Frege and Necessary Beings

Interview by Richard Marshall.

'About the reality of numbers: Crispin Wright and I do want to claim that our abstractionist approach upholds a form of platonism about numbers. But it bears emphasis that it is a very modest, deflationary kind of platonism.'

Bob Haleis the philosopher who broods on modality and metaphysics. Much of his work has been in the philosophy of mathematics. During a long and continuing collaboration with Crispin Wright, he has defended an approach to the foundations of mathematics inspired by Frege, combining a deflationary version of platonism with a form of logicism, according to which mathematical knowledge can be grounded in logical knowledge together with definitions of fundamental concepts. His research as British Academy Reader was mainly in this area, one product of it being the first published neo-Fregean construction of the real numbers. His other main research interest is in the metaphysics and epistemology of modality - the theory of necessity and possibility and related notions.Here he discusses Frege's philosophy of mathematics, Platonism, Logicism, Russell's paradox, his efforts with Crispin Wright to reverse Frege's abandonment of his logicist aim, how Frege answers some puzzles of the philosophy of maths, the Caesar Problem, absolute modality, Quine's dislike of higher-order logic, Dummett's two questions about absolute modality, the interdependence of modality and ontology, the relation between actuality and possibility, the difference between logical and metaphysical modality, what reality looks like and why he favours an essentialist theory of modality.

3:AM:What made you become a philosopher?

Bob Hale:A stroke of great good fortune. 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 copy of the Meditiations, and when I had devoured that, decided I was ready for the British 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.

3:AM:Frege looms large in your work. Beginning with the philosophy of mathematicsthere are two central components of his approach to maths, Platonism on the one hand, and logicism on the other. These are terms of art for the philosophers – can you sketch what they are?

BH:The term ‘Platonism’ derives, unsurprisingly, from Plato’s doctine 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 arithemetic, 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.

3:AM:Frege worked to prove that his Platonism and logicism worked but famously ran into a snag when Russell wrote and pointed out a problem. Russell’s paradox was taken by the 1970s to have been fatal and interest in Frege’s work on maths was considered even by Fregean scholars like Michael Dummett to be of purely historical interest. Can you say something about what Russell did to bring about what seemed to Frege and to many philosophers following the drama a catastrophe for Frege’s project?

BH:If he was to be able to prove the basic laws of arithmetic (in essence, the Dedekind-Peano Axioms, which assert that 0 is a natural number, that ev- ery natural number has another natural number as its immediate successor, etc.), Frege needed to define the primitive terms – ‘natural number (or finite cardinal)’, ‘0’, and ‘successor’. After considering and rejecting proposals to define number contextually – that is, not by providing a defining phrase synonymous with ‘natural number’ which could always be substituted for it, but by giving the meaning of a complete sentence containing the term to be defined by paraphrasing the sentence as a whole) – Frege finally opted to give an explicit definition in terms of classes (more exactly, what he called extensions of concepts): the number of Fs = the class of concepts equinumerous with the concept F. (‘equinumerous’ makes the definition sound a bit circular, but Frege is able to define it in purely logical terms: concepts F and G are equinumerous if and only if there is a one-one correspondence between the F s and the G s. Of course, that still looks circular, beause of the ‘one-one’, but the appearance is superficial – it can be defined in turn, using just quantifiers and the identity predicate ‘=’).

The important point for present purposes is that having defined numbers in terms of classes, when he came to attempt to carry out the rigorous derivation of the fundamental laws of arithmetic, Frege needed an underlying theory of classes. The key principle of his theory is Basic Law V, which asserts, roughly, that the class of Fs = the class of Gs if and only if F and G are co-extensive (i.e. every F is a G and vice versa). What Russell realized was that Basic Law V enables you to define a class corresponding to any given concept (so it works like a version of the principle of Naive Comprehension, which asserts that to every property there corresponds a class of all and only those things having that property). But then you can take the concept: concept which does not apply to the class to which it corresponds, and define and establish the existence of the class corresponding to that concept. You can prove then that that concept applies to the class corresponding to it if and only if it doesn’t. This is a version of Russell’s paradox of the class of all classes which do not belong themselves – this class belongs to itself if and only if it doesn’t. From this we can derive the contradiction that it both does and does not belong to itself.

Frege at first tried to restrict Basic Law V to block the paradox, but he subsequently realized that his remedy was no good, and abandoned his logicist aim.

3:AM:You (and Crispin Wright) reverse this judgement and argue that it’s possible to settle the landscape after the Russellian blast. How have you neo-Fregeans managed to find optimism for his approach using a contextual explanation and ‘abstraction principle’?

BH:In essence, our idea was that Frege gave up too quickly on his second attempt at a contextual definition of number, which does not involve classes or extensions, and so avoids the reliance on Basic Law V which proved fatal to Frege’s own systematic development of his position. The alternative he considers, but rejects, is to define ‘the number of Fs’ by means of the equivalence:
the number of Fs = the number of Gs if and only if the Fs correspond one-one with the Gs

The idea behind this proposal, as Wright and I understand it, is that we can implicitly or contextually define the operator ‘the number of ...’ – which can be applied to a predicate to form a complex term which purports to stand for a number – by fixing the truth-conditions for identity-statements in linking a pair of such terms. An implicit definition, as we understand it, works by taking the definiendum (the expression to be defined) and embedding it in a sentence which otherwise consists only of expressions whose meaning is already understood. The idea is then that the definiendum is to mean just what it needs to mean in order for the sentence as a whole to say something true. To give another, simpler and less controversial, example: we can define ‘or’ implicitly by stipulating that it means just what it needs to mean for the sentence: ‘A or B if and only if it is not the case both that not-A and that not-B’ to be true (where A and B are any sentences).

If the equivalence we use to define the number operator (widely known as Hume’s principle because Frege rather generously attributes the idea to David Hume) is adjoined to a suitable underlying logic, one can prove the Dedekind-Peano Axioms as theorems. Furthermore, the resulting system – sometimes called Frege Arithmetic (FA) – is known to be consistent (more precisely, to be consistent if and only if second-order arithmetic is consistent, which non one seriously doubts).

Of course, many people (as well as Frege himself) have questioned the legitimacy of our proposed definition. But we have done our best to answer the doubts which have been raised, and remain optimistic that they can be answered.

3:AM:You say that by approaching Frege in this way many deep ontological and epistemological questions in the philosophy of maths can be answered. So can you say something about what puzzles are being answered by Frege about the reality of numbers and reason’s role in knowing them? And does your position broaden out to a general Fregean meta-ontological position from this?

BH:This is a large and complex question, so my answer must be somewhat sketchy and unargued.

First of all, if elementary arithmetic can be given an epistemological foundation using Hume’s principle in the way we propose, it is plausible that similar abstraction principles – as they are called – could be devised which can serve as the foundation for other mathematical theories, such as real number theory and perhaps a form of set theory. In fact, quite a while ago, I myself devised an abstraction principle which could be used to define the real numbers as ratios of quantities, corresponding to Frege’s own incomplete theory but avoiding his reliance on set theory. Soon afterwards my friend Stewart Shapiro presented a rather simpler abstractive definition, corresponding more closely to the usual Dedekind construction. There has since been quite a lot of work on an abstractionist form of set theory, but thus far with less conspicuous success.

Second, about the reality of numbers: Crispin Wright and I do want to claim that our abstractionist approach upholds a form of platonism about numbers. But it bears emphasis that it is a very modest, deflationary kind of platonism. Firstly, the Fregean conception of object it espouses is a very general and undemanding one: to be an object is just to be something to which we could refer by means of a singular term, and second, it is enough for a singular term to have reference that there are true sentences of a suitable sort embedding it – so that accepting that there exist numbers is just the objective correlate, so to speak, of accepting that sentences such as ‘3+4=7’ and ‘the number of natural satellites of the Earth = 1’ are true, when taken at face-value. Of course, not all such sentences can be known by reason alone, but some of them, like the first, can be so known, and knowledge of its truth suffices for knowledge of the existence of some numbers.

Finally, on the wider question about meta-ontology: Yes, I think our position does amount to a broader meta-ontological stance. Part of that stance is the deflationary, metaphysically lightweight treatment of questions about the existence of objects of a given kind to which I just alluded. But for me at least, and probably for Crispin as well, it extends beyond that to agreement with what is plausibly taken to be Frege’s view, that more generally, ontologically basic notions like those of object, property, relation, etc., are best explained in terms of the logico-syntactically different types of expression – singular terms, predicates, and various other sorts of functional expression – to which they correspond, and that questions about the existence of these various types of entity can and should be answered by seeing whether there are true statements of an appropriate kind incorporating expressions of the relevant types – so that, to take a very simple example, it suffices for the existence of the property of being a human being that one of the statements ‘Donald Trump is human’ and its negation is true. Of course, some philosophers dislike this approach, because they think it makes what exists somehow mind- or language-dependent, but I believe that this and other objections can be answered. (I’ve tried to answer the main objections in Necessary Beingsch.1)

3:AM:What is the Caesar Problem and why is it important that neo-Fregeans can solve it? Can they solve it and if so how?

BH:The Caesar problem is important because it is what led Frege to think that the kind of contextual definition of the number operator we favour cannot work, and if we are right and he was wrong, it needs to be shown that the problem can be solved, unless there is a way of simply avoiding it altogether.

Briefly, the problem comes about like this. The idea underpinning the contextual definition using Hume’s principle is that we can fix the sense of the number operator by providing a condition for the identity of numbers (i.e. the objects to which numerical terms formed by means of the number operator refer). It aims to do so by telling us that the numbers denoted by ‘the number of Fs’ and ‘the number of Gs’ are the same just in case there is a one-one correspondence between the Fs and the Gs. Frege’s complaint was that while this explanation works well enough when the terms which may or may not refer to the same number have the canonical form ‘the number of Fs’, it tells us nothing about how to settle the question whether, for example, the number of Jupiter’s moons = Julius Caesar. Of course, we may suppose that it is somehow obvious that Julius isn’t a number. Frege’s point is that that is no thanks to the proposed definition, which appears not to get a grip on the question. This is the Caesar problem. It is sometimes thought to show that what it reveals is that Hume’s principle might give us a criterion of identity for numbers, but that it fails to provide what Dummett calls a criterion of application – i.e. a criterion for deciding whether or not something specified in other terms is or isn’t a number.

We – that is Crispin and I – think the problem can be solved. Our basic idea, I would say, is that the charge that Hume’s principle fails to supply a criterion of application is overstated, because one can hold that to be a number is precisely to be something for which questions of identity and distinctness can be settled by reference to facts about one-one correspondence. Since Julius Caesar is not something for which questions of identity and distinctness can be so settled, he is not a number, and hence not identical with the number of Jupiter’s moons. This is, so to speak, the Urgedanke in our ap- proach, which we’ve tried to develop more fully in various papers. Of course, many people remain unpersuaded, but we continue to believe it is fundamentally sound.

3:AM:This leads to broader concerns of metaphysics, logic and epistemology. One of the things you notice and address is that analytic understanding of modal claims in philosophyhas generated a range of extreme proposals. So firstly, what’s at stake in this area of philosophy and what are the pressures that have led to such extremes being proposed? And why focus on absolute modality – wasn’t Quine skeptical about all this?

BH:The extreme proposals range from David Lewis’s ‘modal realism’ (the proposal to eliminate modality in favour of quantification over a huge plurality of ‘worlds’ – the label is misleading, because Lewis isn’t a realist about modality) to the non-cognitivist claim that there are really no objective facts about necessity and possibility, only facts about our imaginative abilities which we ‘project’ onto the world, much as Hume thought we project our own habits of expectation onto the world when we think there are causally necessary connections. What has led to these proposals is a deep-seated unwillingness to accept modal notions and/or modal facts as basic, and the belief that they have to be explained in other terms or just explained away. Quine was, as you say, vigorously sceptical about modality – in effect, he favoured rejecting it altogether (in spite of one or two concessive remarks in less prominent essays). Of course, I’ve tried to argue that Quine’s scepticism is unwarranted, and that it actually destabilizes, but it is for others to judge whether my arguments are any good.

One simple reason for the focus on absolute modalities is that it is at least very plausible that other, relative kinds of modality can, and should, be analysed in terms of it – so that the main question about their legitimacy reduces to a question about absolute modalities.

3:AM:Quine disliked higher-order logic because it carried what you call ‘massive ex- istential commitments.’ And his charge doesn’t lose force even if we ignore his assimilation of such logic to set theory. How do you see off Quine’s challenge?

BH:Quine makes a number of objections to higher-order logic, of which perhaps the best known is his charge that it is really ‘set theory in sheeps’ clothing’. I think that charge is fair enough, as applied to higher-order logics interpreted in accordance with the standard semantics, which takes the second-order variables to range over the full power set of the first-order domain. But as Charles Parsons observed somwhat before Quine’s attack on higher-order logic, the charge that it involves ‘massive existential commitments’ (which is actually Quine’s term, not mine) can just as well be brought even if the second-order variables are taken to range over Fregean concepts, which are not objects, and so not sets.

My own view is we should interpret the second-order variables as ranging over properties, individuated intensionally (roughly, in terms of the meanings of predicates), and that the source of the existential commitments lies in taking the entities over which they range to be individuated extensionally, just as sets are. Of course, Quine would reject any such interpretation, because he was sceptical about meaning and intensions. That is also the source of his well-known objection to properties (or attributes) – that they lack decent identity-conditions (’No entity without identity’). Given his scepticism about meanings, that objection is of course correct. But I think we can and should reject that scepticism too.

3:AM:Dummett asked two questions that you suggest may be considered the two fundamental questions for absolute modality: what’s its source and how do we recognize it? How do you tackle these two questions?

BH:The two main answers to the question about source which philosophers have advanced in recent times have been that necessity is just truth by convention or in virtue of meaning and that it is truth at each one of a vast range of worlds (themselves to be characterized in non- modal terms). Both are reductionist answers – they seek to explain necessity away, in terms of something non-modal. But Dummett’s question doesn’t have to be answered reductively. I think necessity has its source in the nature of things, or essences, but that this isn’t a reductive answer, because the notion of nature or essence can’t be explained non-modally. Of course, I am not the first philosopher to take this position. It is possible that Aristotle held something like this view, although it is difficult to be sure, because what he says on the matter is terse and difficult to interpret. In our own time, Kit Finehas promoted a similar view. My position differs from Kit’s in some important respects, but there is broad agreement here.

The epistemological question is harder to answer briefly. Many philosophers see imaginability or conceivability as a basic source of knowledge of possibility, but while I think the imagination can have a role to play here, I don’t think it can be a primary source of modal knowledge, because to the extent that it can help, it must be suitably constrained, and I don’t think the constraints can be justified without relying on some prior knowledge of necessity. Given my first answer (about source), the question is: how do we get knowledge of the natures of things? I think that some of our knowledge of essence can be a priori, based on definitions of the relevant things which can also be taken as definitions of the corresponding words or concepts. For example, I know that to be a square is to be a plane figure with four sides of equal length meeting at right angles, and I know that because that is what the word ‘square’ means, and I am competent in its use. But there is also a lot of a posteriori knowledge of essence – for example, that water is H2O, that gold is an element, etc. The explanation how we can have such knowledge is more complicated, but I think Kripke provided a good model for it: we know a posteriori that water is necessarily H2O by an inference from that premise that water is H2O (known by empirical investigation) together with the conditional premise that if water is H2O it is necessarily so.

The crucial question is how we know the conditional premise. Kripke suggests that we know it ‘a priori, by philosophical analysis’. I think that’s a good start, but it leaves a lot to be explained. Clearly what is really needed is knowledge of some general principles from which such conditionals may be inferred – principles like: chemical substances necessarily have whatever chemical composition they have, and: individuals essentially belong to kinds, and so on. Whether we can know such principles a priori is not so clear – it may be that the best we can do is give a kind of abductive argument for them. I tried to develop such an argument in Necessary Beingsch.11, but it needs further work.

3:AM:Is it your contention then that we can’t know what kind of things there are (ontology) before we know about modality, nor can we know about modality without drawing on ontology and that neither ontology nor modality is more important than the other – isn’t there a threat of circularity in all this?

BH:I do think that modality and ontology are inderdependent, but my position is not quite as you suggest. I think there probably would be a vicious circularity involved in claiming that all modal knowledge depends upon prior knowledge of ontology, and that all ontological knowledge depends upon prior modal knowledge – it’s not clear that there would be, if ‘all’ were replaced by ‘some’. But anyway, my claim isn’t really epistemological – I think there is clearly a lot of very ordinary modal knowledge that doesn’t presuppose any significant ontological knowledge – I know a lot of things such as that I might have had a boiled egg for breakfast this morning, although I didn’t, and that if they hadn’t been misled by Johnson and Gove’s ‘promises’ about the alleged £350m a week, many people who voted to leave the EU would not have done so (some who did have already realized their mistake). What I do think is that we can’t properly explain what is required for the existence of objects, properties, relations, functions, etc., in wholly non-modal terms. This connects with my answer above. Although it suffices, on my deflationary conception of ontology, for the existence of properties that there are well-understood predicates with appropriate satisfaction conditions, it is not necessary – it is enough that there could be such predicates. More generally, the nature and existence-conditions of various types of entity cannot be explained in non-modal terms.

There would still be a damaging circularity if the relevant modal notions had to be explained in terms of the concepts of those types of entity. But I don’t think that’s so. In fact, I don’t think the modal concepts can or need to be explained in other terms at all. They are fundamental and irreducible. The dependence of modality on ontology is rather different, and as far as I can see, unproblematic – it consists in the fact that the best explanation of the source of ground of modal facts is provided by the essentialist theory, and a detailed exposition of that theory makes use of ontological notions such as object, property, etc.

3:AM:On the face of it it doesn’t seem straightforwardly or intuitively the case that facts about what is actually the case depend on facts about possibility. What’s wrong with thinking that this intuition is right? Can’t we get rid of modal facts – explain them away somehow – see them as being dependent on non-modal facts, say, or mind-dependent?

BH:To answer your question backwards, as it were: many have tried to give reductive explanations of modal facts, tried to view them as dependent on non-modal facts, or tried to explain them away altogether – I don’t think any of these explanations works, and try to explain why, especially in Necessary Beingsch.3.

I agree that people often find the suggestion that facts about what is actually so depend upon facts about what is possible odd or counter-intuitive. That is in itself somewhat surprising, given that it is obviously true that if it had not been possible for things to travel faster than sound, no plane would have broken the sound barrier – isn’t that a case of something actual (planes travelling faster than sound) depending on a possibility (the possibility of their doing so)? However, I think there is a more general reason why people find the suggestion of dependence of the actual on the merely pos- sible unpalatable – this is that they think that modal facts must admit of some kind of reductive explanation. I think they are right that if modal facts have to be reductively explained, we can’t have actual facts depending on them – but I think they’re wrong about the need to explain modal facts reductively.

3:AM:There seem to be different kinds of modality – alethic, epistemic, deontic and absolute – you tend to focus, as we mentioned earlier, on absolute modality and also you discuss metaphysical and logical and analytic modality. Are these last three all the same and how do all these different types of modality connect up, if they do?

BH:I think we should certainly distinguish between logical and metaphysical modality. In my view, which not everyone shares, both are kinds of absolute modality, but they not co-ordinate, competing modalities. Logical necessities are true in virtue of the natures of logical entities (the various logical functions, such as negation, conjunc- tion, etc.) and so are a species of metaphysical necessities, these being those necessities which hold in virtue of the natures of things in general. Oppositely, metaphysical possibilites are a species of logical possibilities – everything metaphysically possible is logically possible (but not vice-versa – Aristotle’s being a sheet of tissue paper is not ruled out by the natures of any logical functions, but it isn’t metaphysically possible). So in one sense, logical necessity is stronger than metaphysical necessity – everything logically necessary is metaphysically so, but not vice-versa – and logical possibility is weaker than metaphysical.

Analyticity is another matter. Those who think that some necessities have their source in meanings will regard it as a kind of necessity. My own view is somewhat different. I think analyticity has a role to play, but it is a role in explaining how we can know some necessities a priori, rather than in explaining their source or basis.

3:AM:So as a kind of take-home thing, can you sketch out what answers you give to the questions that people like me think about quite a bit, things like: what are numbers, could Wittgenstein have had children, would an omnipotent and omniscient God have the same notion of possibility and necessity was humans, and so on. From your work, what is reality looking like?

BH:I’ve said a bit about numbers already, but there is a bit more I’d like to add. I’m still inclined to believe that numbers are abstract objects, but there’s been quite a bit of work recently, partly by people who know a lot more about linguisticsthan I do, arguing that contrary to some appearances, number-words are best understood as predicates rather than names or singular terms. Given my broadly Fregean view of the relations between syntactic and ontological categories (see my earlier answer above), this might be the basis of an argument for viewing numbers as properties rather than objects. I used to believe that there was a compelling argument for taking them to be (abstract) objects, but now I am not really sure about this. What seems to me far more important is not whether numbers are objects rather than properties, but whether, and in what sense, they exist.

I know of no reason to think that Wittgenstein could not have had children. It is at least metaphysically possible, even if there is some reason of which I’m ignorant to believe it was not biologically possible. Whether he could have failed to be the child of Karl and Leopoldine is another, and of course much more controversial matter – I am somewhat inclined to think that isn’t possible, but I don’t know of a completely convincing argument for that view. I think one can give a quite strong argument for the essentiality of kind membership, from which it would follow that he was essentially a human being, and since to be a human being involves having a certain kind of life cycle, starting with a fertilized egg, it would further follow that he necessarily developed from one such. But there is still a gap to close before one gets the essentiality of biological origin.

God’s omniscience would mean that for Him, unlike us, there would be no epis- temic possibilities other than those which he already knew to be realized. This wouldn’t mean his notion of epistemic possibility would be different from ours, but it would make it fairly idle. If the question is taken to concern metaphysical possibility and necessity, I think we should answer rather differently. Since I think the metaphysically impossible is absolutely impossible, I don’t think God’s omnipotence would mean that He could do anything that isn’t independently metaphysically possible – that need not make Him less than fully omnipotent, on a reasonable construal of omnipotence, any more than His inability to act against the laws of logic would.

If some of the arguments I’ve tried to support are good, reality is a mixture of abstract and concrete entities. I think it is consistent with everything for which I’ve argued, but not entailed by anything for which I’ve argued, that reality is a rich mix of necessities and contingencies – not only lots of contingent facts as well as necessary ones, but quite a lot of contingent existents as well as a lot of necessary ones, such as numbers and purely general properties and relations.

3:AM:Does this work on modality connect with metaphysical claims about essences – which seems kind of medievalin a way but also an exciting place to be, very cutting edge and daring in its claims. Is this a good time for philosophy and logic and given that there are prominent scientists (and even a few philosophers) who question why we should listen to philosophy, what would you say to these people?

BH:Yes, as I said in answer to question 9, I favour an essentialist theory of modality – that is, a theory on which the source of necessity lies in the essences or natures of things. It is certainly true that the notion of essence, which has come much more into fashion in recent decades, partly as a result of work by Ruth Barcan Marcusand by Saul Kripke, and more recently Kit Fine’s important papers on the subject, was for a long time very much out of fashion in philosophy, and had a bad name in philosophy of science. The orthodoxy when I started to study philosophy was that the whole idea that you could define anything other than words or concepts, and with it the idea that things have natures or essences, was a terrible aberration, and in so far as people sought to account for necessities, they viewed them as the product of meanings or conventions of language. I have almost the opposite view. Defining words is a special case of defining things in general, and describing what a word means is giving its (semantic) essence. Further, not much, if any, necessity is to be explained by appeal to meanings or conventions.

I also think that the association of essence with bad, pre-Galilean medieval ‘Aristotelian’ science, which helped to give it such a bad name, was quite unfair (as well as being unfair to Aristotle, who was, of course, wrong about a lot of things (especially about motion), but was a good empirical scientist). Contrary to the impression often given by some modern historians of ideas, modern science does have a place for the notion of essence, although modern scientists don’t often use the word ‘essence’ – but they do, for instance, investigate the nature of light, or heat, or (other forms of) energy, etc. There is nothing intrinsically unscientific about the notion of essence, i.e. of what it is to be something. Of course, the philosopher’s interest in essence is different from the scientist’s, and it is, more generally, really important to keep philosophical questions separate from theoretical and empirical scientific ones. If philosophers try to answer scientific questions from the armchair, the scientists are quite right to dismiss what they say. But scientists aren’t above making philosophical claims, especially broadly metaphysical ones, even if they are sometimes unaware that they are doing so, and they have no reason to be contemptuous of philosophy, provided it doesn’t confuse philosophical questions with scientific ones. Unlike Quine, I think there is a pretty clear line between philosophy and science. Much the same is true, I think, with regard to mathematics and philosophy – there are philosophical questions about mathematics, and there are purely mathematical questions. It seems to me that it is generally pretty clear which are which.

3:AM:And for the readers at 3:AM who want to get further into your philosophical world are there five books you could recommend that would help them?

BH:Excluding stuff I’ve written myself, and assuming you don’t just mean introductory books, my five – in no significant order – would be:

Saul Kripke Naming and Necessity

W.V.Quine From a Logical Point of View

Gottlob Frege Foundations of Arithmetic

Paul Benacerraf & Hilary Putnam Philosophy of Mathematics Selected Readings(2nd Edn)

Kit Fine Modality and Tense

I feel bad about not including a couple more of the great dead – the two I was most strongly inclined to include are Aristotle (especially some books of the Metaphysicsand the Posterior Analytics, although both are hard going), and Bolzano (Wissenschaftslehre), whom I find an increasingly interesting and rewarding philosopher. I came to both of these philosophers well on in my own career, and wish now I’d got into their work earlier on. They are full of very interesting ideas, and both do philosophy in a way a I really admire (as do the writers in my five selections). I haven’t mentioned my long-standing friend and frequent co-author, Crispin Wright, because anyone interested in my work in philosophy of mathematics would pretty certainly be reading his stuff too. I myself have benefited enormously from reading it, and even more from working directly with him.

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