W.S. McCulloch

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There are several problems which have bothered me for a long time. At first they seemed unrelated, but as the years have gone by in repetitions of attempts at analysis, each time from a wider base, there begins to appear a similarity in some cases and an interdependence in others that makes me suspect some common factor in the etiology of these, my disease.

First, it seems to be necessary to consider what constitutes an “analysis” as opposed to an “abstraction.” In the former some “given” is “taken apart” in some way, and both the “arts” and the “way” are “preserved” or “re-presented”, whereas in the latter an analysis is begun but some whatnots are “lost”, “disregarded,” or “not represented.” The difference between these procedures appears the minute one attempts to “resynthesize” the original “given.” When one has both relata and relations, the procedure is reversible. When one has only abstracts, it is not reversible. This distinction seems to have been clear to Descartes when he took position and motion to analyze geometrical form, thus creating “analytical geometry.” Short of an *actual* *infinity* of points, each having position only, no other *actual* *analytical* *procedure* appeared to him or has yet been found. With the development of the calculus to describe motion (Fluxions of Newton) or a comparable relation of other “extended magnitudes” the question of whether a given mathematical description was or was not analytical became vital to physics and mathematics in a more extended sense, i.e. if position and motion were analytical of form, then acceleration bore the same relation mathematically to motion that motion did to form, and change of acceleration bore, again, that same relation to acceleration.

Thus, to obtain form from motion one needed also position; to obtain motion from acceleration one needed also a so-called constant of integration, etc. The motion was called the first fluxion of the form of the first derivative, the acceleration was called the second fluxion of the form or the second derivative, etc. So long as the motions considered were smooth, orderly paths such as might well be attributed to particles without sudden contacts, all went fairly well; but when velocities were considered to change a finite amount in no time — i.e., when the acceleration was “infinite” — and when the imaginary was introduced giving the complex variable (having a real and an imaginary part) the question of whether or not a particular mathematical formula was analytical — and if so, in what domain — became extremely complicated, so much so that Weierstrass by *abstracting* the real part of an expression involving a complex variable produced a formula which had a real finite value for every value of the argument, but had no integral at all — i.e., was analytical of nothing.

Throughout this whole development the original meaning of the statement that a given formula was or was not analytical survives unchanged, though we now may ask — in what region? — are there singular points? — or is there some process of analytical continuation beyond a given region? But for my present purpose the most important contribution of Function Theory, the most clearly indicative of the nature of this theory as well as that one from which I shall seek implications later, is this; that if a function is analytical in a given region, then its integral around a closed path in that region is zero — or, on returning to the starting point, one has just that which he had originally.

But it is with another aspect of analytical geometry as anlage of the calculus that we are now concerned. Analysis of form into position and motion is an analysis not into comparable parts, but into relata and relations. When Aristotle created the logic of genus and species he observed the appearance of terms for differentia in his own discourse. These were not comparable to genus and species, but were relations among the others as relata. Yet language compelled him to refer to them as in some genus, a metabasis aes allogenus *in speech*, in logic, and probably in thought. Though Newton invented fluxions to deal with motions, he felt some lack of rigor in the logical structure and preferred where possible the established method of exhaustion. Cantor felt it much more keenly. To put analytical geometry and the calculus on a logical foundation he considered it necessary to have a more formal analysis of form which could be only achieved by asserting an actual infinity of points, an actual division without limit.

To the algebraic mind, to the logical followers of the logic of numbers generated by addition and its inverse (subtraction), a shorthand way of writing it (multiplication) and its inverse (division), a shorthand way of writing multiplication (powers) and its inverse (roots), analytical geometry may always appear “algebraic” geometry. All the points defined by these operations merely need to be supposed to be actual and a calculus can be defined rigorously.

This is in essence what Cantor did. He preferred, because it was logically manageable, an actual infinity of points to motion. A line becomes a class of points and motion (and with it changes, for no generation is considered here) disappears from the universe of discourse. The logic of classes — and of infinite classes at that — appears, but the calculus as an analytical tool ceases to advance. But that is not all. An infinite class is one a part of which can be put into one-to-one-correspondence with all of that class, and we therefore may have a class of all classes which, being a class, is a member of itself. This, to my mind, is a far more astounding paradox than the one usually proposed concerning the class of all those classes which are not members of themselves.

Quite apart from the incredulity of a biologist — a neurophysiologist — as to the possibility of an organism ever experiencing or imagining or actually conceiving in any way an infinity of points in any line, in short, quite apart from my suspicion that this infinity of points is a symbol with no actual meaning, the logical difficulty is apparent, for it leads to contradiction at the logical level. Nor do I believe that it can be explained away or defined out of existence and leave a rigorous calculus, very simply, because it was not arrived at by analysis of analytical geometry and its calculus, but by abstraction that left change (be it motion or generation) out of account.

The __Principia Mathematica__ of Whitehead and Russell in its original and its altered form bears witness to this difficulty and offers two suggestions: a theory of types to prevent classes from becoming wrongly re-entrant, and the phrasing of an additional law for logic, that that which is common to all cannot be one of them. Moreover, both of these notions seem to indicate that something is going on that is lost to logic as heretofore conceived. I strongly suspect that what is going on is thinking, and that thoughts or ideas at the beginning of that thinking are confused with or mistaken for thoughts and ideas at the end of that thinking.

In his Ph.D. thesis, E. Von Domarus has said even more. Namely, that what was occurring was a metabasis aes allogenus, or substantiation in thought of what was an attribute of a substance before that thinking. In symbolic logic Fichte, without having to indicate what was occurring, prevents confusion by carrying an increasing complex of symbols as he progresses. He arrives at no paradoxes, nor at an infinity of points. What then is to be done about logic to make it applicable to the calculus? If the calculus is, as I think, not a collection of thoughts related by thoughts, but of thoughts related by thinking, we must have a logic in which thoughts are related by thinking. We must do to logic, by making thinking primary, what Descartes did to geometry by making motion primary.

A man who had no nouns except the names of things, no verbs except those for doings, no adjectives except for the sensed qualities of things, no prepositions except for “of” in the sense of “a part of,” and no conjunctions except “therefore” would of necessity have a simple metaphysics. The mathematician is burdened with such symbolic wealth that he can scarcely ever have any metaphysics, for his symbols are reiffications of any aspect of his world; each represents some abstraction from, rather than a relata or relation in, an analysis of that world. He can at will assemble a group of such representations and abstract a common aspect of their meanings. No wonder that such an abstractum *from* the group and the abstractum *of* the groups are confused, for both are reiffications of abstracta and so categorically indistinguishable.

Here in mathematics, where the theme is its own theme, no fixed system of categories stands the test of time. One is forever confronted with the metabasis aes allogenus, which multiplies the world by reiffication of abstracta. No wonder, then, that even the greatest of mathematical philosophers have multiplied their worlds, making things of the forms, extensions, productions or meanings of things. It would be as easy for a billiard ball to go crazy as for a man with a concrete vocabulary to go so metaphysically paranoid, and for the same reason. Thinking, not in the sense of mere analysis and resynthesis, but in the sense of Nous — excogitation, creative thinking depending upon abstraction and reiffication — is precluded for both.

What is to be done for metaphysics to keep the multiplied worlds somehow in order and in proper relation to the great world that has generated them by abstraction and reiffication? If the disorder has arisen because of the very nature of thinking and because here the theme is its own theme, we are confronted with a double problem; first — an adequate analysis of thinking itself, and second — the problem of all circularities within it. That circularities — e.g. mathematics — arise indicates that here we are not dealing with analysis, with representation of relata and relations awaiting resynthesis, for with this process any circularity would have to carry the whole world with it and generate nothing new. But Nous or excogitation — e.g. in mathematics — is circular and does generate essential novelty. Wherefore I am convinced that the analysis of creative thinking of E. V. Domarus is correct.

We are concerned with abstraction and reiffication, a procedure which never was and never will be analytical. What is subsumed in any system of categories is first and foremost that the system is complete rather than that it is in process of generation; I mean that what is subsumed initially is the same as what is subsumed finally, or that no new subsumption, or categories, arise due to thinking. Experience has been otherwise. It seems, therefore, that if categorical schemata are to keep pace with creative thinking it would be wiser to examine how categories arise than to attempt to delimit them all. They are generated to handle a process which, being non-analytical, may return by a closed path to its origin irrevocably altered, and that essential novelty may require new subsumptions. Generation into essence is no new problem, nor does it necessarily entail the fallacy of misplaced concreteness. But it does, as I see it, call for a searching analysis of how the new is actually induced in science, in mathematics, in metaphysics itself.

To a biologist, above all to a neurophysiologist, who has to admit that the new is actually induced in the evolution of species and in the brains of men, that word “actually” implies that there is some process to be examined, perhaps to be analyzed in terms of motion (perhaps only in terms of generation). Abstraction and reiffication must be conceived, identified, and investigated in their biological setting. I for one am convinced that the problem is far from hopeless. In fact, abstraction and reiffication seem to me to be the gist of irritability and responsiveness of all organisms, and most highly differentiated in Man. But I am equally convinced that symbolism, in the Aristotelian or biological sense, plays a leading role in the complicated activity of all thinking. To this I shall return later.

For the present, what has been said is sufficient to indicate that the inadequacy of logic, the paradoxes of mathematics, and the inapplicability of older categorical systems all seem to me to stem from this — that what we have are abstractions and what we seek are analyses. Abstractions produce essential novelty, which may go right or go wrong; analyses give us no essential novelty and go neither right nor wrong, but return us unaltered to the origin. Obviously logic, mathematics, and categorical schemata are not the ideal sources of the required analyses. They have, however, served to make it painfully clear what constituted an analysis as opposed to an abstraction.

But is thinking the sole source of these abstractions? Is science — observational science itself — composed only of abstractions, or is there somewhere within it an analysis? I mean literally does the analysis of science (i.e., of *Knowing*) yield only a collection of non-analytical relations (sensations, etc.) of Knower and Known, or is there to be found some Knowledge (relation) of Known by Knower (relata) analytical of Knowing? In science do we have nothing but “Epistemic Correlations” between reiffied abstracta — or conceptions — in the theory, and initial abstracta — or sensations — in the experience? Please distinguish once and for all between this epistemological problem and that which arises for the scientists who seeks to *analyze* the *Known* (as F_{j} (X_{1}; X_{2}; X_{3}...X_{n})).

Partly to make this more emphatic, and partly to show how different are the consequences of presupposititious answers to this, for me, the central epistemological question, I would like to turn the clock back to that hour when Kant was so rudely awakened from his dogmatic slumber by Hume’s piercing analysis of the theory of Knowing implicit in the mechanical theory of the Known. At that time there was one science of note, Mechanics, which was believed to be analytical of the Known; I mean it was supposed that a competent mathematician, given the position, the time, and the momenta (here relata) and the laws of motion (here relations) could from the present predict the future and then reconstruct its past (here and elsewhere) and have again the present, etc. If by “given” one understands being empirically true to the facts, then the predicted future or reconstructed past would also be true to their facts. Historically what had happened was this: that, despite the Scolia of Newton, time, space and causality had been held to be derived empirically, like the “secondary qualities”, the sensations, etc., i.e. by analysis of experience. When Hume showed that from “secondary qualities” which were supposed to constitute experience one never got what was not there in the first place, i.e., time, space and causality, and hence no science of Mechanics, Kant began his critique. Science was the activity of Knowing the world, but only as it is known in Newtonian Mechanics. That science gave an analysis of physical events in terms of theory and observations. In as much as the theory presupposed time, space and causality, these, not deducible from experience *must* have either underlain experience or been subsumed in theorizing.

To the repercussions of this conclusion (amplified in his theory of the a priori in all experience) upon neurophysiology we must return and for the moment ask not what Kant did but what he was assuming (and subsuming) in doing it. Explicitly he assumed that Knowledge was neither a property of the Known nor an instrumentality of the Knower, but the relation of Knower to Known (relata). For him then, science was analyzed into the relation (Knowledge) and the relata (Knower and Known). This science, it must be remembered, was the physics of Mechanics, Newtonian mechanics of the “particle” type, and it was one science, one in the sense that it was deductively related throughout, only the initial and final values being experimentally determined. Moreover, it was believed — or hoped — that it was the analytical relation of Knower and Known constituting Science (or Knowing). Let us for a moment put ourselves in the place of the men of that day and accept the consequences, Science — one — analyzed by Knowledge — as the one and only relation of Knower and Known, neither otherwise defined. We have therefore the conception of Science as a given, or constant, and its equivalent, Knowledge as an analytical function of the variables, Knower and Known.

These conditions led to the conclusion that the system has one and only one degree of freedom. Given a system of one analytical relation of two variables mathematically it makes no difference which is taken as the independent and which as the dependent variable. In practice the consequence of the selection of one rather than the other determines the direction of the reasoning. The choice of the Knower as independent variable leads to the notion of a “dynamic” or “generating” *ego* in the sense that the Known is then logically determinate. We have the world defined by will and idea and pass down this tradition from Kant through Fichte, Lessing, Schlegel, Schopenhauer, and Nietzsche to the psychoanalytical invention of natural man in cultural attire, the postulated ego being a metaphysically independent variable determining the nature of his known world, the precise ego postulated giving rise to the many varying sects. On the other hand, choice of the Known as independent variable leads to the conclusion that the Knower is determined by this Known. There is then no room for variability of opinion, for whatever the world is it must define its Knowers. We are thus headed for a totalitarian and intolerant state.

Of course, as with the ego as independent variable, so with the Known as independent variable the independent variable can be defined ad lib. One may conceive it to be mere matter and have communistic dialectical materialism, or hail it as idea and have nationalistic socialism. In either of these cases we have come down the tradition from Kant through Hegel whose chief interest was in the nature of Knowledge, as a relation, and as one which developed by thesis, antithesis, and synthesis, the new scheme involving a negation of some premise or premises of all that went before.

And, today, just as Kant in his day negated the assumption that time, space and causality were derived from experience, we who disagree with Kant can do so only by negating some or all of his premises. For example, we might suppose that particle mechanics was not analytical Knowledge. Or we might assume that there were two analytical relations of Knower to Known, which were not interdeducible (without reference to experience), say particle and field physics if you will. This is not the place to compare the merits of these or other negations of Kant’s premises, but it is illuminating to note that if we have two not interdeducible (i.e. independent) analytical relations of Knower and Known, then we have a system of *0* degrees of freedom, or neither the Known nor the Knower can be conceptualized ad lib, but both are determined by the system. (The only function then left to the *Ding an sich* is to answer the question: How does it happen that there is something rather than nothing?)

I strongly suspect that short of Doomsday we will become convinced that Science does exhibit Knowledge as approaching two analytical and not interdeducible relations of Knower and Known, and that we will thus find reason for our obstinate belief that the world is what it is quite apart from our experience of it, and that we are what we are, physical things, quite apart from our happening to be aware of the world. (In short, that Knower and Known are determinate.)

Moreover, there is in Kant’s own writings not only the epistemological self or Knower but the empirical self, for the Knower discovers himself within the Known. Identification of the epistemological self with the empirical self gives the Knower a determinateness which makes it seem at first as though we had here a second relation of Knower to Known, for both are Knowns in the science to be analyzed. The difficulty is with identification, for it is hard indeed to see how this is conceivable at all. We have here not A knows B, nor even A knows A, but A knows that A is A where the first A is the Knower in the original sense, the second is the Knower as an abstraction from the preceding (A knows A), and the third A is the empirical Knower as Known by himself.

Briefly, it is very questionable whether the supposed identification can be held to be an analytical relation between any relata, and if so whether this is not in one case an abstraction and in the other a relation. The problem of identification must wait for a full discussion until we return to analytical relations in Science. The identification served to keep Kant from being merely a physicist and metaphysician. It gave his Knower a determinate character which could theoretically be examined in psychology and neurophysiology. In this connection it is well to recall his famous letter indicating why the cerebrospinal fluid could not be considered as the sensorium communis. Thus, in a special sense not the philosophy which followed him but the Science of the Knower is in the direct Kantian tradition.

In my own case, as with many another, it was reading Kant that turned me for solution of the epistemological impasse to psychology and neurophysiology. But I cannot pass on without emphasizing again the importance of asking whether a given relation between given relata is analytical or not. The significance of time, space and causality in Science (Newtonian Mechanics) was that they were considered analytical relations. Kant’s critique is significant because Knowledge (mechanics) was considered an analytical relation of Knower to Known. The system has therefore only one degree of freedom, and the identification of epistemological self with empirical self does not remove this, unless it can be shown that this identification is analytical. Thus, Science and the critique of Science, and the conclusions therefrom which contribute to future political “ideologies” their apparent reason, all hinge on one pivot — Are the fundamental procedures involved analytic or merely abstractive?

Now it is well to remember that even when this mechanics was in conception its author already realized that it was not an analysis of a phenomenon but of a conceptualized real supposed to underlie phenomena. The phenomena were held to be appearances corresponding to some abstractum (*heat* to *random motion*) from the physical real. That Galileo was troubled by the consequent nature of the Knower primarily because he was at heart an Aristotelian is apparent in his own writings. Aristotelian psychology rests on a metaphysics in which the category of substance (a one and a this) permits identification of that which has more than one attribute, or from which a Knower could make more than one abstraction; moreover, there were the common sensibles of position and motion which functioned in the identification of any particular “one and a this.” It would be a great mistake to think that Galileo suffered from Hume’s difficulty, for the Galilean Knower had an Aristotelian sensorium communis, common sensibles, and the category of substance, whereas Hume’s had only “secondary qualities”.

The difficulty for Galileo was how his Knower could be composed of what he, Galileo, though real, and to this Galileo gave the answer — It can not be! And today we are compelled to say the same of the physical world that Galileo thought he had analyzed. Particles and the void are not sufficient; charges, entropy, fields, etc. have been introduced into the attempt to excogitate a world in theory that will correlate (epistemically) with the world that is what it is whether we will or no. Put thus, the difficulty seems to me apparent. The world given in theoretical physics and supposed real is a reiffied abstraction from (and within) the world that is. To mistake it for the real world is to commit the familiar “fallacy of misplaced concreteness.”

From the seductive elegance of Deductive Science it behooves us now to turn to the groping inclusiveness of induction in quest of analyses that are no more abstractions. The most general as well as the most rigorous statement of the procedures involved in inductive science are to be found in Charles Holden Prescott’s Scientific Method and its Extension to Systems of Many Degrees of Freedom. Here one finds a parameter P of the real world expressed P (X_{1}…X_{00}), and that in the theory with which it is correlated F_{j} (X_{1}…X_{n}) differing in general by E for the reason that the theory is an abstraction from which are omitted X_{n} 1 … X_{00} variables of the real world. The advance over previous theories lies first and foremost in the facility with which it indicates the lines of scientific advance, in the direction of converting abstractions into new abstractions nearer to analyses, alternately by closing down on permitted variations in Known variables until E fails to decrease and then by including a new variable.

But for us the most important problem is on what does the method of induction rest? — on correlation! On there being a whatnot that can be identified and has for that identifier two or more properties. These properties may, of course, be abstracta, but the what not must be an identifiable. In his scheme they are “events,” which by whatever properties or abstracta they be qualified, are always extended. Curiously reminiscent these “events” of Aristotelian substances, conveyed somehow by the common sensibles of position and motion, but still more obviously related to the “events” of Whitehead, which are extensions qualified by the ingression of objects and having identity as a *logical necessity*. Thus of the old trio, time, space and causality, the first two are in again, and the last is subsumed in writing P (X_{1}…X_{00}) reminiscent of Whitehead’s scientific object as a hypothetical recognizability so constructed as to explain the casual relations of events, for every X is such an “object.” But Prescott’s scheme is again specific as to the nature of this necessary connection, for F_{j} (X_{1}…X_{n}) can be extended to a new “region of correlation,” (say the future) when the region of correlation originally investigated was not defined in terms of the variables in questions, i.e., when the observed correlation is in no sense a question of the region of correlation.

But inasmuch as F_{j} is an infinity of functions equally true to the fact (for E is finite) the causality involved in determining the more proximal are the new to the observed. In other words to compute this new is to have not a single value but a circle of convergence of values, and to recompute from it back to the original is to have yet a larger circle of convergence. Moreover, the size of this circle is a function of the path, or the theory is such as not to require of necessary connection that it be analytical initially, although in the progress of science these circles of convergence always diminish, or the necessary connections become more nearly analytical.

The article in question does not contain any detailed scrutiny of the “event” as source of correlation, but I have been wondering ever since I read it whether its author might not be able to develop his scheme in terms of an identification which fell short of being fully analytical in terms of extension but was sufficient to establish the equivalent of a substance, a one and a this, which is all that is required for such a theory.

What, then, is the result of a scrutiny of inductive science? The Knowledge is abstract, but it rests on something sufficiently analytical to afford approximate identification of particular what-nots whose properties can then be correlated, and in general its development is toward analysis, by the inclusion in theory of heretofore neglected abstracta as soon as these are shown to be significant in the fact.

That this science falls short of yielding an analysis of the Known does not prevent it from depending upon some relation of Knower to Known sufficiently analytical for identification, i.e. the identity of each multiply qualified bit of extension (“events”) is a logical necessity. Moreover, this identity of that which is qualified in more than one way serves science in a second fashion, namely in giving that common world which we Know in common. Here, in order to avoid gratuitous complications, let us suppose a man takes his gun and his dog and goes after a fox; that the dog (who is color blind) raises the fox (which has to him a characteristic odor) and the man led by the dog to the fox (which is for him a spot of red moving among green things not smelled by him at any time) shoots a fox which the dog then retrieves. Man and dog had in common a world in which was a whatnot (here called a fox) qualified for one by odor and for the other by color, but still identified.

Examination of such problems always reveals that the commonness like the identification rests on extension (time and space) which, while it need not be fully analytical, must be near enough to it to make possible convergence. Thus, the common world of Science is in fact a pious Hope, rather than an achievement. Our worlds are for the most part private, and even our private worlds are not overly common to all our private sciences, the apparent commonness being usually due to a lack of rigor in the identifications involved. Such considerations make me hesitate to assume that the epistemological Knower and Known, when related by Knowledge alone, are analytical of Science.

Science itself seems rather a process for which a common Knower, a common Known and common Knowledge are ends to be approached by the devising of more and more nearly analytical abstractions than an achieved state of affairs to be analyzed in turn into Knower, Known, and Knowledge analytical.

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Wordcloud: Abstraction, Analysis, Analytical, Analyzed, Calculus, Class, Common, Considered, Correlation, Defined, Determined, Epistemological, Experience, Form, Generation, Geometry, Given, Identification, Independent, Kant, Knower, Knowing, Knowledge, Known, Logic, Mathematics, Mechanics, Metaphysics, Motion, Nature, Original, Physics, Points, Position, Problem, Question, Real, Reiffication, Relata, Relation, Science, Sense, Supposed, System, Theory, Thinking, Thoughts, Variable, World

Keywords: Calculus, Wars, Geometry, Points, World, Form, Factor, Mathematics, Position

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