View source for Philosophy:Aristotelian realist philosophy of mathematics

{{short description|Programme in the philosophy of mathematics}}
In the [[Philosophy:Philosophy of mathematics|philosophy of mathematics]], '''Aristotelian realism''' holds that mathematics studies properties such as [[Symmetry in mathematics|symmetry]], continuity and [[Order theory|order]] that can be immanently realized in the physical world (or in any other world there might be). It contrasts with [[Philosophy:Platonism|Platonism]] in holding that the objects of mathematics, such as numbers, do not exist in an "abstract" world but can be physically realized.<ref>{{cite web |url=https://aeon.co/essays/aristotle-was-right-about-mathematics-after-all |title=The mathematical world |last=Franklin |first=James |date=7 April 2014 |website=Aeon |publisher= |access-date=30 June 2021}}</ref> It contrasts with [[Philosophy:Nominalism|nominalism]], [[Philosophy:Fictionalism|fictionalism]], and [[Philosophy:Logicism|logicism]] in holding that mathematics is not about mere names or methods of inference or calculation but about certain real aspects of the world.

Aristotelian realists emphasize [[Applied mathematics|applied mathematics]], especially [[Mathematical model|mathematical modeling]], rather than [[Pure mathematics|pure mathematics]] as philosophically most important. Marc Lange argues that "Aristotelian realism allows mathematical facts to be explainers in distinctively mathematical explanations" in science as mathematical facts are themselves about the physical world.<ref>{{cite journal |last1=Lange |first1=Marc |date=2021 |title=What could mathematics be for it to function in distinctively mathematical scientific explanations? |url=https://www.sciencedirect.com/science/article/abs/pii/S0039368121000157 |journal=Studies in History and Philosophy of Science A |volume=87 |issue= |pages=44–53 |doi=10.1016/j.shpsa.2021.02.002 |pmid=34111822 |s2cid=233545723 |access-date=30 June 2021}}</ref> [[Biography:Paul Thagard|Paul Thagard]] describes Aristotelian realism as "the current philosophy of mathematics that fits best with what is known about [[Cognitive science|minds and science]]."<ref name=Thagard>{{cite book |last=Thagard |first=Paul |date=2019 |title=Natural Philosophy: From Social Brains to Knowledge, Reality, Morality, and Beauty |url=https://books.google.com/books?id=KfeEDwAAQBAJ&pg=PT442 |location=New York |publisher=Oxford University Press |page=442 |isbn=9780190686444}}</ref>

==History== 
Although [[Biography:Aristotle|Aristotle]] did not write extensively on the philosophy of mathematics, his various remarks on the topic exhibit a coherent view of the subject as being both about abstractions and applicable to the real world of space and counting.<ref>{{cite book |last=Bostock |first=D.  |date= 16 August 2012|title=Oxford Handbook of Aristotle |chapter=Aristotle's philosophy of mathematics |editor-last=Shields |editor-first=C.J. |url= |location=Oxford |publisher=Oxford University Press |page= |isbn=9780195187489}}</ref>
Until the eighteenth century, the most common philosophy of mathematics was the Aristotelian view that it is the "science of [[Quantity|quantity]]", with quantity divided into the continuous (studied by [[Geometry|geometry]]) and the discrete (studied by arithmetic).<ref name=Frabook>{{cite book |last=Franklin |first=James |date=2014 |title=An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure |url=https://books.google.com/books?id=0YKEAwAAQBAJ |location=Basingstoke |publisher=Palgrave Macmillan |page=123 |isbn=9781137400727}}</ref>

Aristotelian approaches to the philosophy of mathematics were rare in the twentieth century but were revived by [[Biography:Penelope Maddy|Penelope Maddy]] in ''Realism in Mathematics'' (1990) and by a number of authors since 2000 such as [[Biography:James Franklin (philosopher)|James Franklin]], <ref>{{cite journal |last1=Franklin |first1=James |date=2022 |title=Mathematics as a science of non-abstract reality: Aristotelian realist philosophies of mathematics |url=https://rdcu.be/chatd |journal=Foundations of Science |volume=27 |issue=2 |pages=327–344|doi=10.1007/s10699-021-09786-1 |s2cid=233658181 |access-date=30 June 2021}}</ref> Anne Newstead,<ref> A.G.J. Newstead, (2001). "Aristotle and modern mathematical theories of the continuum", in D. Sfendoni-Mentzou, J. Hattiangadi, and D.M. Johnson (eds), ''Aristotle and Contemporary Science'', Peter Lang, 113-129.</ref> Donald Gillies, and others.

==Numbers and sets==
Aristotelian views of ([[Cardinal number|cardinal]] or counting) numbers begin with Aristotle's observation that the number of a heap or collection is relative to the unit or measure chosen: "'number' means a measured plurality and a plurality of measures ... the measure must always be some identical thing predicable of all the things it measures, e.g. if the things are horses, the measure is 'horse'."<ref>Aristotle, ''Metaphysics'' 1088a4-11.</ref>  Glenn Kessler develops this into the view that a number is a relation between a heap and a [[Philosophy:Universal (metaphysics)|universal]] that divides it into units; for example, the number 4 is realized in the relation between a heap of parrots and the universal "being a parrot" that divides the heap into so many parrots.<ref>{{cite journal |last1=Kessler |first1=Glenn |date=1980 |title=Frege, Mill and the foundations of arithmetic |url=https://www.jstor.org/stable/2025431 |journal=Journal of Philosophy |volume=77 |issue= 2|pages=65–79 |doi= 10.2307/2025431|jstor=2025431 |access-date=30 June 2021}}</ref><ref>{{cite journal |last1=Forrest |first1=Peter |last2=Armstrong |first2=D.M. |author-link2=David Malet Armstrong |date=1987 |title=The nature of number |url=https://www.tandfonline.com/doi/abs/10.1080/05568648709506275 |journal=Philosophical Papers |volume=16 |issue=3 |pages=165–186 |doi=10.1080/05568648709506275 |access-date=30 June 2021}}</ref><ref name=Frabook />{{rp|36-8}}

On an Aristotelian view, ratios are not closely connected to cardinal numbers. They are relations between quantities such as heights. A ratio of two heights may be the same as the relation between two masses or two time intervals.<ref name=Frabook />{{rp|34-5}}

Aristotelians regard sets as well as numbers as instantiated in the physical world (rather than being Platonist entities). Maddy argued that when an egg carton is opened, a set of three eggs is perceived (that is, a mathematical entity realized in the physical world).<ref name=Maddy>{{cite book |last=Maddy |first=Penelope |date=1990 |title=Realism in Mathematics |url= |location=Oxford |publisher=Oxford University Press |page=58-67 |isbn=9780198240358}}</ref> However not all mathematical discourse needs to be interpreted realistically; for example Aristotelians may regard the [[Empty set|empty set]] and zero as fictions,<ref name=Frabook />{{rp|234-40}} and possibly higher infinities.

==Structural properties==
[[File:Königsberg riddle.png|thumb|alt=Diagram of the 7 bridges of Königsberg |The seven bridges of Königsberg, studied by Euler]]
Aristotelians regard non-numerical structural properties like symmetry, continuity and order as equally important as numbers. Such properties are realized in physical reality, and are the subject matter of parts of mathematics. For example [[Group theory|group theory]] classifies the different kinds of symmetry, while the [[Calculus|calculus]] studies continuous variation. Provable results about such structures can apply directly to physical reality. For example [[Biography:Leonhard Euler|Euler]] proved that it was impossible to walk once and once only over the seven bridges of Königsberg.<ref name=Frabook />{{rp|48-56}}

==Epistemology==
Since mathematical properties are realized in the physical world, they can be directly perceived. For example, humans easily perceive [[Facial symmetry|facial symmetry]].

Aristotelians also accord a role to abstraction and idealisation in mathematical thinking. This view goes back to Aristotle's statement in his ''[[Philosophy:Physics (Aristotle)|Physics]]'' that the mind 'separates out' in  thought the properties that it studies in mathematics, considering the timeless properties of bodies apart from the world of change (Physics II.2.193b31-35). 

At the higher levels of mathematics, Aristotelians follow the theory of Aristotle's ''[[Philosophy:Posterior Analytics|Posterior Analytics]]'', according to which the [[Mathematical proof|proof]] of a mathematical proposition ideally allows the reader to understand why the proposition must be true.<ref name=Frabook />{{rp|192-6}}

==Objections to Aristotelian realism==
A problem for Aristotelian realism is what account to give of higher [[Infinity|infinities]], which may not be realized or realizable in the physical world. How to apply Aristotle's theory of [[Philosophy:Potentiality and actuality|Potentiality and actuality]] to Zermelo–Frankel set theory. Mark Balaguer writes:
:"Set theory is committed to the existence of infinite sets that are so huge that they simply dwarf garden variety infinite sets, like the set of all the natural numbers. There is just no plausible way to interpret this talk of gigantic infinite sets as being about physical objects."<ref>{{cite web |url=https://plato.stanford.edu/entries/fictionalism-mathematics/ |title=Fictionalism in the Philosophy of Mathematics |last=Balaguer |first=Mark |date=2018 |website=Stanford Encyclopedia of Philosophy |publisher= |access-date=30 June 2021}}</ref>

Aristotelians reply that sciences can deal with uninstantiated universals; for example the science of color can deal with a shade of blue that happens not to occur on any real object.<ref>{{cite journal |last1=Franklin |first1=James  |date=2015 |title=Uninstantiated properties and semi-Platonist Aristotelianism |url=https://www.academia.edu/11344862 |journal=Review of Metaphysics |volume=69 |pages=25–45 |doi= |access-date=29 June 2021}}</ref> However that does require denying the [[Philosophy:Instantiation principle|instantiation principle]], held by most Aristotelians, which holds that all genuine properties are instantiated. One Aristotelian philosopher of mathematics who denies the instantiation principle on the basis of Frege’s distinction between [[Philosophy:Sense and reference|sense and reference]] is [[Biography:Donald A. Gillies|Donald Gillies]]. He has used this approach to develop a method of dealing with very large [[Transfinite number|transfinite cardinals]] from an Aristotelian point of view.<ref>{{cite book |last=Gillies |first=Donald |date=2015 |editor1-last=Davis |editor1-first=Ernest |editor2-last=Davis |editor2-first=Philip J. |title=Mathematics, Substance and Surmise |publisher=Springer |location=Cham |pages=147–176 |url=https://www.springer.com/gp/book/9783319214726 |chapter=An Aristotelian approach to mathematical ontology |isbn=9783319214726}}</ref>

Another objection to Aristotelianism is that mathematics deals with [[Philosophy:Idealization (philosophy of science)|idealizations]] of the physical world, not with the physical world itself. Aristotle himself was aware of the argument that geometers study perfect circles but hoops in the real world are not perfect circles, so it seems that mathematics must be studying some non-physical (Platonic) world.<ref>Aristotle, ''Metaphysics'' 997b35-998a4.</ref>  Aristotelians reply that applied mathematics studies approximations rather than idealizations and that as a result modern mathematics can study the complex shapes and other mathematical structures of real things.<ref name=Frabook />{{rp|225-9}}<ref>A.Newstead, J. Franklin, (2009). "The Epistemology of Geometry I: The Problem of Exactness", ASCS09 ''Proceedings of the 9th Conference of the Australasian Society for Cognitive Science'', Sydney, 254-260, Article DOI: 10.5096/ASCS200939.</ref>

==References==

{{Reflist}}

==Bibliography==
* John Bigelow, 1988, ''[https://global.oup.com/academic/product/the-reality-of-numbers-9780198249573?cc=au&lang=en& The Reality of Numbers]'', Clarendon, Oxford, {{ISBN|9780198249573}}
* James Franklin, 2014, ''[https://web.maths.unsw.edu.au/~jim/franklinaristotelianrealistphilosophyofmathematics.pdf An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure]'', Palgrave Macmillan, Basingstoke, {{ISBN|9781137400727}}.
* Keith Hossack, 2020, ''[https://books.google.com/books?id=42jDDwAAQBAJ Knowledge and the Philosophy of Number: What Numbers Are and How They Are Known]'', Bloomsbury, London, {{ISBN|9781350102927}}
* Andrew Irvine, 1990. ''[https://books.google.com/books?id=lSPpCAAAQBAJ&dq=irvine+physicalism&pg=PR9 Physicalism in Mathematics]'', Dordrecht, London, {{ISBN|9780792305132}}
* Bob Knapp, 2014, ''[https://mathematicsisabouttheworld.com/ Mathematics is About the World]'', Lexington KY, {{ISBN|9781500551971}}
* Penelope Maddy, 1990, ''[https://oxford.universitypressscholarship.com/view/10.1093/019824035X.001.0001/acprof-9780198240358 Realism in Mathematics]'', Oxford University Press, New York, {{ISBN|9780198240358}}
* Woosuk Park, 2018, ''[https://www.springer.com/gp/book/9783319951461 Philosophy's Loss of Logic to Mathematics]'', Springer, Cham, {{ISBN|9783319951461}}
* Andrew Younan, 2022, ''[https://www.cuapress.org/9780813236124/matter-and-mathematics/ Matter and Mathematics: An Essentialist Account of Laws of Nature]'', Catholic University of America Press, Washington DC, {{ISBN|9780813236124}}

==External links==
* [https://web.maths.unsw.edu.au/~jim/structmath.html The Sydney School in the philosophy of mathematics]
* [https://philpapers.org/browse/mathematical-aristotelianism Philpapers category Mathematical Aristotelianism]


[[Category:Philosophy of mathematics]]
[[Category:Aristotelianism]]

{{Sourceattribution|Aristotelian realist philosophy of mathematics}}