Koo, Alex

English

PhilPapers

Reviews Osmo Pekonen, EditorFeel like writing a review for The MathematicalIntelligencer? Contact the column editor if you wish tosubmit an unsolicited review, or if you would welcomebeing assigned a book to review. Submissions should be uploaded to http://tmin.edmgr.comor to be sent directly to Osmo Pekonen,osmo.pekonen@jyu.fiAn Aristotelian RealistPhilosophy of Mathematicsby James FranklinLONDON: PALGRAVE MACMILLAN, 2014, 320 PP., 63.00 £, ISBN9781137400727REVIEWED BY ALEX KOOWhat is mathematics actually about? To the non-philosopher, this question can seem somewhatabsurd. Mathematics is about things like numbers,sets, and functions. But what, exactly, are these entities? Dothey exist in space–time like protons and electrons? Are theymanmade creations? How do we come to know anythingabout them? When pressed further, two more sophisticatedpositions tend to emerge. Platonism is the view that mathe-matical entities exist in a non–spatio-temporal realmsometimes referred to as ‘‘Plato’s heaven.’’ Things such asnumbers and sets are abstract, mind-independent, and haveno causal powers. The biggest problem facing the Platonist istheir inability to clearly explicate what Plato’s heaven actuallyis, and how we could come to know anything about theabstract entities that exist there without any causal interaction.The second position is mathematical nominalism.Nominalists deny that mathematical entities are real or thatthey exist in any deep sense beyond the way in which weutilize them, such as ink-strokes on a page or verbalutterances. There are many different gradations of nomi-nalism, but all agree that mathematics does not have asubject matter that is abstract or mind-independent. Thebiggest challenges for nominalists are that their positionsseem to conflict with the beliefs of most practicing math-ematicians, and nominalism does not seem able to explainwhy applied mathematics is so successful.In his book An Aristotelian Realist Philosophy ofMathematics, James Franklin argues that Platonism andnominalism represent a false dichotomy in the philosophyof mathematics. There is (at least) one more potentialanswer to fundamental questions about mathematics thatcan evade the drawbacks of the two leading views.Franklin’s position draws from a classical Aristotelian phi-losophy of mathematics. Unlike Platonism and nominalism,classical Aristotelianism says that mathematics is not aboutparticular entities such as the number 2 or the set of naturalnumbers, but instead is about universals that are instanti-ated in the actual physical world. Science is fundamentallyabout universals. Although scientists study particularobjects, what they are actually investigating are universal 2016 Springer Science+Business Media New York, Volume 38, Number 3, 2016 81DOI 10.1007/s00283-016-9627-4properties and relations. Physics is interested in universalssuch as charge or colour, whereas psychology investigatesmental universals like mental states. So what is mathematicsactually about? Aristotelians believe that mathematics is justanother science. Mathematics investigates the real world tolearn about mathematical universals such as symmetry orratio. These universals are mathematical properties andrelations that are as real and physical as any other scientificuniversal.Platonism and nominalism can also accommodate theview that the subject matter of mathematics is about uni-versals and not particulars, but each position has a differenttake on what these universals actually are. For the Platonist,mathematical universals are real and they exist, but they arepurely abstract and noncausal, and do not exist in thephysical world. For the nominalist, universals are not real atall and have no actual properties. The classical Aristotelianposition claims that mathematical universals are real, havecausal powers, and exist in the physical world. Their exis-tence is due to their instantiation in the physical world andnot in some abstract Platonic realm. In this way, the Aris-totelian is able to maintain a realist interpretation ofmathematics but avoid any sort of commitment to an abstract,highly metaphysical realm that we have no causal access to.Classical Aristotelianism has faced two famous and crip-pling challenges. If mathematics is about real physical things,then what about mathematical universals that are uninstan-tiated in the physical world? Consider extremely large finitenumbers, certain complex or real numbers, or the empty set.It is possible that none of these are instantiated in the world.The second challenge is similar but appeals to infinity.Higher-level mathematics makes use of infinite sets of sizesfar larger than the set of natural numbers. Even ifwe allow foran infinite physical universe, or perhaps an infinitely divisi-ble and continuous space–time, the infinite sets with largercardinality used in mathematics do not seem to be instanti-ated (or instantiatable) in the physical world. ClassicalAristotelianism says that mathematics is about instantiateduniversals. If this is so, then all of these uninstantiatedmathematical universals do not exist. This clearly does notmesh well with mathematical practice, and has thus led to theabandonment of the classical Aristotelian view.Franklin’s book is divided into two parts. Part I is ademonstration that Aristotelianism can make sense of allmathematical objects. In Chapters 1 and 2, Franklin intro-duces his own brand of Aristotelianism (Franklin introduceshis position as ‘‘semi-Platonism’’ and ‘‘modal-Aristotelian-ism’’ but eventually abandons these terms in favour of just‘‘Aristotelianism’’) that draws from both Platonist and clas-sical Aristotelian philosophies in order to tackle the problemof the uninstantiated. Franklin claims that mathematics is notjust about instantiated universals, but also about universalsthat are uninstantiated as well. For the most part, the uni-versals that are actually instantiated in the world are acontingent matter. With two notable exceptions, there isnothing about an uninstantiated universal that necessarilyrules it out from being instantiated. Franklin argues that wecan still know about these uninstantiated universals andstudy their necessary interrelations. Whereas classical Aris-totelianism says that universals are exactly those that areinstantiated in the physical world, Franklin’s Aristotelianismsays that universals are those that could be instantiated in thephysical world. Franklin’s slogan for Aristotelianism makesthis clear: ‘‘Instantiation is possible, but not necessary’’.The central aim of the remainder of Part I is todemonstrate that almost all of mathematics can be under-stood as universals that could be instantiated in the physicalworld. Franklin goes through entities such as quantity,ratio, finite sets, geometry, and more to justify his position.The bulk of this presentation does not depart from a tra-ditional classical Aristotelian take. The interesting andnovel sections are where Franklin tackles uninstantiateduniversals that have posed difficulty for classical Aris-totelianism, such as highly abstract mathematical objects(Chapter 4), and objects so large that they cannot be real-ized in anything physical (Chapter 8).In Part II, Franklin argues that Aristotelianism canaccount for mathematical practice. Franklin goes throughmany classic and contemporary issues of mathematicalpractice such as proof, structure, explanation, visualization,and idealization, and claims that all these mathematicaltechniques can explain how we come to know aboutmathematical universals that either are or could be instan-tiated in the physical world. Franklin draws from a wealthof modern philosophy and science including theories ofscientific explanation, artificial intelligence, infant learning,and imagination. Part II is far more speculative comparedto Part I, which is grounded on concrete examples fromboth abstract and applied mathematics.In what follows I will present three specific examplesfrom An Aristotelian Realist Philosophy of Mathematics thatare important cases for the Aristotelian. First are highlyabstract objects that are not instantiated in the physicalworld. Second are objects of extremely large size, such asinfinite sets. Third are zero and the empty set—two familiarobjects that pose a unique challenge and solution. Frank-lin’s treatment of these three issues provides a clearillustration of how his Aristotelianism is different fromclassical Aristotelianism, Platonism, and nominalism.Franklin uses groups as an exemplar for an Aristotelianhandling of highly abstract objects. Consider the cyclicgroup of order 2. This group has many abstract instantia-tions, but it also has physical instantiations such as a CapsLock button, or any other simple toggle. The group SO(3) isa bit more complicated, but it too can be realized in asystem of all physical rotations in three-dimensional space.But what about the infinite system of all possible rotations?For this to be physical, then all of SO(3) would need to berealized at once. But this is impossible because of the(supposed) finite nature of the universe. Franklin notes thatthe universe’s finite size is a contingent fact that could havebeen otherwise. There is nothing about the system of allpossible rotations itself that necessitates it being uninstan-tiated physically. If the world had been different, then thissystem could have a real and physical instantiation. Aris-totelianism ‘‘emphasizes the realizability, rather than actualrealization, of universals’’. Given this, we can still study andlearn about uninstantiated universals by analyzing thingssuch as their structure and relations despite not having aparticular physical instantiation to study.82 THE MATHEMATICAL INTELLIGENCERFranklin’s treatment of infinity is a bit less clear. It iscertainly possible that there are physical instantiations ofsome sizes of infinity such as infinitely many points inspace, but Franklin does not believe that higher-orderinfinite sets can be instantiated in our finite universe. Asmentioned earlier, the finiteness of the universe is a con-tingent fact, and thus higher-order infinite sets could beinstantiated so these sets are real mathematical objects.Quizzically, Franklin seems to be somewhat standoffish inhis commitment to infinite sets. He argues at length that thebulk of mathematics can be done without any infiniteconcepts. The infinite is a ‘‘luxury’’ that makes our calcu-lations easier and smoother, but can be done without.Another assertion Franklin makes is that it is fine for one tobe a realist regarding infinities of countable size, but toreject higher-order infinities. A different argument for thereality of larger infinite sets is needed, at which Franklinonly gestures. So, although it seems that the Aristotelianshould count infinite sets as real mathematical objects,Franklin has laid the groundwork for the Aristotelian tosurvive if we choose to reject infinite sets as well.Unlike the highly abstract or extremely large objectsconsidered earlier that could be instantiated, zero and theempty set are different in that, according to Franklin, theyare both necessarily uninstantiatable. Aristotelianism thusdictates that zero and the empty set are not real mathe-matical objects, but this seems counterintuitive as bothobjects are ubiquitous in mathematical practice. Franklinknows that although it is certainly convenient to use zeroand the empty set, and they may even be indispensable toour understanding of mathematics, this does not mean thatthey are real. Franklin bites the bullet and adopts a nomi-nalist interpretation toward zero and the empty set. No onedoubts their utility, but ‘‘[n]o amount of mathematicalconvenience… can alter metaphysical reality’’. It is possibleand actual instantiation in the physical world that dictatesmathematical reality, and zero and the empty set do notmeet this requirement. Franklin goes to great lengths todemonstrate that all the standard objects of mathematicssuch as the natural numbers, abstract groups, and more areall real according to Aristotelianism, but despite this hisrealism does not extend over zero and the empty set.Franklin does a nice job of developing the Aristotelianposition. He blends a rich philosophical history of Aris-totelianism with modern case studies in both applied andpure mathematics to illustrate how we can conceive ofmathematical objects as physically possibly instantiateduniversals. Although he easily could have stopped there,Franklin recognizes that an important issue for contempo-rary philosophy of mathematics is to give an account ofhow philosophy fits with actual mathematical practice. Theattention paid to the many techniques of mathematicians inPart II provides a nice rounding out of Aristotelianism andis certainly a strength of the book. It is one thing to carveout a unique and consistent philosophical position, but it isan entirely other matter to show it to be both tenable andattractive in practice. Franklin has succeeded in makingsignificant inroads toward both of these goals.Ultimately I find myself somewhat unconvinced. Themost important idea behind Aristotelianism is that of thepotentially instantiable universal. If a mathematical uni-versal is instantiated, then we know that it is real and existsin virtue of this instantiation. We also are able to investigatethe universal just as a scientist would investigate universalslike charge or spin. As stated in the opening line of thebook, ‘‘mathematics is a science of the real world’’ (1).Mathematical objects are ‘‘mind-independent objects whichare spatio-temporal and causal, namely relations such asratios’’. But how does this fit with Franklin’s assertion thatuninstantiated universals are real? They are clearly not inthe ‘‘real world,’’ nor are they spatio-temporal and causal,so what is the nature of their existence? Franklin getsaround this by claiming that these questions are notmeaningful. Uninstantiated universals are not granted anysort of ‘‘existence.’’ But why are questions of existencemeaningful for instantiated universals, yet at the same timemeaningless for the uninstantiated? This is certainly aconvenient move in order to avoid the existence of somesort of nonphysical Platonic realm, but I find it suspiciouslyad hoc and unsatisfying. There is a clear conflict herebetween the physical properties of instantiated universalsand those of the uninstantiated. To make this clear, whatcan it possibly mean for uninstantiated mathematicalobjects to have causal powers? If mathematics containsboth instantiated and uninstantiated objects, then we can-not say that mathematics is a science of the real world at all.Franklin has created a two-class system of mathematicalobjects based around instantiation. The dividing linebetween the two tiers is contingent, as which universals areinstantiated and which are not is entirely contingentaccording to Aristotelianism. For those that fall on theinstantiated side of the line we adopt a physical realistinterpretation, whereas for those that are uninstantiated weseem to adopt a quasirealist interpretation. Even if this issomehow consistent, I have concerns that this quasirealistunderstanding of uninstantiated universals is simply nom-inalism in disguise. Franklin insists that Aristotelians are stillrealist regarding uninstantiated universals, but I struggle tosee how his realism works in these cases beyond mere lip-service.Franklin attempts to evade these problems by claimingthat what really matters is that we ‘‘insist on the reality ofrelations between universals, instantiated or not’’ (29).Unfortunately, avoiding existence questions of mathemati-cal universals by invoking the reality of relations simplyshifts the problem back one level. Suppose I grant thatrelations of instantiated universals are real, such as the ratioof my height to that of my mother. What exactly does itmean for the relations of uninstantiated universals to bereal? What does it mean for these relations to exist insomething other than the physical world? Do these unin-stantiated relations have causal powers? Instead of having atwo-class system of mathematical universals, we now havea two-class system of mathematical relations where theexact same problems emerge.Despite my skepticism, An Aristotelian Realist Philoso-phy of Mathematics is an interesting and challenging book.Franklin is certainly correct in stating that Platonism andnominalism unfairly dominate the landscape in the phi-losophy of mathematics, and developing a rival position is 2016 Springer Science+Business Media New York, Volume 38, Number 3, 2016 83no small achievement. Franklin goes through a wealth ofmaterial in philosophy, mathematics, and science to moti-vate and justify his position. Although this is certainly astrength of the book in that it leaves no stone unturned, it isalso a weakness as a vast amount of prior knowledge isrequired to understand all of Franklin’s arguments.Whereas An Aristotelian Realist Philosophy of Mathematicsmay not be entirely accessible to the novice philosopher ofmathematics, there will still be something for all readers toenjoy due to its large scope and variety of interesting anddetailed examples.Department of PhilosophyUniversity of TorontoToronto, ON M5R 2M8Canadae-mail: alex.koo@utoronto.ca84 THE MATHEMATICAL INTELLIGENCER

_An Aristotelian Realist Philosophy of Mathematics_ by James Franklin

https://philpapers.org/archive/KOOAAR.pdf

An Aristotelian Realist Philosophy of Mathematics by James Franklin

Abstract

Similar works

Full text

Available Versions

PhilPapers