means there is no amount. So there cannot be more, or less, or the same. There can be no comparison whatsoever because what you’re trying to compare is the amount, which does not exist.

Does infinity equal infinity? If you’re intending to compare amounts, the question is again meaningless nonsense. Infinity cannot, in this sense, equal or not equal infinity because you cannot compare them like finite amounts.

As the video shows, Cantor goes on to (rather stupidly) compare two lists. His methods are meaningless because his premise (that infinities can be compared) is already flawed. He then finds with his “diagonalization proof” that you can’t logically define a pairing between every natural number and every real number between 0 and 1, and in the depths of his infinite stupidity thinks that this somehow proves that there are “more” real numbers than natural numbers.

Uh, no it doesn’t, Mr Georg without an “e”. All you’ve shown is that you didn’t actually succeed in defining a pairing. You haven’t proven anything about sizes because infinite sets do not have sizes. Whether or not you can rigorously define a pairing (a one-to-one corresponce) implies nothing at all about sizes. It only proves your definition of the pairing to be paradoxical nonsense. You can’t say “let’s assume we’ve paired all natural numbers to real numbers between 0 and 1” and then say “here’s a real number that can’t be in the list!” That just means we didn’t actually pair the sets to begin with!

The crux of the paradox doesn’t lie in the “sizes” of the sets anyway (which don’t exist). It lies in the inability to express all real numbers with finite decimal places in the decimal system. If we take for granted that we could instead express some otherwise undefined real number with an arbitrary symbol (like, gee I don’t know, a natural number), the paradox is completely resolved. There is nothing to “diagonalize” and the one-to-one corresponse is complete. Logic 1, Cantor 0.

At 6:45: “Cantor’s work was just the latest blow to mathematics…” Perhaps more of a blow to mathematical philosophies than to math itself. Aside from being complete nonsense, it had no implications aside from morons thinking “oh wow, different size infinities sure is amazing, derp!” which is about as meaningful as thinking, “oh wow, five sure is colorful, derp!”

At 7:27: “On the one side were the intuitionists who thought that Cantor’s work was nonsense. They were convinced that math was a pure creation of the human mind and that infinities like Cantor’s weren’t real.”

Perhaps, but whether or not an infinity can be “real” is really not the issue with Cantor’s illogic. Also, his lack of logic in this particular area does not necessarily imply inherent weakness with set theory in general.

The video goes on to speak of set theory’s self-reference paradox. It is indeed a paradox, but is by itself really no weakness of set theory anymore than the existence of paradox itself is somehow a weakness of the human mind that conceives of them. In fact, one could say the ability of a system to define a paradox is actually a strength.  It’s like trying to make a programming language that doesn’t allow for infinite loops by taking out the ability to have any loops at all.

I really like the video’s explanation of Gödel’s work with using; actually, perhaps because it’s visual and tangible, I think it may be the best explanation I’ve seen!

At 31:34: Haha, what is this artsy-fartsy shot? “Look at my back as I gaze at the sky and ponder the deep thoughts of the world…”

The video ends by circling back to the “hole” in math, which is now defined as not being able to know everything with certainty, which seems a rather imprecise way of summing up undecidability and incompleteness as it takes for granted the meaning of “certainty”. I guess we could say: “Hey, Gödel, if math is incomplete, then your proof is incomplete and therefore not a proof! Hyuck hyuck!”

By S P Hannifin, ago
Interesting things

My Solution to the Collatz Conjecture

As promised, here’s my attempted solution to the Collatz Conjecture. My solution is pretty simple, so if you understand the conjecture, you should understand the proof. (I’m not a pro mathematician anyway, just an amateur hobbyist.) I’m eager to get feedback, especially if I somehow missed something subtle (or worse, something really stupid).

PDF of my proof: click here.

If you prefer to watch a video instead, I’ve uploaded myself explaining my solution here:

Here’s to hoping my proof is confirmed!

By S P Hannifin, ago
Philosophy

A problem of chains

As you’re traveling the road, it’s a strange and frightening thing to look around and suddenly realize that most of the world is not walking with you, that what you believe is right and wrong is actually not at all what most of the world accepts or lives by.

At first, it makes the world seem so sad and dark, almost post-apocolyptic, despite the smiles on everyone’s faces. How can I be a part of this world? I can’t live by these standards.

But then it doesn’t seem so bad. You’re just looking at the world the wrong way. You are the enlightened. It’s everyone else that will eventually discover the sad darkness of their world; they’re only smiling because they’re looking down most of the time. They haven’t reached the ends of their chains yet, so they don’t yet realize that they are slaves. But you are not a slave. You know how to look around for chains, and you can’t be chained if you’re always on the look out for them.

Oh, how vain this all sounds! Do I think I’m better than everyone else?

Not intrinsically, and certainly not everyone. They all have the capacity to shed their chains if they want to anyway. But the chains feel good. I’ve been in chains before. I admit that I unfortunately still even let them slip on from time to time. But without them, I can certainly see more than many others. I can understand more. If that sounds vain or prideful, so be it. I am not fooling myself into thinking I can see everything, after all. But I am not going to say that what I see might not be there just because so many others can’t see it. I am not going to disbelieve my eyes for the sake of people who aren’t even looking where I’m looking.

But isn’t it arrogant to think of them as slaves?

No. Why should it be? It would probably seem arrogant to try to help them out of their chains by saying: “Hey, look around! You are a slave!” That wouldn’t help.

And it would be prideful to presume that I’ve got all my chains off, wouldn’t it? Maybe there are chains I can’t see, or chains I’ve let slip back on without noticing. But I can’t deny the existence of chains just because of this. Nor can I stop caring about them. At least I am aware of them and can work to get my own chains off.

Isn’t this a bit wacko? Haven’t I met others who have warned me about chains that are foolish to believe in? Don’t some people see chains where there are none? Yes. They see some real chains, and then they mistake so many other things for chains. “Your shoes are chains!” “Your glasses are chains!” No, they’re not. I think they are mad. And don’t those who don’t see chains at all think I am just as mad? Is there any way to deal with that?

Perhaps not. I can only be honest with myself.

So what about the chains of others? Should I do anything about them? Can I do anything about them? What relationship am I to have with this world that I think is so dark and sad? I certainly can’t force people out of their chains. If only it were so easy. Chains can only be removed by the person in them. Should I ignore the chains of others and just encourage them to look around for them by my own chain-free actions? Perhaps that is the best way. Perhaps the only way. And if they never get out of their chains, so be it. I cannot blame myself for it.

But I still struggle with what I should feel about them. In their chains, they do things that hurt me and each other. They kill (“that’s not a person” or “he deserves death”), they imprison unjustly (“we must force each other to do things”), they lie and steal (“he doesn’t need this; I should have it”). I often feel inclined to hurt them back. But that is the tug of a chain I shouldn’t be bearing. But they don’t have to be in chains! I am so angry with them!

Should I feel sad that they are in chains, or happy that I am not?

I don’t think sadness will get anyone anywhere, as natural as the reaction is. As long as I keep vigilant with my happiness, that is the path I must stick to. If I am happy that I am not in chains, then at least I am keeping the chains away. It is hard, because the chains do slip on sometimes and are comforting, and tearing them off can be painful at first. But the happiness possible without them is a clearer happiness.

A constant struggle, but the road always ends in light, and it will be easier to travel if I keep that in mind, if I am looking forward and not backward.

By S P Hannifin, ago
Philosophy

The forgetting problem

Here’s a question I made up (though I’m sure someone’s asked something like it before… nihil novum sub sole). I don’t know the answer to it, I’m still trying to puzzle it out:

Say you’re trapped in a science lab. The scientists there have invented a way to erase 24 hours worth of memory each night at midnight.

You have several choices on how to spend your day. You may either:

1. Commit suicide
2. Be tortured half the day then play for the other half
3. Be tortured the entire day, giving you the privilege to play all day the next day

Regardless of your choice, you will continue to be stuck in the lab and continue to have your memory erased each night until you die.

The question is not about what you would choose. The question is: does it matter?

UPDATE: Though one could come up with a bunch of variants, what about a guilt-based one? What if you were stuck in the same lab, had the same forgetting-at-midnight scenario going on, only this time the choices are:

1. Be tortured while another science lab prisoner does not suffer for that day
2. Do not suffer while another science lab prisoner is tortured all day

For #2, you might feel guilt, but is that better than the torture itself, especially considering you’ll both completely forget the experience the next day?

Gah, this problem is really annoying me…

By S P Hannifin, ago
Problems

The crime-twins problem

I had a weird dream the other night. (It was random and dream-like, like dreams often are, but for the purposes of this post I’ll pretend like it was more understandable than it really was; the gist of it is the same.) I was the judge of a no-jury court case. It involved a man who had stolen priceless paintings. All of his illegal activities of breaking and entering and stealing these paintings from a museum were all recorded by security cameras. His face could be seen very easily.

Unfortunately he had an identical twin brother. They were both sitting there in the court room before me, both claiming innocence, both with the same solid alibi of walking around a bookstore that night, which was also caught on camera. They both knew exactly what one of them was doing at the bookstore. I questioned them separately (I’m not sure if that’s really something a judge would do, but I did), and they both told the same story, which supported what was on camera.

There was no DNA evidence left behind by either man.

Lastly, I did not have the lie detecting skills of that guy in Lie to Me, so I couldn’t do anything very dramatic involving catching one of them in a lie or giving them the Prisoner’s Dilemma.

And in the dream I thought it was a riddle, as if there definitely was a way to solve the conundrum and I just had to figure it out.

But I woke up with the riddle unsolved.

Is there an answer? Or is that the perfect way to get out of a crime?

The dream made me wonder if DNA would really help such a case anyway. It might! According to this article from Scientific American:

The discovery of this genetic variation gives hope for an obscure but pressing issue in the case of a criminal suspect who is an identical twin. “If one twin is a suspect and the whereabouts of the other twin cannot be determined, then the jury is often left without the ability to find guilt beyond a reasonable doubt” in cases that rely on DNA evidence, says Frederick Bieber, a pathologist at Harvard Medical School.

“If the twin issue comes up in a criminal investigation it’s possible that if there are [copy number variants] that differ between the two twins that might help sort that out,” Bieber says.

Given that there are 80 pairs of identical twins in Virginia’s convicted offender database alone, this might not be as small an issue as it may sound. And such genetic variation also matters to the population at large.

Ha! Not a very original idea for a dream at all, I guess. Subconscious fail.

By S P Hannifin, ago
Problems

Little probability problem

OK, here’s a pretty simple probability problem, but it took me a moment to think about… until my geniusness came up with the solution. The problem is:

You want to choose randomly between 3 choices, but all you have is one coin. What’s the minimum amount of times you need to flip the coin so that you can choose fairly between your 3 choices? (That is, so that each choice has a 33.3…% of being chosen?)

The first person to answer correctly gets the prize of being the first person to answer correctly.

(To complicate the problem, suppose you have two coins, but one has been tainted with iocane powder, and no immunities allowed…)

By S P Hannifin, ago