Why is it that so much in human experience—consciousness, artistic insight, and emotional awareness—is left unaccounted for by mathematics and the physical sciences? And why is it that the faculty of reason becomes self-contradictory at the most exacting levels of scientific investigation? This book explores the underlying limitations of the gift of reason.
Excerpt From The Limits of Reason
This book follows one titled The Immaterial Structure of Human Experience. Though an extension of that work, it makes little direct reference to its point of view. Rather, it is written almost entirely in terms of a representationalist examination of the mind’s operations. It is limited to this viewpoint to avoid reference to the specialized terminology of the previous book. But it does not disagree with its immaterialist perspective.
What is of interest here is not the overall structure of human experience, its character, and its origin, but rather the limits of reason. This is forthrightly demonstrated by an insistence upon the fact that the foundation of mathematical science is empirical. However, this empirical influence is only initial. For mathematics is subsequently drawn into abstraction, where any further buildup of relations is imaginative and logical. But it cannot be overemphasized that the science begins with physical experience.
In keeping with its principle topic, the present work is replete with mathematical references. But its arguments and descriptions are not mathematical. There is no attempt to reason by proofs. And the mathematical concepts employed are elementary in character. What is presented is a general discussion aimed at demonstrating the imaginative origin of these ideas.
The examples which are given need not represent the path actually taken to invent mathematical concepts. They need only show that human imagination, working with the materials of ordinary experience, is all that was needed to erect the imposing edifice of this science. In sum, there need not have been, and indeed are not, innate ideas present in the human mind which transcend material experience.
It is more readily to be observed that physical science is a human creation and not something which might be thought to transcend physical experience in its fundamental principles. Yet, while it adheres to an observational organization of that experience, there is much that is not physical which is built upon the original findings. In addition, it should be noted that there are other aspects of human experience which do not fall under the purview of the scientific method of investigation.
For the above reasons, the present work develops a line of thought which does not accept any form of innate ideas. Nor can it tolerate an approach to the mathematical or physical sciences which suggests an idolatrous attitude toward them. These disciplines are in their entirety human inventions.
They are brilliant inventions. And their practical efficacy is acknowledged with due reverence. For they are viewed instrumentally as powerful tools. However, it is also understood that they are lacking in an access to final truths. These must be approached, if perhaps never fully realized, by other means.
Mathematical reasoning provides a good example of the strengths and limits of the human mind. Take geometry, which underlies much of the thinking in physical science. Prior to the nineteenth century, this geometry was essentially Greek in origin. So it might be asked: if geometry underlies a science which attempts to describe the physical world, how accurately do its geometrical figures represent physical reality? And, by extension, how closely does mathematics in general reflect physical reality?
Geometrical figures are concepts in mathematical science. As such, they are formed in the mind. So what is the relationship of these concepts to the data of what are generally thought of as the senses? Are such concepts concerning the world exact? Or are they inevitably approximations?
To understand this, a closer look should be taken at the relationship between the human mind’s powers of conceptualization and the things this mind is attempting to understand with such concepts. For the concepts, several examples from Euclid’s Elements will do: the circle, the arc, the square, the lines which compose them all, and the angles which make up the square.
Pi is integral to an understanding of the ideal circle. Yet it is an irrational number. It is irrational because it expresses a relationship between two incompatible idealizations which lie within the definition of a circle. These are a fixed radius and the resulting circumference. By an idealization is meant a concept which is created in the mind, rather than from an image which is formed by physical experience. The former can be distinguished from the latter by the fact that its existence as a concept is unique to thought and not found in nature.
The circumference, or uniformly continuous arc, of a circle is such a concept. It is defined by Euclid as equidistantly surrounding a center point. This is determined by a fixed radius.[i] If it should be postulated that this might occur in nature, it cannot be demonstrated that it does. Another similar ideal concept is that of a straight line of fixed length, as in the radius of that circle. Are there any perfectly straight lines in physical experience? And are there multiples lines of exactly the same length? And, again referring to the circumference, how many such radii would be needed to guarantee that it is a uniformly continuous arc?
These concepts stand in contrast to one which is formed directly from the experience of mental impressions like hard, cold, round, and white. These latter impressions are associated together in an image, a snowball, which is considered concrete in character, since the type of the impressions and the manner of their association are directly encountered in physical experience.
When left unaltered, it is this image which supports a concept that is true to that experience. Note that a physical concept such as this tends to be simpler than the ideal concepts just mentioned. That is to say, it is immediately familiar. But this simplicity does not mean that it cannot be a compound of multiple impressions, just as the simplest of the above ideal concepts is.
For a deceptively simple example of idealization, the definition of a line is given as a “breadthless length.”[ii] Numerous impressions make up any image of breadth or length. And the idea of something which is breadthless involves a negation as well: a conceiving of breadth accompanied by a denial of its application to this case.
A straight line is another example which is even more ideal in character. For it is defined as “a line which lies evenly with the points on itself.”[iii] In other words, if the points of this line are marked upon a plane, and the line is removed then returned to match up with at least two of those points, it will match up with any of the rest of those points which lie along its length. This differs from a curved line, which, under the same circumstances, can only make contact with all those of its points in the plane which are covered by its length if it remains in the initial position in which these points were originally laid down.
But in the case of a concrete image matters are different. Since it represents perceptual experience, it comes from nature. For example, the mental image of a snowball arises from a compound of associated impressions which are physical in character. It is a physical object which is hard, cold, round, and white. This is what the image formed in the mind conveys.
The concept of a snowball derived from that image is not markedly different from the image. This is the case so long as such characteristics as the roundness of the snowball remain understood as having been derived from physical experience and not as having been idealized by further embellishments of the imagination. In other words, the initial image will retain its concrete character, until someone deliberately begins to compare it to images drawn from other similar objects and thereby forms a more generalized concept of all of them.
From that point he is likely to proceed to incorporate the associated properties which characterize the object, as well as those which do not, into a more refined definition which no longer accords directly with physical experience. This type of abstraction is well illustrated by the concepts of a perfect circle and a breadthless line. For the perfect circle is derived from multiple instances of less than perfect circles in experience. And the breadthless line is idealized from a line with breadth, which breadth is necessary for it to be physically experienced.
So the simple definition of a snowball references physical experience. And it is understood to be a product of certain conditions such as winter, cold temperatures, and human bodily agency in forming its texture and shape, all of which are elements of physical experience. It is these, along with the light reflective properties of the ice forming the snowball, which make it hard, cold, round, and white.
Now, once again, there are two ideal geometrical concepts which are integral to the definition of a Euclidean circle. They are a continuous, unvarying arc comprising the circumference of that circle and an unvarying straight line of specific length comprising the radius of that circle. These two concepts are not only ideal. They are mutually exclusive of one another.
This is because they are independent creations of the mind. They are not either of them found in physical experience. Such experience might be supposed to supply a connection between them. But it does not. This is because any physically encountered arcs, lines, and the circles they compose are not Euclidean. They lack the precision and uniformity—one might say, the artificiality—of his definitions.
So the sole connection between these two initial Euclidean concepts—the unvarying arc and the straight line of a specific length, which are the circumference and the radius of a perfect circle—lies in the fact that they are both ideal concepts. Ideality is the only thing they have in common.
It is true that ideal concepts like the uniform arc and the straight line of a specific length, or other concepts derived from them like the circle, may be logically connected to each other or to other such concepts in a train of thought. They may even have been built up into an entire system of logically related mathematical propositions, as in Euclid’s Elements. But they are otherwise mutually exclusive. There is no connection between them, other than that of the careful relating of propositional terms which constitutes logic.
Note, for instance, that the perfect circle is offered as a definition and not as a theorem in the Elements. It does not need demonstration, or proof, and is thus given an independent status, like that of an axiom. But, in spite of the fact that it appears to be a self-evident concept, it is in truth a composite product of imagination.
After Archimedes’ work on getting a reasonably accurate, but inexact, number for pi,[iv] it has been made ever more clear that the connection between the circumference and the radius (or the diameter) is not strictly quantitative. Otherwise, today the formula for the Euclidean circle would not be C = πd (or C = 2πr), in which π is an irrational number. So, looking back to Euclid, it can be seen that the concept of a circle is clearly an imaginative invention which is more complex than that employed to define a straight line.
Since it can be asserted that the circumference and radius of a perfect circle are imaginative idealizations with no connection between them but one of imaginative invention, it is clear that they cannot have a connection between them in physical experience. In fact, their ideal character implies that they are themselves more generalized, and thus more highly abstracted, than any concept based directly on physical experience.[v]
This creative contribution of the mind is what it means to develop an idealization, let alone further developing a more complex idealization from the initial idealizations. Consequently, the perfect circle, which is derived from the concepts of the unvarying arc and multiple straight lines of a specific length, can only be an approximation of what is found in nature.
No one has ever seen a circle with a perfectly regular circumference. Nor has anyone seen a perfectly straight line, nor a multiple of lines of the exact same length. Nor, following Euclid’s definition, has anyone seen a single-dimensional line, for that matter. If it is the case that a perfect circle, or a perfectly straight line, or a multiple of lines of the same length, have ever been encountered in experience, it was not known at the time. Or, at best, one could not be certain of the fact. And certainly a breadthless line can neither be encountered nor imagined.
It is this idealization and physical unreality which renders any two geometrical concepts incompatible in their relationship to one another. Since they have no connection in nature, they are utterly distinct and separate idealizations. Hence the existence of pi, the irrational constant which imaginatively connects an unvarying arc and a straight line in an ideal circle. The indeterminate number represented by pi reflects the uncertainty of the relationship in spite of its imaginative invention. It is an unquantifiable relationship.
The justification for saying the relationship is unquantifiable is to be found in Euclid’s definition for an ideal circle, which is stated as:
A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another.[vi] [The italics are added.]
So, to simplify this discussion, let it be said that the straight line—the radius of the circle—is determined at a measure of 1/2. That would make the diameter 1, and thus, using the formula πd, a measure of the circumference which is π is arrived at. Pi is an irrational number, which is indeterminate. Thus the circle’s circumference is indeterminate in this case.
How is this so? It is so because the relationship between the circumference and the radius is by definition always indeterminate. This is because the ideal perfect circle is defined in terms of all the radii falling upon its circumference. The “all” is an indeterminate number of radii, which in turn implies an indeterminate number of points of contact of those radii with the circumference. Thus the circle’s circumference cannot be computed in terms of a rational number.
Why is this? Imagine a kink in this circumference, a sudden angular change of direction in the arc. Such a bend might be microscopically small. It could be indeterminately small. Yet the bend would contradict Euclid’s original definition of a circle. Thus an indeterminate number of radii, and a correspondingly indeterminate number of points in which the radii make contact with the circumference, must be postulated for the definition to hold. This is precisely what the indeterminate, or irrational, number π represents, insofar as it is an irrational number.
But again why, it might be asked, is it possible to have a determinate circumference, which would then require an indeterminate diameter. For instance, this would be the case if the diameter were 1/π. That would make the circumference 1. This occurs because incompatible concepts are being related, one determinate, the other indeterminate.
So, if one of them is assumed to be definitively rational—i.e., determinate—this renders the other indeterminate, or irrational, because there is something indeterminate in the relationship between them. If the diameter is determinate, the circumference is not. If the circumference is determinate, the diameter is not. Hence pi, the constant of indeterminacy which stands as the relationship between them.
It follows from this logical but quantitatively inexact relation that the Euclidean circle is an idealization brought about by the union of two mutually exclusive ideals. In Euclid’s Elements this is accomplished by fiat. It is set down as an initial definition,[vii] thus avoiding the inconvenience of a logical proof. Why the avoidance is not clear. Perhaps the circle seemed too obvious to be a concept which required demonstration. At any rate, its status as a definition obscures its distance from nature.
But let this discussion not stop with the circle. To get a sense of how pervasive this problem is, let a near relation to it be examined: the ideal square. Here too an irrational number will be discovered. It is the square root of two. If a square is constructed with sides of one unit length, by means of the Pythagorean theorem a diagonal which is the square root of two will be obtained. This is very much like pi, inasmuch as it is an irrational number.
The square root of two is always involved in the relationship between the sides and diagonal of a square: = 2, “d” being the diagonal and “a” each of two adjacent sides of the square. Therefore d = a or a = d/. Thus, if a, or each of the sides, is ,the diagonal can be rational. It would be 2. But if the sides of the square are a rational number, the diagonal is irrational.
Now it is clear in some cases that, if the sides of the square are an irrational number other than —say they are —then both the sides and the diagonal can be be irrational at the same time. However, it will not be the case that the sides of the square are rational and the diagonal is also rational.[viii]
So there is an irrational factor, namely , always at play in the relationship between the diagonal and the sides of the square. This presence of an irrational factor in the relationship between the diagonal and sides of a square is similar to that which is found between the diameter and circumference of a circle. . . .
Furthermore, the four corners of a square are right angles. Thus the perimeter of a square encloses an area of 360°, just as does the circumference of the circle that circumscribes it. Though it is true that their areas are different, their bisecting lines are the same. The diagonal and diameter are equivalent. Is it any wonder then that such a diagonal might be represented by an irrational number, either in itself or in its relation to the sides of its square? It certainly has this relationship to the circumference of the circle which circumscribes it.
If the diagonal were a rational number, the sides of the square would not be so. This is simply a result of a transference in the relationship. The square root of two continues to appear either in the sides of the square or in its diagonal. There is always an irrationality in the relationship. This is similar to the relationship between the circumference and diameter of a circle.
It can easily be seen that it is the relationship of the diagonal of the square to the circumference of the circle that is of concern. In other words, what matters is the fact that a square can be inscribed within a circle, thus making its diagonal the diameter of the circumscribing circle. As a result, when relating the sides of the square to its diagonal or the circumference of the circle to its diameter, there is an irrationality in the relationship.
But how is this irrationality caused? It is caused by an indeterminacy which is to be found both in the ideal square and in the perfect circle. This problem of indeterminacy occurs in the perfect circle because what is being attempted is to determine the uniform curvature of an arc by means of an indeterminate plenitude of equivalent straight lines.
The circumference is indirectly defined by Euclid as a perfectly unvarying curve. For, if “all the straight lines falling upon [the circumference] from one point among those lying within the figure are equal to one another,”[ix] then the circumference is a perfectly unvarying curve. The words “perfectly unvarying” may not be his. But they are his meaning.
Can a perfectly unvarying curve be physically demonstrated? It cannot because it is an ideal. It is an ideal abstraction from physical reality. The ideal abstraction has conceptual validity as a thought. And the varying curve of a natural circle (as opposed to “perfectly unvarying”) has physical validity as something in nature. But an ideal thought is not physical. It is a free product of imagination, which rearranges the perceptual content of physical experience. So the leap between physical experience and the ideal cannot form a determined connection.
The same thing occurs with a square because the concept of a square is ideal. It is not found in nature. What specifically is not found is a perfect right angle. Nor can four absolutely equivalent right angles covering the full revolution of changes in direction of the circle’s circumference be found.
Nor can four absolutely equivalent straight lines be found. If any of these things were found, they could not be identified. So the square, composed of four right angles and four absolutely equivalent lines, is one idealization set among the many natural quadrilaterals to be encountered in the world, none of them precisely measurable.
It is in this way that the irrational character of both the numbers under present consideration (pi and the square root of two) reflects the ideality of the perfect circle and the square. An irrational number is specifically an indeterminate number. And there is something indeterminate in the ideal composition of both the circle and the square. But the question remains: why is the relationship between the diagonal and sides of a square indeterminate?
Of course, it is a relationship between two ideal concepts: that of a perfectly straight diagonal line and that of the square’s perimeter with its precisely indicated angles and straight lines of equal length. But this sort of thing is true of all representations of geometrical figures. Any ideal geometrical representation is potentially incompatible in the relation of its parts. For the parts are individually ideal. Thus their quantitative measures and the relationships between them are assumed ideally.
But most of these figures do not exhibit relations which are assumed to be incommensurate by definition. In other words, an incommensurate relationship between the parts of a geometrical figure is generally not explicitly stated in the manner in which it is in Euclid’s definition of a circle. A perfect circle clearly exhibits an incommensurate relationship, when its definition specifies that the number of radii needed to secure an unvaryingly uniform circumference is indeterminate.
So there must be something to further explain the similar relationship between the diagonal and sides of a square. And that something is the fact that the relationship between the diagonal and sides of a square yields an irrational number. It does so because the character of the square results from the nature of its enclosure in the circle which circumscribes it. It results in the square root of two being directly linked to pi.
This link is initially foreshadowed by the equivalence of the diagonal of the square and the diameter of its circumscribing circle. But, in addition to this, the circumference of the circle is proportionately linked to the four equal sides of the circumscribed square. For the sides act as chords apportioning the circumference into four equal arcs. In turn, the four equal arcs together comprise the full circumference of the circle. Thus, as the relationship between the diameter and circumference of the circle is incommensurate, so the relationship between the diagonal and perimeter of the square is also incommensurate.[x]
As already indicated, all geometrical figures are ideal, though they may or may not involve indeterminate relations which are definitionally specified. Nonetheless, the ideal character of their internal relations does imply an uncertainty in the relations of their component elements. For, as previously stated, regardless of the geometrical figure it belongs to, the exact measure of a line is unknown in physical experience. Nor can there be an exact sense of the measure of any angle in physical experience.
The precise character of an angle cannot be physically known because that character is ideal. It is derived by logical deduction from the circumference of a circle, or by a revolution through a Cartesian plane. One complete revolution is 360°. Thus a 90° angle is one fourth of 360°, etc.
Trigonometric functions might appear to contradict this claim about the vagueness of angles because they express angles in terms of a ratio of lines—i.e., the sides of a right triangle. But that merely transfers the problem to the indeterminacy of lines. Moreover, most of these trigonometric values are found to be irrational (or indeterminate) due to the commonly incommensurate character of the sides of a triangle, which only further supports the thesis of ideality.
So, if angles are to be referenced to a circumference in some manner other than an arbitrary division into degrees, the proportions of that circumference can only be determined by the circumference’s relationship to the circle’s radius. Hence the use of radian measure. However, this measure is indeterminate, as it involves pi. So there is a numerical impasse in working out such geometrical concepts, should they be supposed to reflect exact quantitative relations.
In general, it would appear that geometrical concepts belong to their own logical and conceptual reality. Thus they do not arise directly from the physical world. But they are certainly derived from it indirectly. They are idealizations and regularizations of what can be found in irregular plenitude in the physical realm. If mathematical concepts were not thus operatively connected with the physical world, they could not reflect definitive physical results.
Nevertheless, the fact remains that the concepts are ideal. For conceptual thought is made independent of any mental representation of perception through a process of abstraction from nature. And the most highly abstracted type of concept is ideal. It is therefore not directly, but only indirectly, representative of nature.
So, following the implications of this analysis, similar arguments may extend beyond the circle and the square to other geometrical figures, insofar as the ideal nature of such figures is concerned. However, what has been unique to the square and the circle is the definitionally irrational character of the relationship between certain of their constituent parts. This is not the case in a figure like a triangle. For an indeterminate relationship between its components is not definitionally alluded to.
Now let the square’s symmetrical relationship to its circumscribing circle be looked at again. A square is proportionately linked to this circle in both its angles and its sides, not to mention the identity between its diagonal and the circle’s diameter. It is this close multiple relationship which makes a square and a circle so similar in terms of the irrational relationship between their bisecting lines and their perimeters.
Any polygonal figure described by straight lines must be defined by angles. Both the angles and the straight lines would be physically imprecise in character if sought out in experience. It is for this reason that they are considered to be ideal. An attempt to relate angles to lines with quantitative precision is a purely intellectual activity. All of Euclid’s polygons are therefore ideal, though this may not be as apparent as it is in the case of the circle and the square, where one is confronted with a definitionally specified quantitative dissociation of component concepts.
The point in presenting this discussion of arcs, circles, lines, angles, squares, and polygons in general has been to emphasize that they are all idealizations. This can be demonstrated for all of Euclid’s geometrical figures, even if their ideal character is more subtle and must be analyzed in ways other than that of the circle and the square. For example, as can be seen, a rectangle is not readily susceptible to an analysis resembling that of the circle and square. But it does have ideal components, such as its four right angles and its straight lines of specific lengths.
The concept of angles is derived from the circumference of the circle, which is ideal in definition. The lines which constitute the sides of a rectangle are also ideal in definition. They are thus conceptually separate from angles, insofar as their abstraction from the physical world is concerned. They are concepts formed independently from, and without immediate regard to, the concepts of angles. Though, of course, it would not be feasible to execute a graphic representation of an angle without lines.
So angles and lines are not only independent of one another in their initial development as concepts. They are incompatible in a certain sense. For, as ideal abstractions independent of one another, they have no connection to one another in nature. To form a rectangle, they must be brought into union by an increased process of idealization as a result of an imaginative connection being made between them.
For example, it is an intellectual process which informs a person she has encountered an angle in a cliff or shoreline. The concept “angle” does not correspond directly to natural phenomena. Such a concept entails a precision regarding one angle in relation to other angles, which is a product of thought, however primitive in development that thought may be. Without some sort of mental processing, there would simply be a sense of proximity and recession in physical experience. It is the intellectual faculty of the mind which refines primitive experience by means of the development of an overlay of such precision.
So the kinds of geometrical angles and lines seen in Euclid are put together by imagination. In other words, a new geometrical concept, such as a Euclidean triangle or rectangle, which is hewn from the definitions of angles and lines, must be built in the mind without further reference to nature.
That new concept, say the rectangle, is intellectually, not experientially, derived from its constituent ideal parts, the line and the angle. Of course, imperfect trilaterals and quadrilaterals, which resemble triangles and rectangles, are encountered in the physical realm. But they lack the ideal relations of these figures.
All geometrical figures were intended by Euclid and his predecessors to represent the physical world. And they were thought to do so until fairly recent times. Yet these geometrical figures of the mind do not now, nor ever did, exist definitively within the physical world. Rather, it must be said that they have been employed to map it.
But no matter how close to physical experience their representations may be thought to be, however refined they are, they are not exact equivalents of their physical counterparts. They are conceptual approximations of them. Thus a good terrain map will carry a military patrol through unfamiliar country. But the map could not be imagined to stand in for all the features, visible and hidden, of that country. Nor are the map’s terrain features exact equivalents of the few selected physical forms they are intended to represent.
Overall, what this discontinuity between abstract concepts points to is the simple fact that human reason—perhaps especially in its most impressive idealizations, such as the square and the circle—is a fallible instrument. It is makeshift. Needless to say, it works very well in the investigation and control of specific aspects of nature. But what works is not necessarily what is true in an absolute sense.
Mathematical theorems and scientific theories are conceptual and logical in character, since they inevitably involve idealizations and associations between those idealizations. The initial images are representative of objects in the physical world which are refined by imagination (i.e., made simpler and more regular). In this way, each of them is converted into a concept by the precise application of a definition peculiar to its properties.
In the case of mathematics, this is why the elements which contribute to a geometrical theorem must be conceptualized as definitions, and the relations between them as axioms. Subsequent to this initial conceptualization, geometrical figures can be logically (which is to say, associatively) derived from such definitions and axioms, and from other figures already so derived. For the whole of a theorem is abstractly conceptual, both the derived proposition and its constituent terms.
Euclid’s definitions and axioms provide a foundation for his propositions in just this way. The theorems are determined by the logical structure of mathematical science. The logical structure is the set of rules by means of which prior idealizations are put together to create new ones. The rules, which are conceived to be consistent among themselves, determine specified relations of association.
In other words, specific rules apply according to regularized methods of association, as opposed to other rules which might employ other possible methods of association. It is in this way that in arithmetic the assumed equivalence of arithmetical units and the character of the number line are used to determine the numbers 12, 7, and 5, and then, by extended use of these same preliminary assumptions, to further determine that 12 minus 5 should equal 7.
Likewise, a perfect circle must be composed of multiple equal straight lines of a specific length and of a circumference of unvarying curvature because this logical compounding of these prior abstract concepts is how it is defined. The equal straight lines and the unvarying curve are brought into an association with one another by the definition. But none of this gives any of these ideal concepts a special existence either in nature or in some transcendent spiritual realm. They are imaginative idealizations originally derived from abstractions taken from nature.
It is the same with physical science. Experimental verifications of observed phenomena are dependent upon the structure of a theory and the perspective it supplies. Any such theory is logical in character and made up of ideal concepts. It is therefore abstract. A theory has initial components, which are its definitions and axiomatic assumptions. These components and their logical derivatives generate a hypothesis for a further development of the theory. It is this hypothesis which must be tested to get affirmative or negative results.
In fact, the preliminary situation must begin with concepts of the natural components brought under observation. Then the results of experiment must be conceptualized as relations of the observed components. All the concepts and relations will be in some degree ideal, even the most experiential of them. For they are mental constructs conditioned by a theoretical perspective. To the extent that these things are done, a rational structure is established which is parallel, but not equivalent, to nature. This is the experimentally confirmed theory.
For example, if the charge and mass of an electron are measured, as Sir J. J. Thomson did, is it truly known what has been measured? A person does have a conceptual grasp of what has been done. But the physical ground of that research is less certain. For, as regards the relationship between the conceptual and the physical, there must always remain some doubt.
Insofar as any of these concepts, relations, and theories involve idealizations—that is, inasmuch as they involve thoughts that are refined abstractions from physical experience—they are only productive of approximations of what is experienced in the physical world. They are like terrain maps, very sophisticated templates placed over the inscrutable whole of physical reality. Science can get where it is going only because these maps are good enough to get it there, not because science has the fullness of reality in its grasp.
The Right Angle
To begin, let a quadrilateral be imagined. It is desired that this particular quadrilateral should have equal sides. So this will be assumed. And the quadrilateral will be called a rhombus. In addition, the sides of this rhombus will be set perpendicular to one another. This is to say that each side will extend from another in such a manner that, if the base line were extended beyond both sides of its adjoining line, the adjoining line would be no nearer to the base line on one side than on the other.
It can now be noted that, when straight lines are drawn between vertices which are not constructed from the same sides of this quadrilateral, the lines will intersect. And the point of intersection will appear to be equidistant from the four sides. The resulting figure will be called a square. And the intersecting lines will be called diagonals. They are being given these names because they have just been discovered in the manner described.
Furthermore, it can be seen that there are four angles between the diagonal lines at the point of their intersection. Accordingly, there is one of these angles in each of four three-sided parts which together make up the quadrilateral. Let the angles be called central angles. And let the parts be called triangles.
Having done these things, the mind has imaginatively created a square. But it does not yet know conceptually what constitutes a square, other than that the figure has more or less equal sides which are more or less perpendicular to one another. For it is hard to get things exact in a mental picture or a drawing. So let such a figure continue to be imagined and see what is found. . . .
It should be noted as well that each of the equal sides of the four circumscribed triangles is shared by two of the other triangles. Thus there are only four radii among the lines presently drawn within the circle. Because all these sides of the triangles are now accepted as being equal to one another, the possibility that the four adjacent triangles are congruent can be entertained.
Are not the four sides of the square presumed to be equal? These are the bases of the four triangles. That would make all three sides of any one of the triangles equal to the corresponding three sides of the other three, which, from a purely pictorial perspective, makes the triangles appear to be congruent. So the square must be made up of four congruent triangles, each of which has two equal sides. Let them be called isosceles triangles.
It is now fair to assume (not in strictly logical terms, of course) that the four central angles should be equal because the triangles are symmetrical to one another in appearance. If the triangles should be stacked on top of one another with the four central angles at the same end, which has already been suggested by their congruence, it seems likely that the central angles would be equal. For the base sides of the triangles would have to be made comparatively longer or shorter to render these apical angles unequal.
But, as has already been indicated in describing the square, the base sides of the four triangles are, in fact, equal. For they are the equal sides of the square. The same is true of all the base angles of the triangles. They are equal as well. Or at least it would seem that they should be, since the triangles appear to be congruent.
And, the two sides extending from the central angles being equal, the congruency would be the same no matter which face of the triangle is laid against another. So this is precisely what is desired of the square, that it is a quadrilateral composed of four equal parts—parts which are equal to one another in every way.
The fact that the square’s diagonals are equal can now be added with even greater assurance to its provisional character. For the diagonals are diameters of the circumscribing circle. And the diameters are assumed to be equal because each of them is composed of two equal radii proceeding from the center of the circle to its circumference.
That is Euclid’s definition of the circle. But it cannot be a final definition. For a final definition would not only state, but suggest a means of demonstrating, the correctness of the properties involved. These properties have only been imagined and visually compared in this discussion. And they have been definitionally proposed without proof by Euclid. Neither an imaginative supposition nor an axiomatic approach will do for the purpose of accepting a final definition.
So, in addition to what had been previously stated, the square can now be divided into four equal parts, as has been indicated. Therefore, assuming the equality of the diagonals (which are diameters of the circle as well), and imagining as many diameters as may be desired, the regularity of the circle’s circumference may be assumed.
Of course, this regularity is not known to be the case because imagination cannot supply a sufficient quantity of diameters to demonstrate that it is so. But, since all the diameters which can be imagined are properly circumscribed, the arc of the circumference making appropriate contact with the ends of each half diameter, or radius, it is at least pictorially probable that the circumference would form an unvarying arc.
Moreover, it can be seen that the equal bases of each triangle, which are equal sides of the square, are the sides of each triangle which subtend a central angle—i.e., an angle at the center of the circle and square. From this observation, and from the fact that these central angles are presumed to be equal, it can be assumed that the sides of the square demark equal portions of the circle’s circumference.
So, since the lines which make up the sides of the square touch the circumference at both their limits, the sides of the square can be designated as chords. And the demarked portions of the circumference can be called arcs. There are four of these arcs, as there are four sides of the square. Let the areas between the arcs and their respective chords be called segments. They abut one another at the points where their chords touch the circumference. Thus the four arcs together express the full circumference of the circle.
Now, since the chords are equal and each of the arcs is presumed to be uniformly curved in the manner of the others, it is for this reason that the arcs would appear to be equal. But it must be granted that this assumption, as well as others which have been made, involves logical deduction: if such-and-such, then such-and-such.
Such thinking is inescapable. For it occurs in this case at a most fundamental level, which is that of Euclid’s first and fourth common notions:
Things which are equal to the same thing are also equal to one another.
Things which coincide with one another are equal to one another.[xv]
In fact, so fundamental are the common notions, even to imaginative thinking, the use of them appears frequently throughout this discussion.
This is because the common notions are direct associations. And direct associations are the work of imagination. They become principles of reason, or logic, only when defined as such. The simplest of mathematical operations are likewise directly associative in character when performed with small numbers, as can be seen in the next paragraph, where simple division is a matter of subtraction.
It is generally accepted that the circumference of any circle is arbitrarily designated to be 360°. The number 360 is even and therefore separable into equal parts. These equal parts are also even numbered. So they can be separated in the same manner again. Thus the entire operation follows the common notion:
If equals be subtracted from equals, the remainders are equal.”[xvi]
So, if the circumference is 360°, and the four lesser arcs are identical, then each of them must be a quarter of the whole, or 90°.