Frege's 'Third Realm' and Aquinas on 'Divine Ideas'

I. A Question Arising From Kenny

This paper originates in a suggestion by Anthony Kenny towards the end of his book Aquinas on Being (2002). The main aim of the book is to assess Aquinas’ account of being against Fregean criteria. Thomas does reasonably well in this examination but is not awarded a first class honours, mainly because, says Kenny, he is blind to some crucial points made by Frege.

For most of Kenny’s book ‘platonist’ and ‘platonism’ are taken to be ‘bad things’. The cause of Aquinas’s blindness about being is identified as ‘residual platonism’, or even ‘neo-platonism’, a term introduced on the final page of the book (p.194) and not really explained. The introduction of this term, albeit on the final page, does make the reader wonder why Aquinas’s commentaries on neoplatonist texts (of which there are four: Boethius De Trinitate and De Ebdomadibus, Pseudo-Dionyius On the Divine Names, and Liber de Causis) do not figure more prominently throughout Kenny’s book, since they ought to be crucial for studying all this as the main culprits in Aquinas’ failures.

My aim here is not to pursue this particular critique but to take a related but different route. Another thing that happens towards the end of Aquinas on Being is that suddenly, and strikingly, Frege’s ‘platonism’ is spoken of except that now the term is used, if not with approval, at least with nothing like the disapproval it seems to attract from Kenny in Aquinas’s case (p.202). Frege’s ‘platonism’ is with reference to what he calls ‘a third realm’ – Kenny says that if there is anything like this in Aquinas’s work it is the notion of ‘divine ideas’ (p.204).

Hence the concern of this paper, to offer some preliminary thoughts about a comparison which, Kenny concludes, would be the subject-matter of another book.

II. Frege’s ‘third realm’

It is certainly true that the term ‘self-subsistent object’ is a striking, even startling, one. This is how Gottlob Frege describes numbers in his Foundations of Arithmetic (paragraphs 55ff). To anyone acquainted with the Platonist tradition it carries immediate resonances. The forms or ideas of Plato were at times referred to as aut'upostata, self-subsistent things, that level of reality ‘between’ the things of this world which are nevertheless dependent on them for their intelligibility, and some realm of further transcendence which lies above and beyond the multiplicity of such objects.

Proclus, for example, in speaking about what he terms henads, those principles that mediate the transcendent unity of the One to things lower down, describes them as having an autoconstituent existence [he ousia aut'upostata], something between the One which is not constituted in any way and beings which are constituted extrinsically (Elements of Theology 40-53; Commentary on Parmenides Book VII Cousin 1145-1146, 1149 [Dillon, pp.501-02, 504-05]). E.R.Dodds comments that such ‘autoconstituent essences may have a temporal activity even if not a temporal existence’ (Boland, Ideas in God, p.118, note 135).

The neoplatonists, of course, in their interpretation of Plato’s Parmenides, hypostatised the hypotheses, turning what some would regard as logical exercises into ontological levels of reality. The criticisms of Frege’s metaphor or myth of the third realm amount to accusing him of doing something comparable.

But let us not get carried away too quickly with what may be no more than a verbal echo. What does Frege mean by such a description of Number? His German term, selbständig, means independent, autonomous. J.L. Austin’s translation of this, ‘self-subsistent’, may seem a bit loaded philosophically. Or perhaps it is exactly right for capturing what it is Frege wants to say, that numbers are not on the one hand things in the external world, or properties of things in the external world, while on the other hand they are not subjective, not ideas in the imagination, things merely invented by human minds in their engagement with the world (paragraph 93).

Numbers are objective and not subjective. One can see what Frege is anxious to say they are not and one can understand his lucid and at times amusing arguments in support of his views. But it is more difficult to see what they are even while seeing how they need to be as they are if they are to fulfil their function.

One is of course tempted to wonder where the numbers are, a temptation to which I gave in some time ago having the opportunity to ask a professor of mathematics where the number ‘1’ is, to which he gave the encouraging reply ‘that is a very complicated question’. Frege himself considers this question towards the end of his Foundations.

In his 1965 introduction to Frege’s philosophy Jeremy D.B. Walker speaks about the question of his Platonism:

It is sometimes said that Frege was a Platonist, or that his theory of mathematics and logic is a Platonist theory. As it stands this can be seriously misleading; for instance it suggests quite falsely that he attributed some kind of real existence to concepts, and that he construed concepts as objects of some peculiar kind. Frege spent much of his time destroying these two views, which are undoubtedly Platonistic.

But there are two characteristics of Frege’s thought which explain the epithet. The first is his emphasis on the notion of objectivity together with its use in his rejection of psychologism. Even if he distinguished objectivity from existence, the fact that he described concepts, thoughts and senses as objective entities [?] is enough to class him as a philosophical realist. The second is his notion of ‘objects’. The fact that he calls numbers, for instance, objects is not by itself what counts, since we do not know just from this what use he makes of the term ‘object’. It is this use that is significant (p.194).

He goes on to say that this use is explained by pointing to Frege’s view that propositions are not dependent for their truth-values on any psychological event and this amounts to the actual subsistence of true propositions before they come into relation to any mind. Early Frege, says Walker, deploys the notion of objectivity as an alternative to the notions of subjectivity and actuality:

numbers are objective since number-statements are neither about psychological facts nor physical facts, and their truth-values are dependent on reason and not on sensation or intuition (p.195).

Later Frege interprets these negative notions in a positive way, condensing them in what Walker calls ‘the metaphor of the three realms’ (p.195):

Thus objectivity is put on a level with the other two, and comes to seem just a different kind of existence, e.g. subsistence or ‘being’. This is largely the fault of Frege’s language, which is Platonic in the extreme. Objectivity is upheld against subjectivity and actuality … in terms of the independence of truth-values from the occurrence of thoughts (p.195).

Michael Dummett regards Frege’s ‘third realm’ as mythological. For Dummett it is a point at which Frege falls short of his own best aspirations to logical thinking (Frege and Other Philosophers, 1991, chapter 12). In coming to conceive of ‘thoughts’ as independent of language and of thinking beings Frege became guilty of philosophical mythologizing, says Dummett (op.cit., p.250). He describes Frege’s ‘third realm’ as follows:

Whatever the truth is about other philosophers, Frege’s conception of thoughts and their constituent senses is mythological. These eternal, changeless entities inhabit a ‘third realm’, distinct from the physical universe and equally distinct from the inner world of any experiencing subject. Despite their separation from the physical world, many of these thoughts are about that world, and are true or false, not indeed by corresponding to anything in it or failing to do so, but in so far as they are about the external world, in virtue of how things are in that world. Somehow we grasp these thoughts and sometimes judge them to be true or false; indeed, it is only by grasping them that we become aware of the external world, rather than only of our own inner sensations and feelings. Somehow, too, we associate senses with words, and so communicate thoughts and judgements to one another.

As long as this perspective is dominant, all is mysterious. There is no way of explaining how thoughts relate to things in other realms of reality, that is, what makes them about anything. There is no way of explaining how we grasp them: no wonder Frege wrote, ‘this process is perhaps the most mysterious of all’. Above all, there is no way of explaining how we attach senses to words or expressions, that is, what makes them senses of those words and expressions. All this is obscured for us by Frege’s having had very good, if not fully complete, explanations of all these things. It is just that these explanations cannot be reconciled with the mythological picture. When we have Frege’s theory of meaning in view, our perspective has wholly altered: the third realm has receded to infinity (p.251).

It is as if Dummett had said: Frege’s third realm is a confusing distraction from Frege at his best. Some students of Aquinas have said that his doctrine of ‘divine ideas’ is a confusing distraction from Aquinas at his best. In the pages in which he treats of Frege’s mythological third realm, Dummett corrects infelicities in Frege’s work by appeal to his better self, exactly the same corrective process which students of Aquinas have sought to do also. The question is whether we learn something about Aquinas’s mistakes as we eavesdrop on Dummett explaining Frege’s.

III. Aquinas on ‘divine ideas’

I have not mentioned Aquinas for a while and here is an appropriate moment to do so. His ‘third realm’, if we might continue with this phrase for the moment, is the in-between realm of ‘ideas’ which are and are not among things, for things do not have the abstract qualities that these do and yet things depend on them for intelligibility and truth. They are and are not 'in' God. In fact they seem to threaten the simplicity of God so effectively and consistently established by Aquinas in many places.

This was why the question on divine ideas puzzled A.D. Sertillanges who came to regard that question as a hangover from an earlier, Augustinian Christian Platonism, that survived in Aquinas’s works only out of deference to long established tradition. Brian Davies consistently ignores the question on ideas in presenting the thought of Aquinas, Herbert McCabe dismissed its appearance in the Prima Pars of the Summa theologiae as an indication that Aquinas was having a ‘platonic off-day’, and Etienne Gilson respectfully passes it by, not regarding it as contributing anything essential to Aquinas’s thought. None of them, to my knowledge, dismisses it as ‘mythology’ though some of their criticisms are comparable to those Dummett makes of Frege.

Here is an interesting thing, then, that both Aquinas and Frege should be regarded by important interpreters and promoters of their respective works as having gone wrong at what looks like a comparable place: trying to articulate the character of some level of reality which is neither the external, physical world nor the internal, mental world but one that unites the two.

Why then does Aquinas persist in speaking of divine ideas when he clearly rejects what he regards as the platonists’ theory of ideas and in many texts appears to treat of creation, knowledge, truth, etc., in a satisfactory way without reference to this doctrine?

In my book on this question of ideas in God according to Saint Thomas I gathered the evidence to show, I believe, that the doctrine is of more than decorative or pious significance for him, that his theological synthesis requires it (if indeed there is such a theological synthesis – pace Fergus Kerr). A summary of his thinking about this goes as follows (Boland, Ideas in God According to St Thomas Aquinas, 1996, pp.323):

At first Saint Thomas gives arguments from authority for the necessity of ideas in God. Augustine, Dionysius, Averroes: all seem to agree that this is an aspect of how one must speak about God. Later Saint Thomas develops his own arguments for the ideas. It is necessary to speak of the divine ideas for two rea­sons, he says, because God knows all there is and because God is the exemplar of all there is. God’s knowledge is that of the intelligent cause of all things. It is a knowledge which is not gathered from those things but which is available to God in his knowledge of himself. As the source of everything God knows all of which he is capable and all for which he is responsible. The model, pattern, archetype, paradigm or idea for the creation: where can it be except in God and what can it be except God since whatever is in God, is God and whatever God has, God is. Only the divine essence itself, therefore, can be the species by means of which God knows and the exemplar with a view to which God cre­ates.

Note that Thomas for now uses the singular in relation to God. But because the order of the universe is intended by God and is not the accidental result of a succession of agents, God must have the idea of the order of the universe. There cannot be an idea of any whole unless particular ideas are had of those parts of which the whole is made. So it must needs be that in the divine mind there are the proper ideas of all things. It is not repugnant to the simplicity of the divine mind that it understand many things, he goes on. What would be repugnant to its simplicity is the notion that God’s understanding is formed by a plurality of images. Hence many ideas exist in the divine mind as things understood by it. This is because God, knowing His essence perfectly, knows it according to every mode in which it can be known, whether in itself or as it can be participated in by creatures according to some degree of likeness. God knowing his essence as capable of such imitation by any creature, knows it as the particular type and idea of that creature (ST I 15,2).

There is a kind of plurality then (ideas) but whether the whole notion represents the persistence of a piece of Platonising mythology is a fair question. The ideas belong, Thomas adds, not simply to God’s knowledge of things but to God’s knowledge of his own knowledge of things. That by which God understands is one, that which God understands is many (the many things there are as well as the many ‘types’ of things, God’s understanding that he understands many things by His essence) (ST I 15, 2 ad 2).

IV. Aquinas and Frege on Individuals/Units and Knowledge of Them

Frege does not like what is involved in the term ‘idea’ as his term Vorstellung is consistently translated by Austin. This for him is a mental image, the gathered residues of sensation and imagination which cannot be relied upon for certainty and proof. In sentences that recall Descartes, but also Augustine, Frege wants ‘Bestimmtheit und Festigkeit’, definiteness and fixity, to place against the fluctuating and indefinite phases of consciousness which is all that sensations and mental pictures can offer. Frege wants to get beyond approaches which leave everything in flux – historical approaches, for example – and which remove ‘any possibility of getting to know anything about the world’ because ‘everything would be plunged in confusion’ (Foundations, p.VII).

It might seem that Aquinas is implicitly dismissed along with all those empiricists and ‘psychologists’ who make knowledge rely too much on experience and imagination. Thomas after all explicitly argues that there is no human knowing (intellectus) without conversio ad phantasmata (ST I 84,7). This does not get in the way of his believing that human reason is capable of an activity and of a certainty that are free of the limitations and vicissitudes of what is contributed by the changing corporeal organism.

In a very interesting footnote towards the end of his Foundations of Arithmetic, Frege says something very similar. He has just said that the reason’s proper study is itself and that in arithmetic we are concerned with objects given directly to our reason and, as its nearest kin, utterly transparent to it (paragraph 105). The footnote then adds:

By this I do not mean in the least to deny that without sense impressions we should be as stupid as stones, and should know nothing either of numbers or of anything else; but this psychological proposition is not of the slightest concern to us here (p.115).

Frege is aware of the distinction between the ways in which things might be discovered by us (via inventionis) and the kind of ground on which their proof rests (via demonstrationis). Thomas distinguishes a demonstration quia, on the basis of a thing’s effects, from a demonstration propter quid, establishing the nature of something and the reason why it is (ST I 2,2).

In fact ‘ideas’ as used by Frege is closer to ‘phantasmata’ in Aquinas (paragraphs 59-60). In speaking of divine knowledge Thomas is clear that it is not formed by a multiplicity of images, but abstraction for him achieves more than abstraction as Frege understands it. At the same time another Frege footnote, on page 37 of The Foundations of Arithmetic, bears comparison with how Aquinas too understands things.

Something that emerges from Aquinas’s account of divine ideas is the conviction that only God can know individuals as individuals (in their own right, we might say, or even as bearers of proper names?) Human knowing is capable of knowing individuals as instances of what is universal but sensation is then our way of knowing an individual as such. Frege and his commentators sometimes refer to what ‘we humans’ are capable of knowing as if to imply that there might well be other kinds of knowing possible for other kinds of minds. One might wonder also then about Frege’s assertion that ‘reason (human reason?) must be able to embrace all first principles in a survey’ (paragraph 5) and about his quest for a particular kind of knowledge of individual things, namely the Numbers (paragraph 18).

These issues are dealt with at great length in paragraphs 34ff of The Foundations of Arithmetic where various definitions of ‘unit’ are considered. The knowledge of an individual as such – i.e. as not being related to another on some basis that then reduces both to instances,  or ‘units’, of something common – this is his aim in the Foundations, he says (paragraph 34). So he criticises Leibniz, ‘for when he calls each individual object falling under his concept of unitas a unum, this word is being used to signify not the individual object but the concept under which they all fall’ (paragraph 37).

Frege’s solution is to speak in terms of concepts which are not subjective as ideas (in his sense) are (paragraph 47). Identity can be found through the concept, distinguishability through the object (paragraph 54). And these notions, ‘concept’ and ‘object’, become central to Frege’s philosophy. Number is not a property of a concept but an assertion about a concept, an element in the predicate, and so a self-subsistent object (paragraph 57). In Foundations, paragraph 60, Frege explains the sense in which he understands self-subsistence (page 72).

Where then is the number 4, Frege asks (a question I’m sure everybody is eager to ask as they read him). Not anywhere, he says. The fact that the number 4 is exactly the same for everyone who deals with it has nothing to do with its being spatial. ‘Not every objective object has a place’, he concludes (paragraph 61).

He warns against confusing thought and truth (paragraph 78) and there is room here for much further reflection. The question immediately following that on ideas in the Prima Pars of the Summa theologiae is the question on truth (ST I 16).

Frege believes his proof to be, in a wonderful phrase, ‘practically absolutely certain’ (note 1, page 104) and we must leave it at that for now. But not before adding a further wonderful and telling phrase, ‘even the mathematician’, says Frege, ‘cannot create things at will … he can only discover what is there and give it a name’ (paragraph 96).

V. Further Pathways?

There is at least this minimal analogy, then, between Aquinas and Frege, that they may have gone wrong at the same point. If it is correct to put it like that it is, of course, a matter of great interest. To apply an illustration from G.K. Chesterton, on coming across one elephant we say how strange, on meeting a second we say what a coincidence, but if we were to chance upon a third we would begin to suspect a plot.

Can we enter more effectively into the detail of Frege’s thinking and that of Aquinas to see if there is anything more substantial to be added to this minimal analogy, that they both perhaps have gone wrong somewhere along the same road, or at least while thinking in the same direction?

An obvious point from which to continue would be to see how Aquinas speaks about mathematics. One of the key texts is in his commentary on the De Trinitate of Boethius (V 1; V 3; VI 1). In the same commentary are important texts on logic (In Boeth de Trin V 1 ad 2 (with note and references in Maurer's English translation); V 1 ad 4 (also with note)). Another important text is QD de Veritate 1,1 where he speaks about the transcendental properties of being.

 

VI. Concluding Comments

Being – aye, there’s the rub. Does one inevitably return to the question of being? If there are such people as analytic thomists are they people who believe that the views of Frege and Aquinas can be brought together on this? Or is this a point at which a decision needs to be made, either Aquinas on being or Frege on being but not both (as Brian Shanley says from the Aquinas side of the choice and Anthony Kenny says from the Frege side)? I hope I have at least shown that these are questions worth considering, that the work required to answer them will not be easy, but that the issues raised by them are exciting and important.