Doubt: Can we say that Mathematical thoughts (arguments) can be independent of physical (Time-space continuum or material) world since it is an abstract science. In other words can Mathematics be a tool to analyse some immaterial existences (Something like soul, mind, god, love etc,)?


Well, that will depend on your ontological point of view regarding mathematics itself. A necessary prerequisite is that you can somehow formalize the "immaterial entity" you are trying to investigate.

For platonists like Roger Penrose or Gödel mathematics is independent of the human mind, mathematical truths are objective statements about abstract entities. In Penrose words;

Plato made it clear that the mathematical propositions—the things that could be regarded as unassailably true—referred not to actual physical objects (like the approximate squares, triangles, circles, spheres, and cubes that might be constructed from marks in the sand, or from wood or stone) but to certain idealized entities. He envisaged that these ideal entities inhabited a different world, distinct from the physical world. Today, we might refer to this world as the Platonic world of mathematical forms. Physical structures, such as squares, circles, or triangles cut from papyrus, or marked on a flat surface, or perhaps cubes, tetrahedra, or spheres carved from marble, might conform to these ideals very closely, but only approximately. The actual mathematical squares, cubes, circles, spheres, triangles, etc.,

In "The Road to Reality", P.11

For mathematical realists (Radical platonism), theorems have physically real consequences and our world just one among the endless ocean of mathematically possible universes. Kurt Gödel developed an ontological proof for the existence of God, in a system of modal logic, if you are a platonist, a mathematical realist (Or the related concept of modal realism), statements in formal systems can very well be taken to investigate the kind of "immaterial existences" you are referring to.

Of course, what is and what isn't a theorem in mathematics will depend on the axiomatic system being chosen. If "existence" is mathematical realism, the question would be which axiomatic system captures it best.

Intuitionists would reject that mathematics necessarily refer to anything existing objectively in reality. They view mathematics as mental activity. Analysing immaterial existences would not be meaningful for them.

A constructivist may or may not be a (radical) platonist, they reject the law of excluded middle and double negation. Only things which can be constructed from an axiomatic system without Law of excluded middle and double negation exist.

In the end, the things you can make statements about in a formal system will depend on the axioms and inference rules being chosen and your philosophical view on the ontology of mathematics. Gödel's ontological God proof would be unimportant to someone rejecting the axioms and inference rules or if you believe mathematics to be only a mental activity. In case of Gödel's God proof it should be noted that it's not axiomatizable in "Standard Mathematics" (ZF (+C)).

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