Frege and Neo-Fregeanism:Interview with Bob Hale*

Difei Xu

School of Philosophy,Renmin University of China difeixu@ruc.edu.cn

Bob Hale is a philosopher,who has been devoting himself to modality and metaphysics.His research as British Academy Reader was mainly in the philosophy of mathematics.With and during long collaboration with Crispin Wright,he developed and defended a position of neo-Fregeanism,according to which mathematical knowledge can be grounded on logic and definitions of fundamental notions.Neo-Fregeanism has been arousing philosophers’serious thought and discussion since 1980s and still inspires philosophers nowadays in the field of philosophy of mathematics,philosophy of language and philosophy of mind.

1 Personal history and some general questions

Xu:Prof.Hale,I’m honored to have an opportunity to interview you.You and Prof.Wright are protagonists of Neo-Fregeanism,which has significant influence not only to philosophy of mathematics but also to the philosophy of language.Could you tell us what makes you begin this significant philosophical program?

Hale:Crispin Wright and I met in Oxford,in the late 1960s,although we came to know each other much better when he had left Oxford to work in St.Andrews,and I had started to work in Lancaster.Before writing Frege’s Conception,Crispin became very interested in a paper I was writing about singular terms,and made use of the ideas in it in his book,which I read as soon as I could get hold of it.Soon after that,he invited me to write a book about abstract objects,for a series he was editing.As I say in the preface to that book,Crispin read everything I wrote as I was preparing it,and we discussed almost everything in it very closely.It was a tremendously inspiring experience.We have been collaborating ever since then.

Xu:When did you get interested in philosophy?At the beginning when you choose philosophy as your major,were there any special philosophical problems that interested you?

Hale:In fact,I started while still in grammar school.In my lower sixth year,when I was in the library and supposed to be working,I was in fact struggling with a book about existentialism which one of my friends had asked me to look at,because he found it very obscure.The master supervising us,who taught biology,noticed and informed me that I would never make sense of it without first studying Descartes.He very kindly lent me a copyoftheMeditations,andwhenIhaddevouredthat,decidedIwasreadyfortheBritish Empiricists.By the time I was ready to apply for university,I was halfway through the Critique of Pure Reason—with a lot of help from him,I should add!Somewhat tentatively,because I wasn’t very confident about my capacity to be any good at philosophy,I applied for joint degrees with English Literature,and went to Bristol,mainly because the professor there then was Stefan Körner,who had written a book about Kant which I had found useful.Towards the end of my first year,the philosophers persuaded me to switch to full time philosophy.So I really wasn’t settling on special problems interested me.I think the answer was that I was interested in philosophy in quite general way at the beginning.

Xu:During your time in Oxford,what courses and teachers that affected you a lot?

Hale:I studied for most of the time with Gilbert Ryle,and worked mostly on the philosophy of mind.Later,after I had completed the BPhil,I worked for a while with David Pears.Both of them were,of course,a considerable influence on my thinking.I attended classes given by Arthur Prior,John Mackie,Peter Strawson and A.J.Ayer,but wasnotdirectlysupervisedbyanyofthem.TherewasalotofinterestinDavidson’swork at this time.I became interested in Frege,logic,and the philosophy of mathematics,and went to lectures on Frege given by Michael Dummett(these subsequently formed part of his book on Frege’s philosophy of mathematics).They were a real inspiration,and by the time I left Oxford,my interest had become focused on logic and philosophy of mathematics.

Xu:Apartfromphilosophyarethereanyothersubjectsaffectingyourphilosophical thought?

Hale:I really don’t quite know how to answer it,but I have always read quite widely.I think some of what I’ve read in other areas,including some of the great novels,has probably had some effect,but it is hard to identify particular influences.Around the time when I was going to the university I also read quite a bit by Sartre,including his novels,and by Camus,a French writer.I was interested in them,but I think they didn’t really have a lasting impact on my philosophy.I think probably I reread some writings by Sartre,because I found it penetrable would be wrong but I didn’t think he was a guide anyway.

Xu:I know you are also a painter and proficient in music.Art is a totally different field from the work with reasoning and it may involve a different way of thinking.Why you choose to be a philosopher finally instead of a professional artist?

Hale:Istudiedartandarchitectureinmyfinalyearsatschool,anddrewandpainted a lot at that time.However,I did very badly in my art examinations,and I think I may have lost confidence.On the other hand,I did very well in other subjects,and once I had got interested in philosophy,I thought I could probably become reasonably good at it.For many years,I did no painting or drawing,but I have begun to do more in recent years.

Xu:Doyouthinkthatthereisacriterionbywhichwecantellwhatgoodphilosophy is and what bad philosophy is?

Hale:It is difficult to give any simple answer,because the philosophers whose work I most admire are in some ways very different from one another.Clarity and simplicity are great philosophical virtues,and unnecessary complication is to be avoided.But the problems are often complicated,which makes discussing them in simple terms very difficult.But I believe the best philosophers are those who speak and write simply and directly,avoiding needless detours and elaboration.There are,and have been,many excellent philosophers,as well as many more whose work I do not find as rewarding.Among those I read with great pleasure and benefit are Aristotle,Frege,Kripke,Kit Fine,Bolzano,as well,of course,as many others,including my long term collaborator Crispin Wright.I have learned a lot— at least,I hope I have— from studying the work of Quine and Dummett,who have,of course,been very important in the development of philosophy of mathematics and logic.

Xu:There are much fewer female philosophers not only in history but also in modern times.What do you think about women devoting to philosophy?

Hale:Yes,I think it would be good for philosophy if there were more women philosophers.Most of those I have known have been very good—I think they needed to be,in order to get on in what has for long been a male-dominated subject.Ability in philosophy has nothing to do with one’s gender.When I have worked on philosophy with women—as I have done in recent years with my friend and former student Jess Leech—I have always found it very rewarding.

2 Philosophical questions

2.1 Introduction to Neo-Fregeanism in the philosophy of mathematics

Xu:More than 30 years has passed since the publication of Wright’s book Frege’s Conception of Numbers As Objects and your book Abstract Objects,and the topics in Neo-Fregeanism are still hot issues in the philosophy of mathematics.However this field of research is not well known in China.Could you introduce the main theses of Neo-Fregeanism in the philosophy of mathematics to the Chinese readers?What’s the legacy of Frege’s philosophy of mathematics to Neo-Fregeanism?

Hale:Frege took the presence of singular terms in arithmetical statements—such as the numerals‘0’,’1’,‘2’.…and terms for real numbers such as— at face value,and understood them as standing for objects every bit as real as the physical objects we can see and touch,but as abstract rather than concrete.This is his realism or Platonism.And his aim was to show that the fundamental laws of arithmetic are analytic,in the sense that they can be proved using only general logical laws together with definitions of the basic terms,such as‘cardinal number’,‘0’,‘successor’and ‘finite number’.This is his logicism.Neo-Fregeanism retains both of these ideas.Frege’s own attempt to argue for his view broke down in contradiction.His definition of number in terms of extensions of concepts meant that he needed an axiom governing extensions(roughly,classes),and his axiom(Basic Law V)turned out to lead to Russell’s antinomy.Our basic idea is that the route through classes can be avoided,as far as arithmetic goes,by defining the number operator contextually,using Hume’s principle:the number of Fs≡thenumberofGsiffthereisaone-onecorrespondencebetweentheFsandtheGs.Itis known that if Hume’s principle is added to second-order logicwe can prove the Dedekind-Peano axioms for arithmetic,and it is also known that the system is equiconsistent with second-order arithmetic(i.e.arithmetic formalized using the D-P axioms in a second-order language).The issue that remains is whether this technical result has the philosophical significance Wright and I claim—i.e.that we can properly define the basic notions in this way,and that this gives us a route to a priori knowledge of arithmetic.

Xu:In the middle of last century,the research in Frege’s philosophy of mathematics remained largely historical.But circumstances changed after Neo-Fregeanism got on the stage.It is true that philosophy of mathematics has gone past Frege’s horizon.Could you summarize the significance of Frege’s philosophy of mathematics to current philosophical controversy about mathematics?

Hale:It is true that until about 1980,Frege’s work was seen as having largely,if not only,historical interest— because it was widely believed that Russell’s discovery of the contradiction in Frege’s system,along with Gödel’s incompleteness result,showed that Frege’s aims could not be achieved.Wright’s book and our subsequent work has changed that—even if most philosophers are still sceptical about our programme,they take it seriously.Although our way of trying to achieve essentially Frege’s aims is different from his,it owes a good deal to Frege’s own work and ideas,both about how arithmetical concepts such as finite number,successor,the ancestral of a relation,etc.,may be defined,and how the basic laws may be proved,including the crucial theorem thateveryfinitenumberhasasuccessor,sothatthesequenceoffinitenumbersisinfinite.And in analysis,my definition of the real numbers as ratios of quantities is inspired by Frege’s discussion in his Grundgesetze,vol.2.

Xu:When you beginning to launch Neo-Fregeanism Program in the philosophy of mathematics,what problems did you think urgent to solve to defend your philosophical position?

Hale:The main problems requiring solution are certainly the Julius Caesar problem,which was what led Frege himself to scrap the idea of defining the number operator contextually,by means of Hume’s principle,and what is known as the Bad Company problem,which arises because there are abstraction principles having the same general form as Hume’s principle,but which cannot be accepted,either because they are inconsistent(e.g.Basic Law V)or because they are consistent,but inconsistent with Hume’s principle(such as Boolos’s Parities principle).These are both important aspects of the general problem of showing that abstraction principles like Hume’s principle,and the abstraction I use to define the real numbers,are legitimate forms of definition on which a priori knowledge can be based.Of course,there are many more specific problems—we give quite a long list of them in the Reason’s Proper Study.

Xu:How many stages are there in the development of Neo-Fregeanism in the philosophy of mathematics?For each stage what are hot issues?

Hale:When Frege set out to show that the fundamental laws of arithmetic are analytic,what he means by ‘arithmetic’was elementary arithmetic and the arithmetic of the real numbers(i.e.real analysis).Correspondingly,in the neo-Fregean programme,one could think of providing an epistemological foundation for elementary arithmetic in logic plus definitions as the first stage,and doing the same for analysis as the second.But we have also been interested in extending the programme to find an abstractionist basis for some form of set theory,so this could be seen as a third stage.I don’t really think there are any ambitions beyond that.

2.2 Apriority and analyticity

Implicit definitions

Xu:Prof.Hale,in The Reason’s Proper Study(2001),you seem to defend the thesis that the truths in arithmetic are analytic,which are justified by logic and definitions.Unlike Frege,you hold that definitions could be implicit definitions.As far as I know Frege himself did not advocate implicit definition as the foundation of arithmetic(cf.his correspondence to Hilbert),but why Neo-logicists diverge from Frege’s logicism here?Frege’s Theorem is a logical result of Frege’s arithmetic.Does this logical result or the insights of Frege’s philosophy of arithmetic motivate the neo-logicism program?

Hale:ItistruethatFregeeventuallycametorejectimplicitorcontextualdefinitions as a basis for arithmetic,although he certainly considered using them in Grundlagen,even though he finally rejected contextual definition in favor of an explicit one.It seems tomethatimplicitdefinitionsareaquitelegitimatemeansoffixingthemeaningsofsome terms,and are sometimes indispensable,because no explicit definition can be provided.I should probably add that the terms in question one which can’t plausible be regarded having their meanings just simply through ostensive training.I mean obviously there are many words in the language that we learned by examples using context rather than by being defined at all in any way,and definition is only possible because of this basis.Words in the language and practices and constructions that are learned in practice rather than explicit teaching,but there leave room for words that cannot be defined explicitly in the sense of providing a phrase in simply terms that synonymous with them as one can defined,perhaps,spinster by unmarried woman,vixen by female fox.A good example,in some ways,is setting up the propositional logic with the language with the primitive undefined propositional operators,negation and conjunction.You might introduce the truth function “conditional”not by explicit definition,those providing expression which means the same as“if…then…”rather“if p then q”to mean it is not the case both p and not q,so no single word or phrase defines“if…then…”but a whole sentence is transformed into another one.And in this case,the implicit definition would be given by the bicnditional“if p then q if and only if it is not the case that p and not q”,so this seems very unproblematic kind of legitimate implicate definition.I think the idea that the implicit definition somehow is intrinsically not proper doesn’t have much insight for it.Of course,Frege had a quite specific reason for rejecting Hume’s principle as an implicit definition of the number operator—as opposed to an objection to implicit definition in general—namely,the Julius Caesar problem.That is,of course,a very serious problem which needs to be overcome if our procedure is to be acceptable.Of course,the technical result that the D-P axioms can proved in second-order logic+Hume’s principle(which Frege himself more or less proved in Grundgesetze),together with the relative consistency result for this system,is the main mathematical basis for our version of logicism,as far as elementary arithmetic goes.But much of Frege’s importance,not only for our programme,but for philosophy more widely,lies in the great contributions he made to philosophical logic in Begriffsschrift,Grundlagen and Grundgesetze,and in his other philosophical papers,and especially,those on Sense and Reference,Concept and Object,Functions,etc.

Xu:LikeFrege,youseemtoholdthataprioridonotcoincidewithanalyticaltruths.Whenweconsidertheaxiomsofsettheory,especiallylarge-cardinalaxiomstosolveCH,do you think they are a priori?What are the main differences between abstract principles and the other non-logical axioms in other axiomatic systems as a priori?

Hale:Actually,I don’t have a firm view on whether everything which is knowable a priori is analytic.I know of no clear examples of a priori knowledge of non-analytic truths.So unlike Frege,I’m not persuaded that geometry is synthetic but a priori in the way that Kant thought.Frege,at least according to my reading of him,and what I read by other people,thought Kant was right about geometry but wrong about arithmetic in thinking that arithmetic was synthetic a priori.So the disagreement was kind of impartial.I’ve really thought terribly much about geometry,and some of the views are quite crude and simple.But I guess that we need to separate very clearly abstract geometries,which may be Euclid and non-Euclid(various kinds)from physical geometry.They are certainly not a priori when geometry is on physical spaces.The question of the apriority of purely abstract geometrical systems seems to me to be in a wide like group theory.It’s quite plausible to argue that they are analytic.Axioms of group theory can be regarded as simply defining,in some sense,the primitive terms of group theory.About the status of large cardinal axioms I think we can say almost nothing with any confidence.No one has seriously tried to argue that any of them is analytic,as far as I am aware,although Gödel,in a famous discussion of the continuum problem,suggests that the independence resultsindicatethatwehavenotyetachievedafullanalysisoftheconceptofset,andthat further axioms are needed(which would presumably contribute to that analysis).I have to admit that I find the whole development of set theory as a study of higher and higher cardinalities very puzzling,philosophically—it is not intrinsically mathematically puzzling(which is not to say that it isn’t difficult— it is).I think an important difference between(good)abstraction principles and other non-logical axioms,such as the D-P axioms for arithmetic,is that abstraction principles can serve as good implicit definitions,whereas non-logical axioms cannot in general do so,because they typically involve direct assertions of existence which cannot be simply matters of definitional stipulation—they are what Wright and I have called ‘arrogant’.

Xu:Abstract principles provide us epistemic explanations to abstract objects al-though this kind of explanation is not natural procedure in our mind to know the abstract objects.Is it right?

Hale:IamnotentirelysureIunderstandthisquestionfully.Idothinkthatthepossibility of introducing ways of talking about abstract objects through abstraction principles gives us a better understanding of the nature of abstract objects—it helps us,as it were,to get what abstract objects are in a better perspective,from which we can see that recognizing their existence is not,after all,such a massive ontological step.I would not wish to claim that there is anything especially natural about it,however.

Logic

Xu:It seems that Frege did not hold a ‘schematic’conception of logic,according to which logic is a study of logical forms and he did not distinguish logic and metalogic.When he planned to carry out his logicism program,what did his“logic”mean?

Hale:For Frege,logical laws are the laws of truth.It is true enough that he does not make any explicit distinction between logic and metalogic—but I think that to say that he did not distinguish them at all is somewhat misleading.There is a pretty clear distinction,in Grundgesetze for example,between the formal development and the passages of ordinary German in which he explains or comments on them.This does not amount to a metatheory as such.For the purposes of his programme,logic was a system of higher-order logic—in effect,the first such system to be presented.

Xu:From Frege’s Theroem,we know that Neo-logicists take higher-order logics as logic.In your new book,Necessary Beings,you give arguments for the thesis that higher-order logics are not set theory but logic.In your opinion,what is logic?Do you think modal logics are also logic?Is there any criterion,by which we could tell what schemes are logical forms?

Hale:In the broadest sense,logic is the most general investigation of the forms of sound or correct reasoning.That covers both forms of reasoning which are necessarily truth-preservingand forms of reasoning,such as inductivereasoning,which typically are not guaranteed to preserve truth.By this general standard,it seems to me that higherorder logic is clearly logic,and so are modal logics.I think the most plausible mark of logic is its generality,or topic-neutrality,as it is sometimes called—logic,in contrast with physics,or geography,or history,say,is not tied to any particular subject matter;reasoning about anything whatever can be appraised as logically sound,or otherwise.I sort of deliberately avoided saying anything that pacifically addresses that part of logic might be part of mathematics,because whatever the merits of the view that logic is mathematical theory,the strong case could be remained on more or less structure grounds I expect.It seems to me that to think in that way is sort of to forget the kind of what philosophical origin of logic is.Russell had this kind view.It is not accident that philosophy is the subject in which logic has its home.

2.3 Platonism and Logicism

Xu:What does Platonism in the philosophy of mathematics mean?Does this ontological thesis relate to logicism?

Hale:The term ‘Platonism’derives,unsurprisingly,from Plato’s doctrine of the Forms,which he took to be the most truly real entities,lying beyond the reach of the senses but accessible to the trained intellect.In modern philosophy,it is widely used to refer to any view on which there exist,in addition to physical and mental entities,mind-independent and objective abstract entities which have neither spatial or temporal location.In relation to mathematics,Platonists generally hold that the surface grammatical form of mathematical statements,with their apparent reference to objects such as the various kinds of numbers(natural numbers,integers,rationals,reals,and complex numbers),should be accepted at face-value,and hence as involving a commitment to the existence of abstract objects and relations between them.

Logicism is,roughly,the view that mathematics can be grounded in logic.Of course,‘grounded’is somewhat vague,and may be interpreted in a number of ways.Frege was a logicist about arithmetic,which for him meant both elementary number theory and analysis(the theory of real numbers),but not about geometry,which he held,with Kant,to be synthetic a priori.The central claim of his logicism was that arithmetic is analytic,which he took to mean that its fundamental laws could be proved from general logical laws together with definitions.In this sense,logicism is an epistemological thesis.Frege was clear about this.He said very clearly of the notions of analyticity and apriority in the beginning of Grundlagan.The whole set of distinctions between analytic and synthetic and apriori and aposteriori have to do with kind of justification that a judgement could receive.What is going to justification determines whether or not it is analyticorsynthetic,apriorioraposteriori.Thegeneralideathatthesenotionsarelogical notions is very evident in Frege.

Platonism,by contrast,is an ontological thesis.There is no obvious necessary connection between them.One could be a Platonist but not a logicist,and oppositely,one could embrace logicism but not Platonism(Russell,in one phase,seems to have done so,that is being logicist but not being Platonist.).But of course,the two doctrines can be combined,and both Frege and the neo-Fregeans do so,in different ways.

2.4 Necessary beings

Xu:What theses did you want develop and defend in your new book Necessary Beings?I think maybe these theses relate to neo-Fregeanism.Could you introduce the background for this book?

Hale:The central theses in the book are these:

Modalconceptssuchasnecessityandpossibility,andmodalfacts—thatis,factsthe statement of which requires the use of modal notions—are fundamental,indispensable,and irreducible,in the sense that they are required for a philosophically adequate account of reality and cannot be explained,or explained away,in other terms.This is the first thesis.

Some forms of necessity are absolute,in the sense—roughly—that what is absolutely necessary holds unconditionally and with no contingent restrictions,and that this covers not only basic logical necessities,but also some non-logical necessities which are often described as metaphysical.This is the second thesis.The third thesis is:the source or basis of absolute necessities lies not in meanings or concepts,or in facts about socalled possible worlds,but in the natures or essences of things—of objects,properties,relations,functions,and so on.

The main purpose of the book is to explain what I understand by these claims in greater detail,and to argue for them.But of course that leads me to put forward a lot of other claims,e.g.about the nature of properties and relations,about what exists necessarily and what is only contingent,and more generally about ontological questions,which I also try to defend to the best of my ability.

I’ve been working on these topics for quite a number of years,so in some sense in the book,one or two views,I would say,can draw together stuff and develop stuff I’ve been doing for quite some time.But also very important influences like Kit Fine’s work,which I have enormous respect for and find very rewarding although I do differ from Kit in a number of issues,was really quite important.I’m interested both in Aristotle,whose work on definition,necessity,essence,is very interesting.

2.5 Suggestions for further readings

Xu:ThereisagreatdealofworkbeingdoneinthefieldofNeo-Fregeanism.Finally could you recommend some literature to Chinese readers who are interested in Neo-Fregeanism?

Hale:ProbablythebestthingtostartwithistheintroductiontoTheReason’sProper Study(Oxford 2001)which is intended to provide a useful overview of the relevant backgroundandtheneo-Fregeanprogramme.Analternativewouldbethelargelyoverlapping essay by Wright and me on Logicism in the Twenty-First Century,in Stewart Shapiro ed.The Oxford Handbook of Philosophy of Mathematics and Logic(Oxford 2005),which also contains some useful articles on logicism by Peter Clark&Bill Demopoulos by Agustin Rayo.There is,as you say,a lot of work being done—most of it is in articles in journals.Some of the main work Wright and I have done,both separately and together,is collected in The Reason’s Proper Study,but there are more recent papers by one or both of us in various journals and edited collections such as Philosophia Mathematica 2012,Journal of Philosophy 2012,and the collection by David Chalmers and others,entitled Metametaphysics.There is a collection Abstractionism soon to be published by Oxford,editedbyPhilipEbert&MarcusRossberg,whichcontainsalotofusefulpapersonawide range of topics including the semantics,ontology and epistemology of abstraction,the mathematics of abstraction,and issues about the application of pure mathematics.There were special issues of Notre Dame Journal of Formal Logic in 2001 devoted to logicism and of Synthese in 2009 edited by Øystein Linnebo on the Bad Company Problem,and also an issue of Philosophical Books in 2003 containing essays about The Reason’s Proper Study by several critics with replies by Wright and me.

Xu:Thank you very much!