Some have still not understood my contention that mass cannot exist at a point, and that the point cannot exist in physics. Since this contention is at the heart of my critique of the calculus, General Relativity, QED, and string theory, I thought it best to give it a paper of its own.

In my paper on the calculus, I exhaustively showed why the point cannot exist in any equations. Fundamentally this is because mathematical equations—in physics or out of it—consist of numbers, and the numbers we use operationally stand for lengths or intervals or differentials. Any number or variable or function stands for some extension, and therefore we cannot find solutions at points. I showed that it was a very basic definitional problem, and that it had existed since the time of Euclid.

Here I will show in more detail why we cannot have an object or a mass or an event or any material at a point. This should have been proved by my calculus paper, since if you cannot have any math at a point, you can hardly have any physics at a point. But I have further arguments against mass at a point, and it is possible these will be convincing to some readers where my calculus arguments were not.

A singularity is not just math at a point, it is mass at a point. It is believed by modern physicists that objects like black holes can disappear into singularities, and that the universe itself was once a singularity. Since the black hole and the universe are objects with mass, the claim must be that mass equations can have solutions at a point. They can't, simply because no possible mathematical equation can have a solution at a point: we would have to write our equations as the solution of ordinal numbers, and that can't be done. But they can't for other very basic reasons, and those reasons are the ones we will look at here.

Mass is defined as density times volume. In other words, it is a density in a volume. So if your volume goes to zero, your mass must also go to zero. You cannot have mass in a zero volume, by definition of mass and volume. But a point must have zero volume, by definition. Therefore you cannot have mass at a point. And yet QED and GR both try to assign mass to points. The field equations of General Relativity were written by Einstein as the solution of the motions of mass points, and they are still in that general form (as I show below). Some mathematical variations have been applied to the field equations since Einstein, but none of these variations have jettisoned the point from GR. Tensors were, and still are, applied to points in space, even though this is and always has been strictly illogical. A tensor is basically a curved line or vector, and just as with a line, a tensor must logically be applied to an interval or differential. You cannot assign a line to a point, can you? That would be like assigning the color blue to the word “red”. A line is a line and a point is a point, and you cannot assign a line to a point. Likewise, you cannot assign a curved line to a point. A tensor, vector, or line must be applied to a length, or its mathematical equivalent. You cannot assign a length to a point, because a point has zero length.

We find the same problem in QED, where we see physicists applying tensors or events or changes or masses to points. For example, the photon is supposed to be a point particle, even though the two words contradict each other. A particle is a thing, while a point is a nothing. They will say it is OK, because the photon has no rest mass. But the photon does have energy, which by Einstein's equation is a mass equivalent. If you cannot have mass at a point, you cannot have energy at a point, either. This is basic logic. If you cannot have mass or energy or time or volume at a point, then you cannot have an event at a point. And yet modern physicists allow events at points all the time (see below). Gauge math is just like Einstein's math in this regard, since both base their fields or spaces on the same fundamental misconception of the point. Both misunderstand the point because both misunderstand the calculus. Modern physicists and mathematicians treats the point as a unit in their equations, which is like treating the zero as a mathematical unit. They are historically out-of-date in this by about 400 years, since the famous Flemish mathematician Simon Stevin showed in about 1600 that the point must be assigned to zero, not to the number one. The number one is the unit in all modern math, by definition. That is why we use the word “unit”: it comes from the Latin “unus, una, unum” which means “one.” Mathematicians accepted Stevin's analysis, and still do, but they don't appear to have let the idea sink in too far. Modern applied maths totally ignore Stevin's argument, which means we have regressed to the 16th century, before Newton, before Galileo—when they were still burning people for saying that the Earth moved around the Sun.

This is of utmost importance in modern physics because these modern mathematical spaces treat the point as the unit. In trying to create a field as the generator of the motions of mass points, Einstein was trying to assign zero to the unit in his math. His mass points are his units, you see. The numbers we find in solutions apply to them, so they are the units. But points as units is a pre-Stevin idea: it is a 16th century idea. Modern physicists and mathematicians are not more sophisticated than Galileo and Copernicus, they are less sophisticated.

The same can be said of QED, which does the same thing. As in GR, QED lets mass act as a unit. When QED finds mass at a point, it has assigned zero to its unit. This is why its equations have been rebelling from the beginning. This is why QED requires renormalization. When you assign zero to the unit in your math, your math will implode. It will start spitting out zeros and infinities, and this is what causes the need for renormalization. Renormalization is the getting rid of these pesky zeros and infinities, by a sort of mathematical conjuring.

As I have shown, GR has equations that rebel in the same way. The field equations often implode, giving zeros or infinities as solutions. This is precisely what is happening in the math applied to black holes or to the big bang. The math is imploding, just as in QED. But rather than renormalize the field equations, physicists prefer (for some reason) to accept the infinities and zeros, giving them mystical interpretations. In QED, they don't like the singularities and infinities, so they renormalize them away. But in GR, they have found a use for them. In GR, they have made a physical object out of the mathematical failure, selling the singularity as the nut of all massive objects. Singularities are now found all over the place, in many new theories. They are said to explain magnetic monopoles, cosmic strings, domain walls, and even “fuzzballs”. The singularity is even given various dimensions! A domain wall is a two-dimensional singularity, for example. That is somewhat like a three-dimensional line, a unicycle with three wheels, or a ten-armed octopus.