2nd entry from PROMYS

[sniffnoy]

Well, I've already managed to teach a bunch of people Chrononauts and SoC, but I have yet to teach them Gnostica.

You know, being told that you have to *prove* induction - yes, *prove* induction - is certainly rather annoying. Apparently it's not a basic logical principle, like deduction or proof by contradiction (which has become something of a running joke: "How do you prove that proof by contradiction works? Well, first you assume that it doesn't...").

Well, at least I finally managed to rigorously prove that 0*a=0, if a bit more complexly than I had to.

Specifically, my proof was:

1*0=0
(1+1)*0=1*0+1*0=0
(2*0)*a=0*a
2*(0*a)=0*a+0*a
0*a=0*a+0*a
0*a=0

Then Fan pointed out you could just do

0+0=0
0*a=(0+0)*a
And thus arrive directly at my second-to-last line.

Also, I am now officially a dolphin, and not a swordfish as you may have thought. My name is still Sniffnoy the Swordfish, but I am a dolphin.

Hm, some of the people decided to name this useful and apparently true conjecture that I thought of after me (despite the fact that it almost certainly is already proven and has a name) - specifically, since it's a method, they decided to call it The Altman Maneuver, after the Chinese Poker tactic they've named after me (ie stupidly playing all your 2s early). What the two have to do with each other, I don't know.

Without my spam filters, my bergen.org address is certainly filling up quickly. Thank you, Fan.

On another note, we somehow can't seem to get away from the topic of modular buildings. Not buildings where walking straight brings you back to the same place - you just make a big enough ring and spin it really fast - but one in which moving far enough *vertically* takes you back to the same place. Someone pointed out that this violates conservation of energy - drop a ball down the elevator shaft, and it'll just keep accelerating. We finally settled on making the building a spinning ring, but a constantly accelerating - or rather, increasingly accelerating - one. Whether this can be made to work, I don't know, but it sort of makes sense and at least it gets around the conservation of energy problem. But someone noted that centrifugal force will interfere at high velocities - this is why someone pointed out that it has to be increasingly accelerating - although if it's increasingly accelerating, that doesn't make for a very good sort of gravity, does it? I have no clue. I don't know this sort of thing.

Hm... if we generalize that the noun form of something ending in "ous" is to change the "ous" to an "osity" (ie "generosity", "obviosity", "mysteriosity"), this gives us such words as "dangerosity", "rigorosity", and a whole bunch of other funny ones that I can't be bothered to remember right now.

And finally, to restate the obvious: These problem sets are hard!

Comments

[blenrock.livejournal.com]

You get induction for free or the Well Ordering Priciple for free; not both. Tough cookies.

WOP!

Shush

[jonpin.livejournal.com]

They don't know that yet! Don't tell them about WOP or they'll be able to do yesterday's problem set.

Re: Shush

[blenrock.livejournal.com]

Hmm... how are they expected to prove induction without WOP? It certainly doesn't follow from non-WOP axioms. If it does, I'd be very interested in seeing your proof.

Re: Shush

[jonpin.livejournal.com]

They aren't supposed to unless they come up with the necessity of WOP on their own. Yesterday's PS had "Prove there are no integers between 0 and 1."

Re: Shush

[blenrock.livejournal.com]

That used to be one of my favorites. Back at Ross, I think I wrote a poetry proof of that... hehe. I doubt I could still remember how to do it rigorously. At least, I'd have to think very hard if I wanted to remember. I vaguely recall quite a few inequality symbols and then at one point you have some equation or inequality in terms of some variable x.. and then you multiply both sides by x and end up with an x^2, but you're not allowed to use that sort of "fancy" notation yet.. at least not at Ross. Eek.

Okay, now I remember more, but Harry's journal is probably not the best place to post it. =)

Re: Shush

[blenrock.livejournal.com]

And the fact that they're supposed to be proving stuff from axioms of the integers without knowing what the axioms of the integers are bugs me. I understand the "have them reconstruct all of number theory on their own" philosophy, but I just don't like it.

Re: Shush

[blenrock.livejournal.com]

Okay, this has nothing to do with anything, but I was just thinking that it would be really funny if they didn't get up to beloved quadratic reciprocity this year. I don't know why, but I'd find that highly amusing. Ross without QR would be even more amusing.

I guess the reason I thought of that was because it feels like kinda late in the program for the kiddies not to have been exposed to WOP. *shrug*

[sniffnoy.livejournal.com]

So... I either get induction for free or WOP (whatever that is) for free? I can't get there is no integer between 0 and 1 for free? Eh, for all I know, that *is* WOP, and you're not going to tell me anyway, for obvious reasons.

So what was the point of me writing this?

[blenrock.livejournal.com]

No, no matter what you do, I don't think you can take the "no integers between 0 and 1" (*) to be an axiom.

Anyway, it looks like you're playing with one axiom short of a full deck. So until you get another, don't try to prove * rigorously.

Unlike jonpin, I don't really have any reason to care if I give too much away. Let me know if you want me to spoil the excitement of discovery for you. =)

[forcemajeure.livejournal.com]

Gnostica is one of those games where card-counting skills come in handy. Back before casinos made card-counting significantly less effective, my Dad got in trouble for card-counting at the blackjack tables in Nevada, so I suppose I have a genetic advantage of some sort. Or at least that's how others explain away my winning.

[blenrock.livejournal.com]

Wow, your dad can count cards? I've always wanted to be able to do that, but my brain is never willing to cooperate. I guess I haven't really tried since I was eleven or so. Maybe I'll stick card-counting on my list of things to learn to do this summer.

[forcemajeure.livejournal.com]

Learning how used to be more productive. Now the casinos use large numbers of decks mixed together and reshuffle well before they're through it, so the advantage card-counting confers over random chance is greatly diminished. Though apparently some casinos reshuffle at lower points in the deck than others.

Back in Dad's day, casinos looked manually for the signs of card counting -- mainly that a) you were winning and b) you were winning more toward the end of the deck than toward the beginning -- and simply threw you out. They didn't take back your winnings or anything -- there wasn't exactly a law against it -- but they did have the power to tell you never to darken their door again. Serious card-counters had their photos passed around; Dad never got that far. He just did it for a short time with some of his fellow Caltech math geeks.

[blenrock.livejournal.com]

Wow, your dad was a Caltech math geek? I was thisclose to being one of those. On my plane ride there, I remember thinking that I was a couple hours away from seeing the place I'd be spending the next four years of my life. On the plane ride home, I was thinking that I'd probably end up at Harvard. Don't get me wrong - I think Caltech is a wonderful place... but there was just something about it that made me uncomfortable.

And yeah, if I were to try to learn to do the card-counting thing, it would be more for fun than it would be as an effort to make money.

[forcemajeure.livejournal.com]

My Dad was indeed. Though he had mixed feelings about Caltech, and didn't feel, in hindsight, that it was an entirely healthy place to pursue one's education. I visited Caltech, but decided not to apply there; I felt uncomfortable about it too, though I visited MIT and liked it quite a bit, and came very close to going there.

[blenrock.livejournal.com]

Yes, people say MIT and Caltech are so similar, but I couldn't disagree more. I see the obvious similarities, but the environments at the two schools are entirely different, and I'm definitely not just talking about size and location. It's cool that you applied to one and not the other.

I spent last summer at a research program at MIT. I applied there and all that jazz, but I never really loved it. It's an amazing place though, and I'm sure you would've had a fantastic experiece there... but I definitely see you as more of a Yale person. Actually, sebastiennelite and I would be happier if you'd picked Harvard, but oh well. =)

Blah. Enough college talk.

[sniffnoy.livejournal.com]

Maybe it's just me, but I think it would be a bit hard to count cards when every single trump has a different power. And is there really much point to counting the spot cards? Well, yes, there very definitely is, but still...

[blenrock.livejournal.com]

0*a=0*a+0*a
0*a=0

If you were at Ross, they would've made you do that in waaaay more steps.

*pulls out old problem sets*

Lemma: a(0) = 0
Proof:
0 = 0 (equality is reflexive)
0 + 0 = 0 (additive ID)
a(0 + 0) = a(0) (mult. is well defined)
a(0) + a(0) = a(0) (distribution)
a(0) + a(0) + -a(0) = a(0) + -a(0) (add. is w.d.)
a(0) + (a + -a)(0) = (a + -a)(0) (distribution)
a(0) + (0)(0) = (0)(0) (additive inverses)
a(0) = 0 (additive ID)

ughhh... baaaaad memories.

*puts problem sets away*

I was MUCH happier when we got into week two or three and didn't have to be that ridiculously rigorous and the stuff we were proving became less intuitive and annoying.

[sniffnoy.livejournal.com]

Well, I definitely could have said all that, if they asked for it, except for the stuff about well-definedness. We didn't learn that. That's not just substitution property of equality? That's what I always thought it was, and we don't really have to state that, as we're allowed to implicitly assume the properties of equality.

[blenrock.livejournal.com]

Yes, I'm sure you could've done it that way if you had to. I was just whining about how my crazy summer program made me kill more tress than yours. =)

is velocity then very... velous???

[ashleyisachild.livejournal.com]

increasingly accelerating ("jerking") would make for horrible gravity. gravity really is just acceleration, so if you imagine earth's gravity constantly increasing... well, i think you can imagine what it would be like. we eventually wouldn't even be able to stand up, yadda yadda.

ps. my subject line made me think of jealosity. LOL!
pps. or even zealosity? stupendosity? tremendosity? hehe