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In the philosophy of mathematics, **ultrafinitism** (also known as **ultraintuitionism**,^{ [1] }**strict formalism**,^{ [2] }**strict finitism**,^{ [2] }**actualism**,^{ [1] }**predicativism**,^{ [2] }^{ [3] } and **strong finitism**)^{ [2] } 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 **N** of natural numbers.

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 which 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. The logical foundation of ultrafinitism is unclear; in his comprehensive survey *Constructivism in Mathematics* (1988), the constructive logician A. S. Troelstra dismissed it by saying "no satisfactory development exists at present." This was not so much a philosophical objection as it was an admission that, in a rigorous work of mathematical logic, there was simply nothing precise enough to include.

Serious work on ultrafinitism has been led, since 1959, by Alexander Esenin-Volpin, who in 1961 sketched a program for proving the consistency of Zermelo–Fraenkel set theory in ultrafinite mathematics. Other mathematicians who have worked in the topic include Doron Zeilberger, Edward Nelson, Rohit Jivanlal Parikh, and Jean Paul Van Bendegem. The philosophy is also sometimes associated with the beliefs of Ludwig Wittgenstein, Robin Gandy, Petr Vopěnka, and J. Hjelmslev.

Shaughan Lavine has developed a form of set-theoretical ultrafinitism that is consistent with classical mathematics.^{ [4] } Lavine has shown that the basic principles of arithmetic such as "there is no largest natural number" can be upheld, as Lavine allows for the inclusion of "indefinitely large" numbers.^{ [4] }

Other considerations of the possibility of avoiding unwieldy large numbers can be based on computational complexity theory, as in Andras Kornai's work on explicit finitism (which does not deny the existence of large numbers)^{ [5] } and Vladimir Sazonov's notion of feasible number.

There has also been considerable formal development on versions of ultrafinitism that are based on complexity theory, like Samuel Buss's Bounded Arithmetic theories, which capture mathematics associated with various complexity classes like P and PSPACE. Buss's work can be considered the continuation of Edward Nelson's work on predicative arithmetic as bounded arithmetic theories like S12 are interpretable in Raphael Robinson's theory Q and therefore are predicative in Nelson's sense. The power of these theories for developing mathematics is studied in Bounded reverse mathematics as can be found in the works of Stephen A. Cook and Phuong The Nguyen. However these researches are not philosophies of mathematics but rather the study of restricted forms of reasoning similar to reverse mathematics.

- 1 2 International Workshop on Logic and Computational Complexity,
*Logic and Computational Complexity*, Springer, 1995, p. 31. - 1 2 3 4 St. Iwan (2000), "On the Untenability of Nelson's Predicativism",
*Erkenntnis***53**(1–2), pp. 147–154. - ↑ Not to be confused with Russell's predicativism.
- 1 2 "Philosophy of Mathematics (Stanford Encyclopedia of Philosophy)". Plato.stanford.edu. Retrieved 2015-10-07.
- ↑ "Relation to foundations"

**Mathematical logic**, also called **formal logic**, is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, philosophy, and theoretical computer science. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.

In the philosophy of mathematics, **constructivism** asserts that it is necessary to find a mathematical object to prove that it exists. 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. The constructive viewpoint involves a verificational interpretation of the existential quantifier, which is at odds with its classical interpretation.

**Set theory** is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole.

In logic, **Richard's paradox** is a semantical antinomy of set theory and natural language first described by the French mathematician Jules Richard in 1905. The paradox is ordinarily used to motivate the importance of distinguishing carefully between mathematics and metamathematics.

**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 are accepted as legitimate.

**Foundations of mathematics** is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics. In this latter sense, the distinction between foundations of mathematics and philosophy of mathematics turns out to be quite vague. Foundations of mathematics can be conceived as the study of the basic mathematical concepts and how they form hierarchies of more complex structures and concepts, especially the fundamentally important structures that form the language of mathematics also called metamathematical concepts, with an eye to the philosophical aspects and the unity of mathematics. The search for foundations of mathematics is a central question of the philosophy of mathematics; the abstract nature of mathematical objects presents special philosophical challenges.

**Proof theory** is a major branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the logical system. As such, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.

**Edward Nelson** was an American mathematician. He was professor in the Mathematics Department at Princeton University. He was known for his work on mathematical physics and mathematical logic. In mathematical logic, he was noted especially for his internal set theory, and views on ultrafinitism and the consistency of arithmetic. In philosophy of mathematics he advocated the view of formalism rather than platonism or intuitionism. He also wrote on the relationship between religion and mathematics.

In mathematical logic, an **axiom schema** generalizes the notion of axiom.

In the foundations of mathematics, **classical mathematics** refers generally to the mainstream approach to mathematics, which is based on classical logic and ZFC set theory. It stands in contrast to other types of mathematics such as constructive mathematics or predicative mathematics. In practice, the most common non-classical systems are used in constructive mathematics.

In mathematical logic, the **Brouwer–Heyting–Kolmogorov interpretation**, or **BHK interpretation**, of intuitionistic logic was proposed by L. E. J. Brouwer and Arend Heyting, and independently by Andrey Kolmogorov. It is also sometimes called the **realizability interpretation**, because of the connection with the realizability theory of Stephen Kleene.

The following tables list the computational complexity of various algorithms for common mathematical operations.

In proof theory, a discipline within mathematical logic, **double-negation translation**, sometimes called **negative translation**, is a general approach for embedding classical logic into intuitionistic logic, typically by translating formulas to formulas which are classically equivalent but intuitionistically inequivalent. Particular instances of double-negation translation include **Glivenko's translation** for propositional logic, and the **Gödel–Gentzen translation** and **Kuroda's translation** for first-order logic.

**Markov's principle**, named after Andrey Markov Jr, is a conditional existence statement for which there are many equivalent formulations, as discussed below.

**Anne Sjerp Troelstra** was a professor of pure mathematics and foundations of mathematics at the Institute for Logic, Language and Computation (ILLC) of the University of Amsterdam.

In proof theory, **ordinal analysis** assigns ordinals to mathematical theories as a measure of their strength. If theories have the same proof-theoretic ordinal they are often equiconsistent, and if one theory has a larger proof-theoretic ordinal than another it can often prove the consistency of the second theory.

**Geoffrey Hellman** is an American professor and philosopher. He is Professor of Philosophy at the University of Minnesota in Minneapolis, Minnesota. He obtained his B.A. (1965) and Ph.D. (1972) degrees in philosophy from Harvard University. He was elected to the American Academy of Arts and Sciences in 2007.

**Mathematics** is a field of study that investigates topics such as number, space, structure, and change.

**Bounded arithmetic** is a collective name for a family of weak subtheories of Peano arithmetic. Such theories are typically obtained by requiring that quantifiers be bounded in the induction axiom or equivalent postulates. The main purpose is to characterize one or another class of computational complexity in the sense that a function is provably total if and only if it belongs to a given complexity class. Further, theories of bounded arithmetic present uniform counterparts to standard propositional proof systems such as Frege system and are, in particular, useful for constructing polynomial-size proofs in these systems. The characterization of standard complexity classes and correspondence to propositional proof systems allows to interpret theories of bounded arithmetic as formal systems capturing various levels of feasible reasoning.

- Ésénine-Volpine, A. S. (1961), "Le programme ultra-intuitionniste des fondements des mathématiques",
*Infinitistic Methods (Proc. Sympos. Foundations of Math., Warsaw, 1959)*, Oxford: Pergamon, pp. 201–223, MR 0147389 Reviewed by Kreisel, G.; Ehrenfeucht, A. (1967), "Review of Le Programme Ultra-Intuitionniste des Fondements des Mathematiques by A. S. Ésénine-Volpine",*The Journal of Symbolic Logic*, Association for Symbolic Logic,**32**(4): 517, doi:10.2307/2270182, JSTOR 2270182 - Lavine, S., 1994. Understanding the Infinite, Cambridge, MA: Harvard University Press.

- Explicit finitism by Andras Kornai
- On feasible numbers by Vladimir Sazonov
- "Real" Analysis Is A Degenerate Case Of Discrete Analysis by Doron Zeilberger
- Discussion on formal foundations on MathOverflow
- History of constructivism in the 20th century by A. S. Troelstra
- Predicative Arithmetic by Edward Nelson
- Logical Foundations of Proof Complexity by Stephen A. Cook and Phuong The Nguyen
- Bounded Reverse Mathematics by Phuong The Nguyen
- Reading Brian Rotman’s “Ad Infinitum…” by Charles Petzold
- Computational Complexity Theory

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