Roll Over, Pythagoras: the "Holy Grail" of Math May Have Been Found
Ever since antiquity we have been searching for perfect mathematical equations to explain a perfect Universe. Now a Japanese mathematician may have cracked an unsolved problem "at the center of everything."
From 2011-2014, Daniel Honan was the Managing Editor at Big Think. Prior to Big Think, Daniel was Vice President of Production for Plum TV, a niche cable network he helped launch in 2002. The production team he oversaw won over two dozen Emmy awards. Daniel has created numerous shows and documentaries for television, and his film credits include Stealing the Fire, a documentary on the black market for nuclear weapons technology.
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What's the Big Idea?
The so-called "holy grail" of mathematics may have just been found. Shinichi Mochizuki, a professor at the Research Institute for Mathematical Sciences at Kyoto University, released a 500-page proof of the abc conjecture in number theory which involves the relationship between prime numbers, which I will attempt to describe as succinctly as possible below.
A lot of open theory on prime numbers is based on the abc conjecture. So if verified, Mochizuki's proof would mean a great leap forward in our understanding of primes. It could also be used to help simplify other complex proofs, such as Fermat's last theorem.
This is high-level mathematics, which is even inaccessible to many mathematicians. So it is with that caveat that we are providing the following links to you (extremely) intrepid math geeks out there. Here is Mochizuki's study in four parts:
Mochizuki is a math genius who received his Ph.D from Princeton University at the age of 22 and hasn't slowed down since.
So what exactly is the abc conjecture?
The clearest explanation I could find of the abc conjecture is this single paragraph from the ABC@home project from the Mathematical Institute of Leiden University:
The ABC conjecture involves abc-triples: positive integers a,b,c such that a+b=c, a < b < c, a,b,c have no common divisors and c > rad(abc), the so-called radical of abc.* The ABC conjecture says that there are only finitely many a,b,c such that log(c)/log(rad(abc)) > h for any real h > 1. The ABC conjecture is currently one of the greatest open problems in mathematics. If it is proven to be true, a lot of other open problems can be answered directly from it.
*The rad(abc) is the "product of the unique prime factors of a,b, and c"
What's the Significance?
Not surprisingly, there are many other descriptions of the abc conjecture that are completely unwieldy. So to put it as succinctly as possible, here's what Mochizuki had to solve: the conjecture states there is a finite group of triples with the properties described above. Since numbers go up toward infinity, one could find examples of these triples here and there until the end of time. Instead, what Mochizuki was able to establish was a logical reason for this pattern. In other words, he was able to identify exactly why these groups of numbers have always been considered special. If Mochizuki's work is verified, we will be able to finally prove a great deal of theorems largely built upon the abc conjecture in the field of number theory in ways that go even further over my head.
That is why Philip Ball in the journal Nature calls this an "astounding achievement" in the field of Mathematics, perhaps the most significant achievement so far in the 21st century.
Also worth reading is Ivars Peterson's MathTrek blog, from 1997, in which Peterson surveyed other mathematicians on the then-unsolved conjecture's significance. Here's Andrew J. Granville of the University of Georgia in Athens: "This strange conjecture turns out to be equivalent to all the main problems. It's at the center of everything that's been going on."
Peterson also quotes Dorian Goldfeld, who describes the conjecture as a true game-changer in a Math Horizons article:
It is more than utilitarian; to mathematicians it is also a thing of beauty. Seeing so many Diophantine problems unexpectedly encapsulated into a single equation drives home the feeling that all the subdisciplines of mathematics are aspects of a single underlying unity, and that at its heart lie pure language and simple expressibility.
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Swiss researchers identify new dangers of modern cocaine.
- Cocaine cut with anti-worming adulterant levamisole may cause brain damage.
- Levamisole can thin out the prefrontal cortex and affect cognitive skills.
- Government health programs should encourage testing of cocaine for purity.
Civil discourse has fallen to an all time low.
The question that the American populace needs to ask itself now is: how do we fix it?
Discursive fundamentals need to be taught to preserve free expression
In their findings the authors state:
upholding First Amendment ideals.
Talking politics at Thanksgiving dinner
- Progressive Activists: younger, highly engaged, secular, cosmopolitan, angry.
- Traditional Liberals: older, retired, open to compromise, rational, cautious.
- Passive Liberals: unhappy, insecure, distrustful, disillusioned.
- Politically Disengaged: young, low income, distrustful, detached, patriotic, conspiratorial
- Moderates: engaged, civic-minded, middle-of-the-road, pessimistic, Protestant.
- Traditional Conservatives: religious, middle class, patriotic, moralistic.
- Devoted Conservatives: white, retired, highly engaged, uncompromising,
It's interesting to note the authors found that:
"Tribe membership shows strong reliability in predicting views across different political topics."
Here are some statistics on differing viewpoints according to political party:
- 51% of staunch liberals say it's "morally acceptable" to punch Nazis.
- 53% of Republicans favor stripping U.S. citizenship from people who burn the American flag.
- 65% of Republicans say NFL players should be fired if they refuse to stand for the anthem.
- 58% of Democrats say employers should punish employees for offensive Facebook posts.
- 47% of Republicans favor bans on building new mosques.
Here are some guidelines for civic discourse that might come in handy:
- Practice inclusion and listen to who you're speaking to.
Civic discourse in the divisive age
dangerously tribal, fueled by a culture of outrage and taking offense. For the combatants,
the other side can no longer be tolerated, and no price is too high to defeat them.
These tensions are poisoning personal relationships, consuming our politics and
putting our democracy in peril.
Once a country has become tribalized, debates about contested issues from
immigration and trade to economic management, climate change and national security,
become shaped by larger tribal identities. Policy debate gives way to tribal conflicts.
Polarization and tribalism are self-reinforcing and will likely continue to accelerate.
The work of rebuilding our fragmented society needs to start now. It extends from
re-connecting people across the lines of division in local communities all the way to
building a renewed sense of national identity: a bigger story of us."
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