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Please note!:

I) I have tagged "root-systems" and "algebraic-groups" in this question because: i) the UNSTATED background context for the question involves the root-system of $E_8$; ii) the integer 14 plays an obvious and important role in the internal structure of this root-system.

II) This question is closely related to Jyrki Lahtonen's comment on this question:

Any Existing Time-Division Multiplexing Algorithm in which Slot-Allocation is Governed by the $E_8$ Root System?

and in particular, to the link which he provided in this comment to an "executive summary" of some past work of his.

III) This question is further developed by the more recent post here:

Simplest AND largest SUBSET of the $E_8$ root-lattice using only choices (up to 3 dups allowed) of {1,2,4,5,6,9,10,14} which sum to 14?


Background:

To complete our project, my team must evaluate the energetics associated with various decompositions of the integer 14, e.g.

$10+4$

$6+4+4 $

$5+1+4+4 $

$6+5+1+1+1 $

In addition, since we are dealing here with biomolecular energetics, it is reasonable to hypothesize that evolution's process of natural selection INITIALLY chose energetics associated with the "simplest" sum "$10+4$" and then only later chose the progressively less simple sums ($6+4+4$, then $5+1+4+4$, then $6+5+1+1$).

Question:

This evolutionary hypothesis of course makes the assumption that the sum "$10+4$" IS actually somehow "simpler" in a formal sense than the sum "$6+4+4$".

So, does any FORMAL theory of simplicity actually assert that the four sums above can be ranked in terms of their relative simplicity in the obvious way?

If so, please explain in layman's terms and provide a link as well.

Thank you as always for whatever time you can afford to spend considering this matter.

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closed as primarily opinion-based by Matthew Towers, jgon, Stefan4024, Brian Borchers, Leucippus Dec 12 '17 at 0:15

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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I don't think it makes sense to ask for a FORMAL theory if by that you mean something inherent in the logical structure of mathematics.

Since "simple" and "simplest" do not have explicit definitions in ordinary arithmetic, you are free to define them in any consistent way and then do mathematics with those definitions. In your case you seem to want "simpler" to mean "with fewer summands". That's a perfectly respectable definition. Of course you'll need to deal with the fact that neither of the representations $14 = 10 + 4$ and $14 = 9 + 5$ is simpler than the other. The "simplest" expression for $14$ would be $14$.

It's likely that in some sense a computer program would take more time and energy to add more summands.

Whether this definition captures any biological reality is beyond the scope of the mathematics.

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  • $\begingroup$ thanks so much for taking the time to respond. Two things: First, 14 IS actually one of the alternatives we need to evaluate energetically, and we DO presume it is the simplest (I just didn't bother to mention it.) Second, due to a well-motivated and experimentally verifiable overall framework, we are in the fortunate position of NEVER having to evaluate two alternatives with the same number of summands. I'm glad you mentioned this, because we find it remarkable that we never have to do this. $\endgroup$ – David Halitsky Dec 11 '17 at 15:51
  • $\begingroup$ please see my response above - thanks again $\endgroup$ – David Halitsky Dec 11 '17 at 15:52
  • $\begingroup$ @Pippo thanks for the edit ! Much appreciated. $\endgroup$ – David Halitsky Dec 11 '17 at 16:22

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