It is quite the interesting historical phenomenon that the collapse of the Academy into radical skepticism of Carneades is reincarnated again in the body of philosophy, which is manifest in a movement from totalizing claims that attempt to capture the truth into a radical withdrawal from definitive pronouncements which devolves into mere rhetoric rather than philosophy which often are self-refuting as opposed to the totalizing claims being subject to internal inconsistency. Not only was this a single reoccurrence of the example of the Academy in the modern climate but rather a recurring theme that cycled around a few times, each time gaining more momentum, from medieval realism to Ockham’s nomalism, from enlightenment’s rationalism to Kant’s idealism, from Hegel to postmodernism, evident in that each collapse of a system is followed by attempts at strengthening the edifice and correcting the inconsistencies of previous systems leading to more totalizing claims which fall under the weight of their dogmatic correctives which depend on assumptions that are open to critique or unconditional dissent.
The Mathematical Structure of Philosophical Fragmentation
The plurality of additional assumptions needed to bolster the edifice of a system of philosophy becomes the seed of its undoing since either one can simply reject one or more assumptions for all to fall apart or can build another system based on a set of assumptions from the former or can add to that set of assumptions for that goal as well.
Mathematically it goes like this: system A is founded on assumptions $\{P_1, P_2, P_3, P_4\}$, where $P_i$ are axioms from which the system A is built. One can reject all of them and thus have no reason to adopt A, or reject some of them, for instance $P_3$ and have $\{P_1, P_2, P_4\}$ and build system $A_1$, or add a $P_5$ and have $\{P_1, P_2, P_4, P_5\}$ and come out with a different system $A_2$. The issue is made worse the larger the set is, where the number of sub-systems grows exponentially according to the number of assumptions. The order of this growth is presented as $2^N - 1$.
The Self-Refutation of Skepticism
Skepticism in this case in its purely theoretical aspect becomes the rejection of all the assumptions but still the only assumption taken which by the definition of skepticism should not be in the set of assumptions is the assumption that all assumptions are false, which we might represent as this $S = \{P_s\}$ where $P_s = \forall P \in \mathcal{A}: \neg P$ but since since $P_s \in \mathcal{A}$ (as it is itself an assumption), therefore: $P_s$ asserts $\neg P_s$ which is a contradiction.
This leads to the fragmentation of philosophical systems and ideas and the birth of ones that either are also subject to the same process or the skeptical philosophy which collapses under its own assertions of skepticism. This also touches on all other endeavors not only theoretical and abstract philosophies, even scientific, ethical, political currents which are necessarily under-girded by philosophical systems.
Gödel’s Theorems and Philosophical Necessity
Given that we have put this problem within a mathematical and logical context, it is good to bring out our good friend Gödel whose two incompleteness theorems illuminate the issues at hand as well, these state that:
- Any sufficiently complex formal system is either incomplete (cannot prove all truths within its domain) or inconsistent (can prove contradictions).
- No sufficiently complex formal system can prove its own consistency from within itself.
Applied to philosophical systems which are inherently logical in the sense that they construct rational frameworks to account for truth which attempt to be complete and consistent—insofar as such systems can be understood as analogous to formal systems—we get the following: For any philosophical system $S$, there exists a “Gödel sentence” $G_S$ that essentially states “This sentence cannot be proven within system $S$.” and then if $S$ is consistent, then $G_S$ is true but unprovable within $S$ proving its incompleteness and if $S$ proves $G_S$, then $S$ proves the statement to be false, proving its inconsistency.
The $2^N - 1$ scaled fragmentation becomes structurally analogous to Gödelian incompleteness in that each subset of assumptions represents an attempt to complete the original system by adding axioms to capture what was unprovable (increasing $N$), or removing axioms to avoid inconsistency (decreasing $N$). But the theorems demonstrate that this is futile in that every sufficiently complex philosophical system will necessarily generate new undecidable propositions which will merit a skeptical reaction of the whole system.
And also the skeptical formulation $P_s = \forall P \in \mathcal{A}: \neg P$ is analogous to a system attempting to prove its own unreliability, which creates a self-referential paradox similar to Gödelian undecidability. This suggests that the analysis is not just about philosophical contingency but about mathematical necessity. Any philosophical system sophisticated enough to address serious questions about reality will be sophisticated enough to be subject to Gödelian limitations.
This suggests that for philosophical systems aspiring to systematic completeness through rational demonstration alone, no complete philosophical system can exist, no philosophical system can prove its own reliability, and that fragmentation is structurally inevitable.
The Indemonstrable Truths
And that there are these self-referential statements that are true in and of themselves are indemonstrable, and that no philosophical system can find a way to prove them to be true, even though they are, and these cannot be denied, nor can they be demonstrated, nor talked about in a way that fully represent them. The question which should be evoked is that what are these and how do we grasp them if not by reason?
Since as we see every philosophical system is destined to collapse, by virtue of its reliance on solely autonomous rational means removed from unexpressible experience which are the indemonstrable truths.
This invites contemplation and not reasoning, nor can we speak but rather be reverently silent and suspend thought.
Though to clarify I am not arguing for the complete unsolvability of all philosophical system, but only ones which are intending to fully systematize reality based on rationality and logic alone. And I stress that this effort should not be taken to be a systematization itself.