Stupid Patent Of The Month: A Patent On Using Mathematical Proofs

from the break-out-your-wff-'n-proof dept

In some fields, software bugs are more than the proverbial pain in the neck. When software has to ensure that an airplane lands safely, or that a pacemaker keeps operating, there's no room for error.

The idea that mathematical proofs could be used to prove that software is error-free has been around since the 1970s, and is known as "formal verification." But like a lot of technologies that some visionaries saw coming, it took time to develop. In recent years, computing power has become cheap enough for formal verification to become practical for more software applications.

Unfortunately, last month, the field had a monkey wrench thrown into it, in the form of U.S. Patent No. 10,109,010, which the patent office awarded to a U.K.-based company called Aesthetic Integration Ltd.

Claim 1 of the patent describes creating mathematical "axioms"—formal mathematical statements—that describe a computerized trading forum. The patented method then describes analyzing, with a "computer assessment system … the mathematical axioms that describe the operation of the trading forum." In other words, the patent describes using formal proofs to check for bugs in a "computerized trading forum." It's formal verification—just applied to the financial services industry.

Of course, Aesthetic Integration didn't invent formal verification, nor did the company invent the idea of software powering a "trading forum." The company has apparently created software that utilizes formal verification in the financial services space, and that software might be perfectly good. But the Patent Office has effectively allowed the company to patent a whole sector of formal verification.

To be fair, the '010 patent appears to reflect some advanced and difficult programming by Aesthetic Integration. But that does not mean it should be patentable. Consider the following analogy: there are no 50 story buildings in Cincinnati. Building a 50 story building in Cincinnati, and making it compliant with seismic safety standards, would be hard work. It would take many engineers a great deal of effort to apply existing techniques to complete the project. You could write a lengthy paper describing that process, which might include lots of complex charts and diagrams. Still, that does not mean a company that completes such a project should then get a monopoly on tall buildings in Cincinnati.

Aesthetic Integration claims to be the first to apply formal verification to trading software. If that's true, the company may get a well-earned competitive advantage by being the first to this market. But it should not get a 20 year monopoly simply for applying programming techniques that the company itself has described as part of a "mature and effective field of science."

Ultimately, the '010 patent reflects a broader problem with Patent Office's failure to apply a meaningful obviousness standard to software patent applications. We have explained before that the Patent Office is all too willing to hand out patents for using known techniques in a particular field. Flow charts and whirligigs can make a concept look new when it isn't—especially when a patent owner fills its application with obscure language and "patentese." The Federal Circuit has also encouraged this through its hyper-formalistic approach to obviousness. The end result is an arms race where people rush to patent routine software development.

As we've said before, patents are simply a bad fit for software. The Patent Office should stop giving out patents on formal verification, or other well-known software processes.

Republished from the EFF's Stupid Patent of the Month series.

Filed Under: math, patents, proofs, stupid patent of the month

USPTO Suggests That AI Algorithms Are Patentable, Leading To A Whole Host Of IP And Ethics Questions

from the locking-up-life dept

The world is slowly but surely marching towards newer and better forms of artificial intelligence, with some of the world's most prominent technology companies and governments heavily investing in it. While limited or specialist AI is the current focus of many of these companies, building what is essentially single-trick intelligent systems to address limited problems and tasks, the real prize at the end of this rainbow is an artificial general intelligence. When an AGI could be achieved is still squarely up in the air, but many believe this to be a question of when, not if, such an intelligence is created. Surrounding that are questions of ethics that largely center on whether an AGI would be truly sentient and conscious, and what that would imply about our obligations to such a mechanical being.

Strangely, patent law is being forcibly injected into this ethical equation, as the USPTO has come out in favor of the algorithms governing AI and AGI being patentable.

Andrei Iancu, director of the U.S. Patent and Trademark Office (USPTO), says that the courts have strayed on the issue of patent eligibility, including signaling he thought algorithms using artificial intelligence were patentable as a general proposition.

That came in a USPTO oversight hearing Wednesday (April 18) before a generally supportive Senate Judiciary Committee panel.

Both Iancu and the legislators were in agreement that more clarity was needed in the area of computer-related patents, and that PTO needed to provide more precedential opinions when issuing patents so it was not trying to reinvent the wheel each time and to better guide courts.

On some level, even without considering the kind of AI and AGI once thought the stuff of science fiction, the general question of patenting algorithms is absurd. Algorithms, after all, are essentially a manipulated form of math, far different from true technological expression or physical invention. They are a way to make equations for various functions, including, potentially, equations that would both govern AI and allow AI to learn and evolve in a way not so governed. However ingenious they might be, they are most certainly no more invention than would be the process human cells use to pass along DNA yet discovered by human beings. It's far more discovery than invention, if it's invention at all. Man is now trying to organize mathematics in such a way so as to create intelligence, but it is not inventing that math.

Yet both the USPTO and some in government seem to discard this question for arguments based on mere economic practicality.

Sen. Kamala Harris drilled down on those Supreme Court patent eligibility decisions -- Aliceand Mayo, among them -- in which the court suggested algorithms used in artificial intelligence (AI) might be patentable. She suggested that such a finding would provide incentive for inventors to pursue the kind of AI applications being used in important medical research.

Iancu said that generally speaking, algorithms were human made and the result of human ingenuity rather than the mathematical representations of the discoveries of laws of nature -- E=MC2 for example -- which were not patentable. Algorithms are not set from time immemorial or "absolutes," he said. "They depend on human choices, which he said differs from E=MC2 or the Pythagorean theorem, or from a "pattern" being discovered in nature.

Again, this seems to be a misunderstanding of what an algorithm is. The organization and ordering of a series of math equations is not human invention. It is most certainly human ingenuity, but so was the understanding of the Bernouli Principle, which didn't likewise result in a patent on the math that makes airplanes fly. Allowing companies and researchers to lock up the mathematical concepts for artificial intelligence, whatever the expected incentivizing benefits, is pretty clearly beyond the original purpose and scope of patent law.

But let's say the USPTO and other governments ignore that argument. Keep in mind that algorithms that govern the behavior of AI are mirrors of the intelligent processes occurring in human brains. They are that which will make up the "I" for an AI, essentially making it what it is. Once we reach the level of AGI, its reasonable to consider those algorithms to be the equivalent of the brain function and, by some arguments, consciousness of a mechanical or digital being. Were the USPTO to have its way, that consciousness would be patentable. For those that believe we might one day be the creators of some form of digital life or consciousness, that entire concept is absurd, or at least terribly unethical.

Such cavalier conversations about patenting the math behind potentially true AGI probably require far more thought than asserting they are generally patentable.

Filed Under: ai, algorithms, alice, math, patents, uspto

Stories Claiming DNC Hack Was 'Inside Job' Rely Heavily On A Stupid Conversion Error No 'Forensic Expert' Would Make

from the don't-trust-anonymous-sources-unless-you-agree-with-them dept

While we wait for the Mueller investigation to clearly illustrate if and how Russia meddled in the last election, there's no shortage of opinions regarding how deep this particular rabbit hole goes. While it's pretty obvious that Putin used social media and media propaganda to pour some napalm on our existing bonfires of dysfunction, just how much of an impact these efforts had on the election won't be clear until a full postmortem is done. Similarly, while Russian hackers certainly had fun probing our voting systems and may have hacked both political parties, clearly proving state involvement is something else entirely.

Quite fairly, many folks have pushed for caution in terms of waiting for hard evidence to emerge, highlighting the danger in trusting leaks from an intelligence sector with a dismal track record of integrity and honesty. There's also the obvious concern of ramping up tension escalation between two nuclear powers. But last week, many of those same individuals were quick to highlight several new stories that claimed to "completely debunk" Russia's involvement in hacking the DNC ahead of last year's election. The problem? These reports were about as flimsy -- if not flimsier -- than the Russian hacking theories they supposedly supplanted.

In fact, these reports took things one step further by claiming that the hack of the DNC was something committed solely by someone within the DNC itself. This particularly overlong, meandering piece by The Nation, for example, claimed to cite numerous anonymous intelligence sources who have supposedly grown increasingly skeptical over the "Russian hacking narrative." Quite correctly, the report starts out by noting that while there's oodles and oodles of smoke regarding Putin's involvement in the election hacks, the fire (hard evidence) has been hard to come by so far:

"Lost in a year that often appeared to veer into our peculiarly American kind of hysteria is the absence of any credible evidence of what happened last year and who was responsible for it. It is tiresome to note, but none has been made available. Instead, we are urged to accept the word of institutions and senior officials with long records of deception. These officials profess “high confidence” in their “assessment” as to what happened in the spring and summer of last year—this standing as their authoritative judgment.

But it's then that's where things get a little weird. The report repeatedly proclaims that a laundry list of anonymous "forensic investigators, intelligence analysts, system designers, program architects, and computer scientists of long experience and strongly credentialed" have been hard at work "producing evidence disproving the official version of key events last year." But one of the key conclusions by these experts -- and a key cornerstone for of all of these stories -- makes absolutely no sense.

The reports lean heavily on anonymous cybersecurity experts calling themselves "Forensicator" and "Adam Carter," who purportedly took a closer look at the metadata attached to the stolen files. Said metadata, we're breathlessly informed, indisputably proves that the data had to have been transferred from inside of the DNC network and not over the internet, since the internet isn't supposedly capable of such transfer speeds:

"Forensicator’s first decisive findings, made public in the paper dated July 9, concerned the volume of the supposedly hacked material and what is called the transfer rate—the time a remote hack would require. The metadata established several facts in this regard with granular precision: On the evening of July 5, 2016, 1,976 megabytes of data were downloaded from the DNC’s server. The operation took 87 seconds. This yields a transfer rate of 22.7 megabytes per second.

These statistics are matters of record and essential to disproving the hack theory. No Internet service provider, such as a hacker would have had to use in mid-2016, was capable of downloading data at this speed. Compounding this contradiction, Guccifer claimed to have run his hack from Romania, which, for numerous reasons technically called delivery overheads, would slow down the speed of a hack even further from maximum achievable speeds."

That reads like a semi-cogent paragraph, but it's largely nonsense. 22.7 megabytes per second (MB/s) sounds impossibly fast if you don't know any better. But if you do the simple conversion from megabytes per second to megabits per second necessary to determine the actual speed of the connection used, you get a fairly reasonable 180 megabits per second (Mbps). While the report proclaims that "no internet service provider" can provide such speeds, ISPs around the world routinely offer speeds far, far faster -- from 500 Mbps to even 1 Gbps.

And despite the report oddly pooh pooh'ing Romanian broadband's "delivery overheads," many Romanian cities actually have faster internet connectivity than either Russia or in the States (check out Akamai's global broadband rankings). Bernie Sanders learned this last year when he unintentionally pissed off many Romanians when trying to highlight the dismal state of U.S. connectivity. Even then, the hacker in question could have used any number of tricks to hide his or her location and real identity from a high-bandwidth vantage point, so the claim that the hacker couldn't achieve 180 Mbps through a VPN is simply nonsense.

Obviously this raises some questions about what kind of cyber-sleuths we're talking about when they can't do basic conversions or look at some fairly obvious broadband speed availability charts. And it also raises some questions about why reporters thought flimsy anonymous experts were the perfect remedy to the other flimsy anonymous leaks they hoped to debunk. While The Nation couldn't even be bothered to do the simple calculation to determine the speed of the connection used by the hacker was relatively ordinary, in a story titled "Why Some U.S. Ex-Spies Don't Buy the Russia Story," Bloomberg actually did the conversion to get the 180 Mbps speed, and still somehow told readers that such speeds were impossible:

"The VIPS theory relies on forensic findings by independent researchers who go by the pseudonyms "Forensicator" and "Adam Carter." The former found that 1,976 MB of Guccifer's files were copied from a DNC server on July 5 in just 87 seconds, implying a transfer rate of 22.6 megabytes per second -- or, converted to a measure most people use, about 180 megabits per second, a speed not commonly available from U.S. internet providers. Downloading such files this quickly over the internet, especially over a VPN (most hackers would use one), would have been all but impossible because the network infrastructure through which the traffic would have to pass would further slow the traffic."

Yes, all but impossible! Provided you ignore that DOCSIS 3.1 cable upgrades and fiber connections deliver speeds consistently faster than that all around the world every day -- including Romania. False claims and sloppy math aside, after the Bloomberg column ran, several actual, identifiable intelligence experts also came forward doubting the legitimacy of the supposed intelligence sources for these stories altogether:

Surrounded by raised eyebrows, The Nation is now apparently reviewing its story for accuracy after numerous people highlighted that a major cornerstone of the report was little more than fluff and nonsense. Bloomberg has so far failed to follow suit.

So again, there's certainly every reason to not escalate hostility between the United States and Russia with many details still obfuscated and investigations incomplete. And there's also every reason to view reports leaning heavily on anonymous intelligence insiders skeptically after generations of distortions and falsehoods from those same agencies. That said, if you want to debunk the anonymous claims of a growing number of intelligence insiders who claim Russia played pinball with our electoral process, perhaps running into the arms of even more unreliable, anonymous intelligence sources -- without checking your math -- isn't your best path toward the truth.

Filed Under: bandwidth, dnc, election, hack, math, russia

Aussie Prime Minister Says The Laws Of Math Don't Apply In Australia When It Comes To Encryption

from the good-luck-with-that,-mate dept

Oh boy. It's no secret that the Australian government -- led by George Brandis (who has made it abundantly clear he has no clue what a VPN is or what metadata is) -- is pushing strongly for mandated backdoors to encryption. At this point, it's beating a dead horse, but this is a very, very bad idea for a whole host of reasons -- mainly having to do with making absolutely everyone significantly less safe.

And it appears that Brandis' ignorance has moved up the chain of command. Australian Prime Minister Malcolm Turnbull has now put out what may be the single dumbest statement on encryption yet (and that's a pretty high bar). After being told yet again that safe encryption backdoors violate basic mathematics, Turnbull became super patriotic about the ability of Australian law to trump mathematics:

"The laws of Australia prevail in Australia, I can assure you of that," he said on Friday. "The laws of mathematics are very commendable, but the only law that applies in Australia is the law of Australia."

And, then he pulled out the "nerd harder, nerds" argument:

"I'm not a cryptographer, but what we are seeking to do is to secure their assistance," Turnbull said. "They have to face up to their responsibility. They can't just, you know, wash their hands of it and say it's got nothing to do with them."

"I am sure they know morally they should. Morally they should."

So after admitting that he doesn't understand how this works, he's saying that the "moral" responsibility of cryptographers -- who have basically all told him his plan will make people less safe -- is to make people less safe.

Turnbull seems to think he can get around the whole problem by... semantics. You see, if we just redefine things and say we're not asking for "backdoors" then it's fine:

"A back door is typically a flaw in a software program that perhaps the -- you know, the developer of the software program is not aware of and that somebody who knows about it can exploit," he said. "And, you know, if there are flaws in software programs, obviously, that's why you get updates on your phone and your computer all the time."

"So we're not talking about that. We're talking about lawful access."

That bit of word salad suggests that at least a tiny smidgen of actual knowledge made it into his brain. A backdoor is an exploit. But "lawful access" is a backdoor. Pretending they are different suggests a fairly staggering level of ignorance.

Not to be outdone, but Brandis then took his own turn at the podium to spew more ignorance:

Asked how Australia's proposed regime would allow local authorities to read messages sent with either WhatsApp or Signal, Brandis said “Last Wednesday I met with the chief cryptographer at GCHQ ... And he assured me that this was feasible.”

Right. It's pretty well known that intelligence communities can frequently hack into things to get messages, but not because of backdoors to encryption but through other flaws. This includes things like keyloggers or other spyware that effective route around the encryption. But that's entirely different than demanding backdoors. And, of course, this all comes about a week after GCHQ's own former boss argued that attacking the end points was a better strategy than backdoors. It's almost certain that what GCHQ told Brandis is that they can be pretty successful in attacking those endpoints, without undermining encryption -- and that message got twisted in Brandis' mind to believe that it meant that there were already backdoors in Whatsapp and Signal (there are not).

This whole thing is a somewhat tragic comedy of errors with completely clueless politicians making policy badly, potentially putting everyone at risk... while astoundingly claiming that laws can trump basic mathematics. What a joke.

Filed Under: australia, encryption, gchq, george brandis, going dark, malcolm turnbull, math, nerd harder
Companies: signal, whatsapp

Math Is Not A Crime

from the 2+2 dept

Support Techdirt and get Math Is Not A Crime gear! Order by Oct. 3rd »

Our latest Techdirt t-shirt is the Math Is Not A Crime shirt (also available as women's tees, hoodies, v-necks, mugs and stickers), which was initially inspired by some of the debates around outlawing strong encryption -- but also refers to much, much more. One of the key points that defenders of strong encryption have been making in response to attempts to outlaw strong encryption or to demand backdoors is that encryption is just math, and it should never be a crime to do math. But the message actually goes beyond that. It's pointing out that since it is just math, someone else can do the math too and create their own encryption. One of the key legal fights around encryption, Bernstein v. United States, basically centered around the question of whether or not the government could declare a mathematical encryption algorithm a "munition" in order to block its export. Eventually it was determined that source code (and with it, math) are protected by the First Amendment.

But it's not just about encryption. The issue of outlawing math has come up in other contexts that we talk about as well. For example, the patenting of software is one example that has raised concerns about outlawing math. Algorithms are basically just mathematical formulas and not being able to write an algorithm because someone has patented some of it seems particularly stupid. Over a decade ago, Ben Klemens wrote a great book called Math You Can't Use.

And, of course, this enters the copyright realm as well. We're constantly talking about DRM, and a key bit of law that props up bogus DRM systems is Section 1201 of the DMCA, which outlaws any circumvention of DRM. But, many of those circumventions are little more than math as well.

This should be concerning to all of us. Attempting to hold back innovation, privacy and progress by outlawing math is a problem, and that's the reason behind our shirt. Oh, and it's also cool for math teachers and students too. Get yours today, and don't forget to check out the other gear in our super-early holiday sale.

Filed Under: algorithms, drm, encryption, going dark, math, patents

DailyDirt: Cheering For Mathletes

from the urls-we-dig-up dept

It's a no-brainer that math is a critically important subject. Okay, a lot of people probably don't remember much calculus from high school -- or how to do "long division" -- but the exposure to advanced math and encouraging everyone to appreciate math is still laudable. And making sure kids aren't turned off by math will hopefully lead to more people involved in STEM fields in the future.

• Amazon's Education business unit is aiming to get teachers and students to improve their math skills and attitudes towards math. This "With Math I Can" effort is an attempt to encourage girls (and boys) who aren't confident with numbers to stick with math and math-related subjects. [url]
• A study of nearly half a million boys and girls (14-16yo) from 69 countries shows that there's a negligible gender gap when it comes to math skills. A few countries favored boys over girls in math tests, but a couple of countries also had girls beating the boys. So the differences are chalked up to effort and teaching, not any innate ability differences between boys and girls. [url]
• Apparently, girls seem to do more homework and play less with computers than boys do. According to an OECD report, boys test better at math than girls in many countries -- but girls in Shanghai seem to be really good at math. [url]

After you've finished checking out those links, take a look at our Daily Deals for cool gadgets and other awesome stuff.

Filed Under: education, homework, learning, math, stem, teaching
Companies: amazon

DailyDirt: Trust In Math

from the urls-we-dig-up dept

We'll know things are really going wrong when government authorities are trying to innovate their way around math. (And maybe we're already headed that way with backdoors to encryption.) Hopefully, though, we'll be able to trust in math for the foreseeable future, and nevermind about the Banach-Tarski paradox. Math is hard.

• If you thought that quantum cryptography would someday solve all the problems of unauthorized eavesdroppers, some researchers have shown that we'll have to be a bit more careful. Using the Bell inequality to check for attackers isn't necessarily sufficient to certify a quantum key distribution (QKD) as secure, as an eavesdropper could use some quantum hacking to compromise a system with fake detectors. Fortunately, some additional tests can reestablish security and protect against current and future attacks of a specific type. [url]
• Keeping anonymous communications networks anonymous is a bit harder if you can't guarantee there isn't an adversary with the resources to compromise a significant fraction of the network. One solution to this problem might be to include some obfuscation and noise to the network -- and as long as one out of three nodes isn't compromised, messages will be anonymous and untraceable. [url]
• The math challenge of 'graph isomorphism' is supposed to be tough, but if it's downgraded to a less difficult problem, then we might have to move away from number factoring for securing encryption. An algorithmic breakthrough for factoring could allow powerful computers to break into encrypted messages. This wouldn't be the first time we've had to change encryption methods, but this would be a significant development now, given the widespread use of encryption technology that relies on factoring. [url]

If you've been thinking about learning how to code, take a look at our Daily Deals for a collection of online courses to help you program and/or master some professional skills.

Filed Under: backdoors, banach-tarski paradox, cryptography, encryption, graph isomorphism, math, qkd, quantum cryptography, quantum hacking, quantum key distribution, security

DailyDirt: Celebrating Valentine's Day... Or Not

from the urls-we-dig-up dept

Some folks say romance is dead, but maybe it's just in a deep meditative trance. In any case, it's almost that time of year again when chocolates and flowers and possibly awkward marriage proposals are being considered. For the romantic souls, this is an annual tradition that shouldn't be missed. For the more cynical, it's just another commercial holiday to boost sales of various pink-heart themed items.

• The Secretary Problem (aka the Marriage Problem) is a challenge to find an optimization strategy for the situation of having to choose the most satisfactory option among a series of candidates -- without being able to return to a previous candidate and once a candidate is chosen the decision is final. Mathematician Martin Gardner states that the optimal solution is to interview/date the first 36.8% of candidates, and then as soon as you encounter a candidate that appears better than that first group, you hire/marry that person (which may result in getting stuck with the last candidate). QED. Very romantic, yes? [url]
• If you're following the solution to the Secretary Problem, you might want to employ another technique when you've decided. Forget love potions, just answer a few dozen questions with a potential partner and then stare into her eyes for 4 minutes. [url]
• The commercial Anti-Valentine's Day is actually November 11th -- aka Singles Day. It's the world's biggest online retail sales day that dates back to about 1993, and it's really big in China. [url]

If you'd like to read more awesome and interesting stuff, check out this unrelated (but not entirely random!) Techdirt post via StumbleUpon.

Filed Under: love, martin gardner, math, optimization, psychology, retail holiday, romance, secretary problem, singles day, valentine's day

DailyDirt: Waiting In Line Isn't Fun

from the urls-we-dig-up dept

Queuing theory is a subset of applied math that looks into the behavior of waiting in line and algorithms that optimize various aspects of this particular kind of resource allocation. Retailers of all kinds are interested in this kind of math because it can improve customer satisfaction and get more products out the door. Apple reduces long cashier lines with employees who can accept payments anywhere in its stores. Fry's Electronics has the giant single line that feeds into a massive array of cashiers (aka the serpentine line). There are self-checkout lanes at the grocery store, but there's no silver bullet to eliminate waiting in lines. Here are just a few more links on this problem of civilization.

• Ever pick a checkout line at the grocery store and think "I always pick the slowest lane" once you've committed to a particular cashier? The math says that the odds of picking the fastest lane are against you, so you're most likely right that you never pick the fastest line. [url]
• Waiting in line at Disney can be cut short with its Fastpass system, but what does Fastpass really optimize? FYI, Disney doesn't let you hire disabled kids to cut in line anymore. [url]
• The best system for boarding planes is not the one that most airlines use, but Southwest's boarding method does pretty well. A few studies have looked at how to board planes more efficiently, but the Jason Steffen method isn't used by any airline (and there's little incentive for most airlines to change what they're currently doing). [url]

If you'd like to read more awesome and interesting stuff, check out this unrelated (but not entirely random!) Techdirt post via StumbleUpon.

Filed Under: algorithms, boarding, cashiers, fastpass, jason steffen, math, optimization, planes, queuing theory, serpentine line, waiting
Companies: disney, southwest

DailyDirt: Is It Really That Hard To Cut A Cake?

from the urls-we-dig-up dept

Life is filled with small problems. Some more important than others. Mathematicians have attempted to solve some of these conundrums, and apparently one somewhat popular task is cutting things up. Here are just a few (useful?) examples of math applied to the task of cutting a cake.

• This video demonstrates how to cut a cake in a way that maximizes the amount of moist cake that can be eaten if the cake isn't eaten in a single sitting, but over the course of days. This is actually a pretty sad way to eat a cake, assuming you have no friends or don't want to share your cake so that you have to eat it all by yourself. [url]
• If you've ever heard of the Banach-Tarski Paradox, you might think it should be possible to cut up a cake in such a way that you never run out of cake. The proof relies on the Axiom of Choice, but too bad real cake isn't infinitely divisible. [url]
• Everyone knows the classic "you cut, I choose" method for cutting up a cake fairly between two people. Not everyone knows the method for cutting a cake fairly between n number of people.... [url]

If you'd like to read more awesome and interesting stuff, check out this unrelated (but not entirely random!) Techdirt post via StumbleUpon.

Filed Under: axiom of choice, banach-tarski paradox, cake, distribution problems, fairness, food, math, proofs