In the philosophy of mathematics, finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps. The philosophy of mathematics is the branch of Philosophy that studies the philosophical assumptions foundations and implications of Mathematics. In the Philosophy of mathematics In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. Most constructivists, in contrast, allow a countably infinite number of steps. In her book Philosophy of Set Theory, Mary Tiles characterized those who allow countably infinite as Classical Finitists, and those who deny even countably infinite as Strict Finitists.

The most famous proponent of finitism was Leopold Kronecker, who said:

"God created the natural numbers, all else is the work of man. Leopold Kronecker ( December 7, 1823 – December 29, 1891) was a German Mathematician and Logician who argued "

Although most modern constructivists take a weaker view, they can trace the origins of constructivism back to Kronecker's finitist work.

In 1923, Thoralf Skolem published a paper in which he presented a semi-formal system, what is now known as Primitive recursive arithmetic, which is widely taken to be a suitable background for finitist mathematics. Thoralf Albert Skolem ( May 23, 1887 – March 23, 1963) (ˈtɔɾɑlf ˈskuləm was a Norwegian Mathematician known Primitive recursive arithmetic, or PRA, is a Quantifier -free formalization of the natural numbers This was adopted by Hilbert and Bernays as the 'contentual', finitist system for metamathematics, in which a proof of the consistency of other mathematical systems (e. g. full Peano Arithmetic) was to be given. In Mathematical logic, the Peano axioms, also known as the Dedekind-Peano axioms or the Peano postulates, are a set of Axioms for the Natural (See Hilbert's program. Hilbert's program, formulated by German mathematician David Hilbert in the 1920s was to formalize all existing theories to a finite complete set of axioms and provide ) An important thing to understand is that Hilbert's finitism is constrained solely on the length of mathematical proofs. Hilbert did not demand finitism of models, but instead he embraced the very source of transfinitism: "No one shall expel us from the Paradise that Cantor has created for us". It is not hard to understand that an infinitely long proof is impossible: a proof that never ends, is not a proof. Finitists deny the infinity of models too. According to Löwenheim-Skolem theorem LwS, all talk about innumerable infinite models can be substituted by the talk about numerably infinite models. Therefore, LwS is at least somewhat finitist in nature.

Reuben Goodstein is another proponent of finitism. Reuben Louis Goodstein (born 15 December 1912 in London, died 8 March 1985 in Leicester) was an English Some of his work involved building up to analysis from finitist foundations. Although he denied it, much of Ludwig Wittgenstein's writing on mathematics has a strong affinity with finitism. If finitists are contrasted with transfinitists (proponents of e. g. Cantor's hierarchy of infinities), then also Aristotle may be characterized as a Classical Finitist. Georg Ferdinand Ludwig Philipp Cantor ( – January 6 1918) was a German Mathematician, born in Russia. Aristotle (Greek Aristotélēs) (384 BC – 322 BC was a Greek philosopher a student of Plato and teacher of Alexander the Great. Aristotle especially promoted the potential infinity as a middle option between strict finitism and actual infinity. (Note that Aristotle's actual infinity means simply an actualization of something neverending in nature, when in contrast the Cantorist actual infinity means the transfinite cardinal and ordinal numbers, that have nothing to do with the things in nature):

"But on the other hand to suppose that the infinite does not exist in any way leads obviously to many impossible consequences: there will be a beginning and end of time, a magnitude will not be divisible into magnitudes, number will not be infinite. If, then, in view of the above considerations, neither alternative seems possible, an arbiter must be called in;" -Aristotle, Metaphysics, Book 3, Chapter 6.

Even stronger than finitism is ultrafinitism (also known as ultraintuitionism), associated primarily with Alexander Esenin-Volpin. In the Philosophy of mathematics, ultrafinitism, or ultraintuitionism, is a form of Finitism. Alexander Sergeyevich Esenin-Volpin ( Александр Сергеевич Есенин-Вольпин)is a prominent