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 - Filling between two spheres in SphericalPlot3D

Manipulate[ SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, Mesh -> None, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], {n, 0, 1}] I cant' seem to be able to make a filling between two spheres. I've already tried the obvious Filling -> {1 -> {2}} but Mathematica doesn't seem to like that option. Is there any easy way around this or ... Answer There is no built-in filling in SphericalPlot3D . One option is to use ParametricPlot3D to draw the surfaces between the two shells: Manipulate[ Show[SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], ParametricPlot3D[{ r {Sin[t] Cos[1.5 Pi], Sin[t] Sin[1.5 Pi], Cos[t]}, r {Sin[t] Cos[0 Pi], Sin[t] Sin[0 Pi], Cos[t]}}, {r, 1, 2 - n}, {t, 0, Pi}, PlotStyle -> Yellow, Mesh -> {2, 15}]], {n, 0, 1}]

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