Semigroups are like groups but semigroup elements don’t always have inverses, necessarily.
Semigroups obey the associative law:
- a then b, then c = b then c, after a.
but not necessarily the commutative law (3+14=14+3). Aristotle observed that time obeys the associative law.
It is commonly agreed that time moves forward only and not backward. Not invertible means you can’t always undo what was done. (Both groups and semigroups can be noncommutative; order sometimes matters, like whether you put the couch down first or pour the concrete first.) So, John Rhodes says, we should model time with semigroups.
Speaking in terms of sets and sequences, (a,b,c,d,e) is equivalent to (a, ab, abc, abcd, abcde). The two representations (let’s call them events and timelines) serve different functions mathematically but are isomorphic. With this identification in hand, Rhodes launches into a tornadic discussion of groups, commutative and noncommutative:
- groups have fundamental constituent parts;
- we have found all of them;
- we know how they combine ⋊ to form larger actions;
- so we essentially know everything about every group;
- and this bears on Life, The Universe, and Everything.
Rhodes co-invented the wreath product ≀, which explains how to combine the fundamental units of semigroups into any semigroup at all.
If semigroups represent all the logical options that anyone can do with anything, then the total classification of finite simple groups is an achievement with epic implications. It would mean a mathematical theory of all the things that can be done. “Do” and “thing”, that is breaking it down pretty much to the basics.
I read some of The Wild Book at a friend’s house a couple months ago (I haven’t bought a copy yet). Skimming throughout the text, it looks like a really fun read — jumping from abstract algebra to (mathematical) cellular automata to the Krebs cycle to religion.
However, Rhodes draws some unwarranted conclusions, or otherwise demonstrates overly simplistic thinking.
- He equates “religion” with beliefs about the afterlife.
And he’s an Egyptologist or something?
- His example of a semigroup loses all knowledge right away rather than things being gradually more covert depending on ingenuity.
- Does the semigroup therefore model our understanding or the progress of the physical world?
- He says that scientific understanding is equivalent to the introduction of spacetime coordinates. Um, dissection & anatomy? Synthesis of urea? Germ hypothesis, periodic table, polio vaccine?
Also, by existence I think he means experience. Existence does not imply feedback.
- Just because there are many different “models” of time or space, doesn’t reduce the credibility of any religion. Unless the religion specifically stated “The body moves around in ℝ³ forever — throughout time, which is ℝ¹.” I can’t remember reading that in Scripture.
- And semigroups are supposed to be unpopular because they controvert religion? I assume it’s just because they’re abstruse.
Has it really been proven that information is lost when a person dies and is buried or cremated? The smoke from the funeral pyre is in a lower entropic state than the atoms of the nervous system were, but doesn’t the specific configuration parametrise the smoke which affects the wind and so on? Information may be chaotically scrambled but is that the same thing as lost?
I’m not insulting what John Rhodes produced: a rare jewel that looks at scientific and philosophical questions through the lens of abstract algebra. These questions are meant to provoke further discussion of his ideas.