The whole discussion as to the possible existence of space of four or more dimensions is founded on certain technical meanings given to the words "dimension" and "space", and the assumption that mathematical processes necessarily imply corresponding realities. Mathematical formulæ often contain negative quantities, and even impossible quantities, and yet lead to certain conclusions which may be true. But we do not therefore believe that there may be a universe in which all quantities are negative, and another in which all are impossible.

    Space is defined to be "extension as distinct from material substances", but this is hardly general enough, and I think it would be best defined as "that which contains all things both actual and possible," or words to that effect. This is certainly what we all understand when we speak of the necessary infinity of space. It has been tersely said, "space is nothing, but it contains all things"; and again--"space is a sphere whose centre is everywhere and circumference nowhere".

    With such a conception of space as this, the idea that it is of three, or any other number of "dimensions", is absolutely unmeaning, if not absurd and self-contradictory. The so-called "three dimensions" of space, are really only three dimensions in space taken from three axes at right angles to each other supposed to be fixed in position and direction. By means of these, any regular curved lines surfaces or solids can be measured, and defined by means of equations which determine the nature of the curve; and the science which does this is termed Analytical Geometry. But it will be at once seen that the varying coordinates by which this is effected only deal with the measurement of portions of space defined by points lines or surfaces, and have no possible relation to any properties, "dimensions" or limitations of space itself.

    Let us take another example. We can assume a straight line to be of any length in feet or miles &c. Let us call this length a. Now if we multiply a by any other number b we get a number represented algebraically by ab. This is a purely arithmetical result and is used as such in innumerable processes of arithmetic and algebra. But it also gives us a mensurational (or geometrical) result, for, using feet as our unit of measurement, ab will be the number of superficial feet in a rectangular figure a feet long by b feet wide. So, if we multiply abc together, the result is essentially arithmetical, but it also gives us the number of solid or cubic feet in a rectangular solid a feet long, b feet wide, and c feet thick. All this is true for any possible values of a, b, and c. Now a special case of this general rule is, when we take the three quantities a, b, and c, to be equal. Then the result in arithmetic receives a special name, aa is called the square of a, or a²; while aaa is a³; and a multiplied by a three, or any number of times, is said to be raised to the fourth fifth or any higher power of a determined by the number of a's multiplied together. The result will be purely arithmetical, though expressed by a different formula. If we apply these higher "powers" to geometry, we must return to the original arithmetical form. Thus a³, will either give us the number of superficial feet in a rectangular surface aa feet long by a feet wide, or of solid feet in a cube of a feet long wide and high. So a⁴ will give us the area of a square aa feet x aa feet, or in a parallelogram a feet x aaa feet, or of a solid figure (a parallelepiped) a wide, a high, and aa long. So, a⁵ is a mere number, but it will also give us the solid feet in a parallelepiped a feet wide, a feet high and aaa feet long, or of one aa feet wide aa feet high and only a feet long. And so, for any power of a, it may give us either a superficial area or a solid area of rectangular figures whose dimensions multiplied together make up any special power, which we may term aⁿ.

    But according to the mode of reasoning of the mathematicians from their very convenient method of defining curved lines or surfaces, the mere fact that, while a may be used to measure a line, a² to measure a surface, and a³ to measure a solid, therefore a⁴ must be the measure of something that is to a solid, as a solid is to a surface or a surface to a line. And if this argument is sound it implies that there are, or may be, as many distinct categories of bodies or forms (differing in nature as do lines, surfaces, and solids) as there are powers of numbers--in other words that they are infinite.

    Returning now to the alleged different kinds of space, of one, two, or three dimensions, and the argument from this supposed fact, that there may be space of four, five, six, or any number of dimensions, we shall find the conclusion to rest on fallacies of nomenclature and reasoning. A line is said to be "space of one dimension"; a surface "space of two dimensions"; a solid "space of three dimensions". But this is a misuse of terms. What we mean by Space is, that which contains all things. However we limit or divide "space" it must still be capable of containing "something", or it would not be space. A surface, therefore, which has no thickness, is not space, but only the limit of a certain portion of space; and a line, which has neither breadth nor thickness, is not space but merely that which limits a surface or defines the intersection of two surfaces. The phrases, "space of one dimension", of "two dimensions" &c. are actual contradictions in terms. Space, being necessarily infinite in every direction and every portion being necessarily extended in all directions cannot be in the slightest degree affected by any methods we adopt to measure portions of it; and the whole argument from the supposed three kinds of space we know, to a fourth or tenth kind which we do not know and cannot conceive, is of exactly the same nature as that which, from a line, a square, and a cube, argues that there must be something which is to a cube as a cube is to a square, and so on to all infinity.

    The attempt to popularise the argument by supposing intellectual beings to live within surfaces and lines from which they have no senses that enable them to perceive outer space is a ludicrous begging the question. A line or a surface, having by definition no thickness, is not space and cannot contain anything. To suppose an intelligence forming a part of nothing, having a nothing for its dwelling place, which yet confines it and its faculties as with opaque walls of adamant, and to attempt to show what conception of space a being thus situated could or would form, and to reason from this imaginary and impossible being's imaginary conclusions, that our present conception of space is totally erroneous--that it is not infinite and does not contain all things--is certainly the very height of midsummer madness.

    The extraordinary looseness of reasoning of even good writers is shown by the following words in the article Dimension in Chambers Encyclopedia. After stating that a line is of one dimension, a plane surface has two--length and breadth; while a solid is said to be of three dimensions--length, breadth, and thickness, he proceeds-- "Thus it will be seen that, by the term dimension is meant a direction in which extension may be reckoned or measured. The three last-named dimensions are found sufficient to determine all known forms of extension. Hence space is tridimensional."

    The last four words embody a statement as to the nature of space itself, derived from one purely technical formula used by mathematicians to determine the forms or the dimensions of certain portions of space. Logically, there is no connection whatever between the premises and the conclusion.

    I have now shown that all assertions as to the "dimensions of space" (whether two, three, four, or infinite) rest upon confusion as to the meaning of words and deriving results from such incorrect or illogical meanings. Space is said to be, actually, of one, two, or three dimensions; whereas the first two are in no sense space, or even portions of space, but only the limits or surfaces of portions of space. The third, is only a term applied to the mode of determining the shape and size of certain portions of space, and is wholly inapplicable to Space as a general term, which is by definition immeasurable. The other confusion of meaning is as to the word dimension; which, as popularly used refers to the size of any object; but, as used by mathematicians, refers to the varying lengths of one, two, or more co-ordinates as measured from imaginary rectangular axes anywhere in space. Because they can determine the position of lines, surfaces and solids by the use of one two or three such axes, they conclude (without any logical justification whatever) that this particular method of measurement of portions of space, demonstrates a quite unthinkable limitation of Space in general.

Alfred R. Wallace