expand_less ContemporaryPhysicists, when pressed, will assert that the deepest ontology of the world is fields, in particular, quantum fields. They have an equation that still fits on a T-shirt that while summarizing the dynamics of the fundamental fields found in experiment, each part of the equation holds many remaining questions. Can you say you understand the world mathematically if you can't fully decode it, and even if you can't observe all the details, nor calculate solutions for comparison.
What we have is a field that is very far advanced in the current paradigm, what is call the [[Standard Model of Physics]]. The T-shirt equation links to a table of particles that was completed with the discovery of the Higgs Boson. The paradigm allows scientists to build machines and detectors the produce exsquisite results across a spectrum of sciences. What we don't have is a way to think deeply about the real ontology of the universe.
We need to step back from the best contemporary
[[http://en.wikipedia.org/wiki/Theory_of_everything|Theories of Everythng]] (ToEs) areand notlook basedto the processes by which science works and advances. If you take Kuhn's concept of science advancing mostly as "normal science" within a paradigm, then all this talk of comprehensive theories just around the corner is a sign of a paradigm at its end. It is unfortunate that much of Peirce' work remains unfinished, but I think he was on semioticsa path towards a comprehensive foundation for a final theory of the scientific endeavor as a whole.
Philosophically, shouldn't we begin the analysis of theory, understandings of the world, with a level that includes the beings doing the analysis? The anthropic principle says we have to have conditions for life
and eventually being who do notscience considerand philosophy. His taxonomy of the onticfields statusand subfields of signs. knowledge put logic, phylosophy and mathematics in functional relations to one another, and connects it all together with his model of Signs. When you say that the T-shirt equation is a sign, the meaning of that equation is found in the entire spectrum of work that flows from it. Only an inquiry into signs and semiosis can hope to create the permanent foundations to inquiry and discovery of the world in general.
Ideas like Tegmark's [[http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis|mathematical universe hypothesis]] might be better understood with a foundation of triadic signs. signs. Same with Wolfram's group who are theorizing a physics based on simple digital automatons, these theories depend on concepts that say when two "states" are the same informationally in some sense the are "really the same thing" such that they collapse into each other. Since these are computational models of grounding ontologies, you aren't really saying anything different than Peirce saying it is all signs. Similarly, Everettian multi-world theories would need a similar mechanism to fold worlds with the same state arrived at on different branches back together. All of these approaches produce an essentially idealistic and pan-psychic world. If you filter out the occult implications, but you cannot escape the fact that reality, whatever is actually true at the deepest ontological levels is a connected whole. The vey logic of the scientific endeavor assumes it, that we find out the same things if we look deeply enough.
We may not understand the maths and languages of alien intelligence, but it should be possible if we can learn them. All such processes are going to be semiotic, based in signs and relations of signs.
Carlo Roveli speaks of "relational quantum mechanics" in which it might be best to think of the elemental relations as completely constructed of signs.
The way that Feynman diagrams are always constructed of three legs per node is just like Peirce writing that signs need three parts and no more because you can construct higher degree relations from triads, but you can't get a triad from a diad. Of course, each leg of the diagram is a sign, and just like semiosis that constructs meanings from constellations of signs, quantum scientists construct constellations of individual Feynman diagrams that are used to calculate observational expectations.
Physicists characteristically claim what is essentially philisophicalphilosophical territory as their own,own. oftenAs stating that, "Finding a ToEphysical istheory one of theeverything, majorthey [[http://en.wikipedia.org/wiki/Unsolved_problems_in_physics|Unsolved problems in physics]]could".  Thedeclare modestvictory versionand ofgo thishome.  projectTo isgo aboutfurther, integrating theories of the large and small, of gravity and quantum mechanics, and that work is clearly within the bounds of physics.physics, but we are more often in the realms of things that are untestable in practice if not theory as well.   When we get into questions of what really is so in the world, what is real vs. ideal, permanent vs. dynamic or emergent, we are no longer doing physics.  These philisophical arguments often concern the "status of un-observables" in the theory.  TheseThese are objects that appear in the mathematical theory to signify elements of the theory that may be only indirectly observable.  The observables are calculated from mathematical analysis of models in terms of idealized objects from the theory, but if the measurements closely match the calculated predictions we say the theory is (relatively) accurate. The theory only coverse the situations where we observer the phenomena, and whether and how "the same laws apply everywhere" is never an observable. We assume consistency of fundamental law as a given. This is, in fact, the point of departure from magical meanings and superstitions and their ontologies with competing gods and spirits that inhabit the world.
That's what Tegmark's hypothesis is about, assigning reality to ideal objects in the theory and that's why he need signs to ground his ideas. In other words, a comprehensive (mathematical) theory of signs could provide a metaphysics that gives reality to the simulations. Only when you grant ontological status to signs can you claim the simulation has reality. It is precisely the grounding of all of this in embodied experience that grants real status to the abstract entities of science. Roveli's description of properties and observations only being meaningful in relations between interacting systems is a way to say that such interactions are "sign systems" from bottom to top. This is about supporting strong emergence, turtles all the way down. Semiosis of signs vs. elementary partical interactions are just different descriptions, so that complex systems are grounded in the standard model without needing to connect the dots. The emergent properties of higher levels of organization will be better described by sign systems developed by investigators looking at that level. Signs, much in the forms of language including mathematics, are the way all the layers of the analysis are linked such that the realities of each and the whole itself are enacted and experienced.
What these models, signs, relational QM or computational ontologies have in common is a kind of wholistic idealism at the ontological foundations of the world, and the problem for each is how to recover the real from the story. Peirce is precient in his turn towards a much deeper analysis of signs, and I consider his work to be foundational to current and future work on these problems of logic and meaning.  What all of these myopicincomplete theories have in common is that they never consider the reality of the theories themselves and the signs they contain.  They are only concerned about the reality of the objects that the signs in question signify as if semiotic action were not part of the real world.  Sucinctly, the the real clearly includes ideas (signs, meanings) and their beingness, their ontology, is something that must be addressedaddressed. in a true ToE.  Peirce has this to say about the importance of semiotics:

{{The Importance of Semiotics}}

When we pull thinking back from some meta-world into reality, the importance of his project becomse clear.  He is working to bring scientific and philisophical productions into the world and subject it to a rigorous treatment.  Arguments about [[http://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences|the unreasonable success]] of mathematical science aren't careful to distinguish between the [[http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems|limits of mathematics]] and the reality of a complete and absolute mathematics.  To be sure, there is a large measure of faith involved in the leaps necessary to get from our necessarily limited experience and semiotic capacities to any ToE.  We could take Gödel's theorem to suggest that mathematical refers only to this limited space of incompleteness but I see no reason we cannot consider the existence of a complete mathematics, God's math, absolute, complete and consistent.  Consider the related [[http://en.wikipedia.org/wiki/Halting_problem|Halting problem]] which concerns the completeness of sequential computing.  With infinite time or some sort of magic parallelism in an infinite computational capacity, there is a concrete, finite answer to all instances of the halting problem where the answer is "yes, it does halt after N steps".  Nothing says the world, or God cannot do infinite calculations, if our theory claims that this is impossible, we have to say why not.
When we considers his [[http://plato.stanford.edu/entries/peirce-semiotics/#DivInt|final interpretant]], we see a description of the computation of thought and the production of true theories.  The construction of final theories by logical progression according to a rule governed process.  Pierce wouldn't have had Gödel's theorems to challenge his thinking on this, so he seems to project somethng like logical positivism in a not completely analyzed projection that the process is effective, that it will inevitably lead to a complete understanding.  It would have been interesting to see how he responds to these discoveries and the projects in fields like cognitive science to understand intelligence and thinking.  He was so far ahead of his time in so many ways, it would not be surprising if he did develop related ideas, but it also isn't a weakness in his work, just a place he didn't get to yet.
In [[Autopoiesis of Signs and Systems]], the scope of semiotics is expanded to the point it is almost a ToE in its own right.  Although Peirce make use of his semiotics in all analytic projects as described in the quote above, I can't really find any direct evidence that he makes a similar leap.  In all cases, the semiosis is happening in the mind of a thinking being, a human being as the only true example of this, and the signifying is about the systems under inquiry and the interpretants have the character of human thought.  The claim here is that the reality of signs and signifying goes much deeper, that we can model many systems better if we can consider the semiosis in terms of any (sub)systems that have internal organizations and internal states that can be interpretants in a sign process.  In Peirce, chains of interpretants (i.e. where one interpretant is the sign for further interpretation) only lead to more interpretations.  The interpreters are not just symbol processors, they are also objects in the world and the result of many interpretants won't be yet another sign, but an action that responds to the signs present.
In the end, none of this is really a limitation in Peirce, but a reflection of his focus on the history of argument and the logic of thought.  My claim is that his work on signs is a missing element in much current work.  SomeSome of his ideas have infiltrated science through the back door.  Few attribute Peirce as the source of their thinking, but would do well to reframe their work on his foundations.  ItIt seems like some cyberneticists and biologists have come to related insights along their own paths, but unfortunately this foundational work isn't fully integrated with later developments.  MuchMuch current thinking denies the reality of subjective experience and splits the world into the experimenter (observer) and the experiment (world) without any thought to their relationship.  In Peirce, the scientist is in the world and does his work of thinking (semiosis) in the world, about the world.
Development of the theory signs will require the inclusion of the context of interpretation.  Where does the interpretant reside?  Scientific theories are not the possession of individual minds, they contain only what can be the shared understanding of a community.  Peirce is exploring how to think correctly, how to make valid arguments and draw confident conclusions.  He isn't as much concerned with all the automatic things that happen with perception and experience, but where the semiotic approach would be very useful.  When we recognize a face, we are involking an entire evolutionary history where a person's face is their sign in social interactions.  Emotions are expressed and read from facial expressions.  Much of this is measurably independent of culture and training, which demonstrates the instinctiual nature of it.  When the perceptual system recognizes "predetor","predator", the organism goes into evolved automated responses.  These responses can be shared across related groups of organisms, and even share the same signalling systems, specific hormones or neurotransmitters.  The point is that all of this is best analyzed as interactions based on signs, but they aren't anything like the signs and symbols of conscious thought and shared understandings.  For a T-cell in the immune system of an organism, the proteins on the outside of cells and viruses are the signals that they extract the signs of "self" or "alien" (or "self-gone-bad" as in cancer cells).  These signs are interpreted by other immune cells to mount an immune response and defend the system.
This introduction is just a bit of a demonstration of the way that a more complete theory of signs and a foundatonal theory of knowledge production are needed to make progress in many fields. Where we find a mature theory such as the standard model, we should expect the breaking of new ground and many competing theories and lack of clarity on the full implications of any of them. For my money, the ideas that better explain the arrow of time are important. Many of these involve concepts that are hard to accept and require stronger justification, for example Barbour's concept of a Janus point where time just seems to be flowing away from high entropy states.
Which brings up another sticky topic, thermodynamics in contrast to all the rest of physics is not time symetric. I suspect the problem here is again one of how we interpret the signs. If the properties of physical entities only get reality in their relations and interactions, then it doesn't make sense to talk about the micro-states of a physical gas. We can watch systems evolve from unmixed to mixed and go through states covered by the same math as the emergent complexity in Wolfram's digital computational theory, but we can only infer they are equivalent. It is just deeply suggestive of the sort of real complexity that emergest from simple states.
If each interacting entity has a kind of reality as a sign, then all we are seeing in the flow of states towards lower entropy is the evolution by simple rule. With no hierarchy of systems components that can interact, there are no boundaries for a history of interaction to organize and become patterned.