## R.I.P. John Conway

Posted by on Monday, April 13, 2020

We have some incredibly sad news today, with the announcement that the legendary mathematician John Horton Conway has died from COVID-19. He may not be as famous outside of academia as some, but his silly little recreational games have resulted in some of the most interesting fields of mathematical research.

His work is too extensive to review everything, but I would like to briefly cover a few of his more interesting games. (For those who don't know Conway's work, a lot of his mathematical research involved creating these silly little games to play, and then analysing the mathematics behind them)

Conway's biggest claim to fame is the Game of Life - which actually became so successful that he became irritated at people constantly wanting to talk about it. The idea was to randomly place markers on a grid - such as checkers on a checker board - and then evolve the grid according to simple rules. For example, if a piece had one or no neighbours, it was removed from the board. If an empty space had two or three neighbours, a piece was added to it. By evolving the grid through many generations - often using simple software instead of physical pieces - interesting emergent behaviour could be observed. The results often looked like the evolution of a plate of bacteria or other small lifeforms, hence the name. And this simple game resulted in the rapid growth of cellular automata as a serious scientific field.

Another interesting thought that he had was how to define numbers in a slightly different way. Almost all of us learned the numbers through counting - at an early age we learn that a single object is called "one" and when we add another we get "two". This learning goes on for twenty years as we add in fractions and real numbers, but it all begins with counting. Conway tried starting with just pairs of sets - a number being defined by the set of everything smaller and the set of everything larger. He used this method to define all of the integers, fractions, and real numbers, and even developed a new arithmetic for these numbers. But then he went further and defined omega - "the number that is bigger than every integer". And then the number that is larger than omega, and so forth. He defined "the positive number that is smaller than every fraction" and called it 1/omega. These new numbers were dubbed the surreal numbers (and by extension we can also define the surcomplex numbers), and have resulted in serious research in number theory lasting for over thirty years.

And those are just two of his many games that had serious mathematical implications. If he couldn't answer a question himself, he would often place a bounty on the puzzle and offer to pay anyone who could prove one of his theorems. Some of his seemingly trivial problems and conjectures are still unsolved or unproven decades after he made them.

Conway was that unique combination of being incredibly intelligent, and yet joyfully crazy. He was truly eccentric, and built an entire career out of creating and playing children's games that could stump the world's leading minds. Add to that his energy and attitude that inspired generations of students at all levels to pursue a mathematics education, and it becomes even more clear what the world has lost today.

Rest in peace John.

His work is too extensive to review everything, but I would like to briefly cover a few of his more interesting games. (For those who don't know Conway's work, a lot of his mathematical research involved creating these silly little games to play, and then analysing the mathematics behind them)

Conway's biggest claim to fame is the Game of Life - which actually became so successful that he became irritated at people constantly wanting to talk about it. The idea was to randomly place markers on a grid - such as checkers on a checker board - and then evolve the grid according to simple rules. For example, if a piece had one or no neighbours, it was removed from the board. If an empty space had two or three neighbours, a piece was added to it. By evolving the grid through many generations - often using simple software instead of physical pieces - interesting emergent behaviour could be observed. The results often looked like the evolution of a plate of bacteria or other small lifeforms, hence the name. And this simple game resulted in the rapid growth of cellular automata as a serious scientific field.

Another interesting thought that he had was how to define numbers in a slightly different way. Almost all of us learned the numbers through counting - at an early age we learn that a single object is called "one" and when we add another we get "two". This learning goes on for twenty years as we add in fractions and real numbers, but it all begins with counting. Conway tried starting with just pairs of sets - a number being defined by the set of everything smaller and the set of everything larger. He used this method to define all of the integers, fractions, and real numbers, and even developed a new arithmetic for these numbers. But then he went further and defined omega - "the number that is bigger than every integer". And then the number that is larger than omega, and so forth. He defined "the positive number that is smaller than every fraction" and called it 1/omega. These new numbers were dubbed the surreal numbers (and by extension we can also define the surcomplex numbers), and have resulted in serious research in number theory lasting for over thirty years.

And those are just two of his many games that had serious mathematical implications. If he couldn't answer a question himself, he would often place a bounty on the puzzle and offer to pay anyone who could prove one of his theorems. Some of his seemingly trivial problems and conjectures are still unsolved or unproven decades after he made them.

Conway was that unique combination of being incredibly intelligent, and yet joyfully crazy. He was truly eccentric, and built an entire career out of creating and playing children's games that could stump the world's leading minds. Add to that his energy and attitude that inspired generations of students at all levels to pursue a mathematics education, and it becomes even more clear what the world has lost today.

Rest in peace John.