Why did Set Theory start with transfinite numbers?

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I was visiting my mom, and discovered that I didn’t leave my set theory book on the train; I left it at her house. So I’ve been happily reunited with my old text, and I’m going to get back to a few more posts about the beautiful world of set theory.

When you talk about set theory, you’re talking about an extremely abstract notion, one which is capable of representing all sorts of structures: topological spaces, categories, geometries, graphs, functions, relations, and more. And yet, almost every description of set theory plunges straight into the cardinal and ordinal numbers. Why? That’s a question that mystified me for quite a long time. Why do we take this beautiful structure, which can do so many things, and immediately jump in to these odd things about infinities?

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Friday Recipe: Catfish in Dashi Sauce

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This dish is one of my own creations. It’s inspired by reading
Ming Tsai’s cookbooks, and seeing how he combines things. But as far as I
know, he doesn’t do anything like this.

You really need catfish for this. I’ve tried it with other fish, but
it just doesn’t work as well. Catfish has a unique flavor and texture which
is particularly well-suited to this.

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Pathetic Innumeracy – this time from Great Britain

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My fellow SBer Craig McClain sent me a link to yet another an example of how mind-bogglingly innumerate people are. At least, for once it’s not Americans.

The British lottery put out a “scratch-off” game called “Cool Cash”. The idea of
it is that it’s got a target temperature on the card, and to win, you need uncover
only temperatures colder than the target. Simple, right?

Since Britain is on the metric system, they measure temperatures in Celsius. So naturally, some of the temperatures end up being below zero. And that’s where the trouble came in. So many
people didn’t know that below zero, larger numbers are lower and thus colder, that
the lottery had to withdraw the game!

To quote one of the “victims”:

On one of my cards it said I had to find temperatures lower than -8. The numbers I uncovered were -6 and -7 so I thought I had won, and so did the woman in the shop. But when she scanned the card the machine said I hadn’t.

I phoned Camelot and they fobbed me off with some story that -6 is higher – not lower – than -8 but I’m not having it.

I love that “I’m not having it” line. That’s a classic.

What I find particularly surprising is that this isn’t just math – it’s just a basic, minimal awareness of your surroundings. We’re talking
about adults here – people who’ve clearly lived through plenty of winters, where the temperature
in Great Britain routinely drops below zero degrees celsius. That means that these people don’t know that when it’s -10, it’s colder than when it’s -2! To me, this seems to be on about the
same intellectual level as trying to eat wax fruit, because you don’t know he difference between
it and real fruit.

Musical Goofiness with a Mathy-Bent

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Ladies and gentlemen.. For your pleasure and edification, allow me to present… The singing Tesla coils!

Yes, if you’re clever, and you’re willing to do a whole lot of work, you can operate a Tesla coil so that the sparking from the coil produces a particular pitch. Even you’re even more clever, you can vary the way that the coil is run to produce different pitches, and arrange it into a song. And if you’re really remarkably clever, you can set up two singing Tesla coils, and have them play a duet.

Revisiting Old Friends, the Finale

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Now, it’s time for the final chapter in my “visits with old friends” series, which brings us
back to the Good Math/Bad Math all-time reader favorite crackpot: Mr. George Shollenberger.

Last time I mentioned George, a number of readers commented on the fact that it’s cruel to pick on poor George, because the guy is clearly not all there: he’s suffered from a number of medical problems which can cause impaired reasoning, etc. I don’t like to be pointlessly cruel, and in general, I think it’s inappropriate to be harsh with someone who is suffering from medical problems – particularly medical problems that affect the functioning of the mind.

But I don’t cut George any slack. None at all. Because much of what spews from his mouth isn’t the
result of an impaired mind: it’s the product of an arrogant, vile, awful person. Since our last contact
with George, aside from the humorous idiocy, he’s also taken it upon himself to explain how we’ll never
have a peaceful society in America until we get rid of all of those damned foreigners, who have
“unamerican mindsets”. That post was where I really started to despise George. He’s not just a senile
old fool – he’s a disgusting, horrible person, just another of the evil ghouls who used a horrible
event, committed by a severely ill individual, as a cudgel to promote a deeply racist agenda.

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Mathematical Constructions and the Abstraction Barrier

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There was an interesting discussion about mathematical constructions in the comment thread on my post about the professor who doesn’t like infinity, and I thought it was worth turning it into a post of its own.

In the history of this blog, I’ve often talked about the idea of “building mathematics”. I’ve shown several constructions – most often using something based on Peano arithmetic – but I’ve never really gone into great detail about what it means, and how it works.

I’ve also often said that the underlying theory of most modern math is built using set theory. But what does that really mean? That’s the important question, and the subject of this post.

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Friday Random Recipe: Mac and Cheese!

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One of my favorite comfort foods is a mac&cheese tuna casserole. That’s real mac&cheese, not any of that glow in the dark orange garbage. It amazes me just how many people have never actually had a proper, home-made macaroni and cheese! It’s really good eating (unlike the glow in the dark stuff). The most important thing for this is to use good cheddar cheese. The pre-shredded stuff is tasteless – you need a good brick of some kind of high quality, aged cheddar cheese. Of the widely available stuff, my favorite is Cabot extra-sharp.

Ingredients

  • 1 pound of macaroni, cooked according to instructions an the box, then cooled.
  • 3/4 pound shredded cheddar cheese.
  • 2 cups milk.
  • 8 tablespoons butter.
  • 1/2 pound crimini mushrooms, sliced.
  • 1 teaspoon dijon mustard.
  • 1 can of good quality tuna, preferably packed in olive oil, drained
    and crumbled.
  • 1/2 cup bread crumbs.
  • 2 tablespoons of flour.
  • 1 head broccoli, steamed and cut into small pieces.

Instructions

  1. In 2 tbs of butter, brown the sliced mushrooms. Add salt to taste.
  2. In another 2 tbs butter, toast the bread-crumbs until they’re nicely browned.
  3. Put the cooked macaroni, broccoli, mushrooms, and tuna into a large mixing bowl, and mix
    them together.
  4. In a saucepan, melt the remaining 4 tbs butter, and then add the flour. Whisk together
    to form a roux, and cook on medium low heat until it’s about the color of coffee with milk.
  5. Add the milk, and whisk vigourously to dissolve the roux into the milk. Increase the heat
    to medium, and stir until it reaches a boil and thickens.
  6. Add the mustard and 2/3rds of the cheese, and stir until the cheese is melted and the sauce is thick. Add salt and pepper to taste (remember that the salt in the sauce is the only salt for the macaroni.)
  7. Pour the sauce over the pasta mixture in the bowl, and mix it through.
  8. Dump the resulting mixture into a large casserole dish. Top with the toasted bread crumbs
    and the remaining cheese.
  9. Cover with foil, and bake at 350 for about 30 minutes; then remove the foil and let it bake
    for another 20 minutes.

It’s a humble dish, but it’s really awfully good. If you’re adventurous, you can also do all sorts of variations – this dish is great for playing with. For example, replacing the mustard with curry powder
is terrific!

Computing Strongly Connected Components

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As promised, today I’m going to talk about how to compute the strongly connected components
of a directed graph. I’m going to go through one method, called Kosaraju’s algorithm, which is
the easiest to understand. It’s possible to do better that Kosaraju’s by a factor of 2, using
an algorithm called Tarjan’s algorithm, but Tarjan’s is really just a variation on the theme
of Kosaraju’s.

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Friday Not-So-Random 10

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This past week, I discovered a new digital music download site, called Bitmunk. It’s less expensive
than iTunes or Amazon, and has a fantastic selection of obscure bands. Through Bitmunk, I found
a couple of terrific new neo-progressive bands, which has me on a serious prog kick. So for today, I’ve
narrowed the domain of the randomization to just the progressive stuff, and I also cheated a bit to make
sure that the two best of the new bands I found are included in the list.

Just to be clear, I’ve got no connection with Bitmunk, they’re not giving me anything to mention them, etc. I found them by way of a comment in the last FRT here. Someone pointed me at the Bayprog website, which I followed to find a link to Metaphor’s website; after listening to a sample there, I decided to buy their album, and they linked to Bitmunk for digital purchases.

  1. The Mars Volta, “Inertiatic ESP”. The Mars Volta is a recent discovery for me, but not
    via the new site. They’re a sort of hyperkinetic neo-progressive group. The best I can do at describing them is to say that they sound like a cross between King Crimson and Dream Theatre, hopped up on too
    much caffeine. They’re very good – wonderful when I’m in the right mood, but they’re not the easiest
    listen. There’s so much going on, so many fast twists and shifts that it’s easy to get lost. This is a very typical one of their tracks. Weird rhythmic shifts, incredible density. Very cool stuff.
  2. The Flower Kings, “Pioneers of Aviation”. I love the Flower Kings. They are, in my opinion,
    the very best of the neo-progressive bands. Their music is brilliantly written – deep, complex, but still
    melodic, and they’ve got the chops to really pull it off. This is one of my favorite instrumental tracks
    off of their second most recent album. There’s just no way I can say enough about how great the FKs
    are.
  3. Elegant Simplicity, “Time to Breath”. This is one of the two great bands that I discovered
    through Bitmunk. This is the opening track off of their album “The Architect of Light”. I think it’s
    a good introduction to them. The opening is wonderfully strange; a capella voice singing the melody
    that will become the main theme, placed over a strange King Crimsonesque background of tape loop
    and selected noise – with the vocals in a different key than the background, creating a dissonance
    out of what will turn out to be a very smooth melodic theme. They’re clearly very influenced by
    the Flower Kings – they’ve got a very FKish sound; but not derivative, just clearly influenced.
    Very good stuff, I highly recommend it.
  4. Marillion, “Ocean Cloud”. You can’t talk about neo-prog rock without mentioning Marillion. During the dark days of the 80s, they were one of the only bands keeping the progressive flame
    alive. This track is an 18 minute opus off of the “Marbles” double-album, and it’s a great example
    of what I think makes Marillion so great. What they’ve always been best at, to me, is transitions: the best moments in their music are always in the points of change, where they’re shifting between themes or moods. “Ocean Cloud” really shows this off, as it shifts back and forth between gentle, almost lullaby-like delicacy, and roaring intensity.
  5. King Crimson, “FraKctured”. You can’t talk about any kind of progressive rock without mentioning King Crimson. In my opinion they’re just the best progressive group ever, period. They’ve
    gone through many incarnations over the years, from the days when they started off as “The Cheerful Insanity of Giles, Giles, and Fripp”; to the original King Crimson; to the Red era, to the quartet
    with Fripp, Belew, Bruford, and Levin; to the fractalized ProjeCKts; to the current quartet group. Each
    era has been different, all have been amazing.
  6. Spock’s Beard, “Sometimes They Stay, Sometimes They Go”. One of the first really great
    bands in the current wave of neo-progressive. A good track off of their latest album.
  7. Metaphor, “Wheel of the World”. The other of my Bitmunk discoveries. This is from a
    great album, “Entertaining Thanatos”, which the group describes as “Seven lighthearted songs about death”. Metaphor isn’t quite up there with “Elegant Simplicity”, but they’re very good. They’ve got
    some pretty clear influences: there’s a strong sense of King Crimson and the Flower Kings about
    their style. But there’s also some very distinctive and unique stuff. Very good, definitely worth
    listening to.
  8. King Crimson, “Requiem”. More King Crimson. You can never go wrong with more Crimson!
    This is from the second album with Adrian Belew on vocals, but the track is dominated by Fripp’s
    unique guitar.
  9. Porcupine Tree, “The Sky Moves Sideways, Phase 2”. The only neo-progressive group with
    a chance of competing with the Flower Kings for the title of “Best Neo-prog”. I don’t think that
    they quite manage to beat out the Kings, but they come closer than anyone else. This is from their
    most “out there” album. A must listen album.
  10. Pink Floyd, “Astronomy Domine”. And we finish off with something very much not
    Neo. A track from Pink Floyd’s debut album in the 1960s. The version that I’m listening to is the
    1969 live performance from Ummagumma. This version just gives me chills. A mediocre quality recording that’s nearly 40 years old, and it manages to not sound dated at all.

Tag-Teaming with Orac: Bad, Bad Breast Cancer Math in JPANDS

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My friend, fellow ScienceBlogger, and BlogFather Orac asked me to take a look at a paper that purportedly shows that abortion is a
causative risk factor for breast cancer, which he posted about
this morning. When the person who motivated me to start what’s turned out to be a shockingly
successful blog asks for something, how could I possibly say no? Especially when it’s such a great example
of the misuse of mathematics for political purposes?

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