Miles Mathis: A physical point has no dimensions
| Please note that this paper is a simplification by me of a paper or papers written and copyrighted by Miles Mathis on his site. I have replaced "I" and "my" with "MM" to show that he is talking. All links within the papers, not yet simplified, are linked directly to the Miles Mathis site and will appear in another tab. (It will be clear which of these are Miles Mathis originals because they will be still contain "I" and "my".) The original papers on his site are the ultimate and correct source. All contributions to his papers and ordering of his books should be made on his site. (This paper incorporates the the last part of Miles Mathis' calcsimp paper and the first part of Miles Mathis' are paper). |
Background
Many of the problems with Quantum ElectroDynamics are related to the problem the point in modern calculus. QED hit a wall about 20 years ago, and that is why all the big names are now working on string theory. String theory is a horrible mess, one that makes the mess of calculus look like spilled milk. But one of the main reasons it was invented was to save QED from the point. This problem about the point is exactly the same one that cold-cocked QED. All of physics is dependent on calculus and its offshoots, and using calculus with points in the equations has ended up driving everyone a little mad.
The only way that physicists could make the equations of QED keep working is by performing silly operations on them, like the ones that Newton performed in his derivation. These operations in QED are called “renormalization”. That is a big word for fudging. The inventor of renormalization was the same Richard Feynman who I told you about above. His students are still finding new ways to renormalize equations that won’t work in normal ways. Mr. Feynman was a big mess maker, but he did have the honesty to at least admit it, regarding renormalization. He himself called it “hocus pocus” and a “dippy process” that was “not mathematically legitimate.” It would have been nice if Newton or Leibniz or Cauchy had had the intellectual honesty to say the same about the calculus derivation.
The reason this should be interesting to you is that MM's correction to the calculus solves all the problems of QED at one blow, although they haven’t figured that out yet.
The "uncertainty" of quantum mechanics is due (at least in part) to the math and not to the conceptual framework. That is to say, the various difficulties of quantum physics are primarily problems of a misdefined Hilbert space and a misused mathematics (vector algebra), and not problems of probabilities or philosophy. MM's correction to the calculus allows for a fix of all higher maths, spaces, and theories.
That oldest mistake is one that Euclid made. It concerns the definition of the point. Entire library shelves have been filled commenting on Euclid's definitions, but neither he nor anyone since has appeared to notice the gaping hole in that definition. Euclid declined to inform us whether his point was a real point or a diagrammed point.
Most will say that it is a geometric point, and that a geometric point is either both real and diagrammed or it is neither. But all the arguments in that line have been philosophical misdirection. The problem that has to be solved mathematically concerns the dimensions created by the definition. That is, Euclid's hole is not a philosophical or metaphysical one, it is a mechanical and mathematical one. Geometry is mathematics, and mathematics concerns numbers. So the operational question is, can you assign a number to a point, and if you do, what mathematical outcome must there be to that assignment?
It will be shown that you cannot assign a counting number to a real point. A real point is dimensionless; it therefore has no extension in any direction. You can apply an ordinal number to it, but you cannot assign a cardinal number to it. Since mathematics and physics concern cardinal or counting numbers, the point cannot enter their equations.
This is of fundamental contemporary importance, since it means that the point cannot enter calculus equations. It also cannot exit calculus equations. Meaning that you cannot find points as the solutions to any differential or integral problems. There is simply no such thing as a solution at an instant or a point, including a solution that claims to be a velocity, a time, a distance, or an acceleration. Whenever mathematics is applied to physics, the point is not a possible solution or a possible question or axiom. It is not part of the math.
Now, it is true that diagrammed points may be used in mathematics and physics. You can easily assign a number to a diagrammed point. Descartes gave us a very useful graph to use when diagramming them. But these diagrammed points are not physical points and cannot stand for physical points. A physical point has no dimensions, by definition. A diagrammed point must have at least one dimension. In a Cartesian graph, a diagrammed point has two dimensions: it has an x-dimension and a y-dimension. What people have not remembered is that if you enter a series of equations with a certain number of dimensions, you must exit that series of equations with the same number of dimensions. If you assign a variable to a parameter, then that variable must have at least one dimension. It must have at least one dimension because you intend to assign a number to it. That is what a variable is: a potential number. This means that all your variables and all your solutions must have at least one dimension at all times. If they didn't, you couldn't assign numbers to them.
This critical finding has thousands of implications in physics, such as Quantum Electro-Dynamics, since the entire problem of renormalization is caused by this hole in Euclid's definition. Because neither Descartes nor Newton nor Schrodinger nor Feynman saw this hole for what it was, QED has inherited the entire false foundation of the calculus.
Many of the problems of QED, including all the problems of renormalization, come about from infinities and zeroes appearing in equations in strange ways. All these problems are caused by mis-defining variables. The variables in QED start acting strangely when they have one or more dimensions, but the scientists mistakenly assign them zero dimensions. In short, the scientists and mathematicians have insisted on inserting physical points into their equations, and these equations are rebelling.
Mathematical equations of all kinds cannot absorb physical points. They can express intervals only. The calculus is at root a differential calculus, and zero is not a differential. The reason for all of this is not mystical or esoteric; it is simply the one MM has stated above: you cannot assign a number to a point. It is logical and definitional.
This finding is not only useful in physics, it is useful to calculus itself, since it has allowed MM to show that modern derivatives are often wrong. MM has shown that The Derivatives of the Natural Log and of 1/x are Wrong, for instance. MM has also shown that many problems are solved incorrectly with calculus, including very simple problems of acceleration. This finding also intersects with the corrections to special relativity.
The Proof
The most elementary concept that to be analyzed here is the concept of the point. In the Dover edition of Euclid's Elements, we are told, "a point is that which has no part." The Dover edition supplies notes on every possible variation of this definition, both ancient and modern, but it fails to answer the question that is central to my treatise here: that being, "Does the definition apply to a mathematical point or a physical point" Or, to be even more blunt and vivid, "Are we talking about a point in space, or are we talking about a point on a piece of treatise?" This question has never been asked much less answered (until now).
Most will see no point to this question. How is a point on a piece of treatise not the same as a point in space? A point on a piece of treatise is physical: treatise and ink are physical things. So what can Miles Mathis possibly mean?
What it does not mean:
Some readers
will be familiar with the historical arguments on the point, and
it must be clear that MM's question is a completely new one. The
historical question, as argued for more than two millennia now,
concerns the difference between a monad and a point. According to
the ancient definitions a monad was indivisible, but a point was
indivisible and had position. The natural question was "position
where?" The only answer was thought to be "in space, or
in the real world." A thing can have position nowhere else.
A point is therefore an indivisible position in the physical
world. A monad is a generalized "any-point", or the
idea of a point. A point is a specific monad, or the position of
a specific monad.
But my distinction between a mathematical point and a physical point is not this historical distinction between a monad and a point. MM is not concerned with classifications or with existence. It does not matter to MM here, when distinguishing between a physical point and mathematical point, whether one, both, or neither are ideas or objects. What is important is that they are not equivalent. A point in a diagram is neither a physical point nor a monad. A point in a diagram is a specific point; it has (or represents) a definite position. So it is not a monad. But a diagrammed or mathematical point is an abstraction of a physical point; it is not the physical point itself. Its position is different, for one thing. More importantly, whether idea or object, it is one level removed from the physical point, as I will show in some detail below.
The historical question has concerned one sort of abstraction: from the specific to the general. My question concerns a completely different sort of abstraction: the representation of one specific thing by another specific thing. The Dover edition calls its question ontological. My question is operational.
A mathematical point represents a physical point, but it is not equivalent to a physical point since the operation of diagramming creates fields that are not directly transmutable into physical fields.
Applied mathematics must be applied to something. Mathematics is abstract, but applied mathematics cannot be fully abstract or it would be applicable to nothing. Applied mathematics applies to diagrams, or their equivalent. It cannot apply directly to the physical world. And this is why MM calls a diagrammed point a mathematical point. Applied geometry and algebra are applied to mathematical points, which are diagrammed points.
A point on a piece of treatise is a diagram, or the beginning of a diagram. It is a representation of a physical point, not the point itself. When we apply mathematics, we do so by assigning numbers to points or lengths (or velocities, etc.). Physics is applied mathematics. It is meant to apply to the physical world. But the mathematical numbers may not be applied to physical points directly. Mathematics is an abstraction, as everybody knows, and part of what makes it abstract even when it is applied to physics is that the numbers are assigned to diagrams. These diagrams are abstractions.
A Cartesian graph is one such abstraction. The graph represents space, but it is not space itself. A drawing of a circle or a square or a vector or any other physical representation is also an abstraction. The vector represents a velocity, it is not the velocity itself. A circle may represent an orbit, but it is not the orbit itself, and so on. But it not just that the drawing is simplified or scaled up or down that makes it abstract. The basic abstraction is due to the fact that the math applied to the problem, whatever it is, applies to the diagram, not to the space. The numbers are assigned to points on the piece of treatise or in the Cartesian graph, not to points in space.
All this is true even when there is no treatise or pixel diagram used to solve the problem. Whenever math is applied to physics, there is some spatial representation somewhere, even if it is just floating lines in someone's head. The numbers are applied to these mental diagrams in one way or another, since numbers cannot logically be applied directly to physical spaces.
The easiest way to prove the inequivalence of the physical point and the mathematical point is to show that you cannot assign a number to a physical point. We assign numbers to mathematical points all the time. This assignment is the primary operational fact of applied mathematics. Therefore, if you cannot assign a number to a physical point, then a physical point cannot be equivalent to a mathematical point.
Pick a physical point. (Of course, metaphysicians would say that this is impossible. They would give some variation of Kant's argument that whatever point you choose is already a mental construct in your head, not the point itself. You will have chosen a phenomenon, but a physical point is a noumenon. But MM is not interested in metaphysics here; MM is interested in a precise definition, one that has the mathematically required content to do the job. A definition of "point" that does not tell us whether we are dealing with a physical point or a mathematical point cannot fully do its job, and it will lead to purely mathematical problems.)
So, you have picked a point. MM is not even going to be rigorous and make you worry about whether that point is truly dimensionless or indivisible, since, again, that is just quibbling as far as this treatise is concerned. Let us say you have picked the corner of a table as your point. The only thing MM is going to disallow you to do is to think of that point in relation to an origin. You may not put the corner of your table into a graph, not even in your head. The point you have chosen is just what it is, a physical point in space. There are no axes or origins in your room or your world.
OK, now try to assign a number to that point. If you are stubborn you can do it. You can assign the number 5 to it, say, just to vex me. But now try to give that number some mathematical meaning. What about the corner of that table is "5"? Clearly, nothing. If you say it is 5 inches from the center of the table or from your foot, then you have assigned an origin. The center of the table or your foot becomes the origin. MM has disallowed any origins, since origins are mathematical abstractions, not physical things.
The only way to assign a number to your point is to assign the origin to another point, and to set up axes, so that your room becomes a diagram, either in your head or on a piece of treatise. But then the number 5 applies to the diagram, not to the corner of the table.
What does this prove? Euclid's geometry is a form of mathematics. It is unlikely that anyone will argue that geometry is not mathematics. Geometry becomes useful only when we can begin to assign numbers to points, and thereby find lengths, velocities and accelerations. If we assign numbers to points, then those points must be mathematical points. They are not physical points. Euclid's definitions apply to points on pieces of treatise, to diagrams. They do not apply directly to physical points.
I am not going to argue that you cannot translate your mathematical findings to the physical world. That would be nihilistic and idiotic, not to say counter-intuitive. But MM is going to argue here that you must take proper care in doing so. You must differentiate between mathematical points and physical points, because if you do not you will misunderstand all higher math. You will misinterpret the calculus, to begin with, and this will throw off all your other maths, including topology, linear and vector algebra, and the tensor calculus.
To show how all
this applies to the calculus, MM starts with a close analysis
of the curve.
Let us say we are given a curve, but are not given
the corresponding curve equation. To find this equation, we must
import the curve into a graph. This is the traditional way to
"measure" it, using axes and an origin and all the
tools we are familiar with. Each axis acts as a sort of ruler,
and the graph as a whole may be thought of simply as a
two-dimensional yardstick. This analysis may seem self-evident,
but already MM has enumerated several concepts that deserve
special attention.
|
Using the most common variables, it will have an x dimension and a y dimension. This means that any equation with two variables implies two dimensions, which implies two dimensions at every point on the graph and every point on the curve.
If you are mathematician who is chafing under all this "philosophy talk": or anyone else who is the least bit lost among all these words, for whatever reason: there is a reason why MM has bolded the words above. In fact it might be called the central mathematical assertion of this whole treatise: the primary thesis of my analysis:
|
A mathematician who defined a point as a quantity having two dimensions would be an oddity, to say the least. No one in history has proposed that a point has two dimensions. A point is generally understood to have no dimensions. And yet we have no qualms calling a point on a graph a point. This imprecision in terminology has caused terrible problems historically, and it is one of these problems that MM is unwinding here. The historical and current proof of the derivative both treat a point on a graph and on a curve as a zero-dimensional variable. It is not a zero dimension variable; it is a two dimensional variable. A point in space can have no dimensions, but a point on a piece of treatise can have as many dimensions as we want to give it. However, we must keep track of those dimensions at all times. We cannot be sloppy in our language or our assignments. The proof of the calculus has been imprecise in its language and assignments.
- MM clarifies this with an example. Say a bug crawls by on the wall. You mark off its trail. Now you try to apply a curve equation to the trail, without axes. Say the curve just happens to match a curve you are familiar with. Say it looks just like the curve y = x2. OK, try to assign variables to the bug's motion. You can't do it. The reason why you can't is that a curve drawn on the wall, whether by a bug or by Michelangelo, needs three axes to define it. You need x, y and t. The curve may look the same, but it is not the same. A curve on the wall and a curve on the graph are two different things.
- As a second example, say your little brother jumps into his new car and peals off down the street. You run out after him and look at the black marks trailing off behind his car. He's accelerating still, so you should see those marks curve, right? A curve describes an acceleration, right? Not necessarily. The car is going in only one direction, so you can plot x against t. There is no third variable y. But it still doesn't mark off a curve. The car is going in a straight line. Mystifying. The curve for an equation looks like it does only on a graph. Its curve is dependent on the graph. That is, its rate of change is defined by the graph.
All those illustrations and diagrams you have seen in books with curves drawn without graphs are incomplete. Years ago: nobody knows how many years: books stopped drawing the lines, since they got in the way. Even Descartes, who invented the lines, probably let them evaporate as an artistic nuisance. And so we have ended up forgetting that every mathematical curve implies its own graph.
A physical curve and a mathematical curve are not equivalent. They are not mathematically equivalent.
This is of utmost importance for several reasons. The most critical reason is that once you draw a graph, you must assign variables to the axes. Let us say you assign the axes the variables x and y, as is most common. Now, you must define your variables. What do they mean? In physics, such a variable can mean either a distance or a point. What do your variables mean? No doubt you will answer, "my variables are points." You will say that x stands for a point x-distance from the origin. You will go on to say that distances are specified in mathematics by Δx (or some such notation) and that if x were a Δx you would have labeled it as such.
Clearly, this has been the interpretation for all of history. But it turns out that it is wrong. You build a graph so that you can assign numbers to your variables at each point on the graph. But the very act of assigning a number to a variable makes it a distance. You cannot assign a counting number to a point. This will seem metaphysical at first to many people. It will seem like philosophical mush. But if you consider the situation for a moment, it can be seen that it is no more than common sense. There is nothing at all esoteric about it.
Let us say that at a certain point on the graph, y = 5. What does that mean? You will say it means that y is at the point 5 on the graph. But I will repeat, what does that mean? If y is a point, then 5 can't belong to it. What is it about y that has the characteristic "5"? Nothing. A point can have no magnitudes. The number belongs to the graph. The "5" is counting the little boxes. Those boxes are not attributes of y, they are attributes of the graph.
You might answer, "That is just pettifoggery. MM maintains that what he meant is clear: y is at the fifth box, that is all. It doesn't need an explanation."
But the number "5" is not an ordinal. By saying "y is at the fifth box" you imply that 5 is an ordinal. We have always assumed that the numbers in these equations are cardinal numbers numbers [MM uses "cardinal" here in the traditional sense of cardinal versus ordinal. This is not to be confused with Cantor's use of the term cardinal]. The equations could hardly work if we defined the variables as ordinals. The numbers come from the number line, and the number line is made up of cardinals.
The equation y = x2 @ x = 4, doesn't read "the sixteenth thing equals the fourth thing squared." It reads "sixteen things equals four things squared." Four points don't equal anything. You can't add points, just like you can't add ordinals. The fifth thing plus the fourth thing is not the ninth thing. It is just two things with no magnitudes.
The truth is that variables in mathematical equations graphed on axes are cardinal numbers. Furthermore, they are delta variables, by every possible implication. That is, x should be labeled Δx. The equation should read Δy = Δx2. All the variables are distances. They are distances from the origin. x = 5 means that the point on the curve is fives little boxes from the origin. That is a distance. It is also a differential: x = (5 - 0).
Think of it this way. Each axis is a ruler. The numbers on a ruler are distances. They are distances from the end of the ruler. Go to the number "1" on a ruler. Now, what does that tell us? What informational content does that number have? Is it telling us that the line on the ruler is in the first place? No, of course not. It is telling us that that line at the number "1" is one inch from the end of the ruler. We are being told a distance.
You may say, "Well, but even if it is a distance, your number "5" still applies to the boxes, not to the variable. So your argument fails, too."
No, it doesn't. Let's look at the two possible variable assignments:
x = five little boxes or
Δx = five little boxes
The first variable assignment is absurd. How can a point equal five little boxes? A point has no magnitude. But the second variable assignment makes perfect sense. It is a logical statement. Change in x equals five little boxes. A distance is five little boxes in length. If we are physicists, we can then make those boxes meters or seconds or whatever we like. If we are mathematicians, those boxes are just integers.
You can see that this changes everything, regarding a rate of change problem. If each variable is a delta variable, then each point on a curve is defined by two delta variables. The point on the curve does not represent a physical point. Neither variable is a point in space, and the point on the graph is also not a point in space. This is bound to affect applying the calculus to problems in physics. But it also affects the mathematical derivation. Notice that you cannot find the slope or the velocity at some point (x, y) by analyzing the curve equation or the curve on the graph, since neither one has a point x on it or in it. MM has shown that the whole idea is foreign to the preparation of a graph.
No point on the graph stands for a point in space or an instant in time. No point on any possible graph can stand for a point in space or an instant in time. A point graphed on two axes stands for two distances from the origin. To graph a line in space, you would need one axis. To graph a point in space, you would need zero axes. You cannot graph a point in space. Likewise, you cannot graph an instant in time.
Therefore, all the machinations of calculus, all the dx's and dy's and limits, are not applicable. You cannot let x go to zero on a graph, because that would mean you were really taking Δx to zero, which is either meaningless or pointless. It either means you are taking Δx to the origin, which is pointless; or it means you are taking Δx to the point x, which is meaningless (point x does not exist on the graph--you are postulating making the graph disappear, which would also make the curve disappear).
In its own way, the historical derivation of the derivative sometimes understands and admits this. Readers of my treatises like to send MM to the epsilon/delta definition, as an explanation of the limit concept. The epsilon/delta definition is just this: For all ε>0 there is a δ>0 such that whenever |x - x0|<δ then |f(x) - y0|<ε. What MM wants to point out is that |x - x0| is not a point, it is a differential.
The epsilon/delta definition is sometimes simplified as "Whatever number you can choose, I can choose a smaller one." Which might be modified as "You can choose a point as near to zero as you like; but one can choose a point even nearer." But this is not what the formal epsilon/delta proof states, as you see. The formal proof defines both epsilon and delta as differentials. In physics or applied math, that would be a length. Stated in words, the formal epsilon/delta definition would say, "Whatever length you choose, one can choose a shorter one."
Epsilon/delta is dealing with lengths, not points. If you define your numbers or variables or functions as lengths, as here, then you cannot later claim to find solutions at points.
If you are taking differentials or lengths to limits, then all your equations and solutions must be based on lengths. You cannot take a length to a limit and then find a number that applies to a point.
Currently, the calculus uses a proof of the derivative that takes lengths to a limit, as with epsilon/delta. But if you take lengths to a limit, then your solution must also be a length. If you take differentials to a limit, your solution must be a differential. Which is all to say that the calculus contains no points. It contains differentials only. That is why it is called the differential calculus.
All variables and functions in equations are differentials and all solutions are differentials. The only possible point in calculus is at zero, and if that limit is reached then your solution is zero. You cannot find numerical solutions at zero, since the only number at zero is zero.
If this is all true, how is it possible to solve a calculus problem? The calculus has to do with instants and instantaneous things and infinitesimals and limits and near-zero quantities, right? No, the calculus initially had to do with solving areas under curves and tangents to curves, as said above. MM has shown that a curve on a graph has no instants or points on it, therefore if we are going to solve without leaving the graph, we will have to solve without instants or infinitesimals or limits.
It is also worth noting that finding an instantaneous velocity appears to be impossible. A curve on a graph has no instantaneous velocities on it anywhere: therefore it would be foolish to pursue them mathematically by analyzing a curve on a graph.
In conclusion: You cannot analyze a curve on a graph to find an instantaneous value, since there are no instantaneous values on the graph. You cannot analyze a curve off a graph to find an instantaneous value, since a curve off a graph has a different shape than the same curve on the graph. It is a different curve. The given curve equation will not apply to it.
Some will say here, "There is a simple third alternative to the two in this summation. Take a curve off the graph, a physical curve: like that bug crawling or your brother in his car: and assign the curve equation to it directly. Do not import some curve equation from a graph. Just get the right curve equation to start with."
First of all, it should be clear that we cannot use the car as a real-life curve equation, since it is not curving. How about the bug? Again, three variables where we need two. Won't work. To my dissenters one replies, find a physical curve that has two variables and MM can use the calculus to analyze without a graph. They simply cannot do it. It is logically impossible.
One of the dissenters may see a way out: "Take the bug's curve and apply an equation to it, with three variables, x, y and t. The t variable is not a constant, but its rate of change is a constant. Time always goes at the same rate! Therefore we can cancel it and we are back to the calculus. What is happening is that the calculus curve is just a simplification of this curve on three axes."
In answer, yes, we can use three axes, but how you are going to apply variables to the curve without putting it on the graph? Calculus is worked upon the curve equation. You must have a curve equation in order to find a derivative. To discover a curve equation that applies to a given curve, you must graph it and plot it.
The dissenter says, "No, no. Let us say we have the equation first. We are given a three-variable curve equation, and we just draw it on the wall, like the bug did. Nothing mysterious about that."
MM answers, where is the t axis, in that case? How are you or the bug drawing the t component on the wall? You are not drawing it, you are ignoring it. In that case the given curve equation does not apply to the curve you have drawn on the wall, it applies to some three-variable curve on three axes.
The dissenter says, "Maybe, but the curvature is the same anyway, since t is not changing."
MM asks, is the curve the same? You may have to plot some "points" on a three dimensional graph to see it, but the curve is not the same. Plot any curve, or even a straight line on an (x, y) graph. Now push that graph along a t axis. The slope of the straight line decreases, as does the curvature of any curve. Even a circle is stretched. This has to affect the calculus. If you change the curve you change the areas under the curve and the slope of the tangent at each point.
The dissenter answers, "It does not matter, since we are getting rid of the t-axis. We are going to just ignore that. What we are interested in is just the relationship of x to y, or y to x. It is called a function, my friend. If it is a simplification or abstraction, so what? That is what mathematics is."
To that MM answers, fine, but you still have not explained two things:
- If you are talking of functions, you are back on the two-variable graph, and your curve looks the way it does only there. To build that graph you must assign an origin to the movement of your little bug, in which case your two variables become delta variables. In which case you have no points or instants to work on. The calculus is useless.
- Even if you somehow find values for your curve, they will not apply to the bug, since his curve is not your curve. His acceleration is determined by his movement in the continuum x, y, t. You have analyzed his movement in the continuum x, y which is not equivalent.
The dissenter will say, "Whatever. Apply my curve to your brother's car, if you want. It does not matter what his tire tracks look like. What matters is the curve given by the curve equation. An x, t graph will then be an abstraction of his motion, and the values generated by the calculus on that graph will be perfectly applicable to him."
But then are we are back to square one? You either apply the calculus to the real-life curve, where there are points in space, or you apply it to the curve on the graph, where there are not. In real life, where there are points, there is no curvature. On the graph, where there is curvature, there are no points. If my dissenter does not see this as a problem, he is seriously deluded.