数学の哲学
出典: フリー百科事典『ウィキペディア（Wikipedia）』
数学の哲学（すうがくのてつがく）とは、数学の哲学的前提や基礎、含意について研究する哲学の一分野である。
繰り返し取り上げられるテーマとして次のようなものがある:
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- 数学的主題の源は、何か？
What are the sources of mathematical subject matter? - 数学的実在の存在論的身分は、何か？
What is the ontological status of mathematical entities? - 数学的対象を指示するといことが意味しているのは、何か？
What does it mean to refer to a mathematical object? - 数学的命題の性質は、何か？
What is the character of a mathematical proposition? - 数学と論理学の関係は、何か？
What is the relation between logic and mathematics? - 数学における解釈学の役割は、何か？
What is the role of Hermeneutics in mathematics? - 数学で役割を果たしているのは、どのような種類の研究か？
What kinds of inquiry play a role in mathematics? - 数学が研究する対象は、何か？
What are the objectives of mathematical inquiry? - 数学へ経験への結びつきをもたらしているのは、何か？
What gives mathematics its hold on experience? - 数学の背後にある人間の特性は、何か？
What are the human traits behind mathematics? - 数学美とは、何か？
What is mathematical beauty? - 数学的真理の源と本性は、何か？
What is the source and nature of mathematical truth? - 数学の抽象的世界と物質的宇宙との関係は、何か？
What is the relationship between the abstract world of mathematics and the material universe?
- 数学的主題の源は、何か？
数学の哲学（英：philosophy of mathematics）と数学的哲学（英：mathematical philosophy）の用語は、しばしば同義語として使われる[1]。後者は、しかしながら、少なくとも三つの異なる意味を持っている。一つは、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、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、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実践している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はいくつかは、バートランド・ラッセルが彼の書籍『Introduction to Mathematical Philosophy』[2]でとったアプローチを暗示している。
The terms philosophy of mathematics and mathematical philosophy are frequently used as synonyms.[3] 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.
[編集] 歴史の概略（Historical overview）
多くの思想家が、数学の本性に関する彼らのアイデアを提供してきた。今日、一部の数学の哲学者達は、この形の研究とその成果に、彼らの研究の基礎としての説明を与えようとしている。しかし、他の哲学者達は、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する。
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。一時、ギリシア人は1（一）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。この理解は、ギリシアの大いに幾何学的な「直線・辺・コンパス」の視点に由来している。その視点とは、つまり、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、任意の初めの「数」あるいは「一」との比で測定された値で数となる、というものである。これらの初めのころのギリシアの数の概念は、後に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（単位長の）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の再査定がなされた。伝説によれば、ピュタゴラス学派の上位の学徒はこの発見により傷つき、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 geometry. 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 truly was 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、フォーカスは数学と論理の関係へと強力にシフトした。この見方は、フレーゲの時代とバートランド・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.
[編集] 20世紀に於ける数学の哲学（Philosophy of mathematics in the 20th century）
数学の哲学のかわらない問題の一つは、論理と数学の基礎につながる、相互の関係に関わっている。この項目の冒頭で触れたように、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、基礎的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 formal logic, set theory, and foundational issues.
一方で数学的真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 'truthfullness' 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見取り図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、数学は我々のもっと信頼できる知識としての身分をうける資格が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提出した。
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世紀初頭における驚くべき、そして反直感的な形式論理と集合論の発展は、伝統的に数学基礎論と呼ばれるものに関係する疑問へと導いていった。20世紀が進展するにつれ、当初の関心の焦点が、ユークリッドの時代以来、数学の自然な基礎だと受け止められていた公理的手法、数学の根本的な公理の開かれた探求に拡張した。公理、順序、そして集合といった中心的な概念は、新たな強調とともに受け止められた。物理学における数学においては、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。数学理論の無矛盾性の研究は、研究の新たな段階を開発へと導いた。それは、ヒルベルトが超数学（英：metamathematics）又は証明論（英：proof theory）と呼んだ、「数学的研究の対象それ自体」の再検討に数学理論がさらされる反省的批判である。
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 as the natural basis for mathematics. Core concepts such as axiom, order, and set received fresh emphasis. In mathematics as in physics, new and unexpected ideas had arisen and significant changes were coming. Inquiries into the consistency of mathematical theories led to the development of a new level of study, a reflective critique in which the theory under review "becomes itself the object of a mathematical study", what Hilbert called metamathematics or proof theory (Kleene, 55).
20世紀中葉、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。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に関する疑問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 midpoint 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、真の困難であることを考えることができない。そして、私は、我々が提案されている両手いっぱい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。
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摘示し、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.
[編集] 現代の学派（Contemporary schools of thought）
[編集] 数学的実在論（Mathematical realism）
数学的実在論は、一般的な実在論のように、 数学的実体が人間精神から独立に実在すると考える。したがって、数学は人間が発明したのではなく、発見したのであり、宇宙の他の知的生命体もおそらく同様にしているだろうと考える。この視点に立てば、見つけることのできる数学の種類は真にひとつである。例えば、三角形は人間の精神によって作られたのではなく、実体のあるものである。
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、擬似経験主義的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.
実在論においては、数学的対象たる存在者はどのようなものかと、それらを我々がどのように知りうるかの区別がある。
Within realism, there are distinctions depending on what sort of existence one takes mathematical entities to have, and how we know about them.
[編集] プラトニズム（Platonism）
プラトニズムは、数学的実体が抽象的であり、時空や因果の特性を持たず、永遠不変であることを提案する実在論の形態である。これは、最多数の人々が数について持っている自然な観点だと、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、プラトンの日常的世界はその不完全な近似であるに過ぎない普遍かつ究極的な実在である「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。おそらくプラトンは彼の理解を、世界は文字通り数から生まれたと信じていた古代ギリシアのピタゴラス教団から受け継いでいるので、二つのアイデアの間には、意味ある、たんに表面的でない関係がある。
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 naive 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", an unchanging ultimate reality that the everyday world can only imperfectly approximate. The two ideas have a meaningful, not just a superficial connection, because Plato probably derived his understanding from the 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問題は、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 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』[4]）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（下記）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 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.
[編集] 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：
- 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。
- 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においてGaのとき、かつそのときに限りFaでのあるならば、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 en: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。
But Frege's construction was flawed. Russell discovered that Basic Law V is inconsistent (this is en: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.
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のような現代の論理主義者は、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シフト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.
[編集] 経験主義（Empiricism）
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 en: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 en: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.
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 en: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.
Putnam strongly rejected the term "en:Platonist" as implying an overly-specific en:ontology that was not necessary to en:mathematical practice in any real sense. He advocated a form of "pure realism" that rejected mystical notions of en:truth and accepted much en:quasi-empiricism in mathematics. Putnam was involved in coining the term "pure realism" (see below).
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. 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.
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 en:Penelope Maddy's Realism in Mathematics. Another example of a realist theory is the en:embodied mind theory (see below).
[編集] 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 en: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.
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 en:Rudolf Carnap, en:Alfred Tarski and en:Haskell Curry, considered mathematics to be the investigation of formal axiom systems. en:Mathematical logicians study formal systems but are just as often realists as they are formalists.
Formalists are usually very 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 minute string manipulation games mentioned above. While published proofs (if correct) could in principle be formulated in terms of these games, the effort required in space and time would be prohibitive (witness en:Principia Mathematica.) In addition, the rules are certainly not substantial to the initial creation of those proofs. Formalism is also silent to the question of which axiom systems ought to be studied.
[編集] 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.
[編集] 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.
[編集] Fictionalism
Fictionalism was introduced in 1980 when en: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 en:Newtonian mechanics that didn't reference numbers or functions at all. He started with the "betweenness" of en:Hilbert's axioms to characterize space without coordinatizing it, and then added extra relations between points to do the work formerly done by en: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 en: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 "en:Sherlock Holmes lived at 22b 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 en: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 en:second-order logic to carry out his reduction, and because the statement of conservativity seems to require quantification over abstract models or deductions.
[編集] 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 en:Where Mathematics Comes From, by en:George Lakoff and en:Rafael E. Núñez. In addition, mathematician en:Keith Devlin has investigated similar concepts with his book en:The Math Instinct. For more on the science that inspired this perspective, see en:cognitive science of mathematics.
[編集] Social constructivism or social realism
Social constructivism or social realism theories see mathematics primarily as a en: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 en: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 en:folk mathematics not enough, due to an over-emphasis on axiomatic proof and peer review as practices.
The social nature of mathematics is highlighted in its en: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 en:epistemic community and often has great difficulty communicating, or motivating the investigation of en: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 en:cognitive bias, or of mathematician's en: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 en:racism and en:ethnocentrism. Some of these ideas are close to en:postmodernism.
Contributions to this school have been made by en:Imre Lakatos and en:Thomas Tymoczko, although it is not clear that either would endorse the title. More recently en:Paul Ernest has explicitly formulated a social constructivist philosophy of mathematics. [1] Some consider the work of en: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 en:Erdős number. en: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, en:Philip J. Davis, has expressed sympathy for the social view as well.
[編集] 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 en:truth, or even on practices unique to mathematicians such as the proof, a growing movement from the en:1960s to the en: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 en:Eugene Wigner's famous en:1960 paper en: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.
[編集] Quasi-empiricism
One parallel concern that does not actually challenge the schools directly but instead questions their focus is the notion of en:quasi-empiricism in mathematics. This grew from the increasingly popular assertion in the late 20th century that no one en: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. en:Quasi-empiricism was developed by en:Imre Lakatos, inspired by the philosophy of science of en:Karl Popper.
Lakatos's 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 en: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.
en: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).
[編集] Action
Some practitioners and scholars who are not engaged primarily in proof-oriented approaches have suggested an interesting and important theory about the nature of mathematics. For example, en:Judea Pearl claimed that all of mathematics as presently understood was based on an algebra of seeing - and proposed an algebra of doing to complement it - this is a central concern of the en:philosophy of action and other studies of how en:knowledge relates to action. The most important output of this was new theories of en:truth, notably those appropriate to en:activism and grounding en:empirical methods.
[編集] Unification
Few philosophers are able to penetrate mathematical notations and culture to relate conventional notions of en: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.
[編集] Language
Innovations in the en:philosophy of language during the 20th century renewed interest in the question as to 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 en: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 en: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 en:Gottlob Frege and en:Alfred Tarski for the study of mathematical language have been extended greatly by Tarski's student en:Richard Montague and other linguists working in en: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.
[編集] 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 en: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 en:literature.
Philip 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 en:contradiction, ascribed to en:Euclid; the second is a more direct proof involving the en: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.
en:Paul Erdős was well-known for his notion of a hypothetical "Book" containing the most elegant or beautiful mathematical proofs. en:Gregory Chaitin rejected Erdős's book. By way of example, he provided three separate proofs of the infinitude of primes. The first was Euclid's, the second was based on the en:Riemann zeta function, and the third was Chaitin's own, derived from en:algorithmic information theory. Chaitin then argued that each one was as beautiful as the others, because all three reveal different aspects of the same problem.
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[要出典], 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.[要出典]
[編集] 脚注（Notes）
- ^ 例えば、Edward Maziarsが1969年の書評で、「to distinguish philosophical mathematics (which is primarily a specialized task for a mathematician) from mathematical philosophy (which ordinarily may be the philosopher's metier)」と提案するとき、彼は、mathematical philosophy（数学的哲学）をphilosophy of mathematics（数学の哲学）の同義語として使っている。 (Maziars, Edward A. (1969). “Problems in the Philosophy of Mathematics (Book Review)”. Philosophy of Science 36 (3): p. 325.)
- ^ 部分的に『数学の原理』の概念のをより技術的でない手法で解説しているもの。
- ^ For example, when Edward Maziars proposes in a 1969 book review "to distinguish philosophical mathematics (which is primarily a specialized task for a mathematician) from mathematical philosophy (which ordinarily may be the philosopher's metier)", he uses the term mathematical philosophy as being synonymous with philosophy of mathematics. (Maziars, Edward A. (1969). “Problems in the Philosophy of Mathematics (Book Review)”. Philosophy of Science 36 (3): p. 325.)
- ^ P.J. デービス・R. ヘルシュ著、柴垣和三雄、田中裕、清水邦夫訳 『数学的経験』 森北出版、1986年。ISBN 978-4627052109。
[編集] 関連項目（See also）
[編集] 関係するトピック（Related topics）
[編集] 関係する仕事（Related works）
[編集] 歴史的トピック（Historical topics）
[編集] 参考文献（Further reading）
- The London Philosophy Study Guide offers many suggestions on what to read, depending on the student's familiarity with the subject:
- 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), en:The Mathematical Experience, Mariner Books, New York, NY.
- 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), en: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.