symbolic - What is a type in Wolfram Mathematica programming language?

"Everything is an expression" is a popular citation from many Mathematica guidebooks. So, what is type in Mathematica? How does it relate to common types from Haskell, for example?

I did some simulation of dependent types:

DependentType::illegal= "Value is illegal for dependent types with constraints";

FixedSizedVector[n_?(Positive[#]&)] :=
Module[{type = Symbol["Vector" <> ToString[n]]},
    type[dat_] := Message[DependentType::illegal] /; VectorQ[dat] && Length[dat] != n;
    type
];
types =
Map[
    FixedSizedVector[#]&,
    Range[1, 25]
];

I have received an array types of symbols that are constrained with pattern checking rules. Is it a family of types?

Another point is Head replacing. For example, for list Range[1, 5] I can just type Plus @@ Range[1, 5] and get Integer[15]. From the point of view of type, what is Apply?

Answer

The nearest Mathematica has to "types" are Heads of expressions that are Atoms. For example:

Through[{AtomQ, Head}[2]]

{True, Integer}

Through[{AtomQ, Head}[2 + I]]


{True, Complex}

Through[{AtomQ, Head}["cat"]]

{True, String}

and so on...

There are also somewhat different "types" in the context of Compile.

Comments

plotting - Plot 4D data with color as 4th dimension

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plotting - Mathematica: 3D plot based on combined 2D graphs

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