The principle of plenitude is a bit strengthened "Platonist" principle about the existence of mathematical structures. Aside from many other claims, Plato is celebrated for the idea that all the structures in mathematics should be considered as "really existing", although outside the physical world. However, Kragh attributes a stronger statement to Plato, namely that these ideal structures are all realized in Nature. That's very interesting and I am not sure whether I would believe it or agree with it.
Whether it's true also depends on the definition of the "ideal mathematical structures". If you define one as "being so great that even Nature couldn't resist and had to use it", then all ideal structures are tautologically included in reality – by definition. ;-) Is there a definition of an ideal structure that isn't just some "subjective arts" and that could make the question whether Plato's principle of plenitude is true meaningful?
I am not sure.
At any rate, Kragh and Siegfried say that Gell-Mann just mildly generalized Plato when he formulated the totalitarian principle in 1956: everything that is not forbidden is mandatory. Siegfried says that Gell-Mann's paper was just about the strong nuclear force but the principle was the same as Plato's plenitude, anyway.
I am afraid that I mostly disagree with both assertions. Gell-Mann focused on the strong force in a paper but he surely meant the principle to be more general – applicable in similar but not identical questions all over fundamental physics. The point of the totalitarian principle is that there may be laws and general principles such as symmetries and these may prohibit some behavior (e.g. the violation of the charge conservation), like the totalitarian society bans e.g. speech about the evils of Islam. So when they're assumed to be banned, they're non-existent.
But if there is no ban, it is not only true that the effect may exist or not. It must exist. If something is not banned, it's unavoidable that someone does it. In totalitarian societies, it's a rule. If you're not banned from praising the Nazis, the communist party, or the progressives, of course you have to do so. If you could choose, you would have some freedom and the freedom is always dangerous for an idealized totalitarian society.
Gell-Mann's principle is really a predecessor of our thinking about naturalness in the technical sense and has a similar explanation. By "something" that is either forbidden or mandatory, he mostly meant effects – some interactions – and whether they exist or not is determined by the coefficients in front of these interactions, typically in the equations of motion or the Lagrangian. Gell-Mann's point was that if there is no principle that may be used to neatly prove that the coefficients have to be zero (a proof assuming a consistency, locality, symmetry, absence of ghosts etc.), then the coefficients must be nonzero!
Technical naturalness strengthens it by saying that "the coefficients must even be of order one" in some appropriate units. We expect this to be true by Bayesian inference. If there are no principles that would force the coefficients to be zero, very close to zero, or very special in some other way, then the coefficients will almost certainly not be zero, near zero, or very near some very special values! It's just infinitely unlikely for the numbers to have special properties.
(A big possible defect of any such argument is that "there is no principle" isn't quite the same as "we don't know of any such principle now". We may only be certain enough about the latter but not the former because we keep on discovering various principles.)
The argument is probabilistic, not "rigorous", but it's damn rational, anyway. Every good modern theoretical physicist understands that Gell-Mann's totalitarian principle or naturalness isn't just some random hypothesis uncorrelated with the truth – which is equally true or untrue as its negation. It's a statement that is "partially or softly proven" and a statement that has been successful in certain cases (many cases). We can't really ignore it. A theoretical physicist undoubtedly needs to use some general guides of this kind. Some of the physicists find naturalness or Gell-Mann's principle very important, some find it less important – but they have "something else that is conceptual" if they are doing some non-mechanical, deep enough things, at all.
There is surely some very general similarity between Platonism and Gell-Mann's principle. (The latter is also called "the anarchic principle" because it is really the "anarchy" that also underlies the totalitarian societies that makes the assorted things unavoidable. The contradiction between "anarchy" and "totalitarianism" – which may sometimes be considered opposite to each other – is just an illusion, both in politics and in physics. "Imposing" anarchy on the society is only possible through totalitarianism – most of the free and at least slightly rational people just don't want to turn their environment into complete chaos! Free people generally create lots of structures that protect them from mess and chaos. Complete chaos deserving to be called anarchy is, like a complete equality, a totally unnatural and undesirable state of the human society which is why it may only be achieved through totalitarianism.)
But there is also a big difference between Plato's and Gell-Mann's statements. First of all, Gell-Mann was talking about "terms in the Lagrangian" – which is a rough but pretty good description of what he claimed. It is not quite accurate but it's almost accurate and surely way more accurate a description of Gell-Mann's intended principle than if we describe it as "the same what Plato said". Plato wasn't talking about the existence of terms in the Lagrangian but "existence of mathematical structures" as wholes. These are rather different things.
Well, yes, in some perspectives, they are closely related. For example, there may be processes mediated by magnetic monopoles, e.g. proton decay. Processes that depend on magnetic monopoles are expected to exist through Gell-Mann's logic. On the other hand, the existence of these processes also needs a nice mathematical structure, the solitonic solution for the magnetic monopole, to exist mathematically as well as in the real world which is Plato's domain.
Whether Plato's and Gell-Mann's statements are closely related depends on the degree of one's general, unified thinking; as well as his resolution. Both of them are "virtues" of a theoretical physicist. A good theoretical physicist wants to be able to unify things – and therefore see ways or examples of situations in which Plato's and Gell-Mann's statements were related if not equivalent. On the other hand, a good theoretical physicist must still preserve a good and sharp resolution – and see that these things are unequivalent, different, or very different in some important aspects or in most situations or in general.
Even if Gell-Mann were building on the shoulders of a giant named Plato, what he added was of course nontrivial – partly because no physicist in the 1950s would have the insane idea to evolve physics according to the words assigned to a 2,000-years-old dude statue. ;-) Gell-Mann surely deserves a lot of credit for that principle and it's just damn stupid to give all the credit to Plato because Plato just accidentally made some similarly sounding statements but he knew nothing about particle physics so even if the words are positively correlated with the truth, it's a coincidence – a broken clock is right twice a day.
I also want to address the following statement by Siegfried and/or Kragh:
In spite of such fruitful results from applying the totalitarian principle, it remains a mere guideline for scientific pursuits, not a guarantee of success.Well, yes and no. Yes... because it's just some vague probabilistic argument if not a philosophical recommendation or strategy how to think and those may be equally right and wrong in physics. The agreement of the predictions with experiments measures the success – and it's surely not clear whether some far-reaching extrapolation of Plato's comments will lead to an agreement between the theory and experiment.
On the other hand: No... At least in Gell-Mann's form, the statement isn't just some random guess or an unbacked hypothesis. It is backed by a partly provable, albeit vague and statistical, reasoning. Also, it's a misunderstanding that "something completely different, the agreement between the theory and experiment, is what exclusively determines the success of physical theories."
Why is it a misunderstanding? It's a misunderstanding because the agreement with principles such as Gell-Mann's principle is also a measurement of a term contributing to the success of (more detailed) scientific hypotheses or theories or models. In fact, this statement doesn't even contradict the extreme principle that "all success in science boils down to the empirical tests". Why is there no real contradiction? Because Gell-Mann's totalitarian principle is largely an essence extracted from many observations, too! So it's surely considered an advantage for a new physical theory to agree with this generalized pattern. For a less controversial example, it's an advantage for a new theory to agree with the postulates of special relativity – even before we discuss the question whether the particular potential violation of the postulates has been ruled out by actual experiments.
Physics – and, to a lesser extent, other natural sciences – always deals with some kind of generalization and extrapolation. The theory that just extrapolates the known laws and patterns further is always the "simpler one" according to some Occam's razor logic, and it's a reason why we may want to view such "straightforward extrapolations" to be the default null hypotheses to be trusted and tested, until we find some problem with them. And the general assumption that Gell-Mann's principle works is an example of a law that good physicists expect to keep on being true in some sense.
They may stop holding but then it's a paradigm shift, some new addition – or replacement laws – must be substituted, and when they're substituted, it's already a non-minimal choice that makes the theory more contrived according to Occam's criteria.
My point is that when we evaluate two theories or models that "otherwise" agree with the empirical data to a similar extent, it's totally healthy to prefer the theory that is aligned with naturalness or Gell-Mann's principle or perhaps some other similar principles and their variations – simply because these principles are also some lessons extracted from many experiments, observations, and calculations and thinking applied on them! So a theory's being more natural simply is making a theory more likely to be true, and looking for theories that are more likely to be true is the definition of success in theoretical physics.
In particular, theoretical physics and natural science never "quite prove that some theory is totally and precisely true". It's really not possible. We only look for theories and statements etc. that are not sharply excluded, that are more true or more accurate than others, that are capable of surviving for a longer time, and that may do so more naturally, with less fine-tuning etc. These are the criteria to rate theories and all of them boil down to some thinking plus empirical evidence which is why the statement that the "determination of the success is completely independent from the agreement with Gell-Mann's principle or naturalness or similar principles" is simply wrong.
Such a statement is just an expression of the fact that the speaker wants to ban any thinking, extrapolation, or generalization in physics – and imagine that physics is just stamp-collecting, like all the other sciences and "sciences". But as Lord Rutherford has emphasized, physics isn't just stamp-collecting (which is what all the other sciences were according to the Sir) and the identification of principles – sometimes even philosophers' principles – that sound much more general and far-reaching than statements about "very particular experiments" is a top reason why physics is harder and deeper than (almost?) all other sciences. That's why the ability of physics to work with similar far-reaching principles – and to evaluate them, filter them, and refine them – is a vital and almost defining trait of physics. The ability to discuss previously "unimaginably metaphysically deep and almost religious – and increasingly deep" questions is a reason why the most intelligent people are actually attracted to physics. Some questions in religions etc. are wrong or meaningless but others aren't quite meaningless and physics is actually capable of making choices.
As cosmologist Michael Turner has said, physics depends on the fine balance between hot and speculative philosophical ideas (on one side) and the cold boring data (on the other side). When this balance breaks down, physics degenerates either to philosophy or botany, respectively. All the people who claim that "the criteria of success in physics don't depend on things like the agreement with Gell-Mann-like principles" are implicitly assuming that physics is like botany (something that always discusses just some very isolated facts) or they want physics to be degraded to another part of botany. I will never allow such a thing to happen.
And that's the memo.