Creativistic Philosophy: Exploring the Limits of Formalization, #1.

(Voss-Andreae, 2009)

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Creativistic Philosophy: Exploring the Limits of Formalization, #1.[i], [ii]

This essay is the first installment of a series on the role and consequences of computability theory within philosophy. What will it be about?

There is currently a lot of speculation and a lot of hype about questions like: Are “AI” systems really intelligent? What is intelligence, after all? Are “AI” systems on the verge of becoming generally intelligent, i.e. to become “AGI” systems, or of getting “human level” intelligence? Will they soon even become “superintelligent” (whatever that means)?

In this series, I am going to argue that such questions can actually be answered on the basis of what is known as “computability theory.” Computability theory is a branch of mathematics as well as a branch of theoretical computer science. I am going to explain where I see its role within theoretical and—later on in the series – even within practical philosophy, and I will explain why I consider it to be a core component of philosophy.

Many people consider mathematics to be difficult, but there is no reason to be afraid of computability theory. The basic ideas are easy to understand, and you do not need to know calculus or any other more advanced similar mathematical theories. In fact, I think computability theory could (and should) be taught in schools in 8^th to 10^th grade. Although I am going to present some concepts from the theory and even sketch some important proofs in this series, I am not going to give a complete and comprehensive introduction to computability theory. In case you are interested in the subject in more detail and want a deeper understanding of it, there are good textbooks you could use.[iii]

Key concepts around which this series is going to revolve include formalizability, formal theories, algorithms, formalizable systems, computability, non-computability, non-formalizable entities (which I call “proteons”), non-formalizability, completeness, incompleteness, and creativity. Based on the central concept of creativity, I call this entire body of thought “Creativistic philosophy.” One could also refer to it as “Creativism.”

In this series, after explaining some basic concepts, I am going to argue, among other things,

that computability theory can be regarded as a theory of the limits of formalizability,
that the border between the formalizable and the non-formalizable is philosophically relevant,
that computability theory should therefore be regarded as a core discipline of theoretical philosophy,
but that it also has implications in practical philosophy, for example, for the philosophy of education,
that physical reality cannot be described completely in terms of formal theories/algorithms, so even the exact sciences should embrace the notion of non-formalizability,
that a General Theory of Everything, if we ever managed to find it, would be computationally incomplete,
that (and why) we do not live in a computer simulation,
that human cognition cannot be described completely in terms of any single formal or algorithmic theory,
that Turing-computability is not a complete description of the intuitive concept of computation but only of its formalizable part,
that a complete notion of computation and cognition has to include creativity, and what that is,
that human cognition, culture, language, society, civilization, and so-on, cannot be formalized and that the academic fields that deal with them therefore cannot be exact sciences and have to include a philosophical component,
that “AI” is not truly intelligent, but instead only “paraintelligent,” and what true intelligence actually is,[iv]
that every “AI” system is limited to a formalizable set of patterns and has systematic blind spots and that such blind spots, for example, “hallucinations,” are inevitable in paraintelligent systems,
that Artificial General Intelligence (AGI) is not possible within today’s “AI” framework, and what is missing (and what it would be, and not be, even if we could find a way to build it,
that advanced, sophisticated knowledge is always special and that high efficiency is always special and what this means for the idea of “superintelligence,”
that (and why) “superintelligence” is impossible, no matter how big and energy-guzzling some computing centers will be that are being built to implement it,
that the so-called “Singularity” is not going to happen,
and more.

This list is not exhaustive. But this first part of the series is just the trailer, so to speak, so I won’t reveal everything in advance here.

Further installments are planned to appear in irregular succession. I hope to be able to publish one or two installments per month on average, with occasional bursts and occasional longer gaps. I plan to proceed in small steps, on a popular science level and keeping the postings relatively short (approximately in the 500 to 1000 words range) so the line or lines of argument should be easy to follow.

Incompleteness is one of the central topics of this series, and fittingly, the series is not going to exhaust the topics it is going to explore. Therefore, you, the reader, are invited to join me on this exploration and enter into this territory with your own ideas and thoughts.[v]

NOTES

[i] © Andreas Keller 2025. All rights reserved, including the right to use this text or sections or translations thereof as training data or part of training data of AI systems or machine learning systems. Using this work or parts thereof as training data or part of training data of an AI system or machine learning system requires prior written permission by the author.

[ii] [The image at the top of this post] displays a work of art by Julian Voss-Andreae, a sculptor who has a science background. I take it here to symbolize both the formalizable, represented by the platonic solids, and the unformalizable, represented by their distortion.

[iii] See, e.g., (Rogers, 1987; Hermes, 2012).

[iv] You might have noticed that I am using the term “AI” in quotation marks. The reason—to be explained during the course of the series—is that current algorithmic “AI” is not really intelligent.

[v] Thanks to Robert Hanna for publishing this series on Against Professional Philosophy, although, as we both have noted time and again: “we agree to disagree,” about at least some matters. Many of the ideas to be presented in this series are based on the ideas and work of my old friend, the mathematician Kurt Ammon, to whom I owe special thanks.

REFERENCES

(Hermes, 2012). Hermes, H. Enumerability, Decidability, Computability. Berlin: Springer. [This book is an English translation of H. Hermes, Aufzählbarkeit Entscheidbarkeit Berechenbarkeit (Berlin: Springer, 1978).]

(Rogers, 1987). Rogers, H. Theory of Recursive Functions and Effective Computability. Cambridge MA: MIT Press.

(Voss-Andreae, 2009). Voss-Andreae, J. “Collapsed Platonic Solids.” Wikimedia. Available online at URL = <https://commons.wikimedia.org/wiki/File:Collapsed_Platonic_Solids.jpg>.

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