Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Ultrafinitism
Summary
In the philosophy of mathematics, ultrafinitism (also known as ultraintuitionism, strict formalism, strict finitism, actualism, predicativism, and strong finitism) is a form of finitism and intuitionism. There are various philosophies of mathematics that are called ultrafinitism. A major identifying property common among most of these philosophies is their objections to totality of number theoretic functions like exponentiation over natural numbers.
Like other finitists, ultrafinitists deny the existence of the infinite set of natural numbers, on the basis that it can never be completed. ( i.e. there is a largest natural number).
In addition, some ultrafinitists are concerned with acceptance of objects in mathematics that no one can construct in practice because of physical restrictions in constructing large finite mathematical objects.
Thus some ultrafinitists will deny or refrain from accepting the existence of large numbers, for example, the floor of the first Skewes's number, which is a huge number defined using the exponential function as exp(exp(exp(79))), or
The reason is that nobody has yet calculated what natural number is the floor of this real number, and it may not even be physically possible to do so. Similarly, (in Knuth's up-arrow notation) would be considered only a formal expression that does not correspond to a natural number. The brand of ultrafinitism concerned with physical realizability of mathematics is often called actualism.
Edward Nelson criticized the classical conception of natural numbers because of the circularity of its definition. In classical mathematics the natural numbers are defined as 0 and numbers obtained by the iterative applications of the successor function to 0. But the concept of natural number is already assumed for the iteration. In other words, to obtain a number like one needs to perform the successor function iteratively (in fact, exactly times) to 0.
Some versions of ultrafinitism are forms of constructivism, but most constructivists view the philosophy as unworkably extreme.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.