Ultrafinitism

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.

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Ultrafinitism

Finitism

Finitism is a philosophy of mathematics that accepts the existence only of finite mathematical objects. It is best understood in comparison to the mainstream philosophy of mathematics where infinite mathematical objects (e.g., infinite sets) are accepted as legitimate. The main idea of finitistic mathematics is not accepting the existence of infinite objects such as infinite sets. While all natural numbers are accepted as existing, the set of all natural numbers is not considered to exist as a mathematical object.

Constructivism (philosophy of mathematics)

In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove the existence of a mathematical object without "finding" that object explicitly, by assuming its non-existence and then deriving a contradiction from that assumption. Such a proof by contradiction might be called non-constructive, and a constructivist might reject it.