利用者:Mizusumashi/数学の哲学

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?

数学の哲学（英: philosophy of mathematics）と数学的哲学（英: mathematical philosophy）という用語は、しばしば同義語として使われる。しかしながら、後者は、別の意味を少なくとも三つ持っている。一つは、例えばスコラ学の神学者の仕事やライプニッツやスピノザの体系的目標といった、美学、倫理学、論理学、形而上学、神学といった哲学的主題の諸問題を、その主張するところでは、より正確かつ厳密な形へと形式化するプロジェクトを意味する。もう一つは、個々の数学の実践者や、考えかたの似た現場の数学者の共同体の、仕事上の哲学を意味する。加えて、 数学的哲学という用語の理解のいくつかは、バートランド・ラッセルが彼の書籍 Introduction to Mathematical Philosophy でとったアプローチを示唆している。

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 formalizing a philosophical subject matter, say, aesthetics, ethics, logic, metaphysics, or theology, in a purportedly more exact and rigorous form, as for example the labors of Scholastic theologians, or the systematic aims of Leibniz and Spinoza. 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 Russell in his book Introduction to Mathematical Philosophy.

歴史の概略（Historical overview）

歴史上、多くの思想家が、数学の本性に関する彼らの考えを明らかにしてきた。今日の一部の数学の哲学者達は、この形の研究とその成果を、彼らの研究の基礎として説明しようとする。しかし、他の哲学者達は、それらの単純な解釈を超えて、批判的分析へと進むべき彼ら自身の役割を強調している。

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.

西洋哲学と東洋哲学の両方に、数学的哲学の伝統がある。西洋の数学の哲学者は、数学的対象の存在論的身分を研究したプラトンと、論理と現実態と可能態の無限に関する問題を研究したアリストテレスにまで遡る。数学に関するギリシア哲学は、彼らの幾何学の研究の強い影響の下にあった。ギリシア人は1は数ではなく、むしろ任意の長さの単位であるという意見を持っていた。ある数は、比であると定義された。それゆえ、例えば、3は、単位長との比を表しており、本当のいみの数ではなかった。この理解は、「直線・辺・コンパス」という、たぶんに幾何学的なギリシアの視点に由来している。その視点とは、幾何学的問題において描かれた直線は、初めに描かれた任意の長さの直線との比で測定され、初めの「数」、つまり1との比で測定された値で数となる、というものである。これらの初期のギリシアの数の概念は、2の平方根が無理数であるという発見によって、打ち倒された。ピュタゴラスの門人であるヒッパソスは、単位正方形の対角線は、その辺と通約不能であることを示した：換言すると、彼は、単位正方形の対角線とその辺の比を正確にあらわす（有理）数が存在しないことを証明した。これが原因となり、ギリシアの数学の哲学は再検討されることとなった。伝説によれば、この発見によって傷つけられたピュタゴラス学派の学徒達は、ヒッパソスが彼の異端な概念を広めるのを防ぐために、彼を殺害した。

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, and Aristotle, who studied logic and issues related to infinity (actual versus potential). Greek philosophy on mathematics was strongly influenced by their study of en:geometry. At one time, the Greeks held the opinion that 1 (one) was not a en: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 truly was not 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 number 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 traumatized by this discovery that they murdered Hippasus to stop him from spreading his heretical idea.

ライプニッツとともに、焦点は数学と論理の関係へと、強力に移動した。この見方は、フレーゲとラッセルの時代を通して、数学の哲学を支配したが、19世紀終期と20世紀初頭における発展によって、疑問を付されるようになった。

Beginning with Leibniz, the focus shifted strongly to the relationship between mathematics and logic. This view 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.

20世紀における数学の哲学（Philosophy of mathematics in the 20th century）

数学の哲学のかわらない課題の一つは、論理と数学の双方の基礎につながる、相互の関係に関わっている。20世紀の哲学者はこの問題を問い続ける一方で、20世紀の数学の哲学は形式論理、集合論、基礎付けの問題への目立った関心によって特徴付けられる。

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 characterized by a predominant interest in en:formal logic, en:set theory, and foundational issues.

一方で数学的真理が避けがたく必然的であるように思え、他方でその「真理性」の源泉がとらえどころがないままなのは、深遠なパズルである。この問題の研究は、数学の基礎付けのプログラムとして知られる。

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 mathematics program.

この世紀の初め、数学の哲学者はすでに、これら全ての問題に関して、数学の認識論と存在論の概略の明白な違いによって、多様な学派に分かれていた。三つの学派、形式主義、直観主義、論理主義は、部分的には、数学をそれ自体として基礎付けることと、とくに解析学が当然のことと考えられていた確実性と厳密性の基準に答えることができないという広がりつつあった困難とへの応答として、このとき現れた。問題の解決を試みるのであれ、我々のもっと信頼できる知識としての地位をうける資格が数学にはないと主張するのであれ、どの学派もこの問題をこのとき前面に押し出した。

At the start of the 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 rigor 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.

20世紀の初めにおける驚くべき、そして反直感的な形式論理と集合論の発展は、伝統的に「数学の基礎」と呼ばれるものに関係する疑問へと導いていった。紀元前300年前後のユークリッドの時代以来、公理に基づく手法は、数学の自然な基点だと受け止められていたが、20世紀が進むにつれ、当初の関心の焦点が拡張され、数学の基礎的な公理に対する制限のない探求へと至るようになった。公理、順序、そして集合といった観念、そしてまた数学的対象の命題の真理についての観念は、定式化され、数学的に扱うことが許されるようになった。ツェルメロ＝フレンケルの公理系は、多くの数学的議論を解釈する概念的枠組みを提供するものとして集合論を形式化した。物理学におけるのと同様に、数学においても、新しい、予期しないアイデアが登場し、特筆すべき変化が訪れた。ゲーデル数によって、数学理論の無矛盾性の研究が可能となった。その中で理論が再吟味されるこのような反省的批判「それ自体が」ヒルベルトが「超数学」（英: metamathematics）又は「証明論」（英: proof theory）と呼んだような「数学的研究対象となる」。

Surprising and counterintuitive 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 formalized, 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 en: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.

20世紀の中ごろ、圏論として知られる新たな数学理論が、数学的思考の自然な言語としての新たな競争者として登場した（Mac Lane 1998）。しかしながら、20世紀が進むにつれ、まさに当初提起された基礎付けに関する疑問自体が如何によく基礎付けられるのか、というところへ哲学的関心は広がっていった。ヒラリー・パトナムは、20世紀の最後の1/3の状況についての一つの共通見解を、次のように要約した：

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:

哲学が科学における誤りを発見したときは、しばしば、科学は変わらざるを得ない。ラッセルのパラドックスが思い浮かぶし、バークリーの現実の無限小への批判がそうであるように。しかし、変らなければならないのは哲学であることのほうが多い。私は、哲学が今日の古典的数学に発見する困難が、真の困難とは思えない。そして、私は、我々が提案されている無数の数学の哲学的解釈は誤っており、「哲学的解釈」はまさに数学が必要としていないものだ、と考えている。(Putnam, 169-170).

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 actual infinitesimal — 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).

数学の哲学は、数学の哲学者、論理学者、数学者によっていくつもの異なる研究の方向にそって進んでおり、この主題に関する多くの学派が存在する。次のセクションで、これらの学派を個別に取り上げ、彼らの仮説を説明する。

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.

現代の学派（Contemporary schools of thought）

数学的実在論（Mathematical realism）

Mathematical realism, like realism in general, holds that mathematical entities exist independently of the human mind. Thus humans do not invent mathematics, but rather discover it, and any other intelligent beings in the universe would presumably do the same. In this point of view, there is really one sort of mathematics that can be discovered: Triangles, for example, are real entities, not the creations of the human mind.

多くの現場の数学者は、数学的実在論者であった； 彼らは、彼ら自身を自然に存在する対象の発見者だとみなしている。この例には、ポール・エルデシュやクルト・ゲーデルが含まれる。ゲーデルは、ある意味で感覚的知覚と同様に知覚されうる客観的な数学的実在を信じていた。確実な原理（例えば、任意の二つの対象について、正確にその二つの対象によって構成される対象のコレクションが存在する）は真だと直接的にみなされうる。しかし、連続体仮説のようないくつかの予測は、そのような原理の基礎だけによっては決定不可能であると証明されるかもしれない。ゲーデルは、準経験的な方法論がそのような予測を、合理的に受け入れる十分な証拠を提供するだろうと示唆した。

Many working mathematicians have been mathematical realists; they see themselves as discoverers of naturally occurring objects. Examples include Paul Erdős and Kurt Gödel. Gödel believed in an objective mathematical reality that could be perceived in a manner analogous to sense perception. Certain principles (e.g., for any two objects, there is a collection of objects consisting of precisely those two objects) could be directly seen to be true, but some conjectures, like the continuum hypothesis, might prove undecidable just on the basis of such principles. Gödel suggested that quasi-empirical methodology could be used to provide sufficient evidence to be able to reasonably assume such a conjecture.

数学的対象たる存在者はどのようなものか、また、それらを我々がどのように知りうるかについて、実在論の内部にいくつもの異なる立場が存在する。

Within realism, there are distinctions depending on what sort of existence one takes mathematical entities to have, and how we know about them.

プラトニズム（Platonism）

これは、最多数の人々が数について持っている観点だと、しばしば主張される。プラトニズムという用語は、この観点が、日常的世界は不変かつ究極的な実在の不完全な近似であるに過ぎないという、プラトンの（「プラトンの洞窟」のたとえで表される）「イデア界」の教説とパラレルであるように見えることに由来する。古代ギリシアの非常に人気のあったピタゴラス教団が世界は文字通り数から生まれたと信じており、これがプラトンに先行し、おそらくプラトンの考えはこれに影響されたものであるので、「プラトンの洞窟」と「プラトニズム」の間には意味ある、たんに表面的でないつながりがある。

Platonism is the form of realism that suggests that mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging. This is often claimed to be the view most people have of numbers. The term Platonism is used because such a view is seen to parallel Plato's belief in a "World of Ideas" (typified by Plato's cave): the everyday world can only imperfectly approximate of an unchanging, ultimate reality. Both Plato's cave and Platonism have meaningful, not just a superficial connections, because Plato's ideas were preceded and probably influenced by the hugely popular Pythagoreans of ancient Greece, who believed that the world was, quite literally, generated by numbers.

数学的プラトニズムの主張は主要な問題点は、次のようなものである：数学的対象は、正確にどこに、またどのように存在するのか、我々はそれをどのように知りうるのか？　我々の物理的世界と完全に分離され、数学的対象によって占有された世界があるのか？　どのように我々はその分離された世界へのアクセスを得、その対象についての真理を発見するのか？　一つの答えは、数学的に存在する構造はまたそれ自体の世界において物理的に存在する仮定する理論であるUltimate Ensembleかもしれない。

The major problem of mathematical platonism is this: precisely where and how do the mathematical entities exist, and how do we know about them? Is there a world, completely separate from our physical one, which is occupied by the mathematical entities? How can we gain access to this separate world and discover truths about the entities? One answer might be en:Ultimate ensemble, which is a theory that postulates all structures that exist mathematically also exist physically in their own universe.

ゲーデルのプラトニズムは、我々を数学的対象の直接的な知覚へと導く、特別な種類の数学的直観を仮定する。（この考えかたは、フッサールが数学について語った多くのことと類似性を持ち、数学〔的知識〕は総合的かつアプリオリであるとするカントのアイデアを支持する。）デービス（Philip J. Davis）とヘルシュ（Reuben Hersh）は、彼らの著書 The Mathematical Experience（日本語訳『数学的経験』）で、多くの数学者は、慎重にその立場を表明するときには彼らは形式主義（後述）に後退するにもかかわらず、プラトニストであるかのように振舞っていると指摘した。

Gödel's platonism postulates a special kind of mathematical intuition that lets us perceive mathematical objects directly. (This view bears resemblances to many things en:Husserl said about mathematics, and supports Kant's idea that mathematics is synthetic a priori.) Davis and Hersh have suggested in their book The Mathematical Experience that most mathematicians act as though they are Platonists, even though, if pressed to defend the position carefully, they may retreat to formalism (see below).

何人かの数学者は、さらに微妙に異なるバージョンのプラトニズムと同様の見解を持っている。これらのアイデアは、ときにネオ・プラトニズム（英: Neo-Platonism）と呼ばれる。

Some mathematicians hold opinions that amount to more nuanced versions of Platonism. These ideas are sometimes described as Neo-Platonism.

論理主義（Logicism）

論理主義は、数学は論理に還元可能で、ゆえに数学は論理の一部以外の何者でもないというテーゼである（Carnap 1931/1883, 41）。論理主義者は、数学はアプリオリに知りうると考えるが、しかし我々の数学の知識は我々の一般的論理の知識のたんなる部分であり、そのため分析的であって、いかなる数学的直観の特別な能力も不要であると主張する。この観点からは、論理は数学の固有の基礎であり、全ての数学的言明は必然的な論理的真理である。

Logicism is the thesis that mathematics is reducible to logic, and hence nothing but a part of logic (Carnap 1931/1883, 41). Logicists hold that mathematics can be known a priori, but suggest that our knowledge of mathematics is just part of our knowledge of logic in general, and is thus analytic, not requiring any special faculty of mathematical intuition. In this view, logic is the proper foundation of mathematics, and all mathematical statements are necessary logical truths.

ルドルフ・カルナップ（1931）は、論理主義のテーゼを二つの部分で提示した：

1. 数学の概念は、論理的概念から明示的な定義をとおして導きうる。
2. 数学の定理は、論理的公理から純粋に論理的な演繹によって導きうる。

Rudolf Carnap (1931) presents the logicist thesis in two parts:

1.	The concepts of mathematics can be derived from logical concepts through explicit definitions.
2.	The theorems of mathematics can be derived from logical axioms through purely logical deduction.

ゴットロープ・フレーゲは、論理主義の創始者である。彼の影響力のあるDie Grundgesetze der Arithmetik（『算術の基本法則』）の中で、彼が「基本ルールⅤ」（概念FとGにおいて、全ての対象aについてGaのときかつそのときに限りFaであるならば、またそのときに限って、Fの外延とGの外延は等しい）と呼び、彼が論理の妥当な一部と考えた原理である内包性の一般原理を持った論理系から算術体系を作り上げた。

Gottlob Frege was the founder of logicism. In his seminal Die Grundgesetze der Arithmetik (Basic Laws of Arithmetic) he built up arithmetic from a system of logic with a general principle of comprehension, which he called "Basic Law V" (for concepts F and G, the extension of F equals the extension of G if and only if for all objects a, Fa if and only if Ga), a principle that he took to be acceptable as part of logic.

しかし、フレーゲの構成には欠陥があった。ラッセルは、「基本ルールⅤ」は矛盾をはらむことを発見した。（これが、ラッセルのパラドックスである。）この後しばらくして、フレーゲは彼の論理主義のプログラムを捨てたが、ラッセルとホワイトヘッドによって継続された。彼らは、このパラドックスを「悪循環」に由来するものとし、これを扱うために彼らが「分岐した型の理論」（英: ramified type theory）と呼んだものを作り上げた。この体系において、彼らはついに近代数学の多くの部分を作り上げたが、しかし非常に複雑な形式となった（例えば、それぞれに型に異なる自然数があり、無限に多くの型が存在する）。彼らはまた、数学のかなり多くを構築するために、いくらかの「還元公理」（英: axiom of reducibility）のような妥協をしなくてはならなかった。ラッセルでさえ、この公理は論理に本当に属するものではない、と述べている。

Frege's construction was flawed. Russell discovered that Basic Law V is inconsistent. (This is Russell's paradox.) Frege abandoned his logicist program soon after this, but it was continued by Russell and Whitehead. They attributed the paradox to "vicious circularity" and built up what they called ramified type theory to deal with it. In this system, they were eventually able to build up much of modern mathematics but in an altered, and excessively complex, form (for example, there were different natural numbers in each type, and there were infinitely many types). They also had to make several compromises in order to develop so much of mathematics, such as an "axiom of reducibility". Even Russell said that this axiom did not really belong to logic.

現代の論理主義者は（ボブ・ヘイル（Bob Hale）やクリスピン・ライト（Crispin Wright）、おそらくは他の人々も）、フレーゲのものに近いプログラムに回帰している。彼らは基本法則Ⅴを捨ててしまって、ヒュームの原理（英: Hume's principle。概念Fの下にある対象の数は、概念Gの下にある対象の数と、Fの外延とGの外延が一対一対応させられるとき、かつそのときに限り、等しい。）のような抽象原理を支持している。フレーゲは数の明示的な定義のために基本法則Ⅴを必要としたが、数の全ての性質はヒュームの原理から導き出せる。これはフレーゲにとって十分ではなかっただろう。（彼の言葉を換言すれば）数3が事実上ジュリアス・シーザーであることを排除できないからである。加えて、彼らが基本法則Ⅴを置き換えるために採用せざるをえなかった弱められた原理の多くは、明白に分析とも、したがって純粋に論理的ともみなせない。

Modern logicists (like Bob Hale, Crispin Wright, and perhaps others) have returned to a program closer to Frege's. They have abandoned Basic Law V in favour of abstraction principles such as Hume's principle (the number of objects falling under the concept F equals the number of objects falling under the concept G if and only if the extension of F and the extension of G can be put into one-to-one correspondence). Frege required Basic Law V to be able to give an explicit definition of the numbers, but all the properties of numbers can be derived from Hume's principle. This would not have been enough for Frege because (to paraphrase him) it does not exclude the possibility that the number 3 is in fact Julius Caesar. In addition, many of the weakened principles that they have had to adopt to replace Basic Law V no longer seem so obviously analytic, and thus purely logical.

もし、数学が論理の一部分であるならば、数学的対象に関する疑問は、論理的対象への疑問へと還元される。しかし、こう尋ねる人もいるかもしれない、論理的概念の対象とは何なのか？　この視点からは、論理主義は、完全な回答を与えることなく、数学の哲学に関する疑問を論理に関する疑問に移動させたようにみえるかもしれない。

If mathematics is a part of logic, then questions about mathematical objects reduce to questions about logical objects. But what, one might ask, are the objects of logical concepts? In this sense, logicism can be seen as shifting questions about the philosophy of mathematics to questions about logic without fully answering them.

経験主義（Empiricism）

経験主義とは、数学はアプリオリに知られうるということを全く拒否する、実在論の一種である。経験主義は、ちょうどすべての他の科学の事実がそうであるように、我々は経験的な探求によって数学的事実を発見する、という。経験主義は、20世紀初等に唱導された古典的な三つの立場の一つではなく、同世紀中葉に最初に立ち上がった。しかしながら、このような見解の重要な所期の支持者の一人は、ジョン・スチュワート・ミルである。ミルの見解は広く批判された。なぜなら、その見解は「2 + 2 = 4」のような文を不確実で、我々が二個の事物がが二組合わさって四つとなることを観察することからのみ学ぶことができるような偶然的な真理とするからである。

Empiricism is a form of realism that denies that mathematics can be known a priori at all. It says that we discover mathematical facts by empirical research, just like facts in any of the other sciences. It is not one of the classical three positions advocated in the early 20th century, but primarily arose in the middle of the century. However, an important early proponent of a view like this was John Stuart Mill. Mill's view was widely criticized, because it makes statements like "2 + 2 = 4" come out as uncertain, contingent truths, which we can only learn by observing instances of two pairs coming together and forming a quartet.

クワインとパトナムによって定式化された現代の数学的経験主義の主な論拠は、不可欠性論法（英: indispensability argument）である。これは、数学は全ての経験科学にとって不可欠であり、もし我々がその科学によって記述される現象の実在性を信じたいのであれば、我々はその記述のために必要とされるそれらの事物の実在性もまた信じなくてはならない。つまり、電球があのように振舞うのは何故なのか述べるために物理学は電子に言及しなければならないのだから、電子は実在しなくてはならない。科学がその説明を提供するのに数について語る必要があるのだから、数は実在しなくてはならない。クワインとパトナムの哲学全体からは、これは自然主義的な議論である。この立場は数学的対象の存在を経験の最善の説明として論じ、そのようにして、数学からそれを他の科学から区別しているものを剥ぎ取る。

Contemporary mathematical empiricism, formulated by Quine and Putnam, is primarily supported by the indispensability argument: mathematics is indispensable to all empirical sciences, and if we want to believe in the reality of the phenomena described by the sciences, we ought also believe in the reality of those entities required for this description. That is, since physics needs to talk about electrons to say why light bulbs behave as they do, then electrons must exist. Since physics needs to talk about numbers in offering any of its explanations, then numbers must exist. In keeping with Quine and Putnam's overall philosophies, this is a naturalistic argument. It argues for the existence of mathematical entities as the best explanation for experience, thus stripping mathematics of some of its distinctness from the other sciences.

パトナムは「プラトニスト」という言葉を、いかなる本当の意味での数学的実践にも必要とされない特定の存在論を示唆する言葉として、強く拒否した。彼は、真理の神秘的な通知を拒否し、数学における準経験主義から多くを受け入れた、ある形態の「純粋な実在論」（英: pure realism）を擁護した。彼は、「純粋な実在論」という言葉を生み出すことにかかわった（後述）。

Putnam strongly rejected the term "Platonist" as implying an overly-specific ontology that was not necessary to mathematical practice in any real sense. He advocated a form of "pure realism" that rejected mystical notions of truth and accepted much quasi-empiricism in mathematics. Putnam was involved in coining the term "pure realism" (see below).

数学についての経験的な見解へのもっとも重要な批判は、ミルに対して提起されたものと、おおよそ同じである。もし数学が他の科学と同じだけ経験的ならば、そのことは数学の結果も他の科学の結果と同じだけ誤りやすく、同じだけ偶然的であることを意味している。ミルの場合は経験的正当化は直接的になされたが、クワインの場合は間接的で、科学理論全体の整合性（エドワード・オズボーン・ウィルソンのいうところのコンシリエンス）を通してなされる。クワインが指摘するところでは、数学が完全に確実なようにみえるのは、数学が演じている役割が我々の信念の網の非常に中央にあるからであり、それを修正することは我々にとってとてつもなく困難ではあるが、不可能ではない。

The most important criticism of empirical views of mathematics is approximately the same as that raised against Mill. If mathematics is just as empirical as the other sciences, then this suggests that its results are just as fallible as theirs, and just as contingent. In Mill's case the empirical justification comes directly, while in Quine's case it comes indirectly, through the coherence of our scientific theory as a whole, i.e. consilience after E O Wilson. Quine suggests that mathematics seems completely certain because the role it plays in our web of belief is incredibly central, and that it would be extremely difficult for us to revise it, though not impossible.

クワインとゲーデルのアプローチの欠点をそれぞれの面から克服しようと試みる数学の哲学については、Penelope MaddyのRealism in Mathematicsを参照せよ。実在論の理論の他の一つの例は、embodied mind theoryである（後述）。

For a philosophy of mathematics that attempts to overcome some of the shortcomings of Quine and Gödel's approaches by taking aspects of each see Penelope Maddy's Realism in Mathematics. Another example of a realist theory is the embodied mind theory (see below).

このことを示す経験的な証拠として、一歳児は基礎的な算数を行うことができる。

For experimental evidence suggesting that one-day-old babies can do elementary arithmetic, see Brian Butterworth.

形式主義（Formalism）

詳細は「形式主義 (数学)」を参照

Formalism holds that mathematical statements may be thought of as statements about the consequences of certain string manipulation rules. For example, in the "game" of en:Euclidean geometry (which is seen as consisting of some strings called "axioms", and some "rules of inference" to generate new strings from given ones), one can prove that the en:Pythagorean theorem holds (that is, you can generate the string corresponding to the Pythagorean theorem). Mathematical truths are not about numbers and sets and triangles and the like — in fact, they aren't "about" anything at all!

Another version of formalism is often known as deductivism. In deductivism, the Pythagorean theorem is not an absolute truth, but a relative one: if you assign meaning to the strings in such a way that the rules of the game become true (ie, true statements are assigned to the axioms and the rules of inference are truth-preserving), then you have to accept the theorem, or, rather, the interpretation you have given it must be a true statement. The same is held to be true for all other mathematical statements. Thus, formalism need not mean that mathematics is nothing more than a meaningless symbolic game. It is usually hoped that there exists some interpretation in which the rules of the game hold. (Compare this position to structuralism.) But it does allow the working mathematician to continue in his or her work and leave such problems to the philosopher or scientist. Many formalists would say that in practice, the axiom systems to be studied will be suggested by the demands of science or other areas of mathematics.

ダフィット・ヒルベルト

A major early proponent of formalism was en:David Hilbert, whose program was intended to be a complete and consistent axiomatization of all of mathematics. ("Consistent" here means that no contradictions can be derived from the system.) Hilbert aimed to show the consistency of mathematical systems from the assumption that the "finitary arithmetic" (a subsystem of the usual en:arithmetic of the positive en:integers, chosen to be philosophically uncontroversial) was consistent. Hilbert's goals of creating a system of mathematics that is both complete and consistent was dealt a fatal blow by the second of en:Gödel's incompleteness theorems, which states that sufficiently expressive consistent axiom systems can never prove their own consistency. Since any such axiom system would contain the finitary arithmetic as a subsystem, Gödel's theorem implied that it would be impossible to prove the system's consistency relative to that (since it would then prove its own consistency, which Gödel had shown was impossible). Thus, in order to show that any axiomatic system of mathematics is in fact consistent, one needs to first assume the consistency of a system of mathematics that is in a sense stronger than the system to be proven consistent.

Hilbert was initially a deductivist, but, as may be clear from above, 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.

Other formalists, such as Rudolf Carnap, Alfred Tarski and Haskell Curry, considered mathematics to be the investigation of formal axiom systems. Mathematical logicians study formal systems but are just as often realists as they are formalists.

Formalists are relatively tolerant and inviting to new approaches to logic, non-standard number systems, new set theories etc. The more games we study, the better. However, in all three of these examples, motivation is drawn from existing mathematical or philosophical concerns. The "games" are usually not arbitrary.

The main critique of formalism is that the actual mathematical ideas that occupy mathematicians are far removed from the string manipulation games mentioned above. Formalism is thus silent to the question of which axiom systems ought to be studied, as none is more meaningful than another from a formalistic point of view.

Recently, some formalist mathematicians have proposed that all of our formal mathematical knowledge should be systematically encoded in computer-readable formats, so as to facilitate automated proof checking of mathematical proofs and the use of interactive theorem proving in the development of mathematical theories and computer software. Because of their close connection with computer science, this idea is also advocated by mathematical intuitionists and constructivists in the "computability" tradition (see below). See QED project for a general overview.

直観主義（Intuitionism）

詳細は「数学的直観主義」を参照

In mathematics, intuitionism is a program of methodological reform whose motto is that "there are no non-experienced mathematical truths" (L.E.J. Brouwer). From this springboard, intuitionists seek to reconstruct what they consider to be the corrigible portion of mathematics in accordance with Kantian concepts of being, becoming, intuition, and knowledge. Brouwer, the founder of the movement, held that mathematical objects arise from the a priori forms of the volitions that inform the perception of empirical objects. (CDP, 542)

en:Leopold Kronecker said: "The natural numbers come from God, everything else is man's work." A major force behind Intuitionism was en:L.E.J. Brouwer, who rejected the usefulness of formalized logic of any sort for mathematics. His student en:Arend Heyting postulated an en:intuitionistic logic, different from the classical en:Aristotelian logic; this logic does not contain the law of the excluded middle and therefore frowns upon proofs by contradiction. The en:axiom of choice is also rejected in most intuitionistic set theories, though in some versions it is accepted. Important work was later done by en:Errett Bishop, who managed to prove versions of the most important theorems in en:real analysis within this framework.

In intuitionism, the term "explicit construction" is not cleanly defined, and that has led to criticisms. Attempts have been made to use the concepts of en:Turing machine or en:computable function to fill this gap, leading to the claim that only questions regarding the behavior of finite en:algorithms are meaningful and should be investigated in mathematics. This has led to the study of the en:computable numbers, first introduced by en:Alan Turing. Not surprisingly, then, this approach to mathematics is sometimes associated with theoretical en:computer science.

構成主義（Constructivism）

詳細は「Mathematical constructivism」を参照

Like intuitionism, constructivism involves the regulative principle that only mathematical entities which can be explicitly constructed in a certain sense should be admitted to mathematical discourse. In this view, mathematics is an exercise of the human intuition, not a game played with meaningless symbols. Instead, it is about entities that we can create directly through mental activity. In addition, some adherents of these schools reject non-constructive proofs, such as a proof by contradiction.

フィクショナリズム（Fictionalism）

Fictionalism in mathematics was brought to fame in 1980 when Hartry Field published Science Without Numbers, which rejected and in fact reversed Quine's indispensability argument. Where Quine suggested that mathematics was indispensable for our best scientific theories, and therefore should be accepted as a body of truths talking about independently existing entities, Field suggested that mathematics was dispensable, and therefore should be considered as a body of falsehoods not talking about anything real. He did this by giving a complete axiomatization of Newtonian mechanics that didn't reference numbers or functions at all. He started with the "betweenness" of Hilbert's axioms to characterize space without coordinatizing it, and then added extra relations between points to do the work formerly done by vector fields. Hilbert's geometry is mathematical, because it talks about abstract points, but in Field's theory, these points are the concrete points of physical space, so no special mathematical objects at all are needed.

Having shown how to do science without using mathematics, he proceeded to rehabilitate mathematics as a kind of useful fiction. He showed that mathematical physics is a conservative extension of his non-mathematical physics (that is, every physical fact provable in mathematical physics is already provable from his system), so that the mathematics is a reliable process whose physical applications are all true, even though its own statements are false. Thus, when doing mathematics, we can see ourselves as telling a sort of story, talking as if numbers existed. For Field, a statement like "2 + 2 = 4" is just as false as "Sherlock Holmes lived at 221B Baker Street" — but both are true according to the relevant fictions.

By this account, there are no metaphysical or epistemological problems special to mathematics. The only worries left are the general worries about non-mathematical physics, and about fiction in general. Field's approach has been very influential, but is widely rejected. This is in part because of the requirement of strong fragments of second-order logic to carry out his reduction, and because the statement of conservativity seems to require quantification over abstract models or deductions. Another objection is that it is not clear how one could have certain results in science, such as quantum theory or the periodic table, without mathematics. If what distinguishes one element from another is precisely the number of electrons, neutrons and protons, how does one distinguish between elements without a concept of number?

Embodied mind theories

Embodied mind theories hold that mathematical thought is a natural outgrowth of the human cognitive apparatus which finds itself in our physical universe. For example, the abstract concept of en:number springs from the experience of counting discrete objects. It is held that mathematics is not universal and does not exist in any real sense, other than in human brains. Humans construct, but do not discover, mathematics.

With this view, the physical universe can thus be seen as the ultimate foundation of mathematics: it guided the evolution of the brain and later determined which questions this brain would find worthy of investigation. However, the human mind has no special claim on reality or approaches to it built out of math. If such constructs as en:Euler's identity are true then they are true as a map of the human mind and en:cognition.

Embodied mind theorists thus explain the effectiveness of mathematics — mathematics was constructed by the brain in order to be effective in this universe.

The most accessible, famous, and infamous treatment of this perspective is Where Mathematics Comes From, by George Lakoff and Rafael E. Núñez. In addition, mathematician Keith Devlin has investigated similar concepts with his book The Math Instinct. For more on the philosophical ideas that inspired this perspective, see cognitive science of mathematics.

社会構築主義・社会的実在主義（Social constructivism or social realism）

Social constructivism or social realism theories see mathematics primarily as a social construct, as a product of culture, subject to correction and change. Like the other sciences, mathematics is viewed as an empirical endeavor whose results are constantly evaluated and may be discarded. However, while on an empiricist view the evaluation is some sort of comparison with "reality", social constructivists emphasize that the direction of mathematical research is dictated by the fashions of the social group performing it or by the needs of the society financing it. However, although such external forces may change the direction of some mathematical research, there are strong internal constraints — the mathematical traditions, methods, problems, meanings and values into which mathematicians are enculturated — that work to conserve the historically defined discipline.

This runs counter to the traditional beliefs of working mathematicians, that mathematics is somehow pure or objective. But social constructivists argue that mathematics is in fact grounded by much uncertainty: as mathematical practice evolves, the status of previous mathematics is cast into doubt, and is corrected to the degree it is required or desired by the current mathematical community. This can be seen in the development of analysis from reexamination of the calculus of Leibniz and Newton. They argue further that finished mathematics is often accorded too much status, and folk mathematics not enough, due to an over-emphasis on axiomatic proof and peer review as practices. However, this might be seen as merely saying that rigorously proven results are overemphasized, and then "look how chaotic and uncertain the rest of it all is!"

The social nature of mathematics is highlighted in its subcultures. Major discoveries can be made in one branch of mathematics and be relevant to another, yet the relationship goes undiscovered for lack of social contact between mathematicians. Social constructivists argue each speciality forms its own epistemic community and often has great difficulty communicating, or motivating the investigation of unifying conjectures that might relate different areas of mathematics. Social constructivists see the process of "doing mathematics" as actually creating the meaning, while social realists see a deficiency either of human capacity to abstractify, or of human's cognitive bias, or of mathematicians' collective intelligence as preventing the comprehension of a real universe of mathematical objects. Social constructivists sometimes reject the search for foundations of mathematics as bound to fail, as pointless or even meaningless. Some social scientists also argue that mathematics is not real or objective at all, but is affected by racism and ethnocentrism. Some of these ideas are close to postmodernism.

Contributions to this school have been made by Imre Lakatos and Thomas Tymoczko, although it is not clear that either would endorse the title. More recently Paul Ernest has explicitly formulated a social constructivist philosophy of mathematics. [1] Some consider the work of Paul Erdős as a whole to have advanced this view (although he personally rejected it) because of his uniquely broad collaborations, which prompted others to see and study "mathematics as a social activity", e.g., via the Erdős number. Reuben Hersh has also promoted the social view of mathematics, calling it a "humanistic" approach [2], similar to but not quite the same as that associated with Alvin White [3]; one of Hersh's co-authors, Philip J. Davis, has expressed sympathy for the social view as well.

A criticism of this approach is that it is trivial, based on the trivial observation that mathematics is a human activity. To observe that rigorous proof comes only after unrigorous conjecture, experimentation and speculation is true, but it is trivial and no-one would deny this. So it's a bit of a stretch to characterize a philosophy of mathematics in this way, on something trivially true. The calculus of Leibniz and Newton was reexamined by mathematicians such as Weierstrass in order to rigorously prove the theorems thereof. There is nothing special or interesting about this, as it fits in with the more general trend of unrigorous ideas which are later made rigorous. There needs to be a clear distinction between the objects of study of mathematics and the study of the objects of study of mathematics. The former doesn't seem to change a great deal; the latter is forever in flux. The latter is what the Social theory is about, and the former is what Platonism et al. are about.

However, this criticism is rejected by supporters of the social constructivist perspective because it misses the point that the very objects of mathematics are social constructs. These objects, it asserts, are primarily semiotic objects existing in the sphere of human culture, sustained by social practices (after Wittgenstein) that utilize physically embodied signs and give rise to intrapersonal (mental) constructs. Social constructivists view the reification of the sphere of human culture into a Platonic realm, or some other heaven-like domain of existence beyond the physical world, a long standing category error.

伝統的学派を超えて（Beyond the traditional schools）

Rather than focus on narrow debates about the true nature of mathematical truth, or even on practices unique to mathematicians such as the proof, a growing movement from the 1960s to the 1990s began to question the idea of seeking foundations or finding any one right answer to why mathematics works. The starting point for this was Eugene Wigner's famous 1960 paper The Unreasonable Effectiveness of Mathematics in the Natural Sciences, in which he argued that the happy coincidence of mathematics and physics being so well matched seemed to be unreasonable and hard to explain.

The embodied-mind or cognitive school and the social school were responses to this challenge, but the debates raised were difficult to confine to those.

準経験論（Quasi-empiricism）

One parallel concern that does not actually challenge the schools directly but instead questions their focus is the notion of quasi-empiricism in mathematics. This grew from the increasingly popular assertion in the late 20th century that no one foundation of mathematics could be ever proven to exist. It is also sometimes called "postmodernism in mathematics" although that term is considered overloaded by some and insulting by others. Quasi-empiricism argues that in doing their research, mathematicians test hypotheses as well as proving theorems. A mathematical argument can transmit falsity from the conclusion to the premises just as well as it can transmit truth from the premises to the conclusion. Quasi-empiricism was developed by Imre Lakatos, inspired by the philosophy of science of Karl Popper.

Lakatos' philosophy of mathematics is sometimes regarded as a kind of social constructivism, but this was not his intention.

Such methods have always been part of folk mathematics by which great feats of calculation and measurement are sometimes achieved. Indeed, such methods may be the only notion of proof a culture has.

Hilary Putnam has argued that any theory of mathematical realism would include quasi-empirical methods. He proposed that an alien species doing mathematics might well rely on quasi-empirical methods primarily, being willing often to forgo rigorous and axiomatic proofs, and still be doing mathematics — at perhaps a somewhat greater risk of failure of their calculations. He gave a detailed argument for this in New Directions (ed. Tymockzo, 1998).

Unification

Few philosophers are able to penetrate mathematical notations and culture to relate conventional notions of metaphysics to the more specialized metaphysical notions of the schools above. This may lead to a disconnection in which some mathematicians continue to profess discredited philosophy as a justification for their continued belief in a world-view promoting their work.

Although the social theories and quasi-empiricism, and especially the embodied mind theory, have focused more attention on the en:epistemology implied by current mathematical practices, they fall far short of actually relating this to ordinary human en:perception and everyday understandings of en:knowledge.

Language

Innovations in the philosophy of language during the 20th century renewed interest in whether mathematics is, as if often said, the language of science. Although most mathematicians and physicists (and many philosophers) would accept the statement "mathematics is a language", linguists believe that the implications of such a statement must be considered. For example, the tools of linguistics are not generally applied to the symbol systems of mathematics, that is, mathematics is studied in a markedly different way than other languages. If mathematics is a language, it is a different type of language than natural languages. Indeed, because of the need for clarity and specificity, the language of mathematics is far more constrained than natural languages studied by linguists. However, the methods developed by Frege and Tarski for the study of mathematical language have been extended greatly by Tarski's student Richard Montague and other linguists working in formal semantics to show that the distinction between mathematical language and natural language may not be as great as it seems.

See also en:philosophy of language.

Aesthetics

Many practising mathematicians have been drawn to their subject because of a sense of beauty they perceive in it. One sometimes hears the sentiment that mathematicians would like to leave philosophy to the philosophers and get back to mathematics — where, presumably, the beauty lies.

In his work on the divine proportion, H. E. Huntley relates the feeling of reading and understanding someone else's proof of a theorem of mathematics to that of a viewer of a masterpiece of art — the reader of a proof has a similar sense of exhilaration at understanding as the original author of the proof, much as, he argues, the viewer of a masterpiece has a sense of exhilaration similar to the original painter or sculptor. Indeed, one can study mathematical and scientific writings as literature.

Philip J. Davis and Reuben Hersh have commented that the sense of mathematical beauty is universal amongst practicing mathematicians. By way of example, they provide two proofs of the irrationality of the √2. The first is the traditional proof by contradiction, ascribed to Euclid; the second is a more direct proof involving the fundamental theorem of arithmetic that, they argue, gets to the heart of the issue. Davis and Hersh argue that mathematicians find the second proof more aesthetically appealing because it gets closer to the nature of the problem.

Paul Erdős was well-known for his notion of a hypothetical "Book" containing the most elegant or beautiful mathematical proofs. There is not universal agreement that a result has one "most elegant" proof; Gregory Chaitin has argued against this idea. Philosophers have sometimes criticized mathematicians' sense of beauty or elegance as being, at best, vaguely stated. By the same token, however, philosophers of mathematics have sought to characterize what makes one proof more desirable than another when both are logically sound.

Another aspect of aesthetics concerning mathematics is mathematicians' views towards the possible uses of mathematics for purposes deemed unethical or inappropriate. The best-known exposition of this view occurs in en:G.H. Hardy's book en:A Mathematician's Apology, in which Hardy argues that pure mathematics is superior in beauty to en:applied mathematics precisely because it cannot be used for war and similar ends. Some later mathematicians have characterized Hardy's views as mildly dated, with the applicability of number theory to modern-day en:cryptography. While this would force Hardy to change his primary example if he were writing today, many practicing mathematicians still subscribe to Hardy's general sentiments.

Mathematics of philosophy

Mathematics of philosophy is the branch of mathematics which, with mathematic methods, attempts to approach philosophic matters.

For instance, in utilitarism, the units of measurements called hedons and dolors may be used in formulas of various complexity in order to get to what is the best action to do in different situations.

関連項目（See also）

関係するトピック（Related topics）

• 公理的集合論
• 公理系
• 圏論
• 形式言語
• 形式体系
• 数学基礎論
• 黄金比
• 数学史
• 直観論理
• 論理学

• Mathematical beauty
• Mathematical constructivism
• 数理論理学
• 証明
• 数学
• Model theory
• Naive set theory
• 超準解析
• 言語哲学
• 科学哲学

• Philosophy of probability
• Proof theory
• 推論規則
• Science studies
• 科学的方法
• 集合論
• 真理
• The Unreasonable Effectiveness of Mathematics in the Natural Sciences

関係する著作（Related works）

• The Analyst
• ユークリッド原論
• Gödel's completeness theorem - ゲーデルの完全性定理
• Introduction to Mathematical Philosophy

• Kaina Stoicheia
• New Foundations
• Principia Mathematica
• The Simplest Mathematics

歴史関連（Historical topics）

• History and philosophy of science
• 数学史
• 哲学史

参考文献

References

• Aristotle, "Prior Analytics", Hugh Tredennick (trans.), pp. 181-531 in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.
• Audi, Robert (ed., 1999), The Cambridge Dictionary of Philosophy, Cambridge University Press, Cambridge, UK, 1995. 2nd edition, 1999. Cited as CDP.
• Benacerraf, Paul, and Putnam, Hilary (eds., 1983), Philosophy of Mathematics, Selected Readings, 1st edition, Prentice-Hall, Englewood Cliffs, NJ, 1964. 2nd edition, Cambridge University Press, Cambridge, UK, 1983.
• Berkeley, George (1734), The Analyst; or, a Discourse Addressed to an Infidel Mathematician. Wherein It is examined whether the Object, Principles, and Inferences of the modern Analysis are more distinctly conceived, or more evidently deduced, than Religious Mysteries and Points of Faith, London & Dublin. Online text, David R. Wilkins (ed.), Eprint.
• Bourbaki, N. (1994), Elements of the History of Mathematics, John Meldrum (trans.), Springer-Verlag, Berlin, Germany.
• Carnap, Rudolf (1931), "Die logizistische Grundlegung der Mathematik", Erkenntnis 2, 91-121. Republished, "The Logicist Foundations of Mathematics", E. Putnam and G.J. Massey (trans.), in Benacerraf and Putnam (1964). Reprinted, pp. 41-52 in Benacerraf and Putnam (1983).
• Chandrasekhar, Subrahmanyan (1987), Truth and Beauty. Aesthetics and Motivations in Science, University of Chicago Press, Chicago, IL.
• Hadamard, Jacques (1949), The Psychology of Invention in the Mathematical Field, 1st edition, Princeton University Press, Princeton, NJ. 2nd edition, 1949. Reprinted, Dover Publications, New York, NY, 1954.
• Hardy, G.H. (1940), A Mathematician's Apology, 1st published, 1940. Reprinted, C.P. Snow (foreword), 1967. Reprinted, Cambridge University Press, Cambridge, UK, 1992.
• Hart, W.D. (ed., 1996), The Philosophy of Mathematics, Oxford University Press, Oxford, UK.
• Hendricks, Vincent F. and Hannes Leitgeb (eds.). Philosophy of Mathematics: 5 Questions, New York: Automatic Press / VIP, 2006. [4]
• Huntley, H.E. (1970), The Divine Proportion: A Study in Mathematical Beauty, Dover Publications, New York, NY.
• Klein, Jacob (1968), Greek Mathematical Thought and the Origin of Algebra, Eva Brann (trans.), MIT Press, Cambridge, MA, 1968. Reprinted, Dover Publications, Mineola, NY, 1992.
• Kline, Morris (1959), Mathematics and the Physical World, Thomas Y. Crowell Company, New York, NY, 1959. Reprinted, Dover Publications, Mineola, NY, 1981.
• Kline, Morris (1972), Mathematical Thought from Ancient to Modern Times, Oxford University Press, New York, NY.
• König, Julius (Gyula) (1905), "Über die Grundlagen der Mengenlehre und das Kontinuumproblem", Mathematische Annalen 61, 156-160. Reprinted, "On the Foundations of Set Theory and the Continuum Problem", Stefan Bauer-Mengelberg (trans.), pp. 145-149 in Jean van Heijenoort (ed., 1967).
• Lakatos, Imre 1976 Proofs and Refutations:The Logic of Mathematical Discovery (Eds) J. Worrall & E. Zahar Cambridge University Press
• Lakatos, Imre 1978 Mathematics, Science and Epistemology: Philosophical Papers Volume 2 (Eds) J.Worrall & G.Currie Cambridge University Press
• Lakatos, Imre 1968 Problems in the Philosophy of Mathematics North Holland
• Leibniz, G.W., Logical Papers (1666-1690), G.H.R. Parkinson (ed., trans.), Oxford University Press, London, UK, 1966.
• Mac Lane, Saunders (1998), Categories for the Working Mathematician, 1st edition, Springer-Verlag, New York, NY, 1971, 2nd edition, Springer-Verlag, New York, NY.
• Maddy, Penelope (1990), Realism in Mathematics, Oxford University Press, Oxford, UK.
• Maddy, Penelope (1997), Naturalism in Mathematics, Oxford University Press, Oxford, UK.
• Maziarz, Edward A., and Greenwood, Thomas (1995), Greek Mathematical Philosophy, Barnes and Noble Books.
• Mount, Matthew, Classical Greek Mathematical Philosophy, .
• Peirce, Benjamin (1870), "Linear Associative Algebra", § 1. See American Journal of Mathematics 4 (1881).
• Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1-6, Charles Hartshorne and Paul Weiss (eds.), vols. 7-8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931 – 1935, 1958. Cited as CP (volume).(paragraph).
• Plato, "The Republic, Volume 1", Paul Shorey (trans.), pp. 1-535 in Plato, Volume 5, Loeb Classical Library, William Heinemann, London, UK, 1930.
• Plato, "The Republic, Volume 2", Paul Shorey (trans.), pp. 1-521 in Plato, Volume 6, Loeb Classical Library, William Heinemann, London, UK, 1935.
• Putnam, Hilary (1967), "Mathematics Without Foundations", Journal of Philosophy 64/1, 5-22. Reprinted, pp. 168-184 in W.D. Hart (ed., 1996). - ヒラリー・パトナム「基礎付けのいらない数学」『リーディングス 数学の哲学―ゲーデル以後』戸田山和久訳、勁草書房、1995年（ISBN 978-4326101047）。
• Robinson, Gilbert de B. (1959), The Foundations of Geometry, University of Toronto Press, Toronto, Canada, 1940, 1946, 1952, 4th edition 1959.
• Russell, Bertrand (1919), Introduction to Mathematical Philosophy, George Allen and Unwin, London, UK. Reprinted, John G. Slater (intro.), Routledge, London, UK, 1993.
• Smullyan, Raymond M. (1993), Recursion Theory for Metamathematics, Oxford University Press, Oxford, UK.
• Strohmeier, John, and Westbrook, Peter (1999), Divine Harmony, The Life and Teachings of Pythagoras, Berkeley Hills Books, Berkeley, CA.
• Styazhkin, N.I. (1969), History of Mathematical Logic from Leibniz to Peano, MIT Press, Cambridge, MA.
• Tait, William W. (1986), "Truth and Proof: The Platonism of Mathematics", Synthese 69 (1986), 341-370. Reprinted, pp. 142-167 in W.D. Hart (ed., 1996).
• Tarski, A. (1983), Logic, Semantics, Metamathematics: Papers from 1923 to 1938, J.H. Woodger (trans.), Oxford University Press, Oxford, UK, 1956. 2nd edition, John Corcoran (ed.), Hackett Publishing, Indianapolis, IN, 1983.
• Tymoczko, Thomas (1998), New Directions in the Philosophy of Mathematics, Catalog entry?
• Ulam, S.M. (1990), Analogies Between Analogies: The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators, A.R. Bednarek and Françoise Ulam (eds.), University of California Press, Berkeley, CA.
• van Heijenoort, Jean (ed. 1967), From Frege To Gödel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, Cambridge, MA.
• Wigner, Eugene (1960), "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", Communications on Pure and Applied Mathematics 13(1): 1-14. Eprint
• 飯田隆編『リーディングス 数学の哲学―ゲーデル以後』勁草書房、1995年（ISBN 978-4326101047）。
• 岡本賢吾「無限」『岩波　哲学・思想辞典』岩波書店、1998年（ISBN 978-4-0008-0089-1）。
• 萩野弘之「ピュタゴラス」「ピュタゴラス学派」『岩波　哲学・思想辞典』岩波書店、1998年（ISBN 978-4-0008-0089-1）。
• 日本数学会編「数学基礎論」『岩波　数学辞典　第4版』岩波書店、2007年（ISBN 978-4-00-080309-0）。
• 大西琢朗 (2006年). “フレーゲの論理主義と数の存在論”. 京都大学学術情報リポジトリ. 2008年1月8日閲覧。 - 哲学論叢33巻、京都大学哲学論叢刊行会、43-54項（未確認）

Further reading

• The London Philosophy Study Guide offers many suggestions on what to read, depending on the student's familiarity with the subject:
  • Philosophy of Mathematics
  • Mathematical Logic
  • Set Theory & Further Logic

• Colyvan, Mark (2004), "Indispensability Arguments in the Philosophy of Mathematics", Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.), Eprint.
• Davis, Philip J. and Hersh, Reuben (1981), The Mathematical Experience, Mariner Books, New York, NY. - P.J. デービス・R. ヘルシュ『数学的経験』柴垣和三雄、田中裕、清水邦夫訳、森北出版、1986年（ISBN 978-4627052109）。
• Devlin, Keith (2005), The Math Instinct: Why You're a Mathematical Genius (Along with Lobsters, Birds, Cats, and Dogs), Thunder's Mouth Press, New York, NY.
• Dummett, Michael (1991 a), Frege, Philosophy of Mathematics, Harvard University Press, Cambridge, MA.
• Dummett, Michael (1991 b), Frege and Other Philosophers, Oxford University Press, Oxford, UK.
• Dummett, Michael (1993), Origins of Analytical Philosophy, Harvard University Press, Cambridge, MA.
• Ernest, Paul (1998), Social Constructivism as a Philosophy of Mathematics, State University of New York Press, Albany, NY.
• George, Alexandre (ed., 1994), Mathematics and Mind, Oxford University Press, Oxford, UK.
• Kline, Morris (1972), Mathematical Thought from Ancient to Modern Times, Oxford University Press, New York, NY.
• Lakoff, George, and Núñez, Rafael E. (2000), Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, Basic Books, New York, NY.
• Peirce, C.S., Bibliography.
• Raymond, Eric S. (1993), "The Utility of Mathematics", Eprint.
• Shapiro, Stewart (2000), Thinking About Mathematics: The Philosophy of Mathematics, Oxford University Press, Oxford, UK.

外部リンク

• Philosophy of Mathematics （英語） - スタンフォード哲学百科事典「Mizusumashi/数学の哲学」の項目。
• R.B. Jones' philosophy of mathematics page
• Mizusumashi/数学の哲学 - DMOZ（英語）
• The Philosophy of Real Mathematics Blog

ジャーナル

• Philosophia Mathematica journal
• The Philosophy of Mathematics Education Journal homepage