Editing Reductionism (section)

== Definitions ==
''[[The Oxford Companion to Philosophy]]'' suggests that reductionism is "one of the most used and abused terms in the philosophical lexicon" and suggests a three-part division:<ref name=Ruse>{{cite book |title=The Oxford Companion to Philosophy |author=Michael Ruse |editor=Ted Honderich |isbn=978-0191037474 |year=2005 |edition=2nd |chapter=Entry for "reductionism" |publisher=Oxford University Press |page=793 |chapter-url=https://books.google.com/books?id=bJFCAwAAQBAJ&pg=PT1884}}</ref>
# '''Ontological reductionism''': a belief that the whole of reality consists of a minimal number of parts.
# '''Methodological reductionism''': the scientific attempt to provide an explanation in terms of ever-smaller entities.
# '''Theory reductionism''': the suggestion that a newer theory does not replace or absorb an older one, but reduces it to more basic terms. Theory reduction itself is divisible into three parts: translation, derivation, and explanation.<ref name=Ney />

Reductionism can be applied to any [[phenomenon]], including [[object (philosophy)|objects]], problems, [[explanation]]s, [[theory|theories]], and meanings.<ref name=Ney /><ref name=Polkinghorne>{{cite encyclopedia |title=Reductionism  |author=John Polkinghorne |url=http://www.disf.org/en/Voci/104.asp |encyclopedia=Interdisciplinary Encyclopedia of Religion and Science|date=2002 |publisher=Advanced School for Interdisciplinary Research; Pontifical University of the Holy Cross}}</ref><ref>For reductionism referred to [[explanation]]s, [[theory|theories]], and meanings, see [[Willard Van Orman Quine]]'s ''[[Two Dogmas of Empiricism]]''. Quine objected to the [[positivism|positivistic]], reductionist "belief that each meaningful statement is equivalent to some logical construct upon terms which refer to immediate experience" as an intractable problem.</ref>

For the sciences, application of methodological reductionism attempts explanation of entire systems in terms of their individual, constituent parts and their interactions. For example, the temperature of a gas is reduced to nothing beyond the average kinetic energy of its molecules in motion. [[Thomas Nagel]] and others speak of 'psychophysical reductionism' (the attempted reduction of psychological phenomena to physics and chemistry), and 'physico-chemical reductionism' (the attempted reduction of biology to physics and chemistry).<ref name=Nagel /> In a very simplified and sometimes contested form, reductionism is said to imply that a system is nothing but the sum of its parts.<ref name=Polkinghorne /><ref name=GodfreySmith />

However, a more nuanced opinion is that a system is composed entirely of its parts, but the system will have features that none of the parts have (which, in essence is the basis of [[emergentism]]).<ref name=Jones /> "The point of mechanistic explanations is usually showing how the higher level features arise from the parts."<ref name=GodfreySmith />

Other definitions are used by other authors. For example, what [[John Polkinghorne]] terms 'conceptual' or 'epistemological' reductionism<ref name=Polkinghorne /> is the definition provided by [[Simon Blackburn]]<ref name=Blackburn>{{cite book |author=Simon Blackburn |title= Oxford Dictionary of Philosophy |chapter=Entry on ‘reductionism’ |date= 2005 |page=311 |publisher= Oxford University Press, UK |isbn= 978-0198610137 |chapter-url=https://books.google.com/books?id=5wTQtwB1NdgC&pg=PA311}}</ref> and by [[Jaegwon Kim]]:<ref name=Kim>{{cite book |author=Jaegwon Kim |title=The Oxford Companion to Philosophy  |editor=Ted Honderich |isbn=978-0191037474 |year=2005 |edition=2nd |chapter=Entry for ‘mental reductionism’ |publisher=Oxford University Press |page=794 |chapter-url=https://books.google.com/books?id=bJFCAwAAQBAJ&pg=PT1885}}</ref> that form of reductionism which concerns a program of replacing the facts or entities involved in one type of discourse with other facts or entities from another type, thereby providing a relationship between them. Richard Jones distinguishes ontological and epistemological reductionism, arguing that many ontological and epistemological reductionists affirm the need for different concepts for different degrees of complexity while affirming a reduction of theories.<ref name=Jones />

The idea of reductionism can be expressed by "levels" of explanation, with higher levels reducible if need be to lower levels. This use of levels of understanding in part expresses our human limitations in remembering detail. However, "most philosophers would insist that our role in conceptualizing reality [our need for a hierarchy of "levels" of understanding] does not change the fact that different levels of organization in reality do have different 'properties'."<ref name=Jones />

Reductionism does not preclude the existence of what might be termed [[Emergence|emergent phenomena]], but it does imply the ability to understand those phenomena completely in terms of the processes from which they are composed. This reductionist understanding is very different from ontological or strong [[emergentism]], which intends that what emerges in "emergence" is more than the sum of the processes from which it emerges, respectively either in the ontological sense or in the epistemological sense.<ref>Axelrod and Cohen "Harnessing Complexity"</ref> 

=== Ontological reductionism ===
Richard Jones divides ontological reductionism into two: the reductionism of substances (e.g., the reduction of mind to matter) and the reduction of the number of structures operating in nature (e.g., the reduction of one physical force to another). This permits scientists and philosophers to affirm the former while being anti-reductionists regarding the latter.<ref>Richard H. Jones (2000), ''Reductionism: Analysis and the Fullness of Reality'', pp. 24—26, 29–31. Lewisburg, Pa.: Bucknell University Press.</ref>

[[Nancey Murphy]] has claimed that there are two species of ontological reductionism: one that claims that wholes are nothing more than their parts; and atomist reductionism, claiming that wholes are not "really real". She admits that the phrase "really real" is apparently senseless but she has tried to explicate the supposed difference between the two.<ref>Nancey Murphy, "Reductionism and Emergence. A Critical Perspective." In ''Human Identity at the Intersection of Science, Technology and Religion''. Edited by Nancey Murphy, and Christopher C. Knight. Burlington, VT: Ashgate, 2010. P. 82.</ref>

Ontological reductionism denies the idea of ontological [[emergence]], and claims that emergence is an [[Epistemology|epistemological]] phenomenon that only exists through analysis or description of a system, and does not exist fundamentally.<ref>{{Cite journal|last1=Silberstein|first1=Michael|last2=McGeever|first2=John|date=April 1999|title=The Search for Ontological Emergence|url=https://academic.oup.com/pq/article-lookup/doi/10.1111/1467-9213.00136|journal=The Philosophical Quarterly|language=en|volume=49|issue=195|pages=201–214|doi=10.1111/1467-9213.00136|issn=0031-8094}}</ref>

In some scientific disciplines, ontological reductionism takes two forms: '''token-identity theory''' and '''type-identity theory'''.<ref>{{cite book | chapter-url=https://plato.stanford.edu/entries/scientific-reduction/#TypIdeThe | title=The Stanford Encyclopedia of Philosophy | chapter=Scientific Reduction | year=2019 | publisher=Metaphysics Research Lab, Stanford University }}</ref> In this case, "token" refers to a biological process.<ref>{{cite book | chapter-url=https://plato.stanford.edu/entries/reduction-biology/ | title=The Stanford Encyclopedia of Philosophy | chapter=Reductionism in Biology | year=2022 | publisher=Metaphysics Research Lab, Stanford University }}</ref> 

Token ontological reductionism is the idea that every item that exists is a sum item. For perceivable items, it affirms that every perceivable item is a sum of items with a lesser degree of complexity. Token ontological reduction of biological things to chemical things is generally accepted.

Type ontological reductionism is the idea that every type of item is a sum type of item, and that every perceivable type of item is a sum of types of items with a lesser degree of complexity. Type ontological reduction of biological things to chemical things is often rejected.

[[Michael Ruse]] has criticized ontological reductionism as an improper argument against [[vitalism]].<ref>{{cite journal|url=http://icb.oxfordjournals.org/cgi/reprint/29/3/1061.pdf |first=Michael |last=Ruse |title=Do Organisms Exist? |journal=Am. Zool. |volume=29 |pages=1061–1066 |year=1989 |issue=3 |doi=10.1093/icb/29.3.1061|archive-url=https://web.archive.org/web/20081002163413/http://icb.oxfordjournals.org/cgi/reprint/29/3/1061.pdf |archive-date=2008-10-02 }}</ref>

=== Methodological reductionism ===
In a biological context, methodological reductionism means attempting to explain all biological phenomena in terms of their underlying biochemical and molecular processes.<ref>{{Cite encyclopedia |title=Reductionism in Biology |encyclopedia=Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |url=https://plato.stanford.edu/archives/spr2017/entries/reduction-biology/ |last1=Brigandt |first1=Ingo |date=2017 |editor-last=Zalta |editor-first=Edward N. |last2=Love |first2=Alan |access-date=2019-04-28}}</ref>

=== In religion ===
Anthropologists [[Edward Burnett Tylor]] and [[James George Frazer]] employed some [[Metatheories of religion in the social sciences#Edward Burnett Tylor and James George Frazer|religious reductionist arguments]].<ref>Strenski, Ivan. "Classic Twentieth-Century Theorist of the Study of Religion: Defending the Inner Sanctum of Religious Experience or Storming It." pp. 176–209 in ''Thinking About Religion: An Historical Introduction to Theories of Religion''. Malden: Blackwell, 2006.</ref>

=== Theory reductionism ===
Theory reduction is the process by which a more general theory absorbs a special theory.<ref name=":0" /> It can be further divided into translation, derivation, and explanation.<ref>{{cite web | url=https://iep.utm.edu/red-ism/#SH1b | title=Reductionism &#124; Internet Encyclopedia of Philosophy }}</ref> For example, both [[Johannes Kepler|Kepler's]] laws of the motion of the [[planet]]s and [[Galileo Galilei|Galileo]]'s theories of motion formulated for terrestrial objects are reducible to Newtonian theories of mechanics because all the explanatory power of the former are contained within the latter. Furthermore, the reduction is considered beneficial because [[Newtonian mechanics]] is a more general theory—that is, it explains more events than Galileo's or Kepler's. Besides scientific theories, theory reduction more generally can be the process by which one explanation subsumes another.

=== In mathematics ===
In [[mathematics]], reductionism can be interpreted as the philosophy that all mathematics can (or ought to) be based on a common foundation, which for modern mathematics is usually [[axiomatic set theory]]. [[Ernst Zermelo]] was one of the major advocates of such an opinion; he also developed much of axiomatic set theory. It has been argued that the generally accepted method of justifying mathematical [[axioms]] by their usefulness in common practice can potentially weaken Zermelo's reductionist claim.<ref>{{cite journal |doi=10.1305/ndjfl/1093633905 |first=R. Gregory |last=Taylor |title=Zermelo, Reductionism, and the Philosophy of Mathematics |journal=Notre Dame Journal of Formal Logic |volume=34 |issue=4 |year=1993 |pages=539–563 |doi-access=free }}</ref>

Jouko Väänänen has argued for [[second-order logic]] as a foundation for mathematics instead of set theory,<ref>{{cite journal |first=J. |last=Väänänen |title=Second-Order Logic and Foundations of Mathematics |journal=Bulletin of Symbolic Logic |volume=7 |issue=4 |pages=504–520 |year=2001 |doi=10.2307/2687796 |jstor=2687796 |s2cid=7465054 }}</ref> whereas others have argued for [[category theory]] as a foundation for certain aspects of mathematics.<ref>{{cite journal |first=S. |last=Awodey |title=Structure in Mathematics and Logic: A Categorical Perspective |journal=Philos. Math. |series=Series III |volume=4 |issue=3 |year=1996 |pages=209–237 |doi=10.1093/philmat/4.3.209 }}</ref><ref>{{cite book |first=F. W. |last=Lawvere |chapter=The Category of Categories as a Foundation for Mathematics |title=Proceedings of the Conference on Categorical Algebra (La Jolla, Calif., 1965) |pages=1–20 |publisher=Springer-Verlag |location=New York |year=1966 }}</ref>

The [[Gödel's incompleteness theorems|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 problem|undecidable propositions]]. For example, the [[continuum hypothesis]] is undecidable in the [[Zermelo–Fraenkel set theory]] as shown by [[Forcing (mathematics)|Cohen]].

=== In science ===
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, the [[Solar System]] is understood in terms of its components (the Sun and the planets) and their interactions.<ref>{{Cite book|last=McCauley|first=Joseph L.|title=Dynamics of Markets: The New Financial Economics, Second Edition|publisher=Cambridge University Press|year=2009|isbn=978-0521429627|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, although it has been argued that reduction in physics 'never goes all the way in practice'.<ref>{{cite book |last1=Simpson |first1=William M. R. |last2=Horsley |first2=Simon A.H. |series=Synthese Library |date=29 March 2022 |volume=451 |editor1-last=Austin |editor1-first=Christopher J.|editor2-last=Marmodoro |editor2-first=Anna |editor3-last=Roselli |editor3-first=Andrea |title=Powers, Time and Free Will |chapter-url=https://link.springer.com/chapter/10.1007/978-3-030-92486-7_2 |publisher=Synthese Library |via=Springer |pages=17–50 |chapter=Toppling the Pyramids: Physics Without Physical State Monism |isbn=9781003125860 |doi=10.1007/978-3-030-92486-7_2}}</ref>

=== In computer science ===
The role of reduction in [[computer science]] can be thought as a precise and unambiguous mathematical formalization of the philosophical idea of "[[#Types|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 "[[NP-hardness|hard]]" to solve as the former.

Reduction in [[theoretical computer science]] is pervasive in both: the mathematical abstract foundations of computation; and in real-world [[Analysis of algorithms|performance or capability analysis of algorithms]]. More specifically, reduction is a foundational and central concept, not only in the realm of mathematical logic and abstract computation in [[Computability theory|computability (or recursive) theory]], where it assumes the form of e.g. [[Turing reduction]], but also in the realm of real-world computation in time (or space) complexity analysis of algorithms, where it assumes the form of e.g. [[polynomial-time reduction]]. Further, in the even more practical domain of software development, reduction can be seen as the inverse of composition and the conceptual process a programmer applies to a problem in order to produce an alogrithm which solves the problem using a composition of existing algorithms (encoded as subroutines, or subclasses).