Formalism (philosophy of mathematics)

Last updated

In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (alphanumeric sequences of symbols, usually as equations) using established manipulation rules. A central idea of formalism "is that mathematics is not a body of propositions representing an abstract sector of reality, but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than ludo or chess." [1] According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other coextensive subject matter — in fact, they aren't "about" anything at all. Rather, mathematical statements are syntactic forms whose shapes and locations have no meaning unless they are given an interpretation (or semantics). In contrast to mathematical realism, logicism, or intuitionism, formalism's contours are less defined due to broad approaches that can be categorized as formalist.

Contents

Along with realism and intuitionism, formalism is one of the main theories in the philosophy of mathematics that developed in the late nineteenth and early twentieth century. Among formalists, David Hilbert was the most prominent advocate. [2]

Early formalism

The early mathematical formalists attempted "to block, avoid, or sidestep (in some way) any ontological commitment to a problematic realm of abstract objects." [1] German mathematicians Eduard Heine and Carl Johannes Thomae are considered early advocates of mathematical formalism. [1] Heine and Thomae's formalism can be found in Gottlob Frege's criticisms in The Foundations of Arithmetic .

According to Alan Weir, the formalism of Heine and Thomae that Frege attacks can be "describe[d] as term formalism or game formalism." [1] Term formalism is the view that mathematical expressions refer to symbols, not numbers. Heine expressed this view as follows: "When it comes to definition, I take a purely formal position, in that I call certain tangible signs numbers, so that the existence of these numbers is not in question." [3]

Thomae is characterized as a game formalist who claimed that "[f]or the formalist, arithmetic is a game with signs which are called empty. That means that they have no other content (in the calculating game) than they are assigned by their behaviour with respect to certain rules of combination (rules of the game)." [4]

Frege provides three criticisms of Heine and Thomae's formalism: "that [formalism] cannot account for the application of mathematics; that it confuses formal theory with metatheory; [and] that it can give no coherent explanation of the concept of an infinite sequence." [5] Frege's criticism of Heine's formalism is that his formalism cannot account for infinite sequences. Dummett argues that more developed accounts of formalism than Heine's account could avoid Frege's objections by claiming they are concerned with abstract symbols rather than concrete objects. [6] Frege objects to the comparison of formalism with that of a game, such as chess. [7] Frege argues that Thomae's formalism fails to distinguish between game and theory.

Hilbert's formalism

David Hilbert Hilbert.jpg
David Hilbert

A major figure of formalism was David Hilbert, whose program was intended to be a complete and consistent axiomatization of all of mathematics. [8] Hilbert aimed to show the consistency of mathematical systems from the assumption that the "finitary arithmetic" (a subsystem of the usual arithmetic of the positive integers, chosen to be philosophically uncontroversial) was consistent (i.e. no contradictions can be derived from the system).

The way that Hilbert tried to show that an axiomatic system was consistent was by formalizing it using a particular language. [9] In order to formalize an axiomatic system, you must first choose a language in which you can express and perform operations within that system. This language must include five components:

By adopting this language, Hilbert thought that we could prove all theorems within any axiomatic system using nothing more than the axioms themselves and the chosen formal language.

Gödel's conclusion in his incompleteness theorems was that you cannot prove consistency within any consistent axiomatic system rich enough to include classical arithmetic. On the one hand, you must use only the formal language chosen to formalize this axiomatic system; on the other hand, it is impossible to prove the consistency of this language in itself. [9] Hilbert was originally frustrated by Gödel's work because it shattered his life's goal to completely formalize everything in number theory. [10] However, Gödel did not feel that he contradicted everything about Hilbert's formalist point of view. [11] After Gödel published his work, it became apparent that proof theory still had some use, the only difference is that it could not be used to prove the consistency of all of number theory as Hilbert had hoped. [10]

Hilbert was initially a deductivist,[ citation needed ] but he considered certain metamathematical methods to yield intrinsically meaningful results and was a realist with respect to the finitary arithmetic. Later, he held the opinion that there was no other meaningful mathematics whatsoever, regardless of interpretation.

Further developments

Other formalists, such as Rudolf Carnap, considered mathematics to be the investigation of formal axiom systems. [12]

Haskell Curry defines mathematics as "the science of formal systems." [13] Curry's formalism is unlike that of term formalists, game formalists, or Hilbert's formalism. For Curry, mathematical formalism is about the formal structure of mathematics and not about a formal system. [13] Stewart Shapiro describes Curry's formalism as starting from the "historical thesis that as a branch of mathematics develops, it becomes more and more rigorous in its methodology, the end-result being the codification of the branch in formal deductive systems." [14]

Criticisms of formalism

Kurt Gödel indicated one of the weak points of formalism by addressing the question of consistency in axiomatic systems.

Bertrand Russell has argued that formalism fails to explain what is meant by the linguistic application of numbers in statements such as "there are three men in the room". [15]

See also

Related Research Articles

An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἀξίωμα (axíōma), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.

Kurt Gödel Mathematical logician and philosopher (1906–1978)

Kurt Friedrich Gödel was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an immense effect upon scientific and philosophical thinking in the 20th century, a time when others such as Bertrand Russell, Alfred North Whitehead, and David Hilbert were using logic and set theory to investigate the foundations of mathematics, building on earlier work by the likes of Richard Dedekind, Georg Cantor and Frege.

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.

Russells paradox Paradox in set theory

In mathematical logic, Russell's paradox is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions. The paradox had already been discovered independently in 1899 by the German mathematician Ernst Zermelo. However, Zermelo did not publish the idea, which remained known only to David Hilbert, Edmund Husserl, and other academics at the University of Göttingen. At the end of the 1890s, Georg Cantor – considered the founder of modern set theory – had already realized that his theory would lead to a contradiction, which he told Hilbert and Richard Dedekind by letter.

The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's lives. The logical and structural nature of mathematics itself makes this study both broad and unique among its philosophical counterparts.

Gottlob Frege German philosopher, logician, and mathematician (1848–1925)

Friedrich Ludwig Gottlob Frege was a German philosopher, logician, and mathematician. He worked as a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philosophy, concentrating on the philosophy of language, logic, and mathematics. Though he was largely ignored during his lifetime, Giuseppe Peano (1858–1932), Bertrand Russell (1872–1970), and, to some extent, Ludwig Wittgenstein (1889–1951) introduced his work to later generations of philosophers. Frege is widely considered to be the greatest logician since Aristotle, and one of the most profound philosophers of mathematics ever.

Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible.

In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a model, i.e., there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead. The syntactic definition states a theory is consistent if there is no formula such that both and its negation are elements of the set of consequences of . Let be a set of closed sentences and the set of closed sentences provable from under some formal deductive system. The set of axioms is consistent when for no formula .

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 the 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.

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. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.

Metamathematics Study of mathematics itself

Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Emphasis on metamathematics owes itself to David Hilbert's attempt to secure the foundations of mathematics in the early part of the 20th century. Metamathematics provides "a rigorous mathematical technique for investigating a great variety of foundation problems for mathematics and logic". An important feature of metamathematics is its emphasis on differentiating between reasoning from inside a system and from outside a system. An informal illustration of this is categorizing the proposition "2+2=4" as belonging to mathematics while categorizing the proposition "'2+2=4' is valid" as belonging to metamathematics.

In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contains an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system. A formal theory is an axiomatic system that describes a set of sentences that is closed under logical implication. A formal proof is a complete rendition of a mathematical proof within a formal system.

George Boolos American philosopher and mathematical logician

George Stephen Boolos was an American philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology.

A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essentially an "axiomatic system".

In mathematics, Hilbert's program, formulated by German mathematician David Hilbert in the early part of the 20th century, 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.

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.

In a controversy over the foundations of mathematics, in twentieth-century mathematics, L. E. J. Brouwer, a proponent of the constructivist school of intuitionism, opposed David Hilbert, a proponent of formalism. The debate concerned fundamental questions about the consistency of axioms and the role of semantics and syntax in mathematics. Much of the controversy took place while both were involved with the prestigious Mathematische Annalen journal, with Hilbert as editor-in-chief and Brouwer as a member of its editorial board.

References

  1. 1 2 3 4 Weir, Alan (2015), "Formalism in the Philosophy of Mathematics", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Spring 2015 ed.), Metaphysics Research Lab, Stanford University, retrieved 2019-05-25
  2. ↑ Simons, Peter (2009). "Formalism". Philosophy of Mathematics. Elsevier. p. 292. ISBN   9780080930589.
  3. ↑ Simons, Peter (2009). Philosophy of Mathematics. Elsevier. p. 293. ISBN   9780080930589.
  4. ↑ Frege, Gottlob (1903). The Foundations of Arithmetic: A Logico-Mathematical Enquiry Into the Concept of Number. Chicago: Northwestern University Press. p. 183.
  5. ↑ Dummett, Michael (1991). Frege: Philosophy of Mathematics. Cambridge: Harvard University Press. p. 252. ISBN   9780674319356.
  6. ↑ Dummett, Michael (1991). Frege: Philosophy of Mathematics. Cambridge: Harvard University Press. p. 253. ISBN   9780674319356.
  7. ↑ Frege, Gottlob; Ebert, Philip A.; Cook, Roy T. (1893). Basic Laws of Arithmetic: Derived using concept-script. Oxford: Oxford University Press (published 2013). pp. § 93. ISBN   9780199281749.
  8. ↑ Zach, Richard (2019), "Hilbert's Program", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Summer 2019 ed.), Metaphysics Research Lab, Stanford University, retrieved 2019-05-25
  9. 1 2 Snapper, Ernst (September 1979). "The Three Crises in Mathematics: Logicism, Intuitionism and Formalism" (PDF). Mathematics Magazine. 52 (4): 207–216. doi:10.1080/0025570X.1979.11976784.
  10. 1 2 Reid, Constance; Weyl, Hermann (1970). Hilbert. Springer-Verlag. p. 198. ISBN   9783662286159.
  11. ↑ Gödel, Kurt (1986). Feferman, Solomon (ed.). Kurt Gödel: Collected Works: Volume I: Publications 1929-1936. Vol. 1. Oxford: Oxford University Press. p. 195. ISBN   9780195039641.
  12. ↑ Carnap, Rudolf (1937). Logical Syntax of Language. Routledge. pp. 325–328. ISBN   9781317830597.
  13. 1 2 Curry, Haskell B. (1951). Outlines of a Formalist Philosophy of Mathematics. Elsevier. p. 56. ISBN   9780444533685.
  14. ↑ Shapiro, Stewart (2005). "Formalism". The Oxford Companion to Philosophy. Honderich, Ted (2nd ed.). Oxford: Oxford University Press. ISBN   9780191532658. OCLC   62563098.
  15. ↑ Bertrand Russell My Philosophical Development, 1959, ch. X.
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.