The Hidden Fruits of Equations

I don’t have a lot of time to write this week and next but I felt a little surge of thoughts gather when I read about the implications of Einstein’s Second Law (m = E/c^2) which is a simple rearrangement of the now famous E = mc^2.  It was in The Lightness of Being by Frank Wilczek.  Here’s what Wilczek says:

Einstein’s first law is of course E = mc^2.  Famously the first law suggests the possibility of getting large amounts of energy from small amounts of mass.  It calls to mind nuclear reactors and nuclear bombs.

Einstein’s second law suggests something quite different.  It suggests the possibility of explaining how mass arises from energy.

This was the thought Einstein pursued.

Wilczek preceded his discussion of Einstein’s concept of energy with an anecdote about Paul Dirac – specifically, that when asked how he discovered new laws of nature, he responded, “I play with equations.”

Although the same equation written different ways may be algebraically or logically equivalent, their different arrangements can suggest very different things.  It was around this point that my own thoughts began to gather, in an almost reflexive way.

To begin with, it’s interesting that two symbolic representations of relationship, each of which can be said to contain exactly the same information, can narrate different stories, or create different images.  In this case, one arrangement of the equation tells the story of getting energy out of matter and the other about the origins of mass.   This says something about how we, human beings, work – about how we can turn something upside down and actually find something else.  It also says something about what mathematics accomplishes, how it captures insurmountable mysteries and re-presents them on paper, where doodles sometimes live, and where they can then be toyed with, investigated, and used as probes.  Ideas about how the universe works do emerge from relationships among numbers.

As bewildering, complex and dense as it may seem, mathematics is everywhere about relationship.  It is an unbounded exploration of how things can be related, numbers, lines, spaces, fields…or, even more to the point, it is the investigation of all of the possibilities for relationship.  This is exactly why its generalities are stripped of every particular and why the few symbols of an equation can do so much.

How mass arises is a question physicists have yet to answer.  I referred to their search for the elusive Higgs particle (the expected answer to that question) in an earlier post. This recent article in Nature expresses the determination of experimentalists at the Large Hadron Collider in Geneva Switzerland to find it if it is there to be found.

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