The Hierarchy of Maths

Two interesting and complementary ideas concerning the interaction of symbols are “fission” and “fusion.”

Fission is the gradual divergence of a new symbol from its parent symbol (that is, from the symbol which served as a template off of which it was copied).

Fusion is what happens when two (or more) originally unrelated symbols participate in a “joint activation”, passing messages so tightly back and forth that they get bound together and the combination can thereafter be addressed as if it were a single symbol.

Fission is a more or less inevitable process. Like a child growing up. Also like the growth of an Instance from a Class of objects. The instance grows like child. Initially without its own ideas or experiences–parroting its parent Class. But gradually, as it interacts more and more with the rest of the world, the child acquires its own idiosyncratic experiences and inevitably splits away from the parents, becoming autonomous. And one day becoming a full-fledged adult. A class, or prototype, in its own right. A kind of “emergent polymorphism”.

Fusion is a subtler thing. A consequence of ‘joint activations’. For instance, how much do we hear “dough” and “nut” when we say “doughnut”.

“The real problem with this notion of ‘fusion’ is that it is very hard to imagine general algorithms which will create meaningful new symbols from colliding symbols… Like two strands of DNA coming together in a viable and meaningful way.” (Douglas Hofstadter)

Now, lets consider an analogy to atoms. Adding neutrons or protons is fusion. Fission is the act of growing the probability density cloud of surrounding electrons until they are ripe for picking into reality.

However, the process of fusion and fission are not independent. Bear with me through this other example: “When a team of football players assemble, the individual players retain their separateness–they do not melt into some composite entity, in which their individuality is lost. Still–and this is important–some processes are going on in their brains which are evoked by the team-context, and which would not go on otherwise, so that in a minor way, the players change identity when they become part of the larger system, the team. This is called a nearly decomposable system.” (H.A. Simon on “The architecture of Complexity”)

This gives a sense of both merging and differentiating. The contradiction, or ambiguity, between fusion and fission is resolved when you view the system on different levels of abstraction: individual vs group.

To step back up to the connection with the initial example: adding protons or neutrons significantly changes the atom, as does the de/ionization of electrons.

This chemical compound of math seems to represent more of a heterarchical view, like the growth of a crystal, than a hierarchical one. But maybe not.

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“Every mathematician has the sense that there is a kind of metric between ideas in mathematics–that all of mathematics is a network of results between which there are enormously many connections. ”

“In that network, some ideas are very closely linked; others require more elaborate pathways to be joined. Sometimes two theorems in mathematics are close because one can be proven easily, given the other. Other times two ideas are close because they are analogous, or even isomorphic. These are two different senses of the word ‘close’ in the domain of mathematics.” (Douglas Hofstadter)

With a fear of sounding silly, maybe there’s a periodic table for maths, which maps to  different qualities of bonding network compounds. Hydrogen is where fusion is likely.  Plutonium for fission.

But in general, we are interested in the network density of the compounds. Which we can get by Filtering for particular ways of viewing the system as a whole (adjectives to describe The Table), and Focusing on promising subsystems (nouns to describe the individual elements).

The hierarchy is imposed on the compound mixtures through this process of throwing out information.

To construct the skeleton of beauty and simplicity through the harmonically distant chords of math.

And if this is all too vague, if this thought carries too much diversity, then maybe that should serve as a clue that concepts on a higher level are involved and needed.

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