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在科学中
Jouko Väänänen认为二阶逻辑是数学的基础，而不是集合论，而其他人则认为范畴论是数学某些方面的基础。
Jouko Väänänen认为二阶逻辑是数学的基础，而不是集合论，而其他人则认为范畴论是数学某些方面的基础。
== In science ==
== 在科学中 ==
{{More citations needed section|date=August 2011}}
{{More citations needed section|date=August 2011}}
The incompleteness theorems of Kurt Gödel, published in 1931, caused doubt about the attainability of an axiomatic foundation for all of mathematics. Any such foundation would have to include axioms powerful enough to describe the arithmetic of the natural numbers (a subset of all mathematics). Yet Gödel proved that, for any consistent recursively enumerable axiomatic system powerful enough to describe the arithmetic of the natural numbers, there are (model-theoretically) true propositions about the natural numbers that cannot be proved from the axioms. Such propositions are known as formally undecidable propositions. For example, the continuum hypothesis is undecidable in the Zermelo–Fraenkel set theory as shown by Cohen.
The incompleteness theorems of Kurt Gödel, published in 1931, caused doubt about the attainability of an axiomatic foundation for all of mathematics. Any such foundation would have to include axioms powerful enough to describe the arithmetic of the natural numbers (a subset of all mathematics). Yet Gödel proved that, for any consistent recursively enumerable axiomatic system powerful enough to describe the arithmetic of the natural numbers, there are (model-theoretically) true propositions about the natural numbers that cannot be proved from the axioms. Such propositions are known as formally undecidable propositions. For example, the continuum hypothesis is undecidable in the Zermelo–Fraenkel set theory as shown by Cohen.
1931年发表的库尔特 · 哥德尔的不完备性定理，引起了对所有数学公理化基础的可达性的怀疑。任何这样的基础都必须包括强大到足以描述自然数算术(所有数学的子集)的公理。然而，g ö del 证明了，对于任何一致的可递归枚举的公理系统，有足够的能力来描述自然数的算术运算，有关于自然数的真命题(模型-理论)是不能从公理中证明的。这样的命题称为形式上不可判定的命题。例如，正如 Cohen 所证明的，在 Zermelo-Fraenkel 集合论中，连续统假设是不可判定的。
1931年发表的库尔特 · 哥德尔（Kurt Gödel）的不完备性定理，引起了对所有数学公理化基础的可达性的怀疑，任何这样的基础都必须包含足够强大的公理来描述所有自然数的算术(所有数学的子集)。然而，哥德尔证明了，对于足以描述自然数算数的任何一致的可递归枚举的公理系统，有关于自然数的真命题(模型-理论)是不能从公理中证明的。这样的命题称为形式上的不可判定的命题。例如，在科恩（Cohen）提出的 Zermelo-Fraenkel 集合论中，连续统假设是不可判定的。
Reductionist thinking and methods form the basis for many of the well-developed topics of modern [[science]], including much of [[physics]], [[chemistry]] and [[molecular biology]]. [[Classical mechanics]] in particular is seen as a reductionist framework. For instance, we understand the solar system in terms of its components (the sun and the planets) and their interactions.<ref name=":8">{{Cite book|last=McCauley|first=Joseph L.|title=Dynamics of Markets: The New Financial Economics, Second Edition|publisher=Cambridge University Press|year=2009|isbn=978-0-521-42962-7|location=Cambridge|pages=241}}</ref> [[Statistical mechanics]] can be considered as a reconciliation of [[macroscopic]] [[thermodynamic laws]] with the reductionist method of explaining macroscopic properties in terms of [[microscopic]] components.
还原论的思想和方法构成了许多现代科学发展良好的主题的基础，包括许多物理、化学和分子生物学。经典力学尤其可以被看作是一种还原论的框架。例如，我们根据太阳系的组成部分（太阳和行星）及其相互作用来理解太阳系<ref name=":8" /> 。统计力学则可以被认为是宏观热力学定律与用微观组分解释宏观性质的还原方法的调和。
In science, reductionism implies that certain topics of study are based on areas that study smaller spatial scales or organizational units. While it is commonly accepted that the foundations of [[chemistry]] are based in [[physics]], and [[molecular biology]] is based on chemistry, similar statements become controversial when one considers less rigorously defined intellectual pursuits. For example, claims that [[sociology]] is based on [[psychology]], or that [[economics]] is based on [[sociology]] and [[psychology]] would be met with reservations. These claims are difficult to substantiate even though there are obvious associations between these topics (for instance, most would agree that [[psychology]] can affect and inform [[economics]]). The limit of reductionism's usefulness stems from [[Emergence#Emergent properties and processes|emergent properties]] of [[complex systems]], which are more common at certain levels of organization. For example, certain aspects of [[evolutionary psychology]] and [[sociobiology]] are rejected by some who claim that complex systems are inherently irreducible and that a [[holistic]] method is needed to understand them.
In science, reductionism implies that certain topics of study are based on areas that study smaller spatial scales or organizational units. While it is commonly accepted that the foundations of [[chemistry]] are based in [[physics]], and [[molecular biology]] is based on chemistry, similar statements become controversial when one considers less rigorously defined intellectual pursuits. For example, claims that [[sociology]] is based on [[psychology]], or that [[economics]] is based on [[sociology]] and [[psychology]] would be met with reservations. These claims are difficult to substantiate even though there are obvious associations between these topics (for instance, most would agree that [[psychology]] can affect and inform [[economics]]). The limit of reductionism's usefulness stems from [[Emergence#Emergent properties and processes|emergent properties]] of [[complex systems]], which are more common at certain levels of organization. For example, certain aspects of [[evolutionary psychology]] and [[sociobiology]] are rejected by some who claim that complex systems are inherently irreducible and that a [[holistic]] method is needed to understand them.
在科学中，还原论意味着某些研究主题是基于研究更小的空间尺度或组织单位的领域。虽然人们普遍认为化学的基础是基于物理，分子生物学是基于化学，但当一个人思考不那么严格定义的知识领域时，类似的陈述就会变得有争议。例如，人们对声称社会学是以心理学为基础，或者经济学是以社会学和心理学为基础的说法往往会持保留意见。尽管这些话题之间存在明显的联系（例如，大多数人会同意心理学可以影响并影响经济学），但这些说法很难得到证实。还原论效用的限制源于复杂系统的涌现特性，这种特性在组织的某些层次上更为常见。例如，一些人声称复杂的系统从本质上是不可简化的，需要一个整体的方法来理解它们，因而不同意进化心理学和社会生物学的某些观点。
The role of reduction in computer science can be thought as a precise and unambiguous mathematical formalization of the philosophical idea of "theory reductionism". In a general sense, a problem (or set) is said to be reducible to another problem (or set), if there is a computable/feasible method to translate the questions of the former into the latter, so that, if one knows how to computably/feasibly solve the latter problem, then one can computably/feasibly solve the former. Thus, the latter can only be at least as "hard" to solve as the former.
The role of reduction in computer science can be thought as a precise and unambiguous mathematical formalization of the philosophical idea of "theory reductionism". In a general sense, a problem (or set) is said to be reducible to another problem (or set), if there is a computable/feasible method to translate the questions of the former into the latter, so that, if one knows how to computably/feasibly solve the latter problem, then one can computably/feasibly solve the former. Thus, the latter can only be at least as "hard" to solve as the former.