Martin Gardner | The Church of the Fourth Dimension

The Church of the Fourth Dimension is a chapter of Martin Gardner‘s book The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems. It gives a nice and (relatively) simple introduction to the mathematical concept of a transcendent, trans-dimensional elsewhere, where perhaps Gods and souls can be found.

From the book’s Amazon page: “[Gardner’s] stunning exploration of The Church of the Fourth Dimension bridges the disparate worlds of religion and science by brilliantly imagining the spatial possibility of God’s presence in the world as a fourth dimension, at once everywhere and nowhere.”

According to Wikipedia, Martin Gardner was a fideistic deist, professing belief in a god as creator, but critical of organized religion. He stated that he would not rule out a priori the possibility that as yet unknown paranormal forces may influence the physical world.

I used to enjoy a lot, as a teen, Gardner’s Scientific American columns on mathematical games and puzzles, but also on “real” mathematics and physics. Of course, I learned much more from him than from my unbelievably boring (and often not correct) school books. He had a knack for making complex scientific concepts intuitive, visualizable, understandable, and fun.

The story is a fictional report of Gardner’s visit to a “Church of the Fourth Dimension” and discussion with Reverend Arthur Slade, Minister.

Excerpt of Slade’s sermon:

“Our cosmos-the world we see, hear, feel-is the three-dimensional “surface” of a vast, four-dimensional sea. The ability to visualize, to comprehend intuitively, this “wholly other” world of higher space is given in each century only to a few chosen seers. For the rest of us, we must approach hyperspace indirectly, by way of analogy.

“Imagine a Flatland, a shadow world of two dimensions like the shadows on the wall of Plato’s famous cave (Republic, Chapter 7). But shadows do not have material substance, so it is best to think of Flatland as possessing an infinitesimal thickness equal to the diameter of one of its fundamental particles. Imagine these particles floating on the smooth surface of a liquid. They dance in obedience to two-dimensional laws. The inhabitants of Flatland, who are made up of these particles, cannot conceive of a third direction perpendicular to the two they know.

“We, however, who live in 3-space can see every particle of Flatland. We see inside its houses, inside the bodies of every Flatlander. We can touch every particle of their world without passing our finger through their space. If we lift a Flatlander out of a locked room, it seems to him a miracle.

“In an analogous way, Slade continued, our world of 3-space floats on the quiet surface of a gigantic hyperocean; perhaps, as Einstein once suggested, on an immense hypersphere. The four-dimensional thickness of our world is approximately the diameter of a fundamental particle. The laws of our world are the “surface tensions” of the hypersea. The surface of this sea is uniform, otherwise our laws would not be uniform. A slight curvature of the sea’s surface accounts for the slight, constant curvature of our space-time. Time exists also in hyperspace. If time is regarded as our fourth coordinate, then the hyperworld is a world of five dimensions. Electromagnetic waves are vibrations on the surface of the hypersea. Only in this way, Slade emphasized, can science escape the paradox of an empty space capable of transmitting energy.

“What lies outside the sea’s surface? The wholly other world of God! No longer is theology embarrassed by the contradiction between God’s immanence and transcendence. Hyperspace touches every point of 3- space. God is closer to us than our breathing. He can see every portion of our world, touch every particle without moving a finger through our space. Yet the Kingdom of God is completely “outside” 3-space, in a direction in which we cannot even point.”

Flatland from above, from What the Bleep Do We Know!?

Image source: What the Bleep Do We Know!?

In the ensuing discussion, Slade outlines other parallels between 4-dimensional mathematics and religion to illustrate the concepts of transcendent reality, afterlife, and resurrection.

“Henry More, the 17th-century Cambridge Platonist, was the first to regard the spiritual world as having four spatial dimensions. Then along came Immanuel Kant, with his recognition of our space and time as subjective lenses, so to speak, through which we view only a thin slice of transcendent reality. After that it is easy to see how the concept of higher space provided a much needed link between modern science and traditional religions. [Slade finds essential truth in all the great world faiths.]”

“Our bodies are simply three-dimensional cross sections of our higher four-dimensional selves. Obviously a man is subject to all the laws of this world, but at the same time his experiences are permanently recorded-stored as information, so to speak-in the 4-space portion of his higher self. When his 3-space body ceases to function, the permanent record remains until it can be attached to a new body for a new cycle of life in some other 3-space continuum.”

Here Slade outlines a possible mechanism for resurrection: our “souls” (defined as the whole of our experiences, memories, thoughts and feelings) are stored beyond the reality that we perceive, and perhaps they can be retrieved (by God, or by future science) and restored to life.

In his final remarks, Gardner says:

“At the time I wrote about the Church of the Fourth Dimension no eminent physicist had ever contended that there might actually be spaces “out there,” higher than our familiar 3-space. (The use of a fourth dimension in relativity theory was no more than a way of handling time in the theory’s equations.)

“Now, however, particle physicists are in a euphoric state over a theory of superstrings in which fundamental particles are not modeled as geometrical points, but as extremely tiny closed loops, of great tensile strength, that vibrate in higher spaces. These higher spaces are “compacted”-curled up into tight little structures too small to be visible or even to be detected by today’s atom smashers. Some physicists regard these higher spaces as mere artifices of the mathematics, but others believe they are just as real as the three spaces we know and love.”

Imagining our 3-dimensional space as a hyperplane in a 4-dimensional space is a useful way to visualize a transcendent reality, as Reverend Slade says. Of course we cannot see in 4 dimensions, and we can only visualize this concept by thinking of our space as a plane in a 3-dimensional space.

But even thinking of a 4-dimensional space is probably too naive, and reality may be much weirder. Now physicists talk of 10-dimensional space, or maybe 11-dimensional, but I don’t see a reason for 11 instead or 10, or 12, and I guess that perhaps real reality has infinite dimensions.

I often use in presentations the two pictures in this article, (source: What the Bleep Do We Know!?). Who is the old gentleman who observes us from above in his higher reality, and eventually lifts us into his higher reality?

Flatland from above, from What the Bleep Do We Know!?

Image source: What the Bleep Do We Know!?

  • Your focus on Gardner brought up a short quote of his from an interview. It is simple, but logical. Gardner invoked superstring(s) from physics:

    “ ..you can defend immortality on the grounds that everything that constitutes our selves or our identity is a mathematical pattern. If superstring theory turns out to be true, they you can ask what are superstrings made up of, and they aren’t made of anything! If all matter is pure mathematics, then you can imagine that an all powerful deity who knew the pattern could reconstruct you. .”

    I quoted Garnder from this link-

  • Giulio Prisco

    @Spud – thanks, very interesting article.

    I am not a Mysterian, because I believe science is in principle able to understand anything. Having said that, I don’t rule out the possibility that the explanation of some phenomena may elude science for a long time, and require physics that we don’t know yet.

    I see a third way (yes, I am addicted to third ways) beyond ultra-rationalism and Mysterianism. If science is in principle able to understand anything, it may never achieve understanding of everything.

    Using Gardner’s metaphor, suppose we find out that some phenomena in our 3D space can only be explained by embedding it in a 4D space with more fundamental physical laws. And then we find out that some phenomena in the 4D space can only be explained by embedding it in a 5D space with more fundamental physical laws. And then 6…

    In this scenario, science gradually understands more and more things without ever running into a barrier, but there are always infinite “more things in heaven and earth” beyond the understanding of current science.

  • Aliaksandr Yemialyanau

    Rotation in four-dimensional space.

    The 5-cell is an analog of the tetrahedron.

    Tesseract is a four-dimensional hypercube – an analog of a cube.

    The 16-cell is an analog of the octahedron.

    The 24-cell is one of the regular polytope.

    The hypersphere is an analog of the sphere.