Reading Notes: August 18th, 2024

Types of Order and the System Σ

By: C.I. Lewis

It is a commonplace of current theory that mathematics and exact science in general is capable of being viewed quite apart from any concrete subject matter or any system of physical facts to which it may usefully be applied Geometry need not appeal to any intuition of spatial complexes or to a supposititious space form; it has no need to rely upon diagrams or make use of ‘constructions’. Arithmetic makes no necessary reference to the sensible character of collections of marbles or of areas. Dynamics does not require the dubious assumption that the ‘moving particles’ of which it treats are possible of experience or verifiable physical entities. The ‘points’ of geometry and kinematics, the ‘numbers’ of arithmetic, and so on are simply terms x’s, y’s, z’s, entities, anything,—and the question what concrete things may be successfully regarded as such x’s and y’s is a question of application of the science, not one which need be considered while the system itself is in process of development.

If considerations of usefulness and of application are important in determining what assumptions shall be made or what systems developed, still such pragmatic considerations are principles of selection amongst actual and possible systems, and not internal to the systems themselves.

An arithmetic, a geometry, a kinematics, is thus capable of being viewed simply as a complex of relations and operations (relations of relations) which obtain amongst entities the nature of which, apart from those properties which follow from the relations assumed, is wholly indifferent. Such a system may in fact admit of various interpretations and applications more or less useful, all of which satisfy the requirement that these relations and operations be valid. As Professor Royce is accustomed to put it: a system of science is a type of order, the distinguishing characteristics of which are the kind of relations—symmetrical or unsymmetrical, transitive or intransitive, etc.,—which obtain among its terms, and the relations of these relations, by means of which the terms are ‘ordered’ and the relations ‘transformed.’

The growing recognition of the advantages of so viewing systems of pure science is one of the prime motives for the present interest in symbolic logic, or logistic. For logistic is the science which treats of types of order. One may reach the particular type of order which it is desired to portray—the arithmetic or geometry—by further specification of that minimum order which must obtain among entities if they are to ‘belong together’ in a set or system—the order of logic. This can be done in a variety of ways, which may be roughly divided into two groups. These two methods are distinguished by the fact that in the one case the ‘numbers’ of arithmetic or ‘points’ of geometry are treated as (conceptual) complexes having a definite internal structure, while in the other the ‘numbers’ or ‘points’ are the simple and indifferent terms, the x’s and y’s of the system. The former mode of procedure is best illustrated by the investigations of Russell’s Principles of Mathematics and Principia Mathematica of Russell and Whitehead. The other method is exemplified by Dedekind’s Was sind und was sollen die Zahlen, by the Ausdehnungslehre of Grassmann, and by the paper of Mr. A.B. Kempe, “On the Relation between the Logical Theory of Classes and the Geometrical Theory of Points.” But this second method appears in its best and clearest form in the paper of Professor Royce on The Relation of the Principles of Logic to theFoundations of Geometry. Each of these procedures has its advantages and its difficulties. Of late, the first method has received a disproportionate share of attention. For this reason, if for no other, I deem it important to call attention to the second method in general and to Professor Royce’s paper—its notable exemplification—in particular.

Professor Royce generalizes upon certain relations previously pointed out by Kempe, in the paper mentioned above,—certain relations which are fundamental both for logic and for geometry. If ac · b represent a triadic relation in which a and c are the ‘even’ members and b is the ‘odd’ member, ac · b is capable of various significant interpretations. If a, b, and c represent areas, ac · b may be taken to symbolize the fact that b includes whatever area is common to a and c, and is itself included in that area which comprises what is either a or c (or both). The same relation may be expressed in symbolic logic as:

ac⥽b⥽(a +c); or; a̅bc̅ + ab̅c = 0.

This relation may be so assumed that it has the essential properties of serial order. Taking it in the form just given and presuming the familiar laws of the algebra of logic, if ac · b and ad · c, then also ad · b and bd · c. Hereupon we may translate ac · b by ‘b is between a and c,’ and the relation will then have the properties of the points, a, b, c, d, in that order. Further, if a be regarded as an origin with reference to which precedence is determined, ac · b may represent ‘b precedes c,’ and ad · c that ‘c precedes d.’ Since ac · b and ad · c together give ad · b, if ‘b precedes c’ and ‘c precedes d,’ then ‘b precedes d.’ Hence this relation has the essential transitivity of serial order, with the added precision that it retains reference to the origin from which ‘precedes’ is determined.

Professor Royce points out to his students that the last mentioned property of this relation makes possible an interpretation of it for logical classes in which it becomes more general than the inclusion relation of ordinary syllogistic reasoning. If there should be inhabitants of Mars whose logical sense coincided with our own—so that any conclusion which we regarded as valid would seem valid to them, and vice versa—but whose psychology was somewhat different from ours, these Martians might prefer to remark that “b is ‘between’ a and c,” rather than to note that “all a is b and all b is c.” These Martians might then carry on successfully all their reasoning in terms of this triadic ‘between’ relation. For ac · b meaning a̅bc̅ + ab̅c = 0 is a general relation which, in the special case where a is the “null” class contained in every class, becomes the familiar “b is included in c” or “all b is c.” By virtue of the transitivity pointed out above, 0c · b and 0d · b, which is the syllogism in Barbara, ‘If all b is c and all c is d, then all b is d.’ Hence these Martians would possess a mode of reasoning more comprehensive than our own and including our own as a special case.

The triadic relation of Kempe is, then, a very powerful one, and capable of representing the most fundamental relations not only in logic, but in all those departments of our systematic thinking where unsymmetrical transitive (serial) relations are important. In terms of these triads, Kempe states the properties of his ‘base system,’ from whose order the relations of logic and geometry both are to be derived. The ‘base system’ consists of an infinite number of homogeneous elements, each having an infinite number of equivalents. It is assumed that triads are disposed in this system according to the following laws.

1. If we have ab · p and cb · q, r exists such that we have aq · r and cp · r.

2. If we have ab · p and cp · r, q exists such that we have aq · r and cb · r.

3. If we have ab · c, and a = b, then c = a = b.

4. If a = b, then we have ac · b and bc · a, whatever entity of the system c may be.

To these, Kempe adds a fifth postulate which he calls the ‘law of continuity’: “No entity is absent from the system which can consistently be present.” From these assumptions and various definitions in terms of the triadic relation, Kempe is able to derive the laws of the symbolic logic of classes and the most fundamental properties of geometrical sets of points.

But there are certain dubious features of Kempe’s procedure. As Professor Royce notes, the ‘law of continuity’ makes postulates 1 and 2 superfluous. And it renders entirely obscure what properties the system may have, beyond those derivable from the other postulates without this. For the negative form of the ‘law of continuity’ makes it impossible to assume the existence of an entity without first investigating all the properties of all the other entities and collections in the system, where some of these other entities and collections exist only at the instance of the ‘law of continuity’ itself. Consequently the existence of any entity or set, not explicitly demanded by the other postulates, can be assumed only at the risk of later inconsistency. Also, in spite of the fact that Kempe has assumed an infinity of elements in the base set, there are certain ambiguities and difficulties about the application of his principles to infinite collections.

In Professor Royce’s paper, we have no such ‘blanket assumptions’ as the ‘law of continuity,’ and the relations defined may be extended without difficulty to any finite or infinite set. We have here, in place of a ‘base system’ and triadic relations, the ‘system Σ’ and “O-collections.”

The system Σ consists of simple and homogenous elements. Collections of these may contain any finite or infinite number of elements; and any element may be repeated any number of times; so that x and x-repeated may be considered a collection, x, x-repeated, and y a collection, and so on. Greek letters will signify determinate collections in Σ. Collections in Σ are either O-collections or E-collections. O(——) signifies that (——) is an O-collection; E(——) that (——) is an E-collection, i.e., that it is not an O-collection. Assuming for the moment the principles of the algebra of logic, O(pqrs …) signifies that pqrs … + p̅q̅r̅s̅ … = 0. [Both the laws of the algebra of logic and the properties of O-collections which render them thus expressible are, of course, derived from the postulates and not assumed in the beginning.] It will be clear that the order of terms in any O-collection may be varied at will. ‘x is equivalent to y’ means that in very collection in which x or y occurs the other may be substituted for it and the collection in question still remain an O-collection.

If two elements in Σ, say p and q, are such that O(pq) is true, then p and q are said to be obverses, each of the other. Since it will follow from the postulates of the system that all the obverses of a given element are mutually equivalent, and that every element has at least one obverse, a ‘unique representative’ of the obverses of x may be chosen and symbolized by x̅. Pairs of obverses will turn out to have the properties of negatives in logic.

Any q such that O(βq) is true, is called a compliment of β.

Any r such that O(βq) and O(qr) are both true is called a resultant of β.

The postulates of the system Σ are as follows:

I. If O(α), then O(αγ), whatever collection γ may be.

II. If, whatever element b_n of β be considered, O(δb_n), and if O(β) is also true, then O(δ).

III. There exists at least one element in Σ.

IV. If an element x of Σ exists, then y exists such that x ≠ y.

V. Whatever pair (p, q) exists such that p ≠ q, r exists such that while both O(rp) and O(rq) are false, O(pqr) is true.

VI. If w exists such that O(θw), then v also exists such that O(θv) and such, too, that whatever element t_n of θ be considered O(vwt_n).

From these assumptions the whole algebra of logic can be derived in such wise that the system Σ has the order of the totality of logical classes. To see this, we must first define the F-relation. If O(pqrs …) to any number of terms, we may represent the same fact by (F(p̅/qsr …), (Fp̅r̅/qs …), (r/Fp̅q̅s̅ …), etc., where the rule for transforming the O-collection into the corresponding F-collections is that we introduce a bar, separating any one or more elements of the O-collection from the remainder, and then replace each of the elements on one (either) side of the bar by its obverse. Since the order of terms in O-collections is indifferent, terms on the same side of the bar in any F-relation are independent of the particular order in which they are written. Also, it follows immediately from the definition of the relation that F(pq/r̅s̅) and F(p̅q̅/rs) are equivalent. Where the F-relation holds for three terms, it turns out to be identical with the triadic relation of Kempe, and the Kempean ac · b is thus a special case of the F-relation, namely F(b/ac), or F(ac/b), or F(a/bc̅), or F(a̅/b̅c), or F(b/ca), etc., all of which are equivalent. We may, then, define the “illative” relation,—“b is included in c” where b and c are classes, “b implies c” where b and c are propositions, “b precedes c,” where b and c are points or terms in one-dimensional array,—as the special case of any of the above F-relation in which a is the “zero element,” or “null class,” or “origin.” But these F-relations are equivalent, by definition, to O(ab̅c) and O(abc̅). Hence b⥽_ac may be defined to mean O(ab̅c) and b⥽c to mean O(0b̅c). Thus in terms of the totally symmetrical O-relation, the unsymmetrical, transitive dyadic relation which characterizes both serial order and syllogistic reasoning can be defined.

As is well known, the entire algebra of logic may be derived from a class K, the idea of negation, and the illative relation, hence also in terms of the system Σ and O-collections. The ‘zero element’ or ‘null class’ is any arbitrarily chosen member with reference to which all illative relations are supposed to be specified. Such an element o itself bears the illative relation to any other, x, since F(ox/o), or O(oo̅x) holds for any element x. The element i, the “universe” of the algebra of logic, may then be defined as the negative or obverse of the o chosen. In the system Σ, o and i do not differ from any other pair of obverses, apart from the arbitrary choice of a reference element for illative relations. The logical product of two terms, x and y, is then definable as any P such that F(ox/P), F(oy/P), and F(xy/P). The logical sum of x and y is definable as any S such that F(ix/S), F(iy/S), and F(xy/S). P, so defined, will be such that P⥽x and P⥽y, while any w such that w⥽x and w⥽y will be also such that w⥽P. For S it will be true that x⥽S and y⥽S, and any v such that x⥽v and y⥽v is also such that S⥽v. S and P are, in fact, the “lower limit” and “upper limit,” with reference to the chosen zero element, of all the F-resultants of x and y, an F-resultant being any z such that F(xy/z). These definitions for the product and sum of two elements may be extended immediately to any number of elements, or any collection of β, if we replace x an y by “any element of β, however chosen.” The usual laws of the algebra of logic, connecting sums and products, terms and their negatives, and the elements o and i may then be verified for the system Σ. This order of logical entities is contained in Σ in an infinite variety of ways, since any pair of obverses may be arbitrarily chosen for i and o. F-relations and O-relations, not confined to dyads and triads, are capable of representing this order in a generalized form.

There is, moreover, a wealth of order in the system which the algebra of logic, even in terms of any polyadic relation, does not require. It is this difference which renders the system Σ capable of being viewed as a generalized space form.

It follows from postulate V that if p ≠ q, then there is an element ‘between’ p and q. The postulate states: Whatever pair (p, q) exists such that p ≠ q, r also exists such that while both O(rp) and O(rq) are false, O(pqr) or F(pq/r̅) gives, by definition of the illative relation, r⥽_qp and r̅⥽_pq) or r is “between” p and q. And r̅ must be distinct from p and q both, for otherwise, it follows from the definition of obverses, one of the two O(r̅p) and O(r̅q) will be true. Hence postulate V may be restated in the form: For every pair of distinct elements, there exists an element, distinct from both, between them. It is at once obvious that if the elements be “points,” and p⥽_oq mean that p is between o and q, postulate V requires that the order of points in Σ should be dense in every direction (with reference to every pair of points). It is further clear that if we take any pair of distinct points, o and z, and postulate t between o and t, v between t and z, and so on. Owing to the transitivity of the illative relation, we are thus required to postulate for every pair (o, z) an infinite number of elements in the order o⥽_or⥽_ot⥽_ov⥽_oz. Such an ordered collection is continuous. We have already seen that it is dense. It remains to see that it satisfies the requirement that every fundamental segment has a limit. Consider two sections from the collection, κ and λ, such that k is any element of κ, every element j such that j⥽_ok belongs to κ, and every element l, such that for every element k of κ l⥽_ok is false, belongs to λ. There is, then, an element, called S, such that for every element k in κ, k⥽_oS, and if l is any element such that, for every element k of κ, k⥽_ol, then S⥽_ol. Such an element S is the ‘sum’ or ‘upper limit’ of κ, defined above. Hence every fundamental segment has a limit. Any collection thus characterized by a transitive unsymmetrical relation and continuous order deserves to be called a ‘line.’ Every pair of distinct elements in Σ determines such a line.

For every pair of distinct points, o and q, there exists p such that F(oq/p) and hence O(oqp̅). By the definition of the F-relation, if O(oqp̅), then F(o̅q̅/p). Hence if o and q determine a line, o … p … q, there exists also a line, o̅ … p̅ … q̅ or q̅ … p̅ … o̅, in which appear the obverses of all the elements in o … p … q. But it also follows from O(oqp̅) that F(op̅/q̅), or q⥽_op̅. Thus if o … l … z be any line determined with reference to an “origin” o, the line containing the obverses of the elements of o … l … z may be determined by reference to the same origin. And if two elements of o … l … z, say m and n, are such that m⥽_on, then n̅⥽_om̅. If we further consider the order of elements in both lines, o … l … z, and z̅ … l̅ … o̅, with reference to the origin o and its obverse o̅, the two lines appear as a single line which passes from o to o̅ through l, and from o̅ back to o through l̅. Let m and n be any two elements of 0 … l … z such that F(on/m). We have m⥽_on. Hence n̅⥽_om̅. But if we have F(on/m), then also O(onm̅) and so F(o̅m/n). Hence n⥽_mo̅. Thus any two elements, m and n, such that m is between o and n, are also such that n is between m and o̅. From the transitivity of the illative relation, m⥽_oo̅. But if m⥽_oo̅, then from the above m⥽_oo. Thus we have the continuous line, o … m … n … o̅ … n̅ … m̅ … o, or o̅ … n̅ … m̅ … o … m … n … o̅, which has so far the character of the projective line with o as origin and o̅ the point at infinity. And if m, n, r, occur in that order in one ‘direction’ from the origin, then m̅, n̅, r̅, occur in that order in the ‘opposite direction’ from the origin.

Certain further characteristics of order in the system may be mentioned briefly. In general, lines such as those considered above may “intersect” any number of times. From the definition of obverses, O(aa̅) and O(cc̅) always hold. But by postulate I, if O(aa̅), then O(aa̅p), and hence F(aa̅/p), for any element p. Similarly, if O(cc̅), then F(cc̅/p). Thus collections consisting of the F-resultants of different pairs may have any number of elements in common. But in terms of such operations as were in question in the definitions of ‘sums’ and ‘products,’ sets of resultants may be determined such that they have one and only one element in common. Thus certain selected lines in the system intersect once and once only. There are any number of such sets.

In general, if any pair of elements in a set are obverses of one another, all the other elements of the set will be resultants of this pair, and their entire array will be “one-dimensional” so far as dimensionality may be attributed to such a collection. The problem of selecting sets suitable for any space form—any n-dimensional array—is the problem of selecting so that O-collections will be excluded. Such sets, containing no obverses, are the ‘flat collections’ of Kempe. As he pointed out, the excluded obverses will form an exactly similar set, so that ‘spaces’ come in pairs somewhat suggesting companion hemispheres. In terms of “flat collections,” one-dimensional, two-dimensional, n-dimensional arrays, may be specified in any number of ways.

Once the order of the system Σ is generated in terms of O-relations and F-relations, the determination of such more specialized types of order is a problem of selection only. In the words of Professor Royce, “Wherever a linear series is in question, wherever an origin of coordinates is employed, wherever ‘cause and effect,’ ‘ground and consequence,’ orientation in space or direction of tendency in time are in question, the dyadic asymmetrical relations involved are essentially the same as the relation here symbolized by p⥽_yq. This expression, then, is due to certain of our best established practical instincts and to some of our best fixed intellectual habits. Yet it is not the only expression for the relations involved. It is in several respects inferior to the more direct expression in terms of O-relations….When, in fact, we attempt to describe the relations of the system Σ merely in terms of the antecedent-consequent relation, we not only limit ourselves to an arbitrary choice of origin, but miss the power to survey at a glance relations of more than a dyadic, or triadic character.”

With this hasty and fragmentary survey of the system Σ, we may turn to considerations of method. It was suggested in the introduction that the procedure here exemplified differs in notable ways from the method of such studies as those of Principia Mathematica. In that work, we are presented at the outset with a simple, though general, order—the order of elementary propositions so related to one another that one is the negative of another, two may be such that at least one of them is true, and so on. In terms of these fundamental relations, more special types of order—various branches of mathematics—are built up by progressive complication. In some respects this is the necessary character of deductive procedures in general; in other respects it is not. In particular, this method differs from that employed by Mr. Kempe and Professor Royce in that terms, as well as relations, of later sections are themselves complexes of the relations at first assumed. The complication thus made necessary can hardly be appreciated by those who would regard a number, for instance, as a simple entity. To illustrate: In Principia Mathematica, the “cardinal number” of x is the class of referents of the relation ‘similar to’ where x is the relatum. The ‘class of referents’ of any relation R is defined as α such that α is identical with x such that, for some y, x has the relation R to y. ‘Relatum’ is similarly defined. ‘m is identical with n’ means that, for any predicative function φ, φm implies φn. I do not pause upon ‘predicative function.’ α is ‘similar to’ β means that, for some one-to-one relation R, α is identical with the class of referents of R and β is identical with the class of relata of R. A ‘one-to-one’ relation is a relation S such that the class of referents of S is contained in i. ‘i’ is defined as α such that, for some x, α is identical with the x. ‘The x’ is my attempt to translate the untranslatable. The attempt to analyze ‘is contained in’ would require much more space than we can afford. But supposing the analysis complete, we discover that the ‘cardinal number of x’ is ——, where —— is the definition first given, with all the terms in it replaced by their definition, the terms in these replaced by their definition, and so on. All this complexity is internal to the terms of arithmetic. And only when this process is complete can any properties or relations of ‘the cardinal number of x’ be demonstrated. An advantage of this method is that the step from one order to another ‘based upon it’ is always such as to make clear the connection between the two. It preserves automatically the hierarchic arrangement of various departments of exact thinking. The process of developing this hierarchy is tedious and taxes our analytic powers, but there is always the prospect of assured success if we can perform the initial analysis involved in the definitions. But the disadvantages of this complexity can hardly be overemphasized. It is forbidding to those whose interests are simply ‘mathematical’ or ‘scientific’ in the ordinary sense. Such a work as Principia Mathematica runs great risk of being much referred to, little read, and less understood.

In contrast with such complexity, we have, by the method of Mr. Kempe and Professor Royce, an order completely generated at the start, and such that the various special orders contained in it may be arrived at simply by selection. Little or no complication within the terms is required. Involved as the structure of the system Σ may seem, it is, by comparison, a marvel of simplicity and compact neatness. With this method, there seems to be no assurance in advance that any hierarchic relations of different orders will be disclosed, but we shall certainly discover, and without difficulty, whatever analogies exist between various orders. Again, this method relies much more upon devices which may be not at all obvious. It may not tax severely the analytic powers, but it is certain to tax the ingenuity.

In another important respect, advantage seems to lie with this method. One would hardly care to invent a new geometry by the hierarchic procedure, or expect to discover one by its use. We have to know where we are going or we shall not get there by this road. By contrast, Professor Royce’s is the method of the path-finder. The prospect of the novel is here much greater. The system Σ may—probably does—contain new continents of order whose existence we do not even suspect. And some chance transformation may put us, suddenly and unexpectedly, in possession of such previously unexplored fields.

Which of the two methods will prove, in the end, more powerful, no one can say at present. The whole subject is too new and undeveloped. Certainly it is to be desired that the direct and exploratory method be increasingly made use of, and that the advantages of studying very general types of order, such as the system Σ, be better understood.

Some Afterthoughts on Illusionism

According to Illusionism, nothing—whether it be an object or an experience—has sensory-qualities; and if an object (or experience) seems to have sensory-qualities, it is because “our brain’s introspective mechanisms” have “misrepresented” the object (or experience) as having them. The Illusionist’s position is “clever” (but only as “clever” as a parlor trick) because it interprets illusions in a way that does not introduce sensory-qualities—it interprets illusions strictly in terms of dispositions (e.g., the disposition to react as if the illusory thing existed; or, again, the disposition to judge/believe that we are aware of sensory-qualities—even though sensory-qualities do not exist, they only “seem” to exist).

As a result, one cannot “refute” Illusionism by saying, “But an experience of sensory-qualities can’t be an illusion because such an illusion would necessarily involve the presence of sensory-qualities.” Nor can one “refute” Illusionism by saying, “If I seem to be experiencing sensory-qualities, then I am experiencing sensory-qualities.” Against such objections, the Illusionist would be correct in saying that the objectors beg the question against the Illusionist position. 

As mentioned above, the Illusionist thinks that we can “misrepresent” objects (or experiences) as having sensory-qualities even if sensory-qualities don’t exist. The Illusionist tries to make this clear by providing examples the he believes are sufficient to establish the Illusionist’s foundation-stone: (i) the content of a representation need not exist in order for a representation of it to exist, and (ii) we have had many instances of observation wherein we have observed a “representation of a particular sensory-quality” (e.g., a representation of a red sensory-quality) as not having the particular determinate (or determinable) sensory-quality it represents (e.g., “A representation of a [sensory-quality] needn’t be qualitative, just as the word ‘red’ needn’t be red.” (Keith Frankish, Twitter/X, Sep 27th, 2019)

However, Illusionism cannot get off the ground unless it can establish the truth or plausibility of the following proposition:

There exist “representations of sensory-qualities” (e.g., a “representation of a red sensory-quality”) that not only do not have the particular determinate (or determinable) sensory-qualities they represent, but are themselves totally devoid of determinate (or determinable) sensory-qualities of any determinable kind.

Now, as I demonstrate in my essay, A Critical Examination of Illusionism, the aforementioned proposition—a vital premise upholding the entirety of Illusionism—is baseless. Furthermore, I show that the Illusionist cannot argue for the proposition without coming face-to-face with an inescapable, two-horned dilemma: he must choose between a petitio principii or a non-sequiturial inference.

Unfortunately, Keith Frankish has never acknowledged the existence of my paper, despite the fact that I have sent it to him and reached out to him several times about it—surely he has time to read a brief article if he has enough time to pick apart “low hanging fruit” objections on Twitter….But I digress. He is, without a doubt, a busy man and is likely bombarded with countless messages on a daily basis).

I won’t bother writing out the entirety of my argument (and my elaborate defense of it) here, but I’ll present one of its main points. The Illusionist cannot validly infer (or posit) the existence of “representations of particular determinate (or determinable) sensory-qualities” that not only do not have the particular determinate (or determinable) sensory-qualities they represent, but are also themselves totally devoid of determinate (or determinable) sensory-qualities of any determinable kind, from the fact that we have had many instances of observation wherein we have observed a “representation of a particular sensory-quality” (e.g., a representation of a red sensory-quality) as not having the particular determinate (or determinable) sensory-quality it represents; the reason being that we have always observed objects (“representations” included) as having or consisting of determinate (or determinable) sensory-qualities of one or more determinable kinds. However, “instances of observation wherein we have observed a “representation of a particular sensory-quality” (e.g., a representation of a red sensory-quality) as not having the particular determinate (or determinable) sensory-quality it represents” is the only “empirical evidence” available to the Illusionist (and such evidence cannot be used to support his “foundation stone” without the Illusionist committing several fallacies). 

Thus, Illusionism’s foundation-stone is an arbitrary and unwarranted assumption. Far from being a viable option, the “coherent research programme” of Illusionism has the structural integrity of a castle in the air...

Unfortunately, Keith Frankish—one of the most well-known Illusionists alive today—has never acknowledged the existence of my paper, despite the fact that I have sent it to him and reached out to him about it—surely he has time to read a brief article if he has enough time to pick apart “low hanging fruit” objections on Twitter….But I digress. He is, without a doubt, a busy man and is likely bombarded with countless messages on a daily basis.

To conclude, I can’t help but notice that Illusionists have a bizarre and disquieting “disposition” to portray their opponents as being “bedazzled” and “tricked” by an apparent “inner magic show.” When I read books and essays by Illusionists, the only “theatrical performance” or “slight of hand” that I “seem” to be aware of—nay I “know” I am aware of—is the spectacle unfolding before my eyes.

Materialism and the “First Breath” of Mentality

According to Materialism, there was, at some point in time, a first-ever mental event. This event was preceded in time solely by non-mental events. How, then, does the Materialist go on to explain the “first breath” of mentality? He invokes an alluring word: “causality.” A reader sympathetic to the Materialist’s case might protest against such a brief statement of his position, so it would only be fair to let the Materialist interject and present his view in his own words:

Modern Materialism holds that mental events are nothing “over and above” physical events; mentality is but a delicate, rare, and ephemeral form of physicality. Old Materialism made an error that Modern Materialism has since corrected; it blundered by declaring a priori that (i) all physical events are non-mental in nature and that (ii) all mental events are non-physical in nature. Despite this categorial difference between mental events (i.e., non-physical events) and physical events (i.e., non-mental events), Old Materialism went on to assert that every mental event located in the series of past, present, and future mental events is the causally-generated “effect,” or the epiphenomenal “by-product,” of a corresponding non-mental event located in the series of past, present, and future non-mental events. Thus, each mental event is causally dependent upon a particular non-mental event located in the total series of non-mental events. Although the total series of mental events was not itself conceived by these Old Materialists as being a “segment” of the total series of non-mental events, it was nevertheless made subordinate to it. While the thread of mental events lacked self-sufficiency, the chain of physical events (i.e., the totality of past, present, and future non-mental events) constituted an independent series—resting on nothing other than itself. The Old Materialists recognized the difficulties in their position (e.g., the relationship between the two series of events), and failed to build a bridge that could intelligibly unite “cause” and “effect.” Modern Materialism, by contrast, rejects the Old Materialist’s dogmatism and refuses to declare a priori that (i) all physical events are non-mental in nature, and that (ii) all mental events are non-physical in nature. Instead of maintaining that the thread of mental events is populated by causally-generated “effects” or epiphenomenal “by-products” of an “independent” series that is exhausted by non-mental events, Modern Materialism holds that when a mental event occurs, this occurrence is identical to a particular physical event located in the totality of physical events. Thus, mental events are no longer viewed as “residual excrescences” that supervene upon the “shock of atoms” in the physical order; they are no longer granted a unique series of their own (e.g., a non-physical series populated by events caused by antecedent or contemporaneous events in a subvening series of physical events). On the contrary, mental events are physical events. This is Modern Materialism. Although Modern Materialism holds that there was, at some point in time, a first-ever mental event, this event was itself a physical event—there being no categorial difference severing said event from any non-mental event that happened to be simultaneous with, or precedent to it, in time. Now, like any other physical event, the first-ever mental event would have been the effect of a preceding physical event, and this prior physical event would have been a non-mental event—but Modern Materialism assures us that there are no difficulties inherent in such a transition. By assimilating mental events into the physical order by way of “identity,” the Modern Materialist has advanced further than any of his predecessors: he has taken a step forward towards explaining the “first breath” of mentality.

Let’s examine the Materialist’s account in detail. The Materialist postulates two temporally-distinct events: (i) a definite, first-ever mental event, M, and (ii) a definite, non-mental event, Pn, that preceded M in time. How, then, are we to conceive the time-series of which these two events, Pn and M, are  occupants? Let’s offer a brief sketch the nature of the time-series by specifying some of its properties:

“A series is continuous when any term divides the whole series unambiguously into two mutually exclusive parts which between them comprise all the terms of the series, and when every term which so divides the series is itself a term of that series. From this second condition it obviously follows that a number of intermediate terms can always be inserted between any two terms whatever of a continuous series; no term of the series has a next term….The whole series of real numbers us continuous [because] every member of the number-series divides it into two classes, so that every number of one is less than every number of the other, and every number which thus divides the series is itself a term of the number-series....From the continuity of the series of real numbers it follows that any other series which corresponds point for point with the terms of the number series will be continuous. Now one such series is that of the successive parts of time. Every moment of time divides the whole series of moments into two mutually exclusive classes, the moments before itself and the moments which are not before itself. And whatever thus divides the time-series is itself a moment in that series.” (Taylor, Elements of Metaphysics, 171-172) 

Thus, it follows from time’s continuity that no two instants in the time-series ever “touch”.¹ When this fact and the Materialist’s proposed connection between Pn and M are brought into focus, we find that Pn and M cannot share an extremital boundary. In other words, the last instant of Pn cannot be simultaneous with the first instant of M. Indeed, if the last instant of Pn was simultaneous with the first instant of M, then we would have a contradiction on our hands: there would be a moment in time when there was mentality present in a world that, ex hypothesi, was exhausted by non-mentality. In light of this, we must ask the Materialist several questions:

Question (i): Was Pn—rather than any preceding non-mental event—the “cause” of M?

The Materialist’s theory requires that he answer question (i) in the affirmative. And so, in response to question (i), the Materialist declares Pn—rather than any preceding non-mental event—to be the “cause” of M.

Question (ii): What was it about Pn that made this particular non-mental event—rather than any preceding non-mental event—the “cause” of M? 

On pain of inconsistency, the Materialist must respond to question (ii) by asserting that Pn possessed certain “special properties” (i.e., a set of characteristics absent from all prior non-mental events), and that Pn’s possession of these “special properties” made Pn—rather than any preceding non-mental event—the “cause” of M.

Question (iii): What were the “special properties” present in Pn and absent from all preceding non-mental events, that made Pn–rather than any preceding non-mental event—the “cause” of M?

In reply to question (iii), the Materialist will likely posit a bunch of features that allegedly capture the essence or identity of these “special properties.” Let’s symbolize the identity of these “special properties” present in Pn—but absent from all preceding non-mental events—as C.

At first glance, all seems fine and well; however, there is a puzzle lurking beneath the surface: a puzzle involving (i) the continuous nature of the time-series, (ii) the Materialist’s identification of Pn—rather than any of the other non-mental events which preceded it—as being the possessor of the aforementioned “special properties,” and (iii) the Materialist’s identification of what these “special properties” actually are. Let’s explore this latent puzzle in the Materialist’s theory.

As we have noted above, the final instant of Pn cannot be simultaneous with the first instant of M—on pain of contradiction. The first instant of mentality in the world must be “later than” any instant during which the Materialist’s world was “exhaustively non-mental.” However, since the continuity of time implies that between any two instants in the time-series there is an intermediate instant, it follows that between Pn and M (or, rather, between the last instant of Pn and the first instant of M) there was another event, X, distinct from both Pn and M and separating the two events. This prompts us to ask the Materialist more questions:

Question (iv): Is X a non-mental event or a mental event?

The Materialist must answer question (iv) by declaring X to be a non-mental event, Pn+1. For consider, if the Materialist asserted X to be a mental event, then he would have fallen into inconsistency; indeed, he would have been mistaken about M being the first-ever mental event because X would have preceded M in time. 

However, if the Materialist answers question (iv) by declaring X to be a non-mental event, Pn+1, then he contradicts his answer to question (i); in other words, since Pn+1 is later than Pn in time, it would not be true that Pn was the “cause” of M. And so, the Materialist must now revise his answer to question (i) and state that Pn+1—rather than any preceding non-mental event (e.g., Pn)—was the “cause” of M.

Question (v): What was it about Pn+1 that made this particular non-mental event—rather than any preceding non-mental event (e.g., Pn)—the “cause” of M?

On pain of inconsistency, the Materialist must answer question (v) by asserting that Pn+1 possessed certain “special properties” (i.e., a set of characteristics absent from all prior non-mental events), and that Pn+1’s possession of these “special properties” made Pn+1—rather than any preceding non-mental event (e.g., Pn)—the “cause” of M.

Question (vi): What were the “special properties” present in Pn+1 and absent from all preceding non-mental events, that made Pn+1—rather than any preceding non-mental event—the “cause” of M?

The Materialist is forced by his own hand to answer question (vi) by positing a bunch of features that allegedly capture the essence or identity of the “special properties” present in Pn+1 (i.e., features present in Pn+1 but absent from all preceding non-mental events) that made Pn+1—rather than any preceding non-mental event (e.g., Pn)—the “cause” of M. Now, the Materialist cannot, on pain of contradiction, supply us with the same list of “special properties” that he provided in his answer to question (iii). If, in response to question (vi), the Materialist simply regurgitated his answer to question (iii), then, ex hypothesi, the “special properties” of Pn+1 would have been present in an earlier non-mental event (i.e., Pn)—thereby contradicting the Materialist’s answer to question (v). Moreover, the Materialist cannot simply provide his answer to question (iii) in response to question (vi) because he himself has admitted, by implication, that the properties of Pn were not of such a nature as to make Pn the cause of M. Let’s symbolize the Materialist’s revision of these “special properties” as C’.

However, another problem arises. In the same way the last instant of Pn had to be earlier in the time-series than the first instant of M, so too must the last instant of Pn+1 be earlier in the time-series than the first instant of M. If this were not so, and the last instant of Pn+1 was simultaneous with the first instant of M, then there would be a moment in time when there was mentality present in a world that, ex hypothesi, was exhausted by non-mentality—and this, of course, is a contradiction. And, as we have seen, since the continuity of time implies that between any two instants in the time-series there is an intermediate instant, it follows that between Pn+1 and M (or, rather, between the last instant of Pn+1 and the first instant of M) there was another event, X’, distinct from both Pn+1 and M and separating the two events. This prompts us to ask the Materialist more questions:

Question (vii): Is X’ a non-mental event or a mental event?

In answer to question (vii), the Materialist must, of course, respond by declaring this particular event, X’, to be a non-mental event, Pn+2. If the Materialist answered by declaring X’ to be a mental event, then he would have fallen into inconsistency; indeed, he would have been mistaken about M being the first-ever mental event since, ex hypothesi, X’ would have preceded M in time. 

The Materialist must answer question (vii) by declaring X’ to be a non-mental event, Pn+2. For consider, if the Materialist asserted X’ to be a mental event, then he would have fallen into inconsistency; indeed, he would have been mistaken about M being the first-ever mental event because X’ would have preceded M in time. However, by declaring X’  to be a non-mental event, Pn+2, the Materialist contradicts his “revised” answer to question (i)—(i.e., that Pn+1 rather than any preceding non-mental event (e.g., Pn) was the “cause” of M). As a result, the Materialist must “revise” his already “revised” answer to question (i) and declare Pn+2—rather than any preceding non-mental event (e.g., Pn or Pn+1)—to be the “cause” of M. In doing so, the Materialist must also revise his answer to question (v) and question (vi). He must assert that Pn+2 possessed certain “special properties” (i.e., a set of characteristics absent from all prior non-mental events) and that Pn+2’s possession of these “special properties” made Pn+2—rather than any preceding non-mental event (e.g., Pn or Pn+1)—the “cause” of M. However, in doing so, the Materialist must specify these “special properties” of Pn+2. Just like before, the Materialist must list a set of features that allegedly capture the essence or identity of the “special properties” present in Pn+2 (i.e., features present in Pn+2 but absent from all preceding non-mental events) that made Pn+2—rather than any preceding non-mental event (e.g., Pn or Pn+1)—the “cause” of M. However, the Materialist cannot, on pain of contradiction, supply us with the same list of “special properties” that he provided in his answer to question (vi). If the Materialist did so, then the “special properties” of Pn+2 would have been present in an earlier non-mental event (i.e., Pn+1); however, he himself has admitted, by implication, that the properties of Pn+1 were not of such a nature as to make Pn+1 the cause of M.  Ergo, the Materialist must supply us with a new set of properties that were allegedly present in Pn+2, and made Pn+2—rather than any preceding non-mental event—the “cause” of M. Let’s symbolize the Materialist’s revision of these “special properties” as C’’.

However, the problem has only been aggravated. As we’ve seen before, the continuity of time implies that between any two instants in the time-series there is an intermediate instant; ergo, it follows that between Pn+2 and M (or, rather, between the last instant of Pn+2 and the first instant of M) there was another event, X’’, distinct from both Pn+2 and M and separating the two events. This prompts us to ask the Materialist another question:

Question (viii): Is X’’ a non-mental event or a mental event?

The Materialist must answer question (viii) by declaring this particular event, X’’, to be a non-mental event, Pn+3. And we know where this will lead us. The Materialist is trapped in a vicious regress; for, inconsistencies in the Materialist’s responses require that he continually revise his answers ad infinitum. He is unable to consistently identify the non-mental event that allegedly gave birth to the first-ever mental event, and he is unable to consistently specify the identity of the alleged “special properties” that would have made this non-mental event—rather than any preceding non-mental event—the cause of the first-ever mental event.

With every step the Materialist takes towards his first-ever mental event, he is forced to take one step back—he is forever barred from receiving his final reward. The Materialist fails to harmonize the “first breath” of mentality within an asphyxiatingly barren, non-mental world.

“However far we pursue the course of the sense-excitation through the nerve, in however many ways we suppose its form changed and converted into ever finer and more delicate movements, we can never prove that it is in the nature of any movement so produced to cease as movement of its own accord, and to reappear as a bright color, as a tone, as a sweet taste. The chasm is never bridged over between the last state of the material elements within our reach and the first rise of the sensation; and scarce anyone will cherish the vain hope that at a higher stage of development science will find a mysterious bridge in a case where it is the impossibility of any sure crossing-over that forces itself on us with the most evident distinctness.” (Lotze, Mikrokosmus, Vol. I, 148)

Footnotes:

^[1] “For it follows from the structure of the time-series (I) that there are an indefinite number of terms of the series between any two members, between which there is a finite interval, and (II) that there is also an indefinite number of terms before or after any given member of the series. Like the series of real numbers, the time-series, because it satisfies the definition of a continuous infinite series, can have neither a first nor a last term, nor can any member of it have a next term. Applying this to the case of Causation, we may reason as follows: The same reasons which lead us to demand a cause A for any event B, and to find that cause in an assemblage of antecedent events, require that A should be similarly determined by another assemblage of antecedent events, and that this cause of A should itself have its own antecedent cause, and so on indefinitely. Thus, the causal principle, logically applied, never yields an intelligible explanation of any event Instead of exhibiting the transition A—B as the logical expression of a coherent principle, it refers us for the explanation of this transition to a previous instance of the same kind of transition, and then to another, and so forth without end. But it is impossible that what is not intelligible in one instance should become intelligible by the mere multiplication of similar unintelligibilities. (II) Similarly, if we look within the transition A—B. This transition, being continuous, must have its intermediate stages. A becomes B because it has already become C, and the transition A—C—B is again “explained” by showing that A became D which became C which became E which became B. And each of these stages, A—D, D—C, C—E, E—B can be once more submitted to the same sort of analysis. But in all this interpolation of immediate stages there is nothing to show the nature of the common principle in virtue of which the stages form a single process. We are, in fact, trying to do what we try to do wherever we establish a relation between terms, to answer a question by repeating it.” (Taylor, Elements of Metaphysics, 177-178) 

Faith and the “Diamond Net” of Reason

Only when reason has been uplifted by faith may it spread its wings and ascend towards reality and truth. To sever the power of reason from the agency of faith is to render reason idle—it is to condemn reason to a state of dormancy and impotence. Far from being opposing and mutually exclusive faculties or departments of human experience, reason and faith reinforce each other—they are abstractly distinguishable factors of a single process that gradually unfolds throughout the course of conscious development on an individual and collective level. Indeed, reason runs on the rails of faith and revelation: a faith in reason’s own efficacy, and reason’s revelation to itself, and through itself, as efficacious. Now, the notion that reason and faith are compatible and not hostile foes demands explanation. Our purpose in this brief essay is to illustrate how these two factors are complementary and mutually reinforcing movements in the life of mind.

Before presenting a case for why reason and faith are in harmony, it is necessary to examine how they relate to ourselves as conscious beings. Augustine writes that “we both are, and know that we are, and take delight in our being and knowing;”¹ and these are hallmarks of our conscious life. They are evidenced by our insatiable thirst for knowledge and understanding. It is a restless impulse that resonates throughout our minds and drives us onwards towards the fulfillment of an ideal end: the knowledge and understanding of what is real and true. The journey is certainly long, and the path is undoubtably treacherous; however, the mind is not alone on this ‘pilgrim’s progress’. The “two-fold force”² of reason and faith—to use Augustine’s expression—accompany him along the way.

How, then, are we to understand reason?”³ Strictly speaking, reason is the power or activity of the mind concerned with the principles of structure, universality, and order. Reason is neither a matter of merely observing and perceiving the world, nor again is reason merely the act of reporting what one observes or perceives; rather, it is the mind’s capacity for determining and grasping relations; it is an intellectual activity by means of which the mind organizes its conception of the world (e.g., its judgments, thoughts, beliefs, etc.) in a way that exhibits universality, systematicity, coherence, and explicability. Reason seeks to establish a system of necessary connections; a system wherein the impulse of thought can glide smoothly without interruption. Reason’s ideal is to spin a coherent, all-encompassing arras of ideas: an intelligible fabric wherein each element is threaded to every other—every junction and knot opening itself up to further analysis and explanation in light of the whole.

Abstractly considered, reason seems to distance itself from anything reminiscent of faith—an innocent seeming which disposes us to judge faith as hostile to reason. Furthermore, this judgment can quickly solidify into an unspoken conscious attitude—an explicit rejection of the possibility of any interpenetration or ecumenism between the two. So understood, any eirenicon between reason and faith threatens to introduce stains, discolorations, and abrasions into the systematic tapestry that reason has crafted for itself.  

Again, at first glance, reason appears to get along just fine without recourse to leaps of faith, ungrounded revelations, or any authorities other than itself. Indeed, reason affects us so deeply that we are left with the impression that it is an all-seeing arbitrator—being the one, lone standard by means of which all authorities are weighed and measured. Nevertheless, these seemings, appearances, and impressions are, upon thorough analysis, delusive—all being products of vicious abstraction. The light of reason and the agency of faith are, in fact, the systolic and diastolic beats of a single pulse in the life of mind. Indeed, as John Scotus Eriugena reminds us, “there can be no doubt that both spring from a common source…”⁴

We began this essay with the assertion that “reason runs on the rails of faith.” This idea has been disputed and challenged by many. In order to better understand reason’s indebtedness to faith, it is necessary to penetrate into the heart of reason’s claim to be the “measure of all things”—requiring neither assistance, nor supplementation, from anything other than itself. This can be done with help from the following illustration:

Take, for instance, a Geiger counter, a thermometer, and a clock. How are we to determine whether or not the measurements of these diverse instruments are accurate as opposed to delusive? The answer is that we must compare the readings, and calibrate the instruments, in relation to a standard. The question then becomes what this standard is. Usually, the calibration of these instruments involves comparing and correlating their respective readings to those of another instrument of the same respective type. Thus, the Geiger counter’s readings are adjusted and corrected in light of the readings of another Geiger counter; the thermometer’s readings in light of the readings of another thermometer; and the clock’s readings (e.g., the position of the hour, minute, and second hand) in light of the readings of another clock (e.g., a digital watch). However, a problem arises as soon as we ask if the instruments belonging to the second set give us accurate measurements of radioactivity, outside temperature, and time of day. There are three possible options available: (a) compare and calibrate the second set of instruments in relation to the first set—thereby treating the first set as the standard, (b) compare and calibrate the second set of instruments in relation to a third set—thereby treating the third set of instruments as the standard, and (c) compare and calibrate the second set of instruments in relation to a standard that has been arbitrarily determined and whose accuracy cannot itself be called into question. Now, (a) entangles us in a vicious circle, and (b) launches us into a vicious regress.⁵ The only option that we have left is (c). To answer our question, “How do we know that the instruments belonging to the second set give us accurate measurements of radioactivity, outside temperature, and time of day,” we would have to expose the Geiger counter to a known and well-defined quantity of radioactive matter, introduce the thermometer into an environment with a known and controlled temperature, and synchronize the hands of the clock with signals from an atomic clock.

These examples illustrate reason’s capacity for “rising above” and “enveloping” the measurement-situation. Reason “transcends” the instruments, their respective readings, the measurable quantities, and the respective standards in terms of which the instrument readings are compared. Without this power, reason would be unable to either recognize, evaluate, or determine whether or not the instruments were providing accurate or delusive measurements.

In light of this vignette, suppose that reason asks the following question: “Is reason capable of knowing and understanding what is real and true?” How can reason go about answering this question? What appropriate steps, methods, or tools does reason have at its disposal? For consider, when reason asked itself whether the Geiger counter, thermometer, and clock were delivering accurate measurements of radioactivity, outside temperature, and time of day, reason could arrive at an answer by comparing and evaluating the instruments’ respective measurements in relation to an independent and predefined standard of reference. In all three cases, the “authoritative” standard, or reference point, is one that reason has defined for its own purpose: this purpose being reason’s attainment of a universalizable, systematic, coherent, and explicable account of innumerable things and processes (e.g., radioactive, thermal, and temporal phenomena, etc.). Furthermore, reason could evaluate the reliability of the three instruments only insofar as the instruments, their readings, the measurable quantities, and the “authoritative” standards were all given together on a level; or, to use a metaphor, all of these phenomena were circumscribed within reason’s “visual field.” However, just as the bodily eye cannot see itself, it seems that the relation between reason and the world (i.e., a relation in virtue of which reason has the capacity to know the world), will forever evade the “diamond net” of reason.

On the one hand, reason cannot prove its own efficacy—reason cannot argue its way to the conclusion that it is capable of knowing and understanding what is real and true. Reason cannot make its case by means of deductive, inductive, or abductive inferences. Indeed, if reason were to do such a thing, it would be arguing in a circle—pointing to instances of its own past and present activity (i.e., hand-picked instances having been evaluated, and deemed appropriate, by reason itself) and appealing to them as demonstrative evidence of reason’s own competence (or lack thereof). This circularity is no different from the attempt to determine the accuracy of the second Geiger counter, the second thermometer, and the second clock by comparing their respective readings to the first Geiger counter, the first thermometer, and the first clock—the first”⁶ set of instruments being the “authoritative” standard in relation to which the accuracy or inaccuracy of the second set of instruments are judged.   

On the other hand, reason cannot criticize its own efficacy—reason cannot argue its way to the conclusion that it is incapable of knowing and understanding what is real and true:

“To do so would require a second species of thought to sit in judgment upon our first or actual thought, and a third thought to test the validity of the verdict thus obtained, and so on ad infinitum—a species of never-ending appeal as wearisome as fruitless.”⁷

This regress is no different from the attempt to determine the accuracy or inaccuracy of the second Geiger counter, the second thermometer, and the second clock by comparing their respective readings to a third set of instruments whose accuracy or inaccuracy can be determined only in relation to a fourth set, and so on, ad infinitum—leaving us forever in the dark as to whether or not the second set of instruments deliver accurate measurements of radioactivity, outside temperature, and time of day.

Reason, unlike a Geiger counter, thermometer, or clock, is not a thing among things. Furthermore, reason does not exist in vacuo—it is nothing apart from its content: it requires a substantial “filling.” In a word, reason demands a world—a world that admits of being subject to reason’s various organizing and systematizing activities; and, prima facie, reason’s demand for a world appears to be fulfilled. However, couldn’t this apparent communion be delusive? After all, the abstract, logical scaffolding of reason’s implicative and entailment relations cannot, of themselves, disclose any harmony (or lack thereof) between reason and reality without either rotating round and round in circles, or shooting off into an infinite regress. And, if this is so, it appears that reason can neither prove, nor disprove, that it harbors within itself the capacity for knowing and understanding the real and the true. What, then, is the powerful force moves reason to confidently declare that it is “the eye with which the universe beholds itself and knows itself divine?”⁸ Indeed, what is the nature of the unfaltering impulse that leads the mind to affirm that reason envelops reality, or, indeed, to exclaim that what is rational is real; and what is real is rational? We must answer that it is a theoretical assumption—an unwavering conviction that cannot be established, justified, or scrutinized by any of the tools that the rational intellect has forged for itself. Although this assumption is of a theoretical sort, it should not be mistaken for a static premise; on the contrary, it is rather a movement—a galvanizing act that is of profound metaphysical significance. As F.H. Bradley and Bernard Bosanquet expertly put it:

“You may call the intellect, if you like, a mere tendency to movement, but you must remember that it is a movement of a very special kind....In thought the standard, you may say, amounts merely to “act so”; but then “act so” means “think so,” and “think so” means “it is.” And the psychological origin and base of this movement, and of this inability to act otherwise, may be anything you please; for that is all utterly irrelevant to the metaphysical issue. Thinking is the attempt to satisfy a special impulse, and the attempt implies an assumption about reality. You may avoid the assumption so far as you decline to think, but, if you sit down to the game, there is only one way of playing. In order to think at all you must subject yourself to a standard, a standard which implies an absolute knowledge of reality; and while you doubt this, you accept to, and obey while you rebel.”⁹ 

“Thought will go forward if it possibly can. It will affirm meanings; an order or connexion which is one side of meaning. This might be called a non-formal principle. The laws of logic and other axioms are merely, I suggest, the expression of our elementary experiments in actual thinking at different points of our experience....If we are not allowed to think, we are not allowed to exercise the act which these pervading laws need for their establishment. They are not premises. They are principles evident throughout our thinking as the manners of its self-assertion....But if thought is forbidden to proceed, the principle cannot be established. This is not a “psychological” necessity. It is a necessity of the nature of reality which is thought’s function and character to reveal. How do we know it is thought’s function and character to do so? Because every act of thought says so. Thought, in asserting, does not say “I think so”. It says “it is so”. “I think so” is merely one case of “it is so,” and is as absolute as any other assertion of a fact about reality.”¹⁰ 

Reason operates through a conviction—a mode of revelation that stands on its own authority and is incapable of being captured within the “diamond net” that reason casts, and cannot help but cast, upon the world. Faith—faith in the affirmation that reality and truth are appreciable by reason—envelops and saturates the entirety of our intellectual life; indeed, it dictates the normative ideal which reason finds itself compelled to follow in its effort to know and understand the world. Augustine writes that “we must first believe whatever great and divine matter we desire to understand. Else would the Prophet have said in error, Except ye believe, ye shall not understand;”¹¹ likewise, we must affirm—through an act of faith—that there is a harmony between reason and the world, in order for us to understand the world.

Footnotes:

^[1] Augustine, “Self-Knowledge and the Three-Fold Nature of Mind,” essay, in The Longman Standard History of Medieval Philosophy, ed. Daniel Kolak and Garrett Thompson (New York, NY: Pearson Education Inc., 2008), 28–34, 29.

^[2] Augustine, The Essential Augustine, ed. Vernon J. Bourke (Indianapolis: Hackett Pub. Co., 1974), 25.

^[3] Cf. Brand Blanshard, Reason and Analysis (La Salle, IL: Open Court, 1964), 25-26.

^[4] John Scotus Eriugena, Philosophy, Faith, and Reason, n.d., 114.

^[5] Since the accuracy, or inaccuracy, of the first set of instruments had been judged by comparing their respective readings to those belonging to the second set, it would be viciously circular to then evaluate the accuracy of the second set of instruments by comparing their respective readings to those belonging to the first set of instruments.

^[6] “One cannot in strictness speak of testing a thing by itself.” Cf. Richard Lewis Nettleship, Philosophical Lectures and Remains of Richard Lewis Nettleship, ed. A.C. Bradley and G.R. Benson, vol. I, II vols. (London, England: Macmillan and Co., Ltd., 1897), 181.

^[7] Andrew Seth Pringle-Pattison, Hegelianism and Personality, 2nd ed. (London, England: William Blackwood and Sons, 1893), 90.

^[8] F.H. Bradley, Appearance and Reality: A Metaphysical Essay (Oxford: Clarendon Press, 2002), 152-153.

^[9] Bernard Bosanquet, “‘This or Nothing,’” Mind, New Series, XXXI, no. 122 (April 1922): 178–184, 180-181.

^[10] Percy Bysshe Shelley, “Hymn of Apollo,” essay, in The Poetical Works of Percy Bysshe Shelley, vol. III (Boston, MA: Little, Brown and Co., 1855), 52–53, 53.

^[11] Augustine, The Essential Augustine, 24.