Someone recently sent me a link to a really terrific crank. This guy really takes the cake. Seriously, no joke, this guy is the most grandiose crank that I’ve ever seen, and I doubt that it’s possible to top him. He claims, among other things, to have:
- Demonstrated that every mathematician since (and including) Euclid was wrong;
- Corrected the problems with relativity;
- Turned relativity into a unification theory by proving that magnetism is part of the relativistic gravitational field;
- Shown that all of gravitational/orbital dynamics is completely, utterly wrong; and, last but not least:
- proved that the one true correct value of is exactly 4.
I’m going to focus on the last one – because it’s the simplest illustration of both his own comical insanity, of of the fundamental error underlying all of his rubbish.
Let’s start at the beginning, with his introduction:
Historically, pi is the numerical relationship between the diameter and circumference of a circle. It is a geometric constant. What do we mean by geometric? Operationally, geometry is the study of drawn figures. The ancients actually drew their figures on paper (and some of us still do). All the concepts of geometry applied to these figures. A line was a drawn line. A circle was a drawn circle. Of course geometry soon invented some other postulates to help with the mathematics. A point was defined as having no extension, a line was defined as having no width, and so on. But the equations were still understood to apply to the figures. Geometry was always only partially abstract.
In this context, pi was assumed to be a dimensionless constant. It transformed one length to another. This is clear from the basic equation: .
You can see that pi takes us from one length to another and therefore we must assume it is dimensionless.
What I will show in this paper is that this assumption is false. I will show that pi is not dimensionless. It is not dimensionless for the basic reason that the circumference is not a length. Nor is it a distance.
Quite a lot of crankery, right there. Some of it actually makes a bit more sense that it appears at first glance. I’m not saying that it actually make sense, but that given his rather peculiar definitions (which he discusses elsewhere), it’s not quite a bogglingly nonsensical as it appears.
One of his fundamental ideas is that Euclid got the concept of a point wrong. According to Mathis, there are two kinds of points: drawn points, and mathematical points. And you can only meaningfully apply numbers to drawn points; any attempt to assign a number to a mathematical point is completely erroneous. As an implication of this, all numbers must, inevitably, have units. There is, in Mathis’s world, no such thing as a dimensionless number: a dimensionless number is an abstract point, which can’t have a number assigned to it.
Back to his words, as he tries to explain some of this in the context of geometry:
Geometry dismisses time as a consideration. Geometry is understood to be taking place at a sort of imaginary instant. For instance, when we are given or shown a radius, we do not consider that it took some time to draw that radius. We do not ask if the radius was drawn at a constant velocity or if the pencil was accelerating when it was drawn. We don’t ask because we really don’t care. It doesn’t seem pertinent. It seems quite intuitive to just postulate a radius, draw it, and then begin asking questions after that.
It turns out that this nonchalance is a mistake. It is a mistake because by ignoring time we have ignored many important subtleties of the problem of circular motion and of circle geometry.
It’s hard to overstate just how completely and utterly wrong this is.
The fundamental goal of mathematics is abstraction: that is, it’s about taking something that you want to study, and focusing narrowly on that, discarding anything that isn’t essential to understand it. Euclidean geometry is concerned with shapes; it doesn’t matter how they were drawn, or whether they were drawn at all.
But Mathis is absolutely obsessed with idea that drawing something is absolutely critically important. A drawn point is meaningful; an abstract mathematical point is not. A drawn line is different from an abstract mathematical line. And, obviously, a drawn circle is fundamentally different from an abstract circle. Per his reasoning, an abstract circle isn’t particularly meaningful. You can’t (according to him) do something like compute the ratio of the circumference to the diameter in an abstract circle – because that would be applying numbers to something formed of abstract points.
Let’s see a bit more of where he goes with this:
As a simple example of this, when we draw a circle on a Cartesian graph, e make an entirely different set of assumptions than the ones above, although few have seemed to notice this. You would think you could draw a Cartesian graph anywhere you wanted and it wouldn’t make any theoretical difference to the geometry. You could draw a graph on the wall, on the floor, on any flat surface. You would think all you are doing is making things a bit easier on yourself as an artist and a geometer. Just as the old artists would square off their paper in order to make drawing a head easier, a geometer squares off a section of the world in order to create a tidy little sub-world where things can be put in order.
But all this is completely false. Drawing the graph changes everything. If you draw a circle without a graph, then you can say to yourself that the line (that is now the circumference of the circle) is a length. As a length, it can have only one dimension. A length is a one-dimensional variable, right? Perhaps you can see where I am going with this, and you say, “Wait, a circle curves, so we must have two dimensions, at least. We must have an x and a y dimension.” Yes, at the least we must have that. You saw this because you began to think in terms of the Cartesian graph and you could see in your head that the curve implied both x and y dimensions. Very good. But you are not halfway there yet. Take the circle and actually put it into a Cartesian graph. What you find is that the curve is now an acceleration. In fact, any curve is an acceleration in a two-dimensional graph. We all learned this in high school, although I don’t think it sunk far in for most of us.
That line that represents a circumference is taking on dimensions very fast now. At first we thought it was just a length. Then we saw that it required two dimensions. Now we can see that it is an acceleration. What next?
We’ve gotten to the depths of his kookiness here. In his world, a curve drawn on a graph isn’t something that can represent an acceleration: a curve is an acceleration. In his reasoning (in so far as I can follow it), this is because a drawn curve must have units (it’s drawn, so it’s real and has units!); and since the units that he chooses to apply to it look like an acceleration, then the curve is an acceleration. Alternatively, if you’re drawing a curve, in order for the drawn curve to be a curve and not a line, your hand must be accelerated. The curve is that acceleration.
Once again, it’s pretty hard to know just where to begin with how wrong this is. A curve can be a representation of a physical phenomena; it can describe a value with units. But a curve, in math, is not the physical phemonenon that it could represent. The process of drawing and the thing drawn are, to Mathis, the same thing. In Mathis’s world, a circle drawn with a compass, and a circle drawn by tracing another circle are fundamentally different, regardless of the fact that they’re indistinguishable.
This pretty much defeats the entire purpose of mathematics. One more quote of his, just because it’s so perfect:
Now let us return to the geometric circle. All the equations of geometry are created by assuming that time is not a factor. You can’t really just ignore time, so what the geometry does is assume that all underlying time intervals are equal. What does that mean, specifically? Well, it must mean that all the lines are understood to have been drawn with the same velocity. We can ignore the velocity since we define it as equivalent. What does that mean?
It means that the radius is a velocity itself.
Mathis is remarkably long-winded; the stuff I’ve shown you so far gives you the gist. He just keeps on with the same basic nonsense, building it up more and more. The circumference of a circle, according to Mathis, isn’t a distance. Because, you see, a distance is one-dimensional – but the circumference of a circle curves, so that it’s two-dimensional (Note that that’s his wording, not mine; in another typical example of his confusion, he doesn’t distinguish between the edge of the circle, and the circumference. To him, the curve, the distance, and the process of drawing it are all the same thing.)
All of his stuff really, ultimately, comes down to one basic problem: he is absolutely unable to distinguish between the act of drawing something, and the nature of the thing that he drew.
So, for example, he goes on at great length about how calculus is all wrong. The reasoning comes down to the fact that he doesn’t believe in mathematical points, because you can’t draw them. But calculus is based ultimately, on the concept of infinitely small points. You the derivative of a curve by finding the slope of that curve at a point. But if you draw a point, it’s not infinitely small. It’s got a finite size; to be able to draw it, it’s got to have a finite size. Therefore, there’s no such thing as a point, and if there are no points, there’s no calculus. (He does go on about reformulating calculus so that it doesn’t require points.)
So where does the stuff come from?
You need to go through a whole lot of craziness to actually get to it, but it’s more of the same. The root concept is that there is no distinction between a circle, and the process of drawing a circle. The radius of a circle isn’t a length, it’s a velocity. (Or
a distance. Sometimes it’s one, sometimes the other. But we won’t worry about that; consistently is the hobgoblin of a small mind!)
So what’s ?
Well… that’s a bit tricky, because he redefines terms like bloody crazy.
But, basically… is a velocity. By which he means an acceleration. Sort-of. Kind of. Maybe. Ish.
He’s rewritten the equations of orbital dynamics as . That makes no real sense, but it doesn’t matter. Just take it as given for the moment.
Now… you can rewrite that equation as . And the circumference of a circle is . So, obviously, , and is an acceleration. Or.. a velocity. Or… something.
See, it can’t be a straightforward acceleration… because, in his universe, an acceleration is a force with a direction. That’s fine. But… there’s no such thing as a point. And in an orbit, because it’s going in a circle, the direction of the centripetal force changes from instant to instant. But there’s no such thing as an instant, because an instant is a point in time. So there’s no such thing as an orbital acceleration, because the acceleration changes from instant to instant. So, therefore, it’s not an acceleration, but rather a velocity. (No, it doesn’t make sense. Don’t worry.)
But… By his reasoning above, we’ve shown that the circumference of a circle is measured in units of velocity squared. So you can’t talk about as the unitless ratio of the circumference to the diameter – because the circumference is measured in units like , and is measured in.. well, I’m not really sure, because he’s so inconsistent about whether it’s a velocity, or an acceleration, or something else…
But if you continue with his reasoning… Well, let’s not really continue with his reasoning – he takes his time, and frankly, this is giving me a headache. The gist of it is: is an acceleration. And if you think of an object going clockwise around a circle: at the top of the circle, it’s got velocity going towards the right. 90 degrees later, it’s got velocity 0 to the right, but down. And 90 degrees after that, it’s got velocity v to the left, but 0 down. and so on. So, around the circle, it’s been accelerated 8 times the initial . And since , that means that , so .
I’ll close with his summary of what he’s “discovered” in this mess.
We have discovered several important things.
- Pi is a centripetal acceleration and has the dimensions of acceleration.
- The circumference of any circle has the dimensions , if written out in full.
- If the radius is treated as a distance, then the circumference has the dimensions .
- Pi is not applicable to orbits or most other physical circles, since the tangential velocity is not equal to the radial velocity. There is no pi in the sky.
- In orbits and all other circular motion . Something may equal , but it isn’t a velocity.
- There is no such thing as orbital velocity. There is only tangential velocity. The curve described by an orbit is not a distance, nor is it a velocity. It has the dimensions , just like the circumference.
Insanity. Sheer, utter, insanity. This is the kind of rubbish that makes me want to poke my own eyes out, just so I don’t need to look at any more of it.