Philosophy of

Mathematics

By: Rey John B. Rebucas

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What is Philosophy of Mathematics?

The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics (Hilbert,1996).

Studies the nature of mathematical truth, mathematical proof, mathematical evidence, mathematical practice, and mathematical explanation (Russell, 1997).

Ontology for Mathematics: “Being”

Ontology studies the nature of the objects of mathematics.

It is the claim that mathematical objects exist independently of their linguistic expression.

“What we are talking about.” What is a number? What is a point? line? What is a set?

Epistemology for Mathematics: “Knowing”

Epistemology studies the acquisition of knowledge of the truth of a mathematical statement.

“whether what we are saying is true.” Does knowledge come from

experience and evidence? Does knowledge come from argument

and proof? Is knowledge relative or absolute?

According to Plato,

knowledge is a subset of

that which is both truths

and beliefs.

Classical Views on

the Nature of

Mathematics

Plato (c.428–347 B.C.)

He included mathematical

entities—numbers and

the objects of pure geometry such as points, lines, and circles—among

the well-defined, independently existing eternal

objects he called Forms.

Aristotle (384–322

B.C.)

He rejected the notion of Forms being separatefrom empirical objects, and maintained

instead that the Forms constitute parts of objects.

LeibnizHe divided all true

propositions, including those of

mathematics, into two types: truths of fact, and truths of reason,

alsoknown as contingent and analytic truths,

respectively.

Immanuel Kant (1724–

1804) He introduced a new classification of(true) propositions:

analytic, and nonanalytic, or

synthetic, which hefurther subdivided

into empirical/truth, or a posteriori, and non-empirical, or a

priori.

PURE MATHEMATICS

is the analysis ofthe structure of pure space and

time, free from empirical material.

APPLIED MATHEMATICS

is the analysis of the structure of space and time,

augmented by empirical material. .

Four Schools of Mathematical

Philosophy In the first decades of the

twentieth century, three non-platonistic accounts of mathematics were developed:

1.logicism, 2. formalism, 3. intuitionism, and4. predicativism

1. LOGICISM

It holds that mathematics is reducible to principles of pure logic.

Richard DedekindDedekind’s “logicism” embraced allmathematical concepts: the concepts of number—natural, rational,real, complex—and geometric concepts such as continuity.

Logicism, a mathematical truth and logical demonstration go hand in hand.

Quine (1995)

Frege (1884)Frege's Basic Law V: {x|Fx}={x|Gx} ≡

∀x(Fx ≡ Gx)

Frege devoted much of his career to trying to show how mathematics can be reduced to logic.

The Foundation of Arithmetic, a logico-mathematical

investigation into the concept of number.

2. FORMALISM is the view that

much or all of mathematics is devoid of content and a purely formal study of strings of mathematical language/

medium of formal symbols.

David Hilbert In 1899, he published his

epoch-making work Grundlagen

der Geometrie (“Foundations of Geometry”). He developed toanalyze the deductive system of Euclidean geometry—we might call itthe rigorized axiomatic method, or the metamathematical method.

On the formalist view, a minimal requirement of formal systems of higher mathematics is that they are at least consistent.

Hilbert aims theprovision of a new foundation for mathematics by representing its essential form within the realm ofconcrete symbols.

For example:2 + 2 = 4

would count as a realproposition, while there exists an odd perfect number would count as an ideal one/real or concrete.

Kurt Gödel He demonstrates his celebrated Incompleteness Theorems, that

there would always be real propositions provable by ideal means whichcannot be proved by concrete means. He allows finite

manipulations of suitablychosen abstract objects in addition to the concrete ones Gödel hoped tostrengthen finitistic metamathematics sufficiently to enable theconsistency of arithmetic to be demonstrable within it.

3. INTUITIONISM It holds that

mathematics is concerned with mental constructions and defends a revision of classical mathematics and logic.

Intuitionism rejects non-constructive existence proofs as ‘theological’ and ‘metaphysical’.

φ ∨ ¬φAccording to intuitionism, mathematics

is essentially an activity of construction. The natural numbers are mental

constructions, the real numbers are mental constructions, proofs and theorems are mental constructions, mathematical meaning is a mental construction.

L.E.J. Brouwer(1882-1966)

Brouwer heldthat mathematical theorems are synthetic a priori truths. He maintained, in

opposition to the logicists (whom he called “formalists”)

that arithmetic, and so all mathematics, must derive from the intuition of time.

Arend Heyting (1898–1980) A mathematical

theorem expresses a purely empirical fact, namely, the success of a certain construction.

“2 + 2 = 3 + 1”

4. PREDICATIVISM There emerged in

the beginning of the twentieth century also a fourth program.

Due to contingent historical circumstances, its true potential was not brought out until the 1960s.

Poincaré-Russell A sound definition of

a collection only refers to entities that exist independently from the defined collection. Such definitions are called predicative.

Paradox about the collection C of all mathematical entities that satisfy ¬x∈ x.

Weyl (1918) He developed a

philosophical stance that is in a sense intermediate between intuitionism and platonism.

He took the collection of natural numbers as unproblematically given.

New Views In Recent Decades

Structuralism

which holds that mathematics is the study of abstract structures.

Non-eliminative version of structuralism holds that there exist such things as abstract structures.

An eliminativeversion tries to make do with concrete objects variously structured.

Nominalism denies that there are any

abstract mathematical objects and tries to reconstruct classical mathematics accordingly. 

Fictionalism  is based on the idea that, although

most mathematical theorems are literally false, there is a non-literal (or fictional) sense in which assertions of them nevertheless count as correct.

Mathematical Naturalism

 urges that mathematics be taken as a sui generis discipline in good scientific standing.

Rey John B.

Rebucas

BEEDGen. Ed.

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