Mathematical models of reality and site swap mathematics

Laboratory 15 has seen a number of different prefaces. Originally I started lab fifteen with "the mathematics stack" starting with counting and moving up through addition, algebra, and into calculus. I then note that there are other fields of mathematics that are not "in the traditional stack" such as statistics, fuzzy math, chaos theory, and site swap notation.


For a couple terms I shifted to presenting site swap notation abstractly on the board, deliberately obfuscating the connection to juggling. I then ask the class if they understand. Someone usually says yes, although everything I have said is gobbledy-gook.


This term the video triple header on black holes, the Higgs boson, and dark matter/shadow universes, preceded earlier in the term on the nature of time, led more naturally to circle back to:

For a physicist mathematics is not just a tool by means of which phenomena can be calculated, it is the main source of concepts and principles by means of which new theories can be created... ...equations are quite miraculous in a certain way. I mean, the fact that nature talks mathematics, I find it miraculous. I mean, I spent my early days calculating very, very precisely how electrons ought to behave. Well, then somebody went into the laboratory and the electron knew the answer. The electron somehow knew it had to resonate at that frequency which I calculated. So that, to me, is something at the basic level we don't understand. Why is nature mathematical? But there's no doubt it's true. And, of course, that was the basis of Einstein's faith. I mean, Einstein talked that mathematical language and found out that nature obeyed his equations, too. – Physicist Freeman Dyson

Perhaps nature is mathematical, but then is the nature of reality mathematical? I quoted from Tegmark's "Is the Universe Made of Math?" Is mathematics the reality? Or should we be guided by quantum mechanics and note that reality is only made apparent when we observe a system. When we go and look for a wave or for a particle. If observation makes reality, then perhaps reality is only what we believe we will find.


Perhaps the problems with dark matter will not be solved by five types of Higgs bosons. Perhaps the universe is not mathematical. Perhaps the universe is not as real as we might perceive it to be. Sure, the floor seems solid, but maybe only because this is what we fervently believe.

Crazy talk? Perhaps. I do not believe I can walk on water, but perhaps if I had the faith of a mustard seed I really could move a mountain. Maybe that was a way of explaining to a group of fishermen and farmers that reality is not what we take reality to be.

Physical science has been a journey of mathematical models, and we tend to count on the mathematicity of the universe.


With that I turned back to the world of mathematical modeling and wrote out site swap notation, not obfuscated, on the board. I used red, green, and blue balls. I noted that while quantum mechanics had left and right handedness, site swap really does have left and right landing sites. I diagrammed 3 first, and then asked if anyone understood the diagram. In each section one person nodded. I then pitched that one cannot really understand math taught from a white board. Math has to be experienced.


I then juggled the 3 and went on to do 51. To show site swapping, I ran a 342 into the 3 diagram and then juggled that.



The class wrapped up with everyone attempting 3 and some showing 51s and 342s.


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