Informally, a system of the kind I'm asking about is an arrangement of physical components that interact causally with each other and with an external environment. For example: a pendulum, a car, a city, a marsh, a brain.

But more formally, what is this? Has anyone come up with a mathematically rigorous idea of what a system is?

Such an idea would need to meet a series of requirements:

  1. It would need to account for the system changing over time and interacting with its environment.
  2. It would need to account for the system gaining or losing parts, such as birds from far away flying into and becoming part of a marsh system.
  3. It would need to account for the system changing its position in relation to its environment, such as a car driving around.
  4. It would need to mathematically embed the system within the universe of which it is a part. For example, the system state could be a function of the universe state - except that some reflection reveals this would fail requirement 3. The problem would be if there are two physically identical objects. The only way to distinguish which is which would be by tracking how they move around; you cannot tell this from a snapshot of the universe state at a particular time.
  5. It should account for when a system ends and is no more.
  6. It should not include anything "extra" - the mathematical model of the system should not depend on information that is not physically present in the universe.

It seems that a mathematical specification of a system could be important to a philosophical understanding of basically anything. How can we speak rigorously about a "brain" or "consciousness" if we cannot rigorously say what class of object a "brain" might be? I have tried but have not come up with a satisfactory answer here. And yet it seems tantalizingly close to something that could be formalized. We use math all the time to model systems. We just don't mathematically have the general concept, as far as I know.

  • en.wikipedia.org/wiki/Systems_theory might be of interest.
    – user4894
    Feb 17 at 18:33
  • 1
    or, for what I'd think is the broadest approach, Cybernetics Feb 17 at 18:54
  • @JulioDiEgidio You'd think cybernetics or systems theory would have a rigorous definition of a system, but if so I have not located it.
    – causative
    Feb 17 at 19:28
  • @causative I do think the definition of system Cybernetics gives is rigorous (then the classifications/properties, and the relationship to an environment, etc.), but I'd have hard times myself giving any references, and I am reluctant to just give an informal definition in my own words. -- I will see if I can find/concoct anything useful, but no promise, resources on Cybernetics proper are scarse to say the least. Feb 17 at 19:38
  • 4
    Mathematics does not describe what a system "is". Math describes how a system behaves. Feb 17 at 20:40

9 Answers 9


Thank you, I think this is a terrific question because it goes to the very heart of what it means for us to understand something. And if I may get straight to the point: I don't think that "mathematical specification" is understanding -- even though the former could be used to communicate the latter.

I think an individual understands something (a system, a process) when they manage to visualize, by assembling in their imagination, an interactive model of it1. As such, understanding itself is visual -- not verbal. Once we are successful at this, we can describe what we see in our minds using either an imprecise language or, if possible, more rigorously with math. This, however, is how we communicate our understanding to others -- it only happens after the fact, and it is not a precondition for achieving the understanding in the first place.

As for whether we have a universal way to describe any system -- can we say that any system is a finite state machine? If so, it can always be described by a graph.

1 And we do it as part of a larger mental simulation of the world that we strive to understand.

  • Well, a state machine, sure - not necessarily a finite one. But saying it's a state machine only accounts for some of the requirements. It is also necessary to say how the state machine is embedded in the larger universe.
    – causative
    Feb 17 at 20:34
  • @causative -- Perhaps the larger Universe (and everything in it) can also be understood as a state machine? And, if spacetime itself is quantized, a finite one too? Feb 17 at 20:50
  • So, blind people can't understand things..? The information in words is kind of a '1 dimensional' stream of sound, visual information is richer & contains a field of information that can be transmitted simultaneously. The Holographic Principle suggests such transmitions are limited to 2D, not 3D.
    – CriglCragl
    Feb 17 at 21:03
  • @CriglCragl -- Oh, the blind people totally can. In fact, they can be it better at it because our eyes are just the sensors. The seeing happens in the visual cortex, and blind people can use other senses (the sense of touch, hearing, the sense of location) to reconstruct and visualize a 3d model of the world around them. And they sure can use the same mental processing to visualize abstract models. Feb 17 at 21:05
  • @CriglCragl -- and yes, words are a 1-dimetntional stream. We serialize those 3-d structures (4-d because they are virtual machines and run in time) that we visualize in a stream of words -- and we do it in hope that the person on the other end will reconstruct the same 4-d structure in their imagination from this stream, so they will see what we see. This is cumbersome, but this is the only way available to us -- the only way we can understand each other. Feb 17 at 21:16

I was surprised to find that a short, respectably published answer to this question was hard to find, so I wrote one myself. If somebody finds one and answers with it later, please comment on this answer so that I can defer to it. You might be able to get a better answer than mine on Physics SE, but I didn't find one with a search. This partially answered question and links therein may be of some use.

The model of a system is a mathematical object which can be mathematically defined. It is the ordered set of interacting mathematical objects and the rules by which they interact.

The system is the part of reality thus modeled, and has no mathematical definition except for the fact that it is thus modeled. You can trip over your cat, you cannot trip over a mathematical object that represents your cat.

The component definitions of a mathematical model of a system are:

  • Simplifying assumptions and equations of state. I have grouped these together because they're the same mathematical objects.
  1. The topological space. (Which includes rules for determining relative positions and orientations, if applicable.)
  2. The complete list of mathematical objects that aren't the elements of the topological space, but whose position or distribution is associated with the elements of the topological space; and their associated rules for their relationships to other elements in the system.
  3. The complete list of variables that are elements of each mathematical object and their rules for their relationships to other elements of the mathematical object. (For instance, in classical physics, an object's mass is that variable which is inversely proportional to the second time derivative of the position vector associated with a particular object.)
  • Boundary conditions. How are we initializing all of our state variables?
  • Domain. What are the rules for where our model is physically meaningful? This could be as simple as a start and end value for the time variable or as weird as "between observation events".
  • Observables. What do you do to the whole mathematical object to make it output a prediction of a particular measurement?

Notably, a model embeds a system in the universe and handles your need-to-account's by defining the whole universe. For example, if you need to account for mass transfer across the boundary, you just add a "mass transfer across the boundary" object to the universe. In a rare convergence of consistent terminology, this makes the model of a system a mathematical universe, which is where you'll find a formal definition in terms of set theory.

The system's domain is the domain in which the person doing the modeling hopes that that definition of the universe is an adequate approximation as regards predicting measurements. Its rules for observables incorporate what the person doing the modeling hopes adequately accounts for the fact that in defining the universe, she has pretended that she doesn't exist - but she still needs to make predictions.

Practical errors enter via:

  • Simplifying assumptions about the universe that turn out to be false.
  • Boundary conditions based on false or irrelevant measurements.
  • Nonphysical equations of state (bad physics / biology / whatever).
  • Incorrect guess about the domain of relevance.
  • Incorrect choice of observable operators.

Cognitive errors confuse results via:

  • Mistaking state variables for measurement predictions without the intercession of observable operators.
  • Mistaking the model for the system.
  • Mistaking what you've actually predicted for what you'd like to be able to predict. (Or just lying about it: P-hacking, etc.)
  • Forgetting that we had to pretend that we didn't exist when we made the model, but actually we do exist.

I'd recommend the first few chapters of Experimentation by D.C. Baird for a practical guide to the methodology, and the first half of Time Reborn by Lee Smolin for a description of the kinds of cognitive mistakes that this methodology makes scientists susceptible to. Experimentation is written for physics students but its early chapters are intelligible to an educated layman. Time Reborn is written for a lay audience.


Why do you expect there to be a rigorous generic definition of a system? You might as well ask for a rigorous definition of circumstances, arrangement, environment, weather and so on, all words whose inherent vagueness is what allows them to be used as convenient umbrella terms to refer to a wide range of broadly similar things.

If you wish to model a particular system, or type of system, you can define it with whatever degree of rigour suits your purposes.

In your question you suggest that a rigorous definition needs to include the location of the system, how it is 'embedded within the Universe', and so on. Why is that necessary? If I am analysing the workings of a clock-work toy, its location and embedment in the Universe might be entirely irrelevant. In physics and engineering, systems are modelled with the whatever degree of rigour you can get away with, on the grounds that unnecessary precision simply increases the effort involved.

Sure, there might be conceptual commonalities between the workings of say, a lawnmower and brain, but seeking a formalism that allows both to be modelled on an identical basis seems to be missing the point that it is their individual characteristics that we are interested in, so specialised models which can be used to given detailed insights about one or the other are more beneficial than generic models that tell us little about either.

  • Well, I've privately come close to a rigorous definition. There are just technical problems that crop up that I have yet to solve satisfactorily. I see no reason to believe it's impossible. Also, given the centrality of the concept, if it could be rigorously proved that my requirements are impossible to satisfy, that could be very enlightening. (And perhaps there would be clues about which requirements must be loosened to yield a coherent definition.)
    – causative
    Feb 18 at 10:00
  • I'm not saying that you can't come up with a general way of defining a system- I am questioning whether it would add any value. Feb 18 at 10:23
  • In your first sentence you do ask me why I expect there to be a rigorous general definition. (About why it adds value - there is always value in making ideas rigorous, if possible. At minimum it allows us to speak unambiguously, and it may allow us also to prove theorems.)
    – causative
    Feb 18 at 10:42
  • @MarcoOcram Location and "embedding" is pretty important for your hypothetical measurements of your hypothetical toy. All we have to do is have your lab assistant hit it with a hammer or hide it in a cupboard and your model is completely useless. And hammers and hidden cupboards are pretty much the same as leaving it completely alone, on the scale of the universe.
    – g s
    Feb 20 at 10:51
  • @gs true. And I suppose if the clockwork was embedded in concrete, that might have an impact on its workings. Feb 20 at 11:23

Systems Analysis

What did the Zen student say to the hot dog vendor? Make me one with everything!

When we analyze the system as a theoretically separate thing then as the analyst we impose the boundary or definition that separates the system from its surroundings. This is well-known in Thermodynamics the analysis begins with the imposition of the boundary and the characterization of the boundary to define an open or closed system. An adiabatic system has a boundary that prevents the flow of heat between the system and its surroundings. The point is we invent the distinction between the system and its surroundings using our human imagination.

In physics a particle has mass but no volume and objects with volume are treated as systems of particles by imposing the system boundary.

Aristotle thought the surroundings were ideal in heaven so a body (with mass) would go in perpetual circular motion. The boundary between the heavens and the earth was somewhere inside the orbit of the moon. Galileo united the heavens and the earth when he said that a body projected at initial speed on an infinite horizontal plane would remain in uniform straight line motion unless it encounters resistance to motion. Then we began to understand that bodies in outer space do not encounter resistive forces caused by friction and fluid dissipation on earth.

Modern causal models are systems of systems that we analyze on distinct levels as described in this talk about causation by George Ellis:


Economics is my favorite non-scientific science no one knows how we set the price level! If there is perfect competition then there are no price setters and if there are not price setters then who sets the price? If there are price setters then they are setting prices in the deal-making networks. They banned me from the Economics SE for a year a while ago because I kept teaching them that their models are mostly toy models for academic ability to solve the math.

  • This is not mathematical at all but I am upvoting because I've been considering a similar idea. It may be that to define a system in a way meeting my requirements, the definition must include not only the system but also a second mind-system that defines the boundary of the first system. I'm not sure if this idea can really be made to work though. How do we define the boundary of the mind-system?
    – causative
    Feb 19 at 2:59

What is a system?

I was very surprised to see how the word system is defined in various disciplines. I will focus on engineering, but Google "What is a system in engineering/biology/physics/math etc." It's eye opening.

  • Engineering: a combination of components that work in synergy to collectively perform a useful function or the combination of elements that function together to produce the capability required to meet a need.

An automobile engine is a system in engineering. This system is described via blueprints and mechanical drawings which are used to create the physical parts in the system. The properties of these parts including how they behave when in motion, are described by physical laws expressed as mathematical functions.

In engineering, it is far easier to model the system with both drawings and mathematics than just mathematics alone. This is the basis for CAD based engineering software that allows the engineer to "see" the system work. Rather than providing just numerical output.


It is one thing to write down a wishlist of requirements for a mathematical specification. But it is quite a different task, to specify and implement a mathematical model which satisfies these requirements. The difficulties are exemplified by your examples. Let’s only take pendulum, car, city, brain.

  1. The mathematical model of the simple pendulum is quite satisfying, also the solution of its dynamics for small amplitudes.

    The physical theory shows clearly how a mathematical model of a system could look like. But even for arbitrary amplitudes of this model we do not have a closed analytic solution.

  2. On the other end of the spectrum, neuroscience attempts to design a mathematical model of conscious processes, e.g., the “Information Integration Theory of consciousness (IIT)”. The gap of the actual model to your list of requirements is considerable.

Hence I consider your requirements rather utopian and out of touch with reality.

Moreover, before one can design a mathematical model one has to have a good understanding of the system in question. This is not just mathematics and should be done as the first step.

  • There's no need for closed analytic solutions. I'm only looking for a rigorous specification.
    – causative
    Feb 17 at 19:11
  • But, we do have closed analytic solutions for a pendulum, as an example of a simple harmonic oscillator en.wikipedia.org/wiki/… And, if oscillations go beyond the start point of movement from equilibrium, you just have rotation. Now, a double pendulum, that shows high sensitivity to initial conditions, & can only be predicted with limited precision. IIT is a computational theory, not a mathematical one. The difference is important.
    – CriglCragl
    Feb 17 at 22:27
  • @CriglCragl The usual solution of the differential equation of the mathematical pendulum approximates the displacement sin x by x, which is admissible for small amplitudes.
    – Jo Wehler
    Feb 18 at 1:08
  • @JoWehler: At large amplitudes below rotation it's just an expansion of the Taylor series.
    – CriglCragl
    Feb 18 at 8:23

"The centre of the system of the world is immovable."


The modern scientific use of the word system was pioneered by Carnot, in developing theories of thermodynamic engines. The Carnot Cycle:

The Carnot Cycle, diagram

It's interesting to think about why Maxwell's Demon isn't possible, which seems to hinge on considering a closed system, and not just the recording of data but also deleting of it to record new data, to close the cycle like an engine considered in light of returning to it's initial state. A system implies something with interacting parts that persists, and it seems important to think about dynamic persistence through completing cycles.

I like the term 'teleonomic matter' used by David Krakauer, Professor of Complex Systems at the Santa Fe Institute, who uses it to define the domaim of complex systems: teleonomic systems get information about their environment, record it, and change their dynamics in some way as a result. It's easy to see all evolutionary algorithms as belonging in this domain, & so memetics. Teleonomic matter also gives a ground-up way to define what an observer is in quantum mechanics: it's about the transmission of information with 'aboutness', that constrains future outcomes of a patch of the world, to a location that persists as a 'subject', ie an engine or an organism, or any system under consideration. We can look at information transfer alone in terms of Shannon channels and Shannon entropy, but systems need a persistent subject, at the end of such a channel.

Gibbs-free energy is a way to think about how engines and organisms 'extract' entropy from an energy flow, to sustain the persistence of their order, at the cost of a net increase in disorder. When a system is capable of arranging itself to better persist, you can get feedback, Strange Loops and Tangled Hierarchies, which define mind-like behaviour and subjectivities that can transmit memes.

So then, the problem with a precise mathematical way of defining and constraining systems that include Strange Loops, is that we know they can get past the Godel Incompleteness of any mathematical way of defining them, to make new systems able to include new truths, and make the definition more complex. So I'd suggest a formal positive definition that includes systems with feedback, is not possible.

Perhaps a key thing to note about Strange Loops, is that predicting their behaviour depends dramatically more on information inside the Strange Loop, than on what it encounters. For instance, knowing a person's character is a far more efficient way to predict them, than knowing the position and velocity of their atoms.


Coming from a system engineering background, the definition of a system is very simple.

  • A system is whatever is within the system boundary.

You might then ask, what constitutes the system boundary?

  • Very often large projects are composed of systems created by multiple partner companies, so the boundaries of those systems are defined by contract
  • For modelling purposes, the boundary is chosen to give useful results for minimum complexity.

For example, the fuel system on a plane is composed of the pipes and pumps and valves and sensors that deliver fuel from the tanks to the engines. A company might have responsibility for integrating that system, other companies for providing the components and assemblies. The mathematical model of the system they use to design it would include models of the tanks and engine and plane orientation, otherwise it won't handle cases where the tanks can't empty fast enough to feed the engine in inverted g, despite those components being outside the system boundary as far as the contract goes. The company manufacturing the valves might consider a valve assembly a system, so for them the system boundary would be smaller.

You also ask what is a subsystem. A subsystem is a system which is part of another system. Usually the distinction between a subsystem and a component is that the subsystem can be decomposed further. A LED light bulb is a component, but also a subsystem composed of several parts of casing, a couple of circuit boards, LEDs, ICs, inductor, resistors, capacitors etc. The distinction between system/subsystem/component got dropped from SysML and other systems engineering ontologies as it adds little value and two engineers from different companies will call the same thing with any of the three labels.

These concepts are constructed to serve a purpose, they don't really exist in the world outside of that purpose. A system boundary constitutes whatever it is useful to include in that system at that time.

  • In asking what a system is, you use the concept of system throughout. It is hard to break in on the circle. I express my own difficulty. This may be only my shortcoming; it will be interesting to see how other users react. In any case, welcome to PSE. Best - Geoffrey.
    – Geoffrey Thomas
    Feb 20 at 15:54
  • @GeoffreyThomas A system is defined only by its boundary, and that boundary is chosen in a way to be useful to the persons defining it. That is the only rigorous definition of a system I know of, and one you'll find in almost every textbook on the matter. I don't see any circularity in there - the boundary is not defined by the system, but chosen by the engineers. Feb 21 at 10:57
  • Pete Kirkham: I am finding light in the answers and comments your question has attracted. Best - Geoffrey.
    – Geoffrey Thomas
    Feb 21 at 14:47

I like to think a system is defined by a cohesive set of fundamental laws. I might posit that the set of possible states a system is capable of expressing is always deterministic. Whatever theoretical framework one is applying to a given problem, it specifies a standard set of rules or laws which predicates every valid expression, and every term is relative to every other term only by virtue of those laws; perhaps a logical system loosely represents a working environment designed to evaluate certain categories of expressions with freedom and ease.

I'm certain that the idea of a closed system is a totally imaginary concept used only for convenience- that the formal definition would describe an ideal which we could never actually realize. "Welcome to Philosophy", eh!

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