I have been developing the ideas here for many years [203]. I worked particularly actively on them in 1995–1998, 2001 and 2004–2005 [148][1]. But they might have languished forever had it not been for Jonathan Gorard and Max Piskunov, who encouraged me to actively work on them again, and who over the past several months have explored them with me, providing extensive help, input and new ideas. For important additional recent help I thank Jeremy Davis, Sushma Kini and Ed Pegg, as well as Roger Dooley, Jesse Friedman, Andrea Gerlach, Charles Pooh, Chris Perardi, Toni Schindler and Jessica Wong. For recent input I thank Elise Cawley, Roger Germundsson, Chip Hurst, Rob Knapp, José Martin-Garcia, Nigel Goldenfeld, Isabella Retter, Oliver Ruebenkoenig, Matthew Szudzik, Michael Trott, Catherine Wolfram and Christopher Wolfram. For important help and input in earlier years, I thank David Hillman, Todd Rowland, Matthew Szudzik and Oyvind Tafjord. I have discussed the background to these ideas for a long time, with a great many people, including: Jan Ambjørn, John Baez, Tommaso Bolognesi, Greg Chaitin, David Deutsch, Richard Feynman, David Finkelstein, Ed Fredkin, Gerard ’t Hooft, John Milnor, John Moussouris, Roger Penrose, David Reiss, Rudy Rucker, Dana Scott, Bill Thurston, Hector Zenil, as well as many others, notably including students at our Wolfram Summer Schools over the past 17 years. My explorations would never have been possible without the Wolfram Language, and I thank everyone at Wolfram Research for their consistent dedication to its development over the past 33 years, as well as our users for their support.

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