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I would say that an algorithm is a private case of model because it describes how something generally works --- just mathematically.

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    Questions about meaning of words are more suitable for English SE. Algorithm is not a model, it is an instruction for performing calculations, and it does not have to be private.
    – Conifold
    Oct 17 at 4:53
  • A model is a mathematical representation of an object or phenomenon, usually a formula or a set of formulas. An algorithm is a step by step systematic (it does not require reflexion at any step) way to compute a solution to a problem. An algorithm can be based on a model, like a game of chess represented as a series of numbers, but those are two very different things.
    – armand
    Oct 17 at 4:57
  • In mathematics a "model" is a collection of "objects" with their properties that satisfy a theory: natural numbers are a model of arithmetic. Oct 17 at 7:51
  • An algorithm is a "procedure" to solve a problem/produce a result,like the rules for addition that can be applied to a pair of natural numbers to compute their sum. Oct 17 at 7:52
  • Just as an observation, as well as what these words mean in mathematics, there is also a colloquial AI usage of "Model" meaning something like "a system of weighted bayesian decision networks after having been updated with training data". In this respect, sophisticated AI algorithms sometimes rely on an underlying bayesian model as part of their process of making unsupervised decisions, but the concept of Algorithm is broader than those that use Bayesian learning systems. Oct 17 at 13:35
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Formally, an algorithm is "a specific set of instructions for carrying out a procedure". Bread is not an algorithm, but a set of instructions on how to bake bread is an algorithm. A formula is not an algorithm, but a set of instructions on how to solve a formula (such as PEMDAS) is an algorithm. Pi is not an algorithm, but a set of instructions on how to calculate pi is an algorithm. None of these algorithms are models.

A model is a known system which has been claimed to behavior similarly to an unknown system. An algorithm can be developed which then becomes a model, or a model can be proposed which is then tested by an algorithm. Conway's Game of Life is an algorithm about how to change a grid of 0's and 1's; it became a model when someone claimed that the 0's and 1's behave similarly to organisms. The Combined Gas Law is a model; an experiment designed to compare the predictions of this law to actual gasses is an algorithm. There are very few model that have no corresponding algorithms because when someone creates a model they usually propose a algorithm that would allow testing of the model. In principle, they can exist though.

When Democritus first proposed his theory of atoms, he was proposing a model. However, an algorithm that would allow comparison of this model to the actual world did not exist at the time. Eventually such algorithms were developed, but for a while there was a model that did not correspond to any algorithms.

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