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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 existing.
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. Therefore quantification over infinite domains is not considered meaningful. The mathematical theory often associated with finitism is Thoralf Skolem's primitive recursive arithmetic.
The introduction of infinite mathematical objects occurred a few centuries ago when the use of infinite objects was already a controversial topic among mathematicians. The issue entered a new phase when Georg Cantor in 1874 introduced what is now called naive set theory and used it as a base for his work on transfinite numbers. When paradoxes such as Russell's paradox, Berry's paradox and the Burali-Forti paradox were discovered in Cantor's naive set theory, the issue became a heated topic among mathematicians.
There were various positions taken by mathematicians. All agreed about finite mathematical objects such as natural numbers. However there were disagreements regarding infinite mathematical objects. One position was the intuitionistic mathematics that was advocated by L. E. J. Brouwer, which rejected the existence of infinite objects until they are constructed.
Another position was endorsed by David Hilbert: finite mathematical objects are concrete objects, infinite mathematical objects are ideal objects, and accepting ideal mathematical objects does not cause a problem regarding finite mathematical objects. More formally, Hilbert believed that it is possible to show that any theorem about finite mathematical objects that can be obtained using ideal infinite objects can be also obtained without them. Therefore allowing infinite mathematical objects would not cause a problem regarding finite objects. This led to Hilbert's program of proving both consistency and completeness of set theory using finitistic means as this would imply that adding ideal mathematical objects is conservative over the finitistic part. Hilbert's views are also associated with the formalist philosophy of mathematics. Hilbert's goal of proving the consistency and completeness of set theory or even arithmetic through finitistic means turned out to be an impossible task due to Kurt Gödel's incompleteness theorems. However, Harvey Friedman's grand conjecture would imply that most mathematical results are provable using finitistic means.
Hilbert did not give a rigorous explanation of what he considered finitistic and referred to as elementary. However, based on his work with Paul Bernays some experts such as Tait (1981) have argued that primitive recursive arithmetic can be considered an upper bound on what Hilbert considered finitistic mathematics. [1]
As a result of Gödel's theorems, as it became clear that there is no hope of proving both the consistency and completeness of mathematics, and with the development of seemingly consistent axiomatic set theories such as Zermelo–Fraenkel set theory, most modern mathematicians do not focus on this topic.
In her book The Philosophy of Set Theory, Mary Tiles characterized those who allow potentially infinite objects as classical finitists, and those who do not allow potentially infinite objects as strict finitists: for example, a classical finitist would allow statements such as "every natural number has a successor" and would accept the meaningfulness of infinite series in the sense of limits of finite partial sums, while a strict finitist would not. Historically, the written history of mathematics was thus classically finitist until Cantor created the hierarchy of transfinite cardinals at the end of the 19th century.
Leopold Kronecker remained a strident opponent to Cantor's set theory: [2]
Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk. God created the integers; all else is the work of man.
Reuben Goodstein was 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. [4]
If finitists are contrasted with transfinitists (proponents of e.g. Georg Cantor's hierarchy of infinities), then also Aristotle may be characterized as a finitist. Aristotle especially promoted the potential infinity as a middle option between strict finitism and actual infinity (the latter being an actualization of something never-ending in nature, in contrast with the Cantorist actual infinity consisting of the transfinite cardinal and ordinal numbers, which 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, Physics, Book 3, Chapter 6
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Ultrafinitism (also known as ultraintuitionism) has an even more conservative attitude towards mathematical objects than finitism, and has objections to the existence of finite mathematical objects when they are too large.
Towards the end of the 20th century John Penn Mayberry developed a system of finitary mathematics which he called "Euclidean Arithmetic". The most striking tenet of his system is a complete and rigorous rejection of the special foundational status normally accorded to iterative processes, including in particular the construction of the natural numbers by the iteration "+1". Consequently Mayberry is in sharp dissent from those who would seek to equate finitary mathematics with Peano arithmetic or any of its fragments such as primitive recursive arithmetic.
In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the case of infinite sets, the infinite cardinal numbers have been introduced, which are often denoted with the Hebrew letter (aleph) marked with subscript indicating their rank among the infinite cardinals.
Georg Ferdinand Ludwig Philipp Cantor was a mathematician who played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. Cantor's method of proof of this theorem implies the existence of an infinity of infinities. He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact he was well aware of.
In the philosophy of mathematics, intuitionism, or neointuitionism, is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied, but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality.
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics.
In the philosophy of mathematics, constructivism asserts that it is necessary to find 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. 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 the philosophy of mathematics, the abstraction of actual infinity, also called completed infinity, involves infinite entities as given, actual and completed objects.
Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship with other human activities.
In mathematics, Hilbert's second problem was posed by David Hilbert in 1900 as one of his 23 problems. It asks for a proof that arithmetic is consistent – free of any internal contradictions. Hilbert stated that the axioms he considered for arithmetic were the ones given in Hilbert (1900), which include a second order completeness axiom.
Foundations of mathematics are the logical and mathematical framework that allows the development of mathematics without generating self-contradictory theories, and, in particular, to have reliable concepts of theorems, proofs, algorithms, etc. This may also include the philosophical study of the relation of this framework with reality.
Proof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.
In mathematics, Hilbert's program, formulated by German mathematician David Hilbert in the early 1920s, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies. As a solution, Hilbert proposed to ground all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent. Hilbert proposed that the consistency of more complicated systems, such as real analysis, could be proven in terms of simpler systems. Ultimately, the consistency of all of mathematics could be reduced to basic arithmetic.
In mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Although this work has become a thoroughly standard fixture of classical set theory, it has been criticized in several areas by mathematicians and philosophers.
Gentzen's consistency proof is a result of proof theory in mathematical logic, published by Gerhard Gentzen in 1936. It shows that the Peano axioms of first-order arithmetic do not contain a contradiction, as long as a certain other system used in the proof does not contain any contradictions either. This other system, today called "primitive recursive arithmetic with the additional principle of quantifier-free transfinite induction up to the ordinal ε0", is neither weaker nor stronger than the system of Peano axioms. Gentzen argued that it avoids the questionable modes of inference contained in Peano arithmetic and that its consistency is therefore less controversial.
Primitive recursive arithmetic (PRA) is a quantifier-free formalization of the natural numbers. It was first proposed by Norwegian mathematician Skolem (1923), as a formalization of his finitistic conception of the foundations of arithmetic, and it is widely agreed that all reasoning of PRA is finitistic. Many also believe that all of finitism is captured by PRA, but others believe finitism can be extended to forms of recursion beyond primitive recursion, up to ε0, which is the proof-theoretic ordinal of Peano arithmetic. PRA's proof theoretic ordinal is ωω, where ω is the smallest transfinite ordinal. PRA is sometimes called Skolem arithmetic, although that has another meaning, see Skolem arithmetic.
Temporal finitism is the doctrine that time is finite in the past. The philosophy of Aristotle, expressed in such works as his Physics, held that although space was finite, with only void existing beyond the outermost sphere of the heavens, time was infinite. This caused problems for mediaeval Islamic, Jewish, and Christian philosophers who, primarily creationist, were unable to reconcile the Aristotelian conception of the eternal with the Genesis creation narrative.
Infinity is something which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Matthias Schirn is a German philosopher and logician.
Beyond Infinity : An Expedition to the Outer Limits of Mathematics is a popular mathematics book by Eugenia Cheng centered on concepts of infinity. It was published by Basic Books and by Profile Books in 2017, and in a paperback edition in 2018. It was shortlisted for the 2017 Royal Society Insight Investment Science Book Prize.
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