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Mathematical models of the phenomena of the world, such as the weather, are used to make predictions about the outcome of the phenomena from an initial state.

These models are applied as computer programs to simulate the phenomena. The output from these computer programs is the information that the mathematical model predicts from the initial conditions.

Engineers use mathematical computer models to predict the outcome of events to structures such as bridges and buildings.

Computer scientists and neuroscientists use mathematical models to corelate brain activity with mental activity, and the goal everybody knows is to create artificial intelligence.

In all of these cases and in general, is a mathematical model limited as an approximation of the phenomena or is it ever possible that the model can replicate the phenomena precisely?

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    No. First, the title question does not match the one in the post. Even if a mathematical model "replicated the phenomenon precisely", it would still only be a precise mathematical representation of it, hence distinguishable from the phenomenon itself. The closest it can come to "indistinguishable" would be in analog models, but even there a phenomenon is typically modeled by another, more accessible, one. In fact, accessibility and ease of manipulation are the points of modeling, so "indistinguishable" models would defeat the purpose.
    – Conifold
    Jan 25 at 23:38
  • You don't need a map if you have the territory itself.
    – Rushi
    Jan 26 at 7:21
  • Yes. If you consider a mathematical model itself to be a "phenomena", then you have the trivial case where a mathematical model is indistinguishable from itself, and of course it is a model of itself. Feb 8 at 16:52

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Our knowledge of the world is holistic so failure of the model to predict certain phenomena might be equally blamed on the measurement. Given, however, that our measurements (our data, input to the model and the standard against which its outputs are compared) are infallible and always reliable, I don't see a problem with giving an exact model of some phenomena. There are however cases (such as true indeterminism) where would expect to never converge on an ideal, ultimate model - we could, in other words, have two models which are equally reliable but don't predict the same phenomena, because the phenomena cannot be predicted given the data we have or even any data we could have. This is a similar case to one in which the data we supply to the model is insufficient to determine the phenomenon (that's why in quantum physics some interpretations assume "hidden variables" which allow to avoid indeterminism on the ontological level - although not on the epistemic level, exactly because the variables are hidden).

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If I interpret your headline question literally, the answer is obviously no. Mathematical models are inherently different because they lack the physical properties of the phenomena they model. A mathematical model of a fire is not hot. A mathematical model of a bridge won't get you across a river. A mathematical model of an LED doesn't give off light. Etc.

If your question is, can a mathematical model quantify all of the properties of some phenomenon to such a degree of accuracy that there are no measurable discrepancies, then the answer depends on the nature and complexity of the phenomenon being modelled. You could probably generate a numerical model of the wave function of an electron in a helium atom, say, that was sufficiently accurate to account for experimental results, the challenge there benefitting from the fact that the system is a simple one and our ability to probe the accuracy of the model through experiment would be limited.

For many systems of interest, we are a trillion miles from being able to model them with complete authenticity. Take weather forecasting, for example. The models used today are necessarily very broad approximations, since it is utterly impossible to model the entire weather system down to the detail of individual molecules. In any case, the point of models is to predict physical quantities to an acceptable level of precision at an acceptable cost, so the challenge in modelling is to adopt the most simplifying assumptions you can get away with given the task at hand.

Note that there are also systems which require such complicated and computer-intensive calculations in order to model them that the precision of the arithmetical operations can become a limiting factor.

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All models are approximate. They are easily distinguished from the phenomena they replicate inasmuch as the model consists of scratchmarks on paper and the phenomenon is for example electrons scattering off of protons in a particle accelerator.

If they are all approximate, how can they be useful? Here is one way:

Imagine we are modeling the deflection of a steel beam when a certain load is placed on it. If the amount of deflection is small compared to the size of the beam, then we can make the simplifying assumption that the overall shape of the beam does not change under the load, and then write a simplified model based on the microscopic deformation of the iron-to-iron atomic bonds inside the beam. If we do this, we find that the predicted deflection matches the actual deflection to one part in a thousand, which is entirely sufficient to model the response of a skyscraper made of steel beams to a wind load.

If the load produces deformations that are too large to model in this simplified way, then we have to rewrite the model to account for the shape changes. And if that load consists of a large amount of jet fuel that starts a fire which heats the beams, then we have to rewrite the model to account for the effects of heat on the microstructure of the iron/carbon mixture that makes up the steel from which the beams were formed.

The designers of the Twin Towers performed a stress analysis with the assumption that the load was that created by striking the tower sideways with a jet airliner. This let them choose beams strong enough so as to prevent the Tower from being knocked over and collapsing if a plane crashed into it. And indeed, the model was correct: neither tower fell from the impact of a jet. Both remained standing!

But that model assumed that the plane hit the tower while maneuvering to land, in which state it would have just a small amount of fuel remaining in its tanks and the burning of that fuel would not last long enough to heat up the beams to their softening point. They'd need to rewrite that model to account for the plane hitting the tower with a full fuel load where the resulting fire stayed hot enough for long enough to cause the beams to lose strength and start to sag and bow sideways, a circumstance which would cause the tower to collapse under its own weight. Which is exactly how both of the towers eventually fell.

Finally, note that this stress analysis was based on the assumption that the plane striking the tower was lost while maneuvering to land, making the event an accident subject to the statistics of airplanes getting lost on landing approach over NYC. The stress analysis was not based on the assumption that the plane struck the tower as a deliberate act while full of fuel.

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You can try to model physical phenomena using mathematics but it will not work always because you can not put God or Devil into the equations. In other words , choice is involved. Choice can be rational , probabilistic, spontaneous or even irrational. It is impossible to model choice using mathematical models.

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