Everything about Finitism totally explained
In the
philosophy of mathematics,
finitism is an extreme form of
constructivism, according to which a mathematical object doesn't exist unless it can be constructed from
natural numbers in a
finite number of steps.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."
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. 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 (for example full
Peano Arithmetic) was to be given. (See
Hilbert's program.) An important thing to understand is that Hilbert's finitism is constrained solely on the length of mathematical proofs. Hilbert didn't 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 isn't hard to understand that an infinitely long proof is impossible: a proof that never ends, isn't 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. 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 for example
Cantor's hierarchy of infinities), then also
Aristotle may be characterized as a Classical Finitist. 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 doesn't exist
in any way leads obviously to many impossible consequences:
there will be a beginning and end of time, a magnitude won't
be divisible into magnitudes, number won't 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.
Further Information
Get more info on 'Finitism'.
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