(Continued: Morning, Sleepy Head! at ALIL)

FURTHER READING

First World Problems
Inglip Comics
Dictators: How to Immunize Yourself Against Revolution

For example, take The Daily Show with Jon Stewart, or The Colbert Report with Stephen Colbert. People are always linking me to clips from these shows, and I used to consider myself a fan, but nowadays I don’t follow those links unless I’m bored or there’s some compelling reason. Why? Because they haven’t changed in years, and I can almost always predict their punchlines in advance (or if not the exact punchline, the overall idea). This isn’t to say the shows are bad at comedy. They’re still quite funny! But I don’t need to view them. Even if it were imperative that I maintain a solid grasp on those shows (e.g., if I hang out with a lot of young white people), I still would not need to view individual clips.

Again, there are some bloggers whose every scribbling I used to devour, but nowadays I rapidly skim their posts or skip them entirely. Based on nothing but the post title, I can pretty reliably predict what they’re gonna say. In fact, I’ll go further than that: given just the title, if I were forced to, I could write an appropriate article myself. And I must emphasize, I’m still talking about some top-notch bloggers here. It’s just that I’ve reached a point where generic articles no longer offer anything unpredictable. To grow, we must expose ourselves to radically new material.

“The Onion” is another great example. It’s quite fun to browse The Onion just looking at headlines. Once you’ve read a few stories (or maybe even before you’ve read a few stories) you can begin absorbing all the humor through the headlines alone, no longer needing to actually open them and read the full articles. I was an Onion reader way back in the early days, before they became bajillionaires (yeah, yeah, call me a hipster). Back then, most the headlines weren’t even clickable. They didn’t even bother writing stories for them, but those headlines were still the best part of the site.

Any content whose cornerstone is parody, satire, or sarcasm, is bound to become predictable fast (unless it’s of the finest subtlety). My deepest respect goes out to writers who keep on reinventing themselves and reinventing themselves, writing from way out of left field, keeping me on the tips of my toes. And to writers whose work is stuffed full of solid content, whether it be mathematical theorems or cocktail recipes. If there’s one thing I’ve learned from all my blogging, it’s a deep appreciation for all things unpredictable.

FURTHER READING

Blogging vs. Academic Writing
Fun with Pop-Evolutionary Biology
Incentives for Creativity

Devious Dilemmas

Harrowing Headaches

Traumatic Trials

Miserable Messes

Tragic Tragedies

Woeful Worries

FURTHER READING

Philosoraptor Adventures
Inglip Comics
Very Serious People

• h(x,0)=f(x)
• h(x,n+1)=g(x,n,h(x,n)).

In short, h(x,n+1) is defined recursively in terms of h(x,n). It is then understood that this operation (together with composition and basic initial functions) suffices to capture all self-referential functions. Throw in unbounded minimization and you have all computable functions whatsoever. But what about the Fibonacci sequence, h(n+1)=h(n)+h(n-1)? This seems to require a form of recursion which is strictly stronger than the weak version just defined. How does one show that the Fibonacci Sequence is primitive recursive?

One solution is to define an intermediate sequence which we might call the “Fibonacci Table Sequence”. The (n-1)th term of the Fibonacci Table Sequence is intended to encode the list of the first n Fibonacci numbers:

• T(0) = < 1 >
• T(1) = < 1, 1 >
• T(2) = < 1, 1, 2 >
• T(3) = < 1, 1, 2, 3 >
• T(4) = < 1, 1, 2, 3, 5 >
• And so on.

Now, even though individual Fibonacci numbers are defined in terms of two previous Fibonacci numbers, elements of the Fibonacci Table Sequence are only defined in terms of one previous element, namely:

• For n>1, T(n) is obtained by taking the two rightmost entries of T(n-1), adding them together, and appending the result to T(n-1).

The coding of sequences can, of course, be carried out by means of the Fundamental Theorem of Arithmetic; the details are very standard computability theory material.

Here is a double-starred exercise for the reader. Let C(n)=3n+1 if n is odd, and let C(n)=n/2 if n is even; this is the Collatz transformation, of course. Recall the Collatz Conjecture, which states that if you take any initial positive integer and apply C repeatedly, you’ll eventually return to 1. (This conjecture is one of the most profound open problems in contemporary mathematics, btw.) Based on this conjecture, it makes sense to define

• L(n) = the number of times C must be applied to n in order to reach 1.

An alternative, self-referential definition of L is

• L(1)=0
• L(n)=L(C(n))+1.

Exercise: Prove that L is computable, by using a modified version of the “Fibonacci Tables” idea and invoking unbounded minimization just one time. (Note, this does not require assuming the Collatz Conjecture… if the conjecture is true, then L has domain N⁺, and if the conjecture is false, then L has a smaller domain)

FURTHER READING

Collatz Recursion
Church’s Thesis
How I visualize certain subfields of math

The same phenomenon happens with mathematical proofs. If I prove a theorem from ZFC, then, in principle, the proof can be “unwrapped” until it’s nothing but a giant list of steps, each of which is either an axiom of ZFC, a tautology, or an appeal to Modus Ponens. If I prove the theorem in a very dumb, brute force way, or if I use some brilliant strategy to make it more followable, the unwrapped version will still just be axioms, tautologies, and Modus Ponens. The fancy strategy– and any definition which it happens to use for the sake of human readability– only “exists” informally.

If a programmer wanted to troll you, he could implement binary trees without ever mentioning them, just purely manipulating pointers in a way which is equivalent but with no user-friendly explanation (and no names like “root”, “parent”, “left” or “right”). This corresponds to the first paragraph above. If you asked him why his bizarre code works, he could give you an extremely opaque but nevertheless legitimate proof using nothing but axioms, tautologies, and Modus Ponens. This corresponds to the second paragraph above.

Of course, this isn’t really a paradox at all; there’s nothing contradictory about it. It’s just something interesting to think about.

FURTHER READING

Obscure Numbers
The Illogician
Newcomb’s Paradox

The previous linkfest was: Linkfest 12

Technical

Harvey Friedman: Invariant Maximal Cliques and Incompleteness (PDF)

Richard Lipton: An Annoying Open Problem (Group Isomorphism)

John Armstrong: (Pseudo)-Riemannian Metrics, Isometries, Inner Products on 1-Forms, The Hodge Star in Coordinates, The Hodge Star on Differential Forms, Inner Products on Differential Forms

Santo D’Agostino: Dandelin Spheres

Terence Tao: The Jordan-Schur Theorem

John Baez: The Inconsistency of Arithmetic, Chaitin’s Theorem and the Surprise Examination Paradox

Noah Snyder: Subfactors of index less than 5

James Colliander: Edinburgh Arnold Memorial Workshop Notes

David Speyer: Random Partitions

Willie Wong: Gauge invariance, geometrically

Jesse Johnson: The generalized Scharlemann-Tomova conjecture

William Gasarch: Candidate for a new Millenium Problem

Scott Aaronson: In Defense of Kolmogorov Complexity

Emmanuel Kowalski: Families of cusp forms, Action Graphs

Akhil Mathew: Theorem A via general formalism

Lance Fortnow: Mahaney’s Theorem

Non-Technical

Jon Purdy: How to be productive, Tricky programming concepts aren’t

Steven Padnick: Batman the Plutocrat

Charlie Brooker: Nightclubs are hell

George Csicsery: Julia Robinson and Hilbert’s Tenth Problem

Sunanda Creagh: Princeton bans academics from giving copyright to journal publishers

Michael Taylor: Peers, review your actions

Hale: Defense Insider Information, Steve Jobs Dies, Market Jumps 183 points, Stupid Statistics

Peter Gross: Beauty and the Beast’s Dark Delusion

Daniel Batchelder: Ersatz Genuineness in Little Mermaid’s “Under the Sea”

Richard Lipton: The Role of Memory in (Computer Science) Theory

William Gasarch: Where do theorems go to die?

Santo D’Agostino: Using failure as a friendly tool for learning

Steve Pavlina: What is Commitment?

Erin Pavlina: The Benefits of having a Challenge Buddy

Nina Paley: Dear Internet, we need better image archives

Dave Winer: Google needs a Google, Can Larry Reboot Google?, Occupying Facebook, The message of Occupy

Sonic Charmer: Misanthropic Philanthropists

John Robb: Resilient Manufacturing, Occupy Wall Street (the theory), Occupy (Insert Your City Here)

Ilkka Kokkarinen: Breaking the characters, For us to live any other way (than with Basic Income) was nuts

Vaslittlecrow: An open letter and warning from a former tea party movement adherent to the Occupy Wall Street movement (via zhai2nan2)

Jon Purdy: Correctness is a boolean

Nicholas Powiull: Occupy Wall Street Global Revolution

David Graeber: Occupy Wall Street rediscovers the radical imagination

Thirtyseven: #Occupy Itself

JSE: Is Arrow’s Theorem Interesting?

Lance Fortnow: Moneyball, Bibliographies, What is Random?

Tanya Khovanova: Star Trek TNG Science Quiz, David Bernstein’s Paradox

Andrew Gelman: Now you too can buy a selection of garbled Wikipedia articles for a mere $1400-$2800 a year!, That advice not to work so hard

Paul Raven: Robot Lawyers, Human Cashiers

Kareem Carr: The Work-Life Balance

Simon Grey: Overpopulation

Sam Shah: Putting ME first, (On teaching about) Absolute Values

Remy Porter: The New Hire’s A Bust

South Bend Seven: Bring on the driverless car!

Khatzumoto: In defense of kanji tattoo typos

Burt Totaro: What the papers say (about EPSRC)

Mike Fenwick: The Recipe Multiplier

Henry: Stories from the past: The Nintendo Website at Summer Camp

Bill Grazier: Laughing Our Way to the Altar

Ruben Berenguel: Being Zen in the middle of an argument

Giovanni Dannato: Genetic and Memetic Legacies

Paul Hartzog: Unemployment is the Cure

Delusion Damage: Skill-Based Life

Ferdinand Bardamu: They’re called “Public” Utilities for a Reason

Darcy Tacoma: A Crash Course in Wine-Making

Art, Photography, Music, Etc.

Telefunker: Egg Factory D (abandoned Belgian egg factory pics)

Bradley Garrett: Space & Grime: Sapping Chicago’s Skyscrapers

Dusty Goodwin: Hounddaug (Fox and the Hound-Dog Remix), Orders from Mordor (LoTR Remix)

D!tto: Lov.E (Wall-E Remix)

Money Concerns

Junior College easily beats University when it comes to affordable tuition. I actually never paid a dime for any of the courses I took (except for textbooks). In San Diego, I took advantage of a program sponsored by the state of California, which allowed high school students to take community college classes for free. Thanks, California! And when I was in the Air Force, of course the Air Force covered my tuition. Even without any of that assistance, junior college remains a lot cheaper than its more prestigious sisters. Compare prices in your area and you might be surprised at the magnitude of the difference.

Cutting Edge Research

The drawback to C.C. is that all the big names in current research tend to be at universities. In fact, most JCs don’t even require a PhD as a job requirement: your professors might only have master’s degrees. The thing is, that doesn’t really matter so much in your first two years. In junior college you take basic courses, things which are classified as freshman or sophomore in university terms. And at that level, young university students don’t gain a lot from cutting-edge research. Of course, there are exceptions: most universities offer students a chance to do research under the supervision of the big names; however, only a miniscule fraction of university students actually take advantage of such programs (and, in some fields, that “research” might amount to doing brute labor in a lab). All in all, the elite research done by a university doesn’t actually add much to the first two years’ education.

Quality of Courses

Paradoxically, courses at junior colleges are often better than their university equivalents. One of the reasons for this is the class size. It’s rare for a junior college to have an auditorium class with hundreds of students. In university, that pretty much describes all the core first-year classes! You can easily do your first two years at a 4-year institution without ever speaking directly with a professor.

Besides course size, what about academic rigor? Here I can only offer anecdotes. But when I took the first-year calculus sequence at Mira Mesa Community College, it was far more rigorous than the equivalent sequence at most universities. In Calculus 2, we rigorously studied Riemann integrals, calculating limits as norms of arbitrary partitions went to zero. In Calculus 3 (one of the biggest struggles I’ve ever had in a math course), we hit things like Green’s Theorem, Div, Curl, and Grad– things that are vanishing from university curricula nowadays, they’re too advanced. When I took Differential Equations at Pima Community College (dressed, half the time, in battle dress uniform), we learned Laplace Transforms: a third-year or higher topic according to many universities.

Flexibility

University students are expected to do university as if it’s a full-time job, and then some. As such, the schools will not flinch to schedule a crucial class at 8 or 9 in the morning, five days a week. Universities are getting better about offering evening and Summer classes, but it’s still very difficult to do Uni while working a regular nine-to-five. Community colleges, on the other hand, go out of their way to be flexible, offering all kinds of crazy accommodations. I needed a Spanish class and the Weather Squadron wanted me working pretty much all the time: I was able to take the class on the weekends, in massive blocks on Saturdays and Sundays.

Social Life

Where universities really shine is the Social Experience. University is a way of life. You live on campus (literally!) and eat, breathe, and date campus. Especially for those crucial first two years. This is the one thing I feel deprived of. I never got the “dorm experience”. Community College is much different: people come to CC from all walks of life, and when class is over, they go right back to those lives. There is no student life in Junior College; University is little else but.

(I had a very different kind of dorm experience: it involved more angry drill sergeants screaming in my face, less house parties and cute girls.)

The Verdict

Based on all the above, my verdict is as follows.

• If you want the Student Experience…
  • Go to university, no questions asked. Live in the dorms, whether it’s required or not. If you’re desperate to make friends, try for a dorm with a roommate– or three! If this is the reason you are going to university, then you should really put in the effort. Party as much as you can. Try out for a fraternity or sorority if you can afford it.
• If you want to devote your life to academics, starting NOW…
  • Go to university, and aggressively take advantage of what you’re paying for. Visit professors’ office hours, not just to talk about homework, but to talk about their research. Ignore the social experience, except where it’s relevant to academics. Ask your advisor about opportunities to do supervised research. Ask around and get somebody to help you write an article for an undergraduate research journal (such as the Rose-Hulman Undergraduate Mathematics Journal). Download cutting-edge journal articles from the Library website (believe it or not, this under-used capability is what a huge portion of your tuition is paying for). Attend talks and conferences, even if they’re totally over your head (I call it Learning by Osmosis).
• Any other case…
  • Go to Junior College. You’ll save money, you’ll get smaller courses which might even be more rigorous than the university’s, you’ll have more flexible options. But before you commit, do your homework: does this college’s units transfer to the universities you want to transfer to? Of course, this is assuming you plan to transfer to a university and do a full BA or BS.

Of course, there are other options, too. You could start your own business…

FURTHER READING

My Time at the University of Arizona
My Time in Air Force Boot Camp
Adulthood Phases
Questions on Work

• Theorem: There is no obscure counting number.
• Proof Assume there is an obscure counting number. Then there is a smallest one. But the fact that it’s the smallest obscure number, is pretty non-obscure! Contradiction.

One way to try to find obscure numbers is to take big databases of numbers and find the smallest missing number. For example, Nathaniel Johnston found that 11630 was the smallest number missing from the Online Encyclopedia of Integer Sequences. However, 11630 later showed up (it happens to be a “5 times centered pentagonal number”) and Nathaniel’s Number was revised to 12067. That, too, soon made an appearance (something to do with generating sequences), and Johnston’s Constant swelled up to 12407. And now 12407 has appeared in the encyclopedia (it is the numerator of Hermite(3,19/26), whatever that means). Johnston apparently gave up, and hasn’t updated his blog post since.

I thought to myself: what’s an even bigger database of numbers than the OEIS? The answer screams itself out: Google. Unfortunately, unlike the OEIS, you can’t just download an exhaustive list of all the strings Google has indexed. The only way to find the smallest obscure number according to Google would be to enter all the numbers in until you finally got a results page with no results. Unfortunately, such a brute force attack would take forever (and you’d probably be banned from Google before you reached your conclusion).

Still, you can use Google to check whether a given number is obscure or not. For example, I randomly chose the number 553634475998 by mashing my keyboard, and sure enough, Google found no matches, confirming the number’s obscurity– at least, until I publish this blog post, at which point its obscurity will vanish!

Still, the Paradox of the Noninteresting Number relies somehow on an omniscient overseer deciding what’s interesting, what’s obscure: in the paradoxical argument, the fact that N is the smallest obscure number, is supposed to somehow make it non-obscure, and indeed, it WOULD, if we could actually detect what this number N is.

There really is (say) a smallest number N such that no human will ever write down N– assuming the human race eventually dies out in a finite amount of time. No human can predict what this number is, though, without breaking it. It is a designator which doesn’t concretely designate anything, at least not to us: it could designate something concrete to a more powerful being who outlives us all.

FURTHER READING

Philosoraptor Adventures
The Morning Star Paradox
Newcomb’s Paradox

Meanwhile, the blogosphere continued to churn out tons of great scholarship, some of which I’ve picked to share with you. Hope you enjoy these links. The previous linkfest was Linkfest 11.

Technical

Terence Tao: The inverse function theorem for everywhere differentiable maps, The Brunn-Minkowski inequality for nilpotent groups

Richard Lipton: Can Quantum Machines Do It All?

John Armstrong: Compatible Orientations, Orientable Atlases, Orientation-Preserving Mappings, Integrals over Manifolds, Integrals and Diffeomorphisms, The tangent space at the boundary, Oriented manifolds with boundary

Timmo Prisk: God and the boulder he can’t lift (or: an amusing reason to insist that worlds in Kripke structures should share a common background language)

Dave Pritchard: Virtual Valuations

JSE: Homology of the Torelli group and negative-dimensional vector spaces

Akhil Mathew: Sudoku and zero-knowledge proofs

Emmanuel Kowalski: What’s special with commutators in the Weyl group of C5?

Hilbertthm90: Deformations of p-divisible groups

Willie Wong: Extensions of (co)vector fields to tangent bundles

Non-Technical

Douglas Rushkoff: Are jobs obsolete?

Mark Lee: “Contagion” and the Rule of Law

John Baez: U.S. Weather Disasters in 2011, Fools Gold (and Platonic Solids), The Malay Archipelago

Gavin Andresen: Defense is a public good. So spend less on it.

Ramaswamy Sudarshan: The modern trend to monetize everything ignores civilizational wisdom

South Bend Seven: Eight Netflix Thoughts

David Brin: Competitive Enterprise or Idolatry of Property, Southern California Goes Dark

William Pitt: The (Tea Party) Cult of Death

Ilkka Kokkarinen: Like a boss (commentary on rage comics)

Alex Papadimoulis: The Non-Deleting Delete, The Cooling Kludge

William Gasarch: Do we use technology before it’s perfected? Should we?, Conventions in Math: just to make rules work or more?

Ulysses: Post-(Republican)-Debate Thoughts

Sonic Charmer: My Most Unlibertarian Position, My Economic Prescription: Maneuver X

Giovanni Dannato: Mindspace, Does Perception Create Linear Time?

Victor Gijsbers: What is the first secondary world?

Andrew Wegman: Some thoughts on academic cheating inspired by Frey, Wegman, Fischer, Hauser, Stapel

Lance Fortnow: The Anti-Privacy Generation, Imagine (what the future will be like with self-driving cars)

Jason Dyer: Getting math problems wrong for cognitive science reasons

Ross Dawson: The implications of the new broader, flatter distribution of music taste

Samir Khuller: On Conference Locations

John Weitzmann: Extension of Copyright Term for Sound Recordings in the EU

Jeffrey Shallit: Robots as Companions

Peter Hicks: Freeing Train Data

Andrés Caicedo: Alan Turing year

Sepp Hasslberger: Fukushima nuclear disaster still active — media is silent

Alexandre Borovik: Vorderman’s Report

George Monbiot: Academic publishers make Murdoch look like a socialist

Remy Porter: Get to the COPPA!, Common Educational Oriented Language

Luke Wolcott: The virtue of verbosity

Tae Kim: Saying more (in Japanese) than “this” and “that”

Dagonet: After the Storm (East Coast Flood)

Pankaj Mishra: The dead end of globalisation looms before our youth

Santo D’Agostino: On the nature of scientific theories

Matthew Belinkie: The Video Game Plot Scale

Suresh Venkat: A way forward on reformatting conferences

Simon Grey: On Patent Reform

Ben Casnocha: Hard-to-define jobs are more secure

Art and Photography

Telefunker: Abandoned Belgian hospital

Default User: Psychotic Architecture

Ken Baker: Hairy circle of spheres

Bradley Garrett: Detroit: Beyond Ruination

Eran Amir: 1,500 Photographs, 500 People, 100 Seconds

[Editor's Note: Philosoraptor is, of course, one of those funny internet memes. The image which inspired it was originally drawn by Sam Smith.]

The Early Years

Our dinosaur began his training extremely early in life. In grade school, he befriended every class clown and wise-ass and begged them to teach him their rebellious knowledge. The names of these early masters are lost to the mists of time but their influence lives on in some of our scholar’s most well-known verses.

The Mad Scientist Years

Immediately after graduating high school, Philosoraptor went to university where he majored in Science. But his adviser turned out to be of the Mad Scientist variety, and our hero would never complete the degree. Nevertheless, analysts have little doubt the crazy adviser left an influence on the future philosopher.

Studying under Tyler Durden

Philosoraptor went through a few very strange years of which we don’t know very much. According to his official biographer: “During our interviews, he couldn’t say much about that time, due to ongoing investigations and a court gag order.” What we do know is that he was a Space Monkey studying under Tyler Durden, and this is reflected in some of his later prose.

Shock-produced fatal cranial expansion

While taking part in Project Mayhem, our raptor was a SPFCE trooper (see the section title above). That’s a long way of saying he specialized in making peoples’ heads explode.

Philosoraptor Goes East

After getting his life sorted out and getting out of Tyler’s army, our protagonist decided to head east and study under the Buddhist masters. He was drilled in Zen and other eastern mysticism. Many historians speculate that it was around this time that he made the decision to devote his life to philosophy.

Heresy

Through his eastern mystic connections, our intrepid velociraptor ended up making friends in the International Brotherhood of Heretics. This would inform large swathes of his teachings on religious matters.

Philosoraptor’s Work on Contradictions

As soon as he returned to the west, he began a rigorous study of the contradictions and inconsistencies in folk wisdom. This was highly controversial: at the time, all the major journals were completely dominated by folk wisdom, and many an esteemed doctor’s reputation was ruined by this rogue philosopher pointing out their errors.

The Pundamentals of Wordplay

Of course, no account of Philosoraptor’s works would be complete without his primary contribution: The Pundamentals. Published posthumously after the painstaking assembly of thousands of stray papers sewn into the philosopher’s coat, this is the lone book which actually bears his name. Over thirty different universities awarded posthumous honorary Pun.h.D’s for the landmark work, which forever changed how mankind views wordplay.

Illogic

In the later years of his life, Philosoraptor began to show an interest in formal illogic. It’s too bad he didn’t live to do more significant work here. Friends say he was very fond of the illogical work of Kurt Gödel.

Final Words from the Deathbed…

FURTHER READING

Troll Dad
Advice Tyler Durden
The Illogician
Inglip Comics