SOURCE: Smith, Jonathan, Lawrence I. Berkove, and Gerald Baker. “A Grammar of Dissent: Flatland, Newman, and the Theology of Probability.” Victorian Studies 39, no. 2 (winter 1996): 129-50.

[In the following essay, Smith, Berkove, and Baker explore Abbott's interest in the writings of John Henry Newman and Bishop Joseph Butler and their theories of analogical and probabilistic reasoning.]

In Philomythus (1891), his belated attack on John Henry Newman's Essay on Ecclesiastical Miracles (1843), Edwin Abbott rejects Newman's claim that accounts of miracles do not require legal proofs: “here faith has no place,” writes Abbott, “and ‘legal proof’ is the best possible proof; and if you cannot get it, you ought to try at least to get something as much like it as possible” (90). In place of Newman's principle of “antecedent probability”—that ecclesiastical miracles should be presumed true on the basis of their Biblical predecessors—Abbott offers a diametrically opposite principle:

[P]eople practically deny, and are quite right in practically denying, the existence of everything of which they have no evidence, direct or indirect. There may be regions of four, five, or fifty dimensions. … But we are so constituted as not to act on any “may be” that is not at least suggested by some evidence. … The right rule is, to regard as non-existent all alleged facts for which there is no evidence direct or indirect; and to regard as antecedently false, or highly improbable, all statements that contradict our knowledge of the fixed and orderly course of things.

(91-92; original emphasis)

Abbott's reference to “regions of four, five, or fifty dimensions” is not a casual one. Although best known in his own day as a successful headmaster and a theological controversialist, he is best known in ours as the author of Flatland (1884), the charming “Romance of Many Dimensions.” Indeed, Flatland's story of a two-dimensional figure struggling to understand and describe three-dimensional space has long fascinated mathematicians and physicists similarly struggling with n-dimensional and non-Euclidean geometries.¹ In contrast, recent scholarship has begun to situate Flatland in relation to Abbott's other writings, particularly his characteristically liberal views on science and religion.² Yet most of this scholarship shares with the work of the scientific and mathematical popularizers the assumption that Flatland celebrates the powers of the imagination. We want to suggest, however, that Flatland is a cautionary tale about the dangers of the imagination when wrongly employed. More specifically, it can be viewed as part of Abbott's longstanding effort to expose what he regarded as the fallacies in Newman's theology, especially his appropriation of the analogical and probabilistic reasoning of Bishop Joseph Butler's Analogy of Religion (1736). Such a perspective further broadens the intellectual, social, and religious context in which Flatland and Abbott's Newman writings are located by demonstrating their role in Victorian debates over the status of Newman and Butler and in the various constructions and uses of probability.

I

Abbott graduated from Cambridge in 1861. Four years later, at twenty-six, he was appointed headmaster of the City of London School (CLS), where he remained until 1889. He quickly established a reputation as a curricular reformer whose school provided a more relevant and practical education for its urban, middle-class clientele.³ During the 1870s, he launched his prolific career as a writer, publishing studies of theology, Shakespeare, and Bacon; textbooks on grammar and composition; and even a Christian historical romance. Flatland appeared, anonymously, in 1884. Five years later, Abbott retired from the CLS to devote the remainder of his life to the exposition of his theological system.

Abbott's overtly public obsession with Newman seems to have begun shortly after his retirement. His first brief critical comments appeared in “Illusion in Religion,” published in the Contemporary Review just three months after Newman's death in 1890. A Toynbee Hall lecture elaborating these comments led him to begin an extensive assessment of Newman's thought that appeared in 1892 as the two-volume Anglican Career of Cardinal Newman. Abbott became so outraged when writing the chapter on Newman's Essay on Ecclesiastical Miracles, however, that he expanded this chapter into a separate book published in 1891 as Philomythus: An Antidote against Credulity. This work produced cries of outrage from Newman's friends and supporters, in particular Wilfrid Ward and Spectator editor R. H. Hutton, the man who in 1864 had played such a decisive role in swinging public opinion behind Newman during the controversy with Kingsley. Hutton described Philomythus as “a sort of theological caning” from the schoolmasterly Abbott, one that “ill becomes him, and … makes him, what it is hard to make a man of Dr. Abbott's power, even ridiculous” (538).⁴ Never one to back away from a fight, Abbott responded both in the pages of the Spectator and in a long preface to the second edition of Philomythus.

Although Flatland was published six years prior to Abbott's first direct public comments on Newman, Abbott had already distinguished his theology and its mode of reasoning from those characteristic, in his view, of Anglo-Catholicism. The centerpiece of his theology, evident long before the attack on Newman's Essay, was a rejection of miracles as a basis for Christian belief. Reviewing his theological writings in his own Apologia (1907), Abbott spoke of his early conviction, grown stronger with time, “that a belief in miracles ought not to be regarded as necessary for the worship of Christ” (x). Accounts of miracles, even New Testament claims about the virgin birth and the resurrection of Christ, should be subject to “historical and scientific tests” (xi). Anxious to reconcile science and religion, Abbott had long claimed that both the study of nature and the study of Scripture involve a patient, rational investigation of appearances. Both Nature and the New Testament, he declared in Through Nature to Christ: The Ascent of Worship through Illusion to the Truth (1877) and The Kernel and the Husk (1887), contain illusions—material signs and wonders of doubtful veracity that nonetheless point toward spiritual truth. As Ptolemaic astronomy should be appreciated for having reflected humanity's importance in a divinely-created cosmos, not dismissed for having wrongly placed the earth at the physical center of the universe, so the resurrection of Christ should be understood in terms of the immortality of the soul, not as a physical event. As he put it in “Illusion in Religion,” “in religion, as well as in science, we must be prepared for illusions, trying to discern the truth beneath them, and to get out of them as much good as we can, until the time arrives when the kernel of truth in them is separable from the husk of error” (721).

Abbott thus sought, from the beginning of his career, his own via media, one that avoided the agnosticism of scientific materialists like T. H. Huxley and the anti-scientific conservatism of both Evangelical literalists and Tractarians. In his Oxford Sermons (1879), Abbott declared that in what he called “Liberal Christianity” miracles are “an interesting but by no means vital question.” They are unnecessary for the worship of Christ, whom Liberal Christians “adore not as a wonder-worker but as the Healer of the souls of men” (xxvii). This view of miracles, Abbott contended, constituted “the radical difference” (xxvii) between Liberal Christianity and its opponents. Abbott's position was, according to Owen Chadwick, consistent with late-Victorian Anglican theology, yet Chadwick also notes that Abbott was controversially forthright in his claim that miracles constituted an impediment to rational belief and hence were unnecessary—even for a clergyman. Chadwick reports that The Kernel and the Husk “gave especial offense” and that “[o]n Trinity Sunday 1887 Charles Gore, regarded by Tractarians as heir to the mantle of Pusey and Liddon, denounced the book from the pulpit of Oxford university” (2: 138-39).

The Tractarians' horror at Abbott's theology was matched by Abbott's horror at theirs, at least as epitomized by Newman's pre-conversion works. In the long Preface to the second edition of Philomythus, Abbott calls Newman's Essay on Ecclesiastical Miracles an “Abomination of intellectual Desolation” (lxvii). He says it gave him a “mental and almost moral shock” (v) and that he read it with “bewildering amazement” and “absolute horror” (lxvii). Abbott charged Newman with “that kind of unconsciously dishonest and conveniently credulous Assent which springs from a misplaced application of Faith to historical facts” (lxxviii). “I have never met,” Abbott wrote,

with such perfect and fascinating instances as are to be found in Newman's writings, of that subtle and delicately-lubricated illative rhetoric by which you are led downwards on an exquisitely elaborated inclined plane, from a truism to a probability, and from a strong probability to a fair probability, and from a fair probability to a pious but most improbable belief.

(32-33)

Newman's Essay, written in 1842-43, was clearly an important document in his move toward Rome. Having defended the truth of Scriptural miracles in an 1826 essay on the ground that such miracles, while violating the system of nature, do not violate the more important moral system (6-20), Newman argued that there is an “antecedent probability” in favor of ecclesiastical miracles because of the precedent of Biblical ones. In assessing how much the differences between ecclesiastical and Scriptural miracles should “prejudice” us against the former, Newman says we must “betak[e] ourselves to the argument from Analogy” (147). God, argues Newman, makes different things for different purposes. Just as we, who are used to domestic animals and shocked by wild beasts, nonetheless do not conclude that wild beasts are inferior, so we should not assume that ecclesiastical miracles are inferior to Scriptural ones:

There is a far greater difference between the appearance of a horse or an eagle and a monkey, or a lion and a mouse, … than between even the most august of the Divine manifestations in Scripture and the meanest and most fanciful of those legends which we are accustomed without further examination to cast aside.

(151)

Although Newman was far more rigorous in weighing the evidence for ecclesiastical miracles than Abbott's rendering of his arguments makes him appear, the weakness of such analogies is readily apparent. In Philomythus, Abbott declared them to be “mere verbal pyramids balanced on their tops” (136).

It is difficult to underestimate the importance of this analogical and probabilistic reasoning to Newman's thought. It was developed not only in the two Essays on miracles, but also in the University Sermons (1826-43) and the Essay on the Development of Christian Doctrine (1845), and it reached its fullest expression in An Essay in Aid of a Grammar of Assent (1870). In the Apologia, Newman summarized the three central claims regarding his use of this reasoning as follows:

[T]hat absolute certitude which we [are] able to possess, whether as to the truths of natural theology, or as to the fact of a revelation, [is] the result of an assemblage of concurring and converging possibilities … ; that probabilities which did not reach to logical certainty, might suffice for a mental certitude; that the certitude thus brought about might equal in measure and strength the certitude which was created by the strictest scientific demonstration.

(29)

It is also difficult to underestimate the importance for Newman of the primary source for these ideas: Butler's Analogy of Religion. In the Apologia, Newman credited his study of Butler's Analogy in the mid 1820s with providing the two “underlying principles” crucial to the development of his religious views. These principles were, first, that “the very idea of an analogy between separate works of God leads to the conclusion that the system which is of less importance is economically or sacramentally connected with the more momentous system,” and, second, that “Probability is the guide of life” (21). Butler had sought to refute contemporary Deists, who accepted the existence of a divine creator of the universe but rejected the identification of that creator with the Christian God. He not only employed the argument from design for God's existence, but he also argued that the analogies between nature and the Scriptures provided evidence of the plausibility that the creator of the universe was indeed the Christian God. Conceding that the imperfect knowledge of humans makes absolute certainty rare, Butler nonetheless contended that probable knowledge is a more than adequate basis for action and belief, whether in everyday life or in the momentous questions of religion, including the evaluation of miracles. Newman, as the above passage suggests, went even further: an assemblage of probabilities may not be enough for logical certainty, but it is adequate for mental certitude.

In Philomythus, Abbott takes issue both with Butler's principle and with the theory of converging probabilities Newman built on it. There is room, he says, for reason but not for probability in religious questions. To place all beliefs, from belief in God to belief in ecclesiastical miracles, on an “Inclined Plane of Probabilities” is to make all beliefs of the same kind (75). In a disparaging characterization of Newman's reasoning, Abbott writes that “if we go on patiently ‘accumulating probabilities,’ God will at last step in, as it were, and, with a magic touch, convert our heap of probabilities into a ‘certitude’” (81). Such a view, he contends, far from strengthening belief, is liable to induce us to jettison our certitudes or to prevent us from ever being certain in the first place.

The Anglican Career offers a broader critique of Newman's theology and even challenges Newman's rendering of his own spiritual development in the Apologia. But all of Abbott's criticisms serve the larger purpose of extending Philomythus's exposure of Newman's “mixture of credulity and sophistry” (1: ix). In The Anglican Career, the Essay on Ecclesiastical Miracles is presented as the clearest example of Newman's most general and fundamental failing: the domination of his Reason by his Imagination (1: viii).

G. B. Tennyson has identified the Doctrine of Analogy and the Doctrine of Reserve as the two great principles, simultaneously aesthetic and religious, of Tractarian poetics. The physical world contains signs and symbols of the spiritual world, so the intense emotional expression of the imagination in art is in fact a religious act. Reserve, however, demands that the communication of religious knowledge or the expression of the religious imagination be accompanied by restraint or indirection befitting the sacredness of the subject. Recently Tennyson has specifically extended this to include Newman's theological writings as well, but has argued that the Doctrine of Reserve is the more important of the two in these works.⁵ Certainly Kingsley, busily attacking Newman's Reserve as an admission that the Catholic clergy embraced lying, would have agreed with Tennyson about its centrality. Abbott, however, saw matters rather differently. For him, Analogy was the key to Newman's theology, and its misuse constituted a logical and aesthetic travesty.

Abbott's theology also depended heavily on the Imagination, but the difference between the two men is striking. For Abbott, writing in The Kernel and the Husk just a few years before his Newman books, Imagination is “the basis of all knowledge,” whether scientific or religious (54). Imagination and Reason are distinct and separate faculties: Imagination, operating not deductively but inductively on a foundation of fact, comes first, “leap[ing] to general conclusions, mostly premature or false, but all containing a truth from which the falsehood must be eliminated” (55). Faith is, for Abbott, a “form of Imagination” (46), but it must be subjected to the test of Reason, which eliminates the falsehood and arrives at truth by pointing out differences, making distinctions, and testing explanations (49).

In Abbott's eyes, Newman violated this process in two ways. First, he did not establish a groundwork of fact but simply used his Imagination to leap directly to general truths from idiosyncratic feelings and impressions or from the dictates of “authority.” Second, he then applied Reason not to test his Imaginative claims but to confirm them. “A single superstitious impression,” says Abbott in The Anglican Career, is for Newman “more powerful than a thousand facts” (1: 59). He complains that Newman's definition of Reason includes Faith, thus making it “reasonable” to accept less evidence than the Reason usually requires. Abbott stresses that whereas for him Faith is always belief based on and tested by evidence, for Newman Faith is belief without proof (2: 110-11). Newman's view of Reason and the Imagination thus made him “the servant of Antiquity (as against Novelty); the servant of Synthesis against Analysis; … the servant of the Likenesses of things (as against their Unlikenesses); the servant of Dogma and Authority (as against Experiment and Induction)” (1: 58).

II

The protagonist/narrator of Flatland, A Square, lives in a two-dimensional world whose classist and sexist social hierarchy is based on the regularity and number of a figure's sides—the greater the number of equal sides, the higher the figure is on the social ladder. Women are straight lines, workers and soldiers are isosceles triangles, members of the middle class are equilateral triangles, professional men and gentlemen are squares and pentagons, the nobility are polygons of six sides or more, and priests are those with so many sides as to be called circles. The second half of Flatland centers on the efforts of various geometrical figures to convince other figures of the existence of higher spatial dimensions than the ones they inhabit. It begins with A Square describing a vision in which he attempts to convince the King of Lineland of the reality of the second dimension. A Square's efforts to describe this second dimension of course fail because the King has no way of conceptualizing it—space to the King is by definition one-dimensional. But A Square's efforts to establish the existence of the second dimension by passing through Lineland prove equally futile, because the King sees him appear and disappear as a point whenever he passes in or out of Lineland, a phenomenon which the King ascribes to “some magic art” (64).

A Square then changes roles when he is himself visited by a Sphere from Spaceland. The Sphere fails to convince A Square of the reality of the third dimension either by words or by the action of passing through Flatland, a process which causes him to appear to A Square first as a point, then as a circle whose circumference increases and decreases, and then as a point again before disappearing entirely. Like the King of Lineland, A Square concludes that his visitor is “some extremely clever juggler” and that there are, after all, “such people as Enchanters and Magicians” (74). The Sphere then employs “the method of Analogy” (74) to describe the appearance of three-dimensional figures: if a point has no sides and one terminal point, if a straight line has two sides and two terminal points, and if a square has four sides and four terminal points, then a cube must have, “in strict accordance with Analogy” (76), six sides and eight terminal points. This demonstration, however, merely enrages A Square. Dismayed, the Sphere is forced to resort to parlor tricks like removing a tablet from a locked cabinet, but this also fails to convince A Square. It is only when the Sphere lifts A Square out of Flatland and into the third dimension, from which A Square can look down into Flatland, that A Square believes.

Part of the obvious satire of Flatland is thus aimed at those who refuse to accept even the possibility of the existence of higher dimensions, those who think that their experiences define reality. A Square is critical of the King of Lineland's view that space itself is linear, but he himself is just as violently committed to the belief that “True Space is a Plane” (62). Prior to A Square's “conversion,” his grandson, in a geometry exercise that also employs analogical thinking, declares that if 3 can be represented geometrically by a line of three inches, and if 3² can be represented geometrically by a square with sides of three inches, then 3³ must also have a geometrical meaning. A Square angrily characterizes this conclusion as “nonsense” because “Geometry has only Two Dimensions” (66). Similarly, when A Square, after his acceptance of “the Gospel of Three Dimensions” (77), presses the Sphere about the Land of Four Dimensions, the Sphere snaps that “there is no such land” (89). Later regretting his anger, the Sphere consoles A Square with a vision of Pointland, “the Abyss of No dimensions” (93), whose King is the apotheosis of egoism and self-satisfaction, believing that “He is himself his own World” (94). As A Square puts it in his Preface, “we are all alike liable to the same errors, all alike the Slaves of our respective Dimensional prejudices” (xii).

But A Square also declares in the Preface that Flatland should commend itself “to those Spacelanders of moderate and modest minds who—speaking of that which is of the highest importance, but lies beyond experience—decline to say on the one hand, ‘This can never be,’ and on the other hand, ‘It must needs be precisely thus, and we know all about it’” (xiii). Readers of Flatland too often miss the satire directed at those who say, “it must needs be precisely thus”; it is here that the criticism of the type of reasoning employed by Newman and Butler is most apparent. By embodying analogy's limitations, Flatland forces its readers to acknowledge the consequences of inappropriate analogical thinking. It becomes itself a model of how analogy should and should not be employed.

First, it must be emphasized that, while figures are wrong to declare the impossibility of higher dimensions, they are not wrong to be skeptical about them. Recall that in the quotation with which we opened, Abbott declared that people are right to deny the existence of everything for which they have no evidence, including “regions of four, five, or fifty dimensions.” Comments about higher dimensions throughout Abbott's works are invariably of this type, ridiculing the belief, not the refusal to believe, in the physical existence of higher dimensions.⁶ The King of Lineland has no evidence that a second dimension exists, and A Square, before he is taken into Spaceland, has no reason to believe in a third. Accepting the initial denials by A Square and the Sphere about the existence of higher dimensions would be tantamount to relying merely on authority, and reliance on authority was of course one of Abbott's chief criticisms of Newman. “True faith,” Abbott later wrote in The Anglican Career, “warns us never to try to believe anything to have happened as to which God has given us definite means of ascertaining whether it happened or not,” whereas the “false faith” of Newman decrees it to be “our duty, imposed on us by God, to believe, in religion, that certain facts have happened which … evidence would either not prove, or would even disprove” (2: 113-14; original emphasis).

Second, both A Square and the King of Lineland are right to try to account for their strange experiences with natural rather than miraculous explanations. Abbott found Newman's “contempt for facts” clearly demonstrated by Newman's argument that natural explanations for supposed miracles do not rule out the possibility of miraculous intervention (Philomythus 14). This suggests that the skepticism of A Square and the King, their refusal to leap to and defend miraculous explanations, is not to be seen as narrow-minded. They dismiss as evidence the intimate knowledge of their worlds that the higher beings display on the reasonable grounds that such knowledge could easily be obtained without recourse to a higher dimension. The King of Lineland, for example, can infer the sex, size, and location of his subjects by sound, so A Square's description of some of the King's neighbors, whom A Square can see directly but the King cannot, fails to convince the King of the existence of the second dimension. Both the King and later A Square also speculate that their visitors are monstrous denizens of their own dimensions rather than figures from higher dimensions (58, 68). The conclusion in both cases that the visitor is some sort of magician or trickster represents, paradoxically, a staunch adherence to the conviction that the apparently miraculous apparition can ultimately be accounted for without recourse to the supernatural.

Third, A Square is both right to reject the Argument from Analogy prior to his physical experience of the third dimension and wrong to employ it so uncritically after that experience. In English Lessons for English People (1871), Abbott and his co-author J. R. Seeley had described the Argument from Analogy in disparaging terms: “so far as it is an argument at all, [it] comes under the head of Induction. Otherwise it is not an argument, but a metaphorical illustration of an argument” (273).⁷ Induction was for Abbott the path to all knowledge, religious or scientific, but it had to be pursued rigorously, and analogy in the vast majority of cases was at best suggestive. Newman, in contrast, placed “considerable stress upon syllogistic and none at all upon inductive logic”; as “the servant of the Likenesses of things” he used induction only in its weakest form, and he used it fancifully (Anglican Career 1: 58, 61). Abbott criticizes Butler's example of analogy as the “loose” notion that “what has been, will be” (English Lessons 272), but Newman's similar argument that “what God did once, he is likely to do again” is for Abbott far worse. It provides the license, complains Abbott, for almost any claim at all:

Because God is supposed to have suspended the Laws of Nature once for a definite purpose, and in certain ways, … therefore it is a likely supposition that He has repeatedly suspended, is suspending, and will suspend, the Laws of Nature, in quite different ways … for quite different purposes, and often … for no purpose at all.

(Philomythus 9-10; original emphasis)

Abbott's distrust of analogy is evident in Flatland's emphasis on its limitations. The Argument from Analogy is repeatedly employed, but it convinces no one. The ability to describe the geometrical characteristics of a three- or four-dimensional object based on an analogy with a corresponding object in fewer dimensions says nothing about whether or not such objects and such dimensions actually exist.⁸ Nor does it enable someone to visualize an object in higher dimensions. As A Square's grandson says, “it must be that a Square of three inches every way, moving somehow parallel to itself (but I don't see how) must make Something else (but I don't see what) of three inches every way—and this must be represented by 3³” (66).

Moreover, logical fallacies, most of them based in analogy, abound in Flatland. A Square repeatedly invokes “Laws,” “Ordinances,” and “Decrees” of Nature to justify what are clearly culturally-constructed ideologies of oppression in which expression is censored, individuality is discouraged, and dissent is systematically crushed (see also Smith 203-06). Flatland's rigid class system is based on the specious analogy between many-sided regularity and moral and intellectual superiority, a view that not only permanently subjugates women and the Isosceles “rabble” (10), but also justifies euthanizing irregular figures. The designation of “Circle” for a many-sided polygon, and the according of perfection to this priestly class, suggests the Catholic doctrine of Papal Infallibility (1870), with its official recognition of perfection in an inherently flawed being. (“It is always assumed, by courtesy,” A Square informs us, “that the Chief Circle for the time being has ten thousand sides” [44].) Circles, however, are the primary exception to the “law of nature” that each son of a regular figure will have one more side than its father. Their sons often advance by fifty or even a hundred sides, but the highest Circles are increasingly infertile and more likely to produce dangerously irregular offspring. And polygonal parents frequently pursue the risky and decidedly unnatural option of surgery to increase the number of their sons' sides. Such cosmetic surgery, if successful, even though it does not change the nature of the child, nonetheless increases its social status appreciably. Flatlanders, enraptured by the imagined rightness (and righteousness) of their class system, ignore these deviations as long as the outward appearance of conformity and regularity is maintained.

It is thus no surprise that the dimensional Argument from Analogy becomes compelling for A Square only after he has been lifted out of Flatland and into Spaceland. But his subsequent application of it leads him into error. In English Lessons, Abbott characterized enumeration—the progression without experiment from one premise to another in an unbroken, and perhaps infinite, series—as “an insecure method of proof” (269), yet this is precisely the error that A Square makes. Extending the Sphere's dimensional analogy, A Square asks, “in Four Dimensions shall not a moving Cube—alas, for Analogy, and alas for the Progress of Truth, if it be not so—shall not, I say, the motion of a divine Cube result in a still more divine Organization?” (90). A Square asks to be taken to the “land of Four Dimensions” (89), which “must exist” (90) because the three-dimensional space described by the Sphere's previous analogy proved to exist. When the Sphere grudgingly allows that the fourth dimension may exist, A Square gets even more carried away:

In that blessed region of Four Dimensions, shall we linger on the threshold of the Fifth, and not enter therein? Ah, no! Let us rather resolve that our ambition shall soar with our corporal ascent. Then, yielding to our intellectual onset, the gates of the Sixth Dimension shall fly open; after that a Seventh, and then an Eighth—.

(91-92)

This infinite series of dimensions, each “more divine” than its predecessor, exposes A Square's limited notion of divinity. Flatland is, in part, about the way we conceive of Deity. A Square's equation of more dimensions with more perfection and more divinity is fallacious. It confuses the material with the spiritual and relies on a human reference frame for defining the divine. These limitations are exposed when A Square initially “worship[s]” his guide to Spaceland with “silent adoration” (80), yet no sooner looks down into Flatland than he declares that he himself has “became as a God” (82). The Sphere points out that if this “omnividence” (82) is a divine attribute, then every Spacelander, including the most depraved criminal, is a God.⁹ A Square fails to understand, however, for even when he focuses on dimensions beyond the third, he wrongly associates these physical spaces with spiritual realms inhabited by the divine:

As you yourself [he tells the Sphere], superior to all Flatland forms, combine many Circles in One, so doubtless there is One above you who combines many Spheres in One Supreme Existence, surpassing even the Solids of Spaceland. And even as we, who are now in Space, look down on Flatland and see the insides of all things, so of a certainty there is yet above us some higher, purer region, whither thou dost surely purpose to lead me—O Thou Whom I shall always call, everywhere and in all Dimensions, my Priest, Philosopher, and Friend—some yet more spacious Space, some more dimensionable Dimensionality, from the vantage-ground of which we shall look down together upon the revealed insides of Solid things.

(87-88)

A Square worships number rather than being, mistakenly believing that higher dimensions must exist and must be spiritually superior. Like Newman, A Square uses Imagination and Analogy to reason his way into a “certainty” that has little foundation.

Also like Newman, A Square commits errors that combine a too-fertile imagination with a too-literal understanding. In The Anglican Career, Abbott complained that Newman's “religion of fear,” his emphasis on an evil world and a wrathful God, was, while “consistent with the analogy of Nature,” a far too literal reading of the New Testament and its precepts (2: 8, 12). Focus on an angry God blinds us to his love and leads us to become obsessed with explaining the purpose of his wrath. Such speculation, like that involved in “imaginative romances” of higher dimensions, serves a limited purpose:

It may be that what our planet suffers, other planets may behold and take warning from it, so that they may not suffer: but all such thoughts as these are as far off and visionary as imaginative romances about four or five dimensions, and the only justification for either is, that they keep us intellectually modest and truthful.

(2: 26)

A Square is hardly “intellectually modest” about higher dimensions, for he learns his most important lesson only by rote. Deciding that the phrase “Upward, not Northward” was “the clue to the whole proof” (96), he repeats these words “mechanically,” like a “charm” (93). He believes it means merely the distinction between what Flatlanders regard as “up” (Northward) and what “up” is in the third dimension—a thrust out of the plane. He fails, in other words, to recognize the spiritual truth in what Abbott would regard as illusion: “upward” means spiritual and intellectual advancement, not simply the crossing of dimensional boundaries without personal development. Not surprisingly, A Square rapidly loses his ability to visualize the physical meaning of this expression after his return to Flatland; despite losing confidence in its efficacy, however, he continues to employ it.

Paradoxically, “upward” actually means looking inward. When the Sphere first appears to A Square, the Sphere attempts to describe the third dimension by saying that A Square might perceive it if he had an eye in his side, that is, an eye inside of himself (70). A Square makes several jokes about the absurdity of having an eye in his stomach, although at times he does seem to glimpse the spiritual significance of such “insight.” When the Sphere explains that the fourth dimension is said by some to exist only in Thought, A Square is enthusiastic (91). When he recognizes the inadequacy of “Upward, not Northward” for convincing others, he composes a treatise entitled “Through Flatland to Thoughtland” in which he speaks “not of a physical Dimension” but of a place, where, theoretically, figures of higher dimensions could exist (99). Even so—and despite the fact that the title of A Square's treatise echoes the title of Abbott's own Through Nature to Christ—A Square begs the Sphere to “take me to that blessed Region” of Thoughtland (91), and he admits that his metaphorical discussion of Thoughtland was written merely to evade the law that prohibited discussion of the third dimension (99). Indeed, Abbott offers the proper interpretation of “Upward, not Northward” and “Through Flatland to Thoughtland” near the end of The Anglican Career, where he says facts are the “divine voices” through which “we pass upward, exchanging the illusive for the less and less illusive, till we draw near to truth itself” (2: 430).

III

Abbott's attack on Newman, and especially his attack, in both Flatland and the Newman books, on analogical and probabilistic reasoning, can also be situated in a broader historical context. At stake was, most immediately, the nature of Newman's legacy, for in choosing to take up the cudgel of the vanquished Kingsley, Abbott revisited not only the terrain of the Apologia, but, through it, of the Oxford Movement and a half-century of religious debate.¹⁰ Abbott was clearly concerned that the defense of Butler's and Newman's theology of probability, much of it coming from figures within Anglicanism, was not only out of touch with contemporary thought but also endangering Christianity itself.

Newman's death, and “the chorus of indiscriminate praise” (Philomythus lxiv) that followed it, both was and was not the impetus for Abbott's assault. He explains in the Preface to the second edition of Philomythus that he had not set out to attack Newman's works (lxiv-lxv), and he claims in the Introduction that such an attack is necessary only because Newman's followers, especially Hutton and Ward, have declared the master's works to be worthy of study and full of logical force. Such “legendary exaggeration,” Abbott complains, has gradually attached itself even to “the popular estimate of Newman's work” (vi). As we suggested above, Abbott's attacks on Newman are part of an ongoing debate with Newman's Catholic theology and the Tractarian thought that led to it, a debate carried on with Anglo-Catholic thinkers rather than just with Newman. In the Preface to The Anglican Career, the context of this broader debate is clearer. The quarter-century of public regard for Newman that followed the Apologia has, according to Abbott, ignored the challenge posed by his conversion: if Newman was right, then shouldn't Anglicans become Catholic (v)? And if Newman was right, then Abbott's “Liberal Christianity” was also in danger.

But there are other specters lurking not far in the background, and those are atheism and agnosticism. Responding to Ward in a letter to the Spectator, Abbott asked, “Cannot Newman's best friends see that they are playing into the hands of his worst enemies (of whom I assuredly am not one) when they … defend him in this way?” (622). For the two years preceding Abbott's letter, T. H. Huxley had engaged in a running controversy over New Testament miracles, with Gladstone as his chief opponent and Newman as his primary whipping boy. In “Agnosticism and Christianity” (1889), he declared that he could construct an effective “Primer of Infidelity” from Newman's Essay on Ecclesiastical Miracles (Science and the Christian Tradition 5: 333n). As Abbott would in Philomythus, Huxley attacked Newman's conception of evidence, his doctrine of antecedent probability, and his apparent contempt for science. “He believed that his arguments led either Romeward, or to what ecclesiastics call ‘Infidelity,’ and I call Agnosticism,” Huxley wrote of Newman's Essay. “I believe he was quite right in this conviction; but while he chooses one alternative, I choose the other” (5: 345). Whereas Hutton defended Newman's reasoning (Cardinal Newman 60-67), Abbott concluded that the futility of such a defense was bound to have the opposite effect. If one reading of Newman led to Rome, another led to Huxley's infidelity.¹¹

This is clearer if we examine Newman's theology in the context of contemporary views of Butler's Analogy and of probabilistic reasoning generally. Historians of probability and mathematics agree that Butler's work is, in the words of Lorraine Daston, “the culmination of the probabilistic tradition of natural theology” associated with the early years of the Royal Society (204-05; see also Hacking 177). Classical probability incorporated both “objective” and “subjective” versions. The objective version involved the identification of statistical regularities in seemingly random phenomena, while the subjective version was concerned with assessing degrees of belief in the face of imperfect knowledge (Daston 188-210; Hacking 12). Butler and others like him employed the objective version in contending, as part of the argument from design, that statistical regularities in natural phenomena provided proof of divinely-ordained order. But Butler also employed the subjective version when he argued that, even in momentous religious and moral questions, a sufficiently high degree of probability justified the belief of a reasonable person even without absolute certainty (Daston 56-67).

This classical interpretation of probability began to show signs of strain in the latter half of the eighteenth century and broke down entirely around 1840. Increasingly, probabilists separated the objective and subjective versions of probability, attacking the latter. Instead of emphasizing the certainty that could arise from uncertainty, they focused on the sources of error that made assessing degrees of belief difficult if not impossible (Daston 210-24; Porter 77-88). At the center of this shift, especially in the period after Butler, was the debate over miracles. According to Daston, Hume's 1748 essay successfully made the uniformity of nature the fundamental assumption of this debate. The violation of natural laws, Hume argued, was intrinsically improbable, and no miracle had sufficient extrinsic evidence in its favor to overcome this. While critics challenged Hume's elevation of natural law and questioned how any amount of past experience could make us certain of the intrinsic improbability of miracles, they were clearly swimming against a cultural tide (Daston 323-26). Thus, although Newman's invocation in the early 1840s of the “antecedent probability” of miracles was not without precedent, it was also characteristically and aggressively opposed to the liberal intellectual spirit of the age. The same was true of his decision to re-issue the two Essays on miracles in 1870, the same year in which The Grammar of Assent appeared, with its mature articulation of the argument that “converging probabilities” yield certitude.

Theodore Porter has demonstrated that, following the breakdown of the classical paradigm, the statistical movement was closely associated in the 1850s and 60s with liberalism and, through the appearance of Buckle's History of Civilization in 1857, determinism (30-35, 57, 149-51, 162-64). Statistical regularity, once viewed reassuringly as a sign of God's ordering hand, instead became evidence of the lack of human freedom. Fixed percentages of the population must die in infancy, commit suicide, and become criminals. An 1861 notebook entry indicates Newman was familiar with Buckle's use of statistical determinism, although Newman scoffed at the idea that “the notion of fixed laws is a new idea of modern times,” seeing in Buckle's work “the old idea of Fate or Destiny” (Philosophical Notebook 2: 155). Republishing his work on miracles and continuing to develop the probabilistic reasoning of Butler in The Grammar of Assent was, in such a climate, a calculated strategy.

By the 1880s, however, Abbott could view this strategy not as a courageous defense of tradition but as either dangerously ignorant or dangerously reactionary. The critical backlash against Buckle had insured that probabilistic thought was not uniformly associated with determinism, but Abbott handled probability gingerly in questions of faith. He was clearly aware of the significance of the shift in the understanding of probability and felt that Newman was not. In Philomythus, he asserts that “[a]ll probability is, at bottom, of a statistical nature” (49); as the measurement of regularities in large collections of data, it has nothing to offer those wrestling with the existence of God or of miracles. Newman, although he “talked so much about, and set such store upon, probabilities,” had “very little notion indeed about them, not having any practical conception of induction” (112). This was indeed evident in Newman's treatment of converging probabilities, which, as Abbott argued in English Lessons, do not lead to certitude but to conclusions that are “less probable” (289). And Newman's reliance on probabilistic theology and his use of Butler played into the hands of agnostics like Huxley.

As Boyd Hilton and Hamish Swanston have demonstrated, Butler's popularity surged in the 1830s, 40s, and 50s but declined suddenly in the 1860s and 70s (Hilton 170-71, 336-39; Swanston 3-12). The timing of these trends was not accidental: the early surge coincided with the Oxford Movement, while the decline followed the publication of The Origin of Species (1859), with its assault on the design argument, and the general rejection of Butler by the liberal theologians of Essays and Reviews (1860). Butler's Analogy was made required reading at Oxford in 1832 but removed from its central place in the curriculum in 1864. Gladstone, whom Hilton calls “the last Butlerian” (340), first became interested in Butler as an Oxford student in the early 1830s, wrote a defense of Butler's use of probability for the Nineteenth Century in 1879, and published an edition of Butler's Works, along with his own Studies Subsidiary to the Works of Bishop Butler, in 1896. He attempted to refute attacks on Butler by a series of prominent critics—beginning with James Martineau in the 1840s and including Walter Bagehot (1854), Sara Hennell (1859), Leslie Stephen (1876) and Matthew Arnold (1877)—most of whom had found the Analogy to be at best inadequate for the nineteenth-century believer and at worst, in Stephen's review of Gladstone, “an incentive to atheism” (115).

Abbott's commentary on Butler in the Newman books is mostly indirect. The tone of his remarks on Newman's use of Butler, however, suggests both a respect for the author of the Analogy and a sense of the book's limitations—dangerous limitations when appropriated by Newman. This is not surprising given Abbott's intellectual coming of age during the period when liberal Anglicans who had shared in the Butlerian revival began to withdraw their enthusiasm, essentially abandoning Butler to the High Church thinkers. In The Anglican Career, Abbott refers to the Analogy as an “able work” and praises its “logical character” (1: 74), but Philomythus suggests that Newman “altered” Butler's argument and concludes that the guide of life is not probability but “faith based upon experience” (52). Like virtually every other nineteenth-century commentator on Butler, including Newman, Abbott was aware that the Analogy was primarily a negative argument, designed to prove that nature and revelation were not incompatible and that the analogy between them should produce a prepossession in favor of, rather than against, Christianity. Significantly, the edition of the Analogy that Abbott owned was edited by Joseph Angus, a Baptist divine, whose long headnote on “Analogy; Its Nature and Use” stresses that analogy “is demonstrative only in proving results or causes not to be improbable. … It defends Christianity, without in the first instance contributing materially to the positive evidence on its side” (xviii; original emphasis).

Newman, of course, had tried to do much more. Aware of the dangerous tendency of Butler's doctrine “to destroy in [many minds] absolute certainty, leading them to consider every conclusion as doubtful,” Newman explained in the Apologia that his argument about the production of certitude from converging probabilities was meant to go beyond Butler (28). In a note added to The Grammar of Assent in 1880, he contended that while Butler's analogical argument merely removed an objection that allowed Christianity a fair hearing, his own analogical argument about the accumulation of probabilities was a positive one because it led to certitude. It helped to establish the truth of Christianity, and in his own case, it led him to Catholicism: “if the doctrine of the Eucharist was not from heaven, why should the doctrine of Original Sin be? If the Athanasian Creed was from heaven, why not the Creed of Pope Pius?” (320). Newman's analogical leap from original sin to transubstantiation, from the Athanasian Creed to Papal Infallibility, is suspiciously similar to A Square's certitude about the physical existence and spiritual superiority of higher dimensions.

IV

In her fine essay, Rosemary Jann rightly sees Flatland as an important part of Abbott's response to both scientific materialism and the “dogmatic orthodoxy” that “unwittingly abetted” it (474). She associates this “dogmatic orthodoxy” with Newman's reliance on authority, an authority characterized by a lack of imagination, an unwillingness to make dimensional leaps (485). Like other commentators who believe that Flatland celebrates analogical reasoning unproblematically, she is thus troubled by the book's “highly ambivalent ending” (488) in which A Square is imprisoned, his evangelical mission a profound failure.

But the ending of Flatland is not ambivalent. If we keep before our minds the book's criticism of those who, in employing analogy uncritically, say “it must needs be precisely thus,” the ending is perfectly consistent. A Square is punished for his imaginative excesses, just as Abbott later excoriated Newman for similar sins. Newman does rely on authority, and in doing so he does abet materialists, but the cause, as Abbott repeatedly stresses in Philomythus and The Anglican Career, is not that Newman has too little imagination, but too much.

In The Grammar of Assent, Newman argued that the accumulation of probabilities leading to unconditional assent is no less the method of mathematics and science than it is of moral philosophy and theology. In a famous analogy explaining his theory, he wrote:

We know that a regular polygon, inscribed in a circle, its sides being continually diminished, tends to become that circle, so that its tendency to be the circle, though ever nearer fulfilment, never in fact gets beyond a tendency. In like manner, the conclusion in a real or concrete question is foreseen or predicted rather than actually attained; foreseen in the number and direction of accumulated premisses, which all converge to it. … [Thus,] the practiced and experienced mind is able to make a sure divination that a conclusion is inevitable, of which his lines of reasoning do not actually put him in possession.

(207-08)

Abbott, too, saw congruity in the methods for attaining scientific and religious knowledge. In The Kernel and the Husk, he declared that “I believe in a perfect circle by Faith” (32). But for Abbott, this is different from declaring a many-sided polygon a circle and, as in Flatland, equating circularity with perfection. A “tendency” should not be confused with the thing itself. Many-sided polygons are not circles, and converging probabilities do not create certitude. While sympathetic with the dilemma of how to communicate spiritual experiences, Abbott felt that Newman's and A Square's methods could lead to the wrong conclusions. “We must surely say,” Abbott remarks on another of Newman's analogies between mathematical and religious inquiry, that Newman “proves his unique competency to write a ‘Grammar of English “thinking,” adapted for the loosest of theological “thinkers”’” (Anglican Career 2: 393; original emphasis).

Notes

1. Flatland was itself both the product of and a participant in efforts to popularize non-Euclidean geometry (Smith 183-86). The first appropriation of Abbott's book came in 1907; for references to subsequent examples see Banchoff, Introduction xxvi-xxxi, and Smith 263n18. The most recent addition to this list is Kaku.

2. See Banchoff, Gilbert, Jann, and Smith.

3. See Banchoff, Introduction; Douglas-Smith.

4. Woodfield (15n) notes that Abbott was virtually unique in seeing Hutton as idolatrous rather than fair-minded in his assessment of Newman. Abbott's willingness to attack Hutton, who shared many of his general positions, is a striking indicator of the importance of this issue for him.

5. Victorian Devotional Poetry Ch. 2; “Removing the Veil” 11-19.

6. See, for example, Kernel 258-59 and Apologia 63.

7. Seeley (1834-95) was cut from the same theological and intellectual cloth as Abbott. He attended both the CLS and Cambridge a few years before his friend. He taught at the CLS from 1859 to 1863, just prior to Abbott's appointment as headmaster, then moved on to University College London as Professor of Latin and finally, in 1869, to Cambridge, where he succeeded Kingsley as Professor of Modern History. His controversial Renan-inspired biography of the human Christ, Ecce Homo, appeared in 1865, and his Natural Religion in 1882.

8. As Smith has argued elsewhere, Abbott also uses this strategy to criticize both spiritualists—who contended that “spiritual” phenomena provided evidence that the fourth dimension did exist—and scientific materialists—who used non-Euclidean geometry to attack conceptions of necessary truth (199-200, 206-08).

9. The Sphere tells A Square that if omnividence does not make a figure “more just, more merciful, less selfish, more loving,” then it cannot make a figure more divine (82). Such a view shocks A Square because these are the “qualities of women,” and “a Circle is a higher Being than a Straight Line, in so far as knowledge and wisdom are more to be esteemed than mere affection” (83).

10. Both Philomythus and The Anglican Career are peppered with references to Kingsley's charges against Newman, often defending and even extending them. As a result, Abbott's critics were quick to link the two attacks (Hutton, “Abbott's Attack”; Ward). One anonymous correspondent to the Spectator (2 May 1891) even regretted “that Dr. Abbott did not assault Newman while he was alive,” for the public would have seen “a grand repetition of the flaying of an assailant, as we saw it when Kingsley attacked.”

11. Abbott's anxieties were perhaps heightened by the fact that Huxley, in his analysis of the authorship of the Synoptic Gospels in “Agnosticism” and “Agnosticism: A Rejoinder,” had praised Abbott's similar analysis in the ninth edition of the Encyclopaedia Britannica and his 1884 Common Tradition of the Synoptic Gospels (see Science and the Christian Tradition 5: 221n, 273n).

Our thanks for helpful commentary and suggestions by Joan Richards and two anonymous reviewers.

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