Is Mathematics Real?

by Mina Basilious

  • What Do we mean by "Is Mathematics Real?"?
  • Mathematical objects are not material objects.
  • The truthness of Mathematics is powerful than any other sciences.
  • Mathematics is not dependent on observations.

What is Mathematics?

Mathematical Formalism

Mathematics is the study and manipulation of formal systems — symbols and rules for transforming them; mathematical activity is syntactic derivation inside agreed-upon formal languages.

In Mathematical Formalism

Mathematical objects do not have independent metaphysical existence — they are tokens or expressions in formal systems; truth is mainly provability within those systems.

Mathematical Logicism

Mathematics is (in principle) reducible to logic; mathematical truths are logical truths or follow from purely logical axioms plus definitions.

In Mathematical Logicism

Mathematical objects are logical or set-theoretic entities (they “live” in the logical universe), and existence is understood via logical definition/derivation.

Gödel’s Completeness Theorem

In first-order logic, everything that is true in all models can be proved.

Unification of Mathematics

Gödel First Incompleteness Theorem

  • Let be a consistent, effectively axiomatizable formal theory that is strong enough to encode basic arithmetic
  • Then there exists a sentence in the language of such that neither nor its negation is provable from .

Mathematical Intuitionism (constructivism)

Mathematics is the constructive activity of the human mind; mathematical statements are true only when we have explicit constructions (proofs) for them.

In Mathematical Intuitionism

Mathematical objects “exist” only as mental constructions or processes of construction; nonconstructive existence claims are rejected or reinterpreted.

The Problem with LEM for Intuitionism

Weak Indirect Proof


Strong Indirect Proof



The big elephant in the room

Platonism

Mathematical Platonism

Mathematics is the discovery of an objective realm of abstract mathematical entities (numbers, sets, functions) and their relations; mathematical practice uncovers truths about that realm.

In Mathematical Platonism

Mathematical objects exist independently of minds and language — they “live” in a timeless abstract realm; proofs reveal (rather than create) facts about these objects.

In Mathematical Platonism

Disscusion and Q&As

Thank you for Listening