Matematik felsefesi, matematiği anlama çabalarını sınıflandırmaya çalışan bir felsefe dalıdır.
Başlıca soruları matematik ve matematiğin konusu olan nesnelerin kaynağı ile ilgilidir. Özellikle doğru bir önermenin özelliklerini inceler:
- Matematiksel konu maddesinin kaynakları nedir ?
- Bir matematiksel nesne ne manasıyla ilgilidir ?
- Bir matematiksel önermenin niteliği nedir ?
- Matematik ile mantık arasındaki ilişki nedir ?
- Hermeneuticlerin matematikteki rolü nedir ?
- Matematikte bir rol hangi türde soruşturmayla oynanır
- Matematiksel soruşturmanın nesnesi nedir ?
- Matematiğin arkasındaki insan özellikleri nedir ?
- Matematiksel güzellik nedir ?
- Matematiksel gerçeğin doğası ve kaynağı nedir ?
- Soyut matematikler dünyası ile materyel evren arasındaki ilişki nedir ?
Diğer önemli bir konu matematiksel bir kuramın gerçekliğidir. Matematik (Doğa Bilimlerinden farklı olarak) deneysel olarak sınanamadığı için belirli bir matematik kuramını gerçek bulmak için nedenler aranmaktadır (Bkz. Epistemoloji). Luitzen Brouwer’in temellerini attığı Sezgici Matematik bu görüşün bilenen temsilcilerindedir. Mantıkçı Matematik yaklaşımı ise Bertrand Russell ve Gottlob Frege tarafından savunulmuştur.David Hilbert, biçimcilik akımının temsilcilerinden sayılmaktadır. Gelenekselcilik mantıkcı
görgücüler (Rudolf Carnap, Alfred Jules Ayer, Carl Hempel) tarafından temsil edilmiştir.
The philosophy of mathematics is the branch of philosophy that studies the philosophical
assumptions, foundations, and implications of mathematics.
Recurrent themes include:
- What are the sources of mathematical subject matter?
- What is the ontological status of mathematical entities?
- What does it mean to refer to a mathematical object?
- What is the character of a mathematical proposition?
- What is the relation between logic and mathematics?
- What is the role of hermeneutics in mathematics?
- What kinds of inquiry play a role in mathematics?
- What are the objectives of mathematical inquiry?
- What gives mathematics its hold on experience?
- What are the human traits behind mathematics?
- What is mathematical beauty?
- What is the source and nature of mathematical truth?
- What is the relationship between the abstract world of mathematics and the material universe?
- Is maths an absolute and universal language? (this has been a common theme in the Sci-Fi genre)
The terms philosophy of mathematics and mathematical philosophy are frequently used as synonyms. The latter, however, may be used to mean at least three other things. One sense refers to a project of formalising a philosophical subject matter, say, aesthetics,ethics, logic, metaphysics, or theology, in a purportedly more exact and rigorous form, as for example the labours of Scholastic theologians, or the systematic aims of Leibniz andSpinoza. Another sense refers to the working philosophy of an individual practitioner or a like-minded community of practicing mathematicians. Additionally, some understand the term mathematical philosophy to be an allusion to the approach taken by Bertrand Russellin his book Introduction to Mathematical Philosophy.
Many thinkers have contributed their ideas concerning the nature of mathematics. Today, some philosophers of mathematics aim to give accounts of this form of inquiry and its products as they stand, while others emphasize a role for themselves that goes beyond simple interpretation to critical analysis. There are traditions of mathematical philosophy in both Western philosophy and Eastern philosophy. Western philosophies of mathematics go as far back as Plato, who studied the ontological status of mathematical objects, andAristotle, who studied logic and issues related to infinity (actual versus potential). Greekphilosophy on mathematics was strongly influenced by their study of geometry. For example, at one time, the Greeks held the opinion that 1 (one) was not a number, but rather a unit of arbitrary length. A number was defined as a multitude. Therefore 3, for example, represented a certain multitude of units, and was thus not "truly" a number. At another point, a similar argument was made that 2 was not a number but a fundamental notion of a pair. These views come from the heavily geometric straight-edge-and-compass viewpoint of the Greeks: just as lines drawn in a geometric problem are measured in proportion to the first arbitrarily drawn line, so too are the numbers on a number line measured in proportional to the arbitrary first "number" or "one." These earlier Greek ideas of numbers were later upended by the discovery of the irrationality of the square root of two. Hippasus, a disciple of Pythagoras, showed that the diagonal of a unit square was incommensurable with its (unit-length) edge: in other words he proved there was no existing (rational) number that accurately depicts the proportion of the diagonal of the unit square to its edge. This caused a significant re-evaluation of Greek philosophy of mathematics. According to legend, fellow Pythagoreans were so traumatised by this discovery that they murdered Hippasus to stop him from spreading his heretical idea. Greek ideas remained dominant until the 17th century. At this time, and beginning withLeibniz, the focus shifted strongly to the relationship between mathematics and logic. This perspective dominated the philosophy of mathematics through the time of Frege and of Russell, but was brought into question by developments in the late 19th and early 20th century.
Philosophy of mathematics in the 20th century
A perennial issue in the philosophy of mathematics concerns the relationship between logic and mathematics at their joint foundations. While 20th century philosophers continued to ask the questions mentioned at the outset of this article, the philosophy of mathematics in the 20th century was characterised by a predominant interest in formal logic, set theory, and foundational issues.
It is a profound puzzle that on the one hand mathematical truths seem to have a compelling inevitability, but on the other hand the source of their "truthfulness" remains elusive. Investigations into this issue are known as the foundations of mathematicsprogram.
At the start of the 20th century, philosophers of mathematics were already beginning to divide into various schools of thought about all these questions, broadly distinguished by their pictures of mathematical epistemology and ontology. Three schools, formalism,intuitionism, and logicism, emerged at this time, partly in response to the increasingly widespread worry that mathematics as it stood, and analysis in particular, did not live up to the standards of certainty and rigour that had been taken for granted. Each school addressed the issues that came to the fore at that time, either attempting to resolve them or claiming that mathematics is not entitled to its status as our most trusted knowledge.
Surprising and counter-intuitive developments in formal logic and set theory early in the 20th century led to new questions concerning what was traditionally called the foundations of mathematics. As the century unfolded, the initial focus of concern expanded to an open exploration of the fundamental axioms of mathematics, the axiomatic approach having been taken for granted since the time of Euclid around 300 BCE as the natural basis for mathematics. Notions of axiom, proposition and proof, as well as the notion of a proposition being true of a mathematical object (see Assignment (mathematical logic)), were formalised, allowing them to be treated mathematically. The Zermelo-Fraenkel axioms for set theory were formulated which provided a conceptual framework in which much mathematical discourse would be interpreted. In mathematics as in physics, new and unexpected ideas had arisen and significant changes were coming. With Gödel numbering, propositions could be interpreted as referring to themselves or other propositions, enabling inquiry into the consistency of mathematical theories. This reflective critique in which the theory under review "becomes itself the object of a mathematical study" led Hilbert to call such study metamathematics or proof theory.
At the middle of the century, a new mathematical theory known as category theory arose as a new contender for the natural language of mathematical thinking (Mac Lane 1998). As the 20th century progressed, however, philosophical opinions diverged as to just how well-founded were the questions about foundations that were raised at its opening. Hilary Putnam summed up one common view of the situation in the last third of the century by saying:
When philosophy discovers something wrong with science, sometimes science has to be changed — Russell's paradox comes to mind, as does Berkeley's attack on the actualinfinitesimal — but more often it is philosophy that has to be changed. I do not think that the difficulties that philosophy finds with classical mathematics today are genuine difficulties; and I think that the philosophical interpretations of mathematics that we are being offered on every hand are wrong, and that "philosophical interpretation" is just what mathematics doesn't need. (Putnam, 169-170).
Philosophy of mathematics today proceeds along several different lines of inquiry, by philosophers of mathematics, logicians, and mathematicians, and there are many schools of thought on the subject. The schools are addressed separately in the next section, and their assumptions explained.