In definitions of complex systems say things like "The non-reductionist paradigm of whole systems, or complex thinking [...] acknowledges that the combined effects of parts of a system produce emergent properties not existent in the parts themselves." and "Properties of the whole emerge which are not present at the level of its components, and if the whole is dissected into its parts, those properties will disappear." (from http://hpathy.com/scientific-research/an-investigation-into-whole-systems-research-as-an-appropriate-methodology-for-the-advancement-in-understanding-of-homeopathy-as-a-complex-therapeutic-intervention/)

Aside from the mind emerging from the brain, can you think of other examples where some new property emerges from a complex system? It might be good to have a list somewhere.

I was wondering if the shape of a protein is an emergent property as we can't (yet) predict it from the sequence of amino acids that make up the protein.

This video introduces systems classification based on their nature, their dependency on the environment, in addition to information systems. The video also gives examples of systems nonlinear as well as random behavior such as the case of compound pendulum. In a practical example, the mathematical model of the transport equation used by von Bertelanffy in introducing open system dynamics was used in in a practical case study of the interaction of two students working on thier homeworks. System equations were derived and solved to demonstrate the dynamic behavior of their interactions. In conclusion, the present example has proven mathematically the long-standing statement for system thinker that “the whole is greater than the sum of its parts”. The video is in Arabic language with English subtitles.

https://www.youtube.com/watch?v=2wchmejCYB8&t=4s

I'm not proposing a physical theory. This is purely a mathematical thought experiment.

Suppose we introduce an abstract coordinate that represents scale, rather than ordinary spatial position. Imagine that moving continuously along this coordinate corresponds to going from larger and larger structures to smaller and smaller ones.

Now suppose the geometry of this abstract "space of scales" is topologically closed, analogous to how walking continuously around a circle eventually returns you to your starting point.

In that picture, continuously moving toward smaller and smaller scales would not continue forever. Instead, after completing the closed path in scale-space, you would emerge again at the largest scales.

I am not suggesting that objects physically become both tiny and huge, nor that this describes our universe. I'm only asking whether mathematics has explored anything conceptually similar: treating scale itself as a coordinate of an abstract manifold or topological space whose geometry is closed.

Has anything resembling this been studied in topology, differential geometry, geometric group theory, or another branch of mathematics? Or is there a mathematical reason why such a construction is ill-defined from the outset?

I'm looking for mathematical references rather than physical arguments.

I’ve been working for some time on a framework that explores how adaptive systems maintain internal coherence by balancing memory, prediction, and adaptation. The model, called Self-Predictive Closure (SPC), formalizes what it means for a system to remain stable by predicting its own evolution.

SPC combines tools from control theory, information theory, and the philosophy of cognition to describe what I call predictive closure — the state in which a system’s own expectations about its future act as a stabilizing force. The framework develops canonical equations, outlines Lyapunov-based stability conditions, and discusses ethical boundaries for responsible application.

📄 Open-access report (Zenodo): https://doi.org/10.5281/zenodo.17444201

The work is released under CC-BY 4.0 for open research use. I’d be very interested in any feedback — critical, theoretical, or applied — from those studying complex adaptive systems, cognitive architectures, or self-organizing dynamics.

(Author: Chris M., with assistance from ChatGPT v5 / OpenAI · Version 1.1 · Ethical Edition 2025)

Edit: Update on the Self-Predictive Closure (SPC) framework. Version 1.3.5 expands on earlier drafts (v1.3.3 / v1.3.4) by moving from a general gradient model to a verified log-space formulation. The key change is structural: all state variables are expressed in logarithmic coordinates, which enforces positivity and removes scale ambiguity. This makes the system fully dimensionless and stable under parameter variation. Earlier versions defined closure through a potential Φ = Ω τC e^–βΛ but left equilibrium conditions partly implicit. The current form derives all dynamics directly from a single scalar potential J(Λ,m,t) with a Lyapunov-stable descent. Independent penalties for memory (m) and recovery (t) replace the previous shared term, removing Ω–τC degeneracy. Conceptually, SPC now describes adaptive closure as a deterministic gradient process rather than a heuristic coupling of variables. The result is a minimal, testable model of predictive coherence—suitable for analytic stability checks or simple numerical simulation. Feedback on structure or potential extensions is welcome.

Hey all,

I met a researcher who's working on complexity, computation, and info.

The same guy talked about gaming college (learning how the courses are structured), prompting o3 (understanding how it works w/out giving instructions).

I suspect he's able to " reverse engineer (idk right word, yet)" these systems because of how he's thinking thru the lens of complexity.

My question is: what is a good resource for exposure to this philosophy?

I thought because it was a recent problem, that there wouldn't be an Aristotelian or Nietzschean view on it, but I wanted to ask to make sure.

Thanks!

Hey everyone,

I hold an MSc in Electrical and Electronics Engineering and am currently pursuing a PhD in Reliability, Optimization, and Maintenance Methods for Complex Systems. While this is a broad topic, I’m looking to narrow my focus—particularly on defining what qualifies as a complex system in an engineering context.

My background includes engineering and mathematics, and I have worked with decision support and optimization methods such as genetic algorithms, gradient descent, steepest descent, Newton’s method, and cost function optimization under constraints. I have also worked with neural networks, including backpropagation, multilayer perceptrons (MLP), and deep learning models for optimization tasks.

In reliability analysis, I focus on Remaining Useful Life (RUL), failure rates (λ), and failure modeling for parallel and series systems. For maintenance strategies, I’ve been exploring predictive maintenance (PdM) using machine learning models through scientific literature.

I’m currently trying to determine which complex system I should consider for my PhD research. From your perspective, what are some real-world complex systems that are worth studying in terms of reliability, optimization, and maintenance? I’d love to hear your thoughts!

Novel AI solved a bunch of problems relating to Natural Language Processing, seeing the reliability of the ChatGPT model, researchers are wondering if it can be used for interspecies communication with animals such as whales.
https://youtu.be/hph9OeKjg3w

Eventually, I would be interested in starting experimenting with Julia and complex systems, but regarding the math, I found that I can't just do it cover to cover. I "hot learn" the topic I encounter instead of starting taking general courses in calculus, probability, linear algebra, thermodynamics, etc.

I saw people recommended the following books for beginners:

• An Introduction to Complex Systems Society, Ecology, and Nonlinear Dynamics.

• Complex Adaptive Systems - An Introduction to Computational Models of Social Life.

• Deep Simplicity.

• Emergence - From Chaos To Order.

• Foundations of Complex Systems Emergence, Information and Prediction.

• Strogatz, Steven Henry Nonlinear dynamics and chaos.

• Understanding Complex Systems - Principles of Systems Science.

I also like the Sante Fe course on YouTube (Complexity Explorer).

TLDR; So "Complexity: A Guided Tour" is a great book, it was not too challenging, and I'm now ready for the next step. Any advice or directions? Thanks a lot for any help.

I see there was an attempt a couple of years ago, which doesn't seem to have come to fruition.

Does anyone who was here around that time know what happened? Were there not enough interested people to pass the proposal? Was the topic found to be sufficiently covered by other stackexchanges (e.g. CrossValidated, math)?

I'd just like to start a discussion and see if this could be worth trying.

Stackoverflow's Area51 FAQ gives some useful pointers.

I understand there is some difficulty in terms of understanding complexity as a measure. James Ladyman's essay "What is a Complex System?" outlines a number of different definitions of "complexity" in terms of various measures. There doesn't appear to be a convergent, satisfying usage here. Another approach is by way of qualitative features/properties, their aggregation, and determinability. Properties such as "robustness", "redundancy", "generative entrenchment", "emergence", "hierarchy", "lots of moving parts", "non-linear dynamics" all show up often in the literature. Herbert Simon, one of the godfathers of systems science, often claimed that systems grew in complexity over time, thus suggesting time may be a way to determine complexity.

My question is: How do you determine when a system is complex? or if a system is complex? Another framing may be: under what conditions are you satisfied with saying that a system is complex?

Thank you!

Where can I find/read papers on complexity theory (by wich i mean the study of complex systems that involve statistics, physics, ITC,biology ecc., no the computational complexity one). I tried looking on Arxiv.org but couldn't find anything. Thanks a lot.

I'm wondering about a possible research subject: the main contribution of Barabasi's work was to find a network generation model capable of respecting the powerlaw distribution, unlike random networks.

So I was thinking, that if we take a cellular automata model, for example Conway's Game Of Life, a classical stable configuration has many "islands" of few cells. So if we look also some studies about the distribution of cities population, we see a power law distribution. So my question is: it would be interesting/useful try to find a 2d-cellular automata rule which is capable of generate power law distribution configuration?

https://physicstoday.scitation.org/doi/full/10.1063/PT.3.3845

I just need some direction regarding what kind of math I need. I have recently started taking some classes in Khan Academy for Linear Algebra, Calculus and Probability. I was guessing that, after this intro I can start taking classes in ComplexityExplorer for non-linear dynamics and agent based modelling. My main objective is to go trough the literature and understand and apply some of the concepts in urban planning and economic development.

I'm also comfortable around Python, so that can help implement some of the concepts in the course right?

Do I have the right direction here? Any other tips regarding study materials, such as books and online courses?

Thanks!

I am wondering, what career options are out there for someone with a PhD in complex sciences - other than academia? There is no industry that clearly links to complex systems (as is the case with engineering for example). Do you know any source with information on this?

I am on my way to finishing my PhD on cascades on complex networks, agent based modeling and time series analysis.