One couldn’t a priori say whether, ontologically, there’s a single formal system. If there were, it seems that a single semi-formal heuristic could model it.
That single semi-formal heuristic, which would relate the distinct nondeterminate & determinate syllogistics of a given system, could, in principle, model a single formal system.
Concretely, it seems, for example, that in a materialist monist ontology, nondeterminate static relations, whether in/finite, un/bounded, un/curved +/- and so on, could constrain otherwise determinate, dynamical activities of an energy plenum. We could only model the nondeterminate relations but not explain them, other than to say they’re necessarily like that, noncausally, by the very axiomatic nature of that given formal system.
Can a TOE be formalized?
I would hold that a TOE most certainly can be delimited formally.
Even within given layers of complexity, we can not only successfully reference but even robustly describe properties, entities & relations. With such descriptions, we are not only able to abductively hypothesize & inductively test but can complete a virtuous cycle of triadic inference, deductively using univocal terms, both semantically & ontologically, achieving a certain explanatory adequacy.
It is when we are methodologically thwarted & ontologically befuddled by the emergence of novel properties, entities, states & systems, confronted by reality’s various aporia, that we are forced to fallback on mere exploratory heuristics in order to make vague, overdetermined possibilities more precise; general, underdetermined probabilities more specific; and ambiguous actualities better defined.
When we are seriously thus thwarted, we can engage in a rather nonvirtuous cycle of abductive hypothesizing & deductive clarifying, unable to interrupt it by inductive testing. That dyadic inferential cycling can be efficacious and hypothetically fecund, if done rigorously, opening our minds to new avenues of exploration. It can also become rationalistically vicious, if we imagine we’re thereby achieving explanatory adequacy, as it instead forecloses on research programs.
At the margins of aporia or horizons of knowledge or interface of novelty, we must engage in semi-formal heuristics because we’re relying on analogies of proportion & attribution, e.g. the quantum and the gravitational have these similarities, but those are outnumbered by these dissimilarities.
There is another sense in which nondeductive processes come into play. We inductively & abductively infer 1st principles and such, but call them self-evident, so must rely on refutation by reductio to convince others they’re wrong about, for example, this or that version of a principle of sufficient reason and, yes, even common sense notions of causation. But the semi-formal in play in this discussion pertains to how we relate nondeterminate & determinate syllogistics, respectively, in their modes of identity & being.
Specifically, the identity relations that I am using refer to the formalization of otherwise ambiguous natural language sentences, with their various meta-logics, predicate logics, propositional logics and term logics. Aristotelian syllogistics employ a term logic. In modeling a syllogistic theory, the “axioms” are just the rules employed for moving from premises to conclusions.
Otherwise, the validity we’re seeking in our nondeterminate and determinate syllogistics applies to arguments. Some arguments regarding non/determinate realities that sound counterintuitive, causing interlocutors to come out of the woodwork with reductios & charges of absurdity, can more easily be demonstrated as sound by disambiguating to which modal category they refer, i.e. a reality’s properties, existents or relations and whether non/determinate, and can thereby be made more intuitive.
We do end up with a single formal system that handles propositions about nondeterminate & determinate realities equally well. When we say semi-formal in this sense, it refers to the theory of predication and syllogistic reasoning that’s being applied to nondeterminate and determinate realities but not to the specific syllogisms employed.
Presumably, where mathematical language is being employed for a putative Theory of Everything , we could run into trouble if such a closed formal symbol system is vulnerable to Godel-like constraints, forced to choose between consistency or completeness.
Hawking believed a TOE would thus be vulnerable. See: http://www.damtp.cam.ac.uk/events/strings02/dirac/hawking/
While there are analogous logical phenomena like undecidability, the halting problem, unfalsifiability, circular referentiality and others, they shouldn’t be facilely conflated and/or applied, such as with Godel’s Theorems, to every attempt at formalization.
In the first place, Godel was talking meta-mathematically, and this is analogous to how I was talking about the semi-formal nature of a theory of predication and syllogistic reasoning for what would otherwise be ambiguous natural language sentences. In the same way that the semi-formal nature of our natural language meta-theory of predication & reasoning would not change the formal nature of the syllogisms under that meta-theory with respect to their ability to deliver formally sound deductive conclusions, Godelian meta-mathematical constraints don’t change the reliability of our mathematical formulas, though they may be variously axiomatized. At least, I have little interest in proceeding through the hundreds of pages of the Principia Mathematica, wherein the axioms required for the arithmetic system that proves 2+2=4 are formulated and proved.
Secondly, Hawking was not abusing Godelian theorems, facilely conflating and applying them to a TOE. He was just giving us an interpretive assist by invoking it as an analogy. The proper take-away may have been, therefore, not that we couldn’t formalize a consistent & complete TOE, but that it would necessarily entail fundamental limit conditions. And he didn’t mean only black hole limits on information concentrations or putative volumetric, geometric, topological or in/finite limits we’re still trying to define. What he was suggesting, some say, is that the TOE’s modeling power would be constrained predictively, since, as occupants within the system, we’d have to self-referentially model ourselves. This is not a constraint, however, on any theoretic ability to formally state a TOE’s axioms or reality’s fundamental principles, even if we remain forever constrained in that regard in terms of practical feasibility.
Finally, even if the Godelian analogy further extended to suggest that our TOE will inescapably express some true statements that could not be deduced from its root axioms, in addition to any phenomena its modeling power could not predict, it would only mean that we couldn’t derive our formal theory from those axioms, thus proving it. That wouldn’t make the formalized TOE, in and of itself, semi-formal or informal, or mean that it couldn’t, in principle, be written out. It only means we wouldn’t know if it’s the authentic TOE, except by tasting & seeing its truths and inductively-abductively inferring the truth of its axioms and their reliability, e.g. in much the same way we overcome solipsism.
Now, maybe we’ve come full circle back to an inherently heuristical nature of our knowledge approaches, but we can distinguish those from our theoretic formulations, in and of themselves, as they formalize our models, even those of Everything. And maybe we’ve discovered that primitive axioms are in a category of givens, where deductive proofs simply do not apply because it’s a category error to treat noncaused, nondeterminate relational aspects of reality as explicable other than in terms of their own nature, simply stated. If that makes all complete descriptions semi-formal and not wholly deducible, that’s just one of their intrinsic features but in no way a defect.
We can’t a priori say whether or not, some day, when an authentic, formal TOE just happens to get written down, whether its truths & reliability will be so patently obvious that – not only will we employ its axioms with all the confidence we now place in those formulated & proved in the Principia, but – we’ll be as disinterested in that TOE’s axioms as most of us are, now, in those that undergird the logic of 2+2=4.
About those Modes of Identity & Being
As an epistemic heuristic, where ontological primitives are bracketed, one might conceive of 3 categories to refer to nondeterminate realities, let’s say, properties, entities & relations, and use the same 3 categories to refer to determinate realities. Those references would be univocal, semantically, but, metaphysically, would be analogical, because we’re distinguishing between nondeterminate & determinate realities. Some would say we’re employing 2 syllogistics, modes of identity for nondeterminate & of being for determinate realities.
For nondeterminate realities, from properties, alone, we can identify the reality, essentially (it acts or does); from exemplifications of an entity, alone, we can identify the reality, existentially (it is); from the nature of its relationality, alone, we can identify the reality, formally (it effects). That’s because a nondeterminate reality always IS what it DOES as known by its EFFECTS. To know something from any of those categories of such a reality is sufficient to identify it.
For determinate realities, however, the only mode of identity is formal identity. To identify a determinate reality, it’s not enough to know that a reality is actual, existentially (that it is), or to know its properties, essentially (how it behaves), we need to know its relations, formally, such as genus & species (what it is). The identification of a determinate reality is irreducibly triadic, requiring us to know something from each of these categories in order to identify it.
These 2 syllogistic modes, together, would comprise a single semi-formal heuristic, epistemically.
When we “unbracket” the primitives, there’s nothing that would a priori commit one, in principle, to applying this heuristic to a monist, dualist or pluralist ontology vis a vis distinct modes of being, each with different primitives & axioms, wholly related or unrelated one to the next.
A monist ontology, though, could represent a single formal system (of nondeterminate & determinate characteristics) modeled by a single semi-formal heuristic.
A pluralist ontology would seem to indicate multiple formal systems (determinate formalities), which may or not indicate singular or multiple nondeterminate nomicities, which may or not indicate singular or multiple semi-formal heuristics. There could be many worlds, variously overlapping, field-like, envisioned by venn diagrams & modeled in part by set theories.
A/theological Conceptions of the Ens Necessarium and the Actus Purus
Among those with various a/theological Peircean stances, some have argued that the architectonic is either inherently atheistic or theistic. For example, certain theists may point to Peirce’s “Neglected Argument for the Reality of God,” wherein he abducted the Ens Necessarium, as evidence that it’s theistic.
As an informal, inductive-abductive inference, to me, Peirce’s Ens Necessarium could justifiably be conceived as closely related to or perhaps even derived from the First Principles. As such, it would intuitively draw a distinction between determinate and nondeterminate realities.
What the abduction of the Ens Necessarium wouldn’t a priori implicate, however, is whether or not the nature of such necessity would merely nomological or also clearly ontological. We could say that the former implicates an Analogia Axiomata, while the latter further intuits an Analogia Entis.
So, the Ens Necessarium conception presents a metaphysical a la carte menu – not only to diverse monist, dualist, pluralist & dipolarist meta-ontologies, but also – to various idealist, materialist, physicalist, naturalist & supernaturalist ontologies.
In other words, conceptually, it’s a bring your own ontology heuristic.
For example, a materialist monist intuition of being, which might locate an Actus Purus in a physical dynamical energy plenum, ontologically & immanently, would conceive any transcendent Ens Necessarium in strictly nomological terms. Such a minimalist conception of transcendence would, therefore, further employ the Peircean methodology solely in terms of a search for reality’s necessary boundary & limit conditions, essentially probing physical reality’s manifold & multiform generalities, probabilities & regularities to specify which might be necessitarian in nature.
A minimalist transcendent methodology would interrogate physical reality, for example, asking such questions as whether it’s NECESSARILY
- volumetrically in/finite,
- geometrically un/bounded or un/closed,
- topologically un/re/curved, spatialized temporally,
- temporalized spatially,
- essentially or emergently spatio-temporal,
- essentially non/inflationary,
- quasi/exponentially expansionary,
- dimensionally 2/3/4/more-D,
- with dimension/less physical constancy,
- with non/universal constancy,
- nomologically im/mutable
- and on and on and on.
Answers to certain of these questions will necessarily implicate answers to certain others.
Methodological stipulations of putative answers generate interpretive models, variously testable, empirically, and variously falsifiable, theoretically and/or practically.
Such an approach to an Actus Purus would necessarily conceive all existents and most regularities nominalistically. The Ens Necessarium, alone, would be conceived essentially per an Analogia Axiomata, accounting for regularity-like necessities.
Or, for example, a supernaturalist intuition of being, conceiving both an Analogia Axiomata, nomologically, and Analogia Entis, ontologically, would conceptually locate an Actus Purus in a robustly transcendent way. There are as many competing interpretive models of this Ens Necessarium conception as there are, well, just for one example, inflationary models of the cosmos:
The problem with inflation isn’t the idea per se, but the overproduction of useless inflationary models. There are literally hundreds of these models, and they are – as the philosophers say – severely underdetermined. This means if one extrapolates the models that fit current data to regimes which are still untested, the result is ambiguous. Different models lead to very different predictions for not-yet made observations. Presently, it is therefore utterly pointless to twiddle with the details of inflation because there are literally infinitely many models that one can think up, giving rise to infinitely many different “predictions.” ~ Sabine Hossenfelder
Regarding, then, those putative Analogia Axiomata & the Analogia Entis
So much turns on whether nature’s contingent existents are mute, brute or fruit vis a vis their primal being (materially), primal support (efficiently), primal ground (formally) & primal goal (finally).
They, thus far, appear mute regarding same, at least, in terms of our modeling attempts, which remain referential & descriptive, exploratorily, but not finally interpreted, explanatorily.
Whether they are brute or fruit turns on whether or not a final mereological interpretation of nature explains a model of reality as a whole as the mere brute sum of its parts, or as either greater than the sum of its parts or even hierarchically transcended by some supernatural reality, of which both the whole of nature and its parts, would be the clear fruit.
Whether nature’s brute or fruit, there must be some necessary aspect transcending any mere regularities. If brute, its models must still specify some nomological necessities. If fruit, its models must specify additional ontological as well as nomological necessities. The vigilant observer will note an implicit explanatory tension that presents, here, in terms of which model properly avoids any fallacy of composition. That cannot be known a priori. While we wouldn’t deny the in principle knowability of the fallacy’s applicability, a posteriori, practically, it could remain extremely problematic indefinitely.
Once stipulating to specific, static, essentialist nomological necessities, if nature as a whole is defined in terms of an eternal dynamical plenum, whether of physical energy and/or energy plus consciousness (as primitives), then its remaining properties, existents & relations will admit only of nominalist characterizations and all of nature’s efficient, material, formal & final causes will be emergently closed, i.e. both epistemologically & ontologically reducible, solely in terms of that plenum’s fundamental nature.
For example, if a physical energy plenum, its forms & ends would be, ultimately, wholly determined & capturable, in principle, in their mathematical formulations, i.e. physical equations, even if not wholly predictive due to intrinsic observer constraints, as the model couldn’t escape the self-referentiality that would inhere due to its modeler being modeled therein.
The physical energy plenum’s dynamical properties & existents would, via a nonstrict identity, functionally & structurally perdure, i.e. efficiently & materially, solely in terms of their relations to sources of energy, i.e. formally appropriating energy & finally dissipating it, entropically.
Even in an idealist model, which would take consciousness as a primitive, a nonstrict identity would apply to the practical (not essentialist) self, efficiently & ideally, solely in terms of its relations to the primal source of consciousness, i.e. formally appropriating it via participatory dynamics & finally dissipating it, entropically, as it substratively & creatively diffuses.
I will not further explicate, for example, Thomist supernaturalist accounts, for example, with their appropriations of Aristotelian hylomorphism. (Other accounts can be semiotic, analytic, process and so forth). Those hardly require the explication that I provided in the causal account, above, providing analogues for efficient, material, formal & final causation for physical & idealist monist conceptions, because many more are already, at least somewhat, familiar with those Thomist approaches.
We mustn’t pretend, however, that the entropic accounts take us any further toward ultimate explanatory adequacy than competing models of nature. They, too, remain mere exploratory heuristics, for, in the very same way that analogous telic realities present vis a vis new qualities of existence, so, too, the concept, entropy, is way too vague to accomplish the explanatory heavy-lifting that some naively imagine they’ve accomplished employing it.
Thermodynamic systems entropy, information systems entropy & evolutionary entropy (rate of organismal environmental energy appropriations & reinvestments in survival & reproduction) are indeed formally homologous & certainly hierarchically interdependent entropieS. We can describe that interdependency but that doesn’t EXPLAIN it!
Actually, I need to edit this one & will do so explicitly to better reinforce the point:
THIS: We can describe that interdependency but that doesn’t EXPLAIN it!
S/H/B: We can successfully REFER to that interdependency but that doesn’t mean we have even successfully DESCRIBED it, yet, much less EXPLAINED it!
And, when we do proffer explanations, that doesn’t mean they can’t be LAYERED (as Jack Haught has accessibly explained for decades). They may even need to be layered in order to make proper contextual sense of it all. (And no, we don’t employ Occam’s Razor to adjudicate competing accounts that don’t already enjoy explanatory adequacy.)