## Posts Tagged ‘**Paul Erdos**’

## “To change something, build a new model that makes the existing model obsolete”*…

There is remarkably little reflection taking place about the state of science today, despite significant challenges, rooted in globalization, the digitization of knowledge, and the growing number of scientists.

At first glance, all of these seem to be positive trends. Globalization connects scientists worldwide, enabling them to avoid duplication and facilitating the development of universal standards and best practices. The creation of digital databases allows for systematic mining of scientific output and offers a broader foundation for new investigations. And the rising number of scientists means that more science is being conducted, accelerating progress.

But these trends are Janus-faced…

Jeremy Baumberg argues that we live in an age of hyper-competitive, trend-driven, and herd-like approach to scientific research: “What Is Threatening Science?“

* Buckminster Fuller

###

**As we rethink research,** we might spare a thought for Paul Erdős; he died on this date in 1996. One of the most prolific mathematicians of the 20th century (he published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed), he is remembered both for his “social practice” of mathematics (he engaged more than 500 collaborators) and for his eccentric lifestyle (he spent his waking hours virtually entirely on math; he would typically show up at a colleague’s doorstep and announce “my brain is open”, staying long enough to collaborate on a few papers before moving on a few days later).

Erdős’s prolific output with co-authors prompted the creation of the Erdős number, the number of steps in the shortest path between a mathematician and Erdős in terms of co-authorships. Low numbers are a badge of pride– and a usual marker of accomplishment: as of 2016, all Fields Medalists have a finite Erdős number, with values that range between 2 and 6, and a median of 3. Physics Nobelists Einstein and Sheldon Glashow have an Erdős number of 2. Baseball Hall of Famer Hank Aaron can be considered to have an Erdős number of 1 because they both autographed the same baseball (for number theorist Carl Pomerance). Natalie Portman’s undergraduate collaboration with a Harvard professor earned her an Erdős number of 5; Danica McKellar (“Winnie Cooper” in *The Wonder Years*) has an Erdős number of 4, for a mathematics paper coauthored while an undergraduate at UCLA.

## “A mathematician is a device for turning coffee into theorems”…

In 1937, Lother Collatz proposed that, when starting with any natural number (positive integer) n, then either dividing it by 2 if n is even, or multiplying n by 3 and adding 1 if n is odd, then continuing this process indefinitely, one will always reach the number 1.

The proposition is pretty broadly known as “The Collatz conjecture”; but as it commanded the energies of a number of other mathematicians, it is also sometimes called “the Ulam conjecture” (after Stanislaw Ulam), “Kakutani’s problem” (after Shizuo Kakutani), “the Thwaites conjecture” (after Sir Bryan Thwaites), “Hasse’s algorithm” (after Helmut Hasse), or “the Syracuse problem.” More generically, it’s often just called the “3n+1 conjecture”… and the sequence that it generates, “the wondrous numbers.”

But as wondrous as the sequence may be, it is no pushover to prove. Indeed, legendary (and legendarily “homeless”) mathematician Paul Erdos– the source of this post’s title quotation– posted a $500 prize for a proof… an offer he never had to make good. Still, most mathematicians who have played with the problem believe the conjecture to be true,as both experimental evidence and heuristic arguments support it.

It is in any case, fascinating– and beautiful. London-based designer **Jason Davies** has created a visualization of the Wondrous Numbers, an animation that runs the Collatz logic **in reverse**— starting with 1 and “growing” a tree of natural numbers:

*click here to watch this grow from a single digit*

**As we remark that there’s a fundamental propriety to everything in the world ultimately coming back to one,** we might recall that it was on this date in 1940, at approximately 11:00 am, that the first Tacoma Narrows suspension bridge collapsed due to wind-induced vibrations. Situated on the Tacoma Narrows in Puget Sound, near the city of Tacoma, Washington, the bridge had only been open for traffic a few months. For those who missed it in high school physics: