A reply to “The Extinction of Pi: The short version”

We’re very proud to have a guest author today. If you’ve been following the comments on Ex Falso… you will have noticed Dan, who kindly offered to write replies to Mathis’ more math-based papers. He took a look at Mathis’ ‘proof’ that \pi=4 (in kinematic situations) and promptly sent us this article. We’re happy to have Dan on board and are looking forward to more articles!


 

In The Extinction of Pi: the short version Mathis claims to have a simple geometric proof that Pi=4.

Before I begin I feel I should comment on Mathis’ method of proof on this one. This is one of the rare occasions where his ideas are easy to understand as there is no excess verbiage, no ill defined terms, only one point where the verbal explanation does not quite match the diagram (and an honest mistake at that), and no appeals to unrelated or ambiguous notions like ‘velocity of the radius’, although you may find this happens more frequently in long version of The Extinction of Pi. Many mathematicians have different requirements before they will call a proof elegant. If all known proofs of a statement require long and intricate arguments or “heavy machinery” and a new proof which relies solely on elementary geometry is found, then they will for the most part agree that it is elegant. All things considered, had his conclusions been accurate or his method sound, then this is the only time I have seen Mathis present what would be called an elegant proof.

Disproof #1

Left: Mathis diagram --- Right: my diagram

To quote Mathis himself:

Given the diagram above, which is part of a circle, we let

AO = AB

This makes the angle at B equal to 45°. Which means that

DB = DC.

This must mean that, by substitution,

AD + DC = AB

Now, the distance AD + DC can be redrawn as any number of sub-distances, as above.

Here we divide the distance into four sub-distances, taking us from A to C in 4 steps.

This new path, drawn by the sub-distances, is equivalent in length to AD + DC, as can be seen by any cursory analysis. It is nearer the path of the arc AC, but it retains the original distance of AD + DC.

If we divide the distance into eight sub-distances instead of four, we approach the path of the arc AC even nearer; but, again, we retain the original distance of AD + DC.

If we take this process to its limit, we take our path to the path of the arc AC.

This must mean that the length of the arc AC is equal to the distance AD + DC.

By substitution, this means that

arcAC = AB

Since 8 such arcs make up the circle, the circumference of the circle is 8AB.

Since AB = AO, and AO is the radius, we have found that

C = 8r

π does not exist in the circle equation. It is extinct.

The above is Mathis’s “proof” that Pi = 4, at first glance it seems plausible, the little ‘steps’ do visually look more and more like the arc of the circle as they are made ever smaller. All computer graphics are done in the same spirit; the pixels an LCD monitors are tiny squares, so any time anything curved is displayed on an LCD monitor it is being approximated visually by these squares. However a visual approximation of a curve is not the same as a curve. The error I mentioned where his explanation does not match the diagram involves his letter C. He uses it once as a point on the graph and then later as the circumference of the circle. This is not a major mistake and does not in any way change his argument or my proof that his method is flawed. I will now refer to Mathis’ method as the ‘step method’ because when drawn; the approximation looks like steps in a staircase. My proof that his method is invalid will now follow.

Definition and Labels

  1. Let a curve be the locus of points satisfying the equation y = f(x) where f is a continuous function or equivalently the set {(x,y) : xεR, y=f(x), f is continuous}. (think about drawing a curve above the x-axis where you never have to lift your pencil and above each x there is only one point on the curve and it never intersects itself)
  2. Let l(Z) denote the length of the curve Z
  3. In my diagram above, let R and B denote the red curve from (0,0) to (1,1) and the straight blue line from (0,0) to (1,1) respectively.
  4. A curve representing the function f(x) is said to be monotonic and strictly increasing if x<y implies f(x) < f(y) (If y is the right of x, then the point on the curve above y is higher than the point on the curve above x)
  5. A curve representing the function f(x) is said to be monotonic and strictly decreasing if x f(y) (If y is the right of x, then the point on the curve above y is lower than the point on the curve above x)
  6. A straight line is the curve between two points with the shortest distance. So if A is a straight line connecting any two points, and B is any curve distinct from A connecting the same two points we have l(A) < l(B) where l(Z) is the function that gives the length of the curve.

Assume that Mathis is correct and the length of a curve can be approximated by considering the lengths of the horizontal and vertical line segments that make up the ‘steps’ in the diagrams. Mathis is correct, no matter how few or how many of these steps we subdivide the interval [0,1] into, the total of their lengths will remain constant. They steps can be made small, medium, large, evenly sized, or unevenly sized yet their total length will remain the same. The below equation demonstrates this for evenly sized steps, which I will point out can only be drawn when the curve you are approximating is an arc of a circle or a straight line.

\lim_{n\to\infty} n\cdot(\frac{1}{n}+\frac{1}{n}) = 2

Now consider the curve R and B in the right side of the diagram. They can be approximated by ‘steps’ as in the Mathis proof. Notice that the big triangle in black has two sides, each of length 1 giving a combined length of 2. Thus the total length of the steps approximating B must be 2 implying the length of B is 2. The steps that approximate R must have total length 2 implying R has total length 2. We have shown that l(B) = l(R) or that the length of B and the length of R are in fact equal. However by assumption since B is a straight line it has the shortest possible length, and since R is a curve distinct from B it must have length greater than B. We have two contradictory statements: l(B) = l(R) and l(B) < l(R). The assumption that the ‘step method’ of calculating length is accurate leads to a contradiction and is therefore invalid.
\Box QED

One can extend the above line of reasoning to show a result of the Mathis method is the absurd idea that if m and n are any two distinct points, and X & Y are two distinct monotonic strictly increasing/decreasing curves joining m to n, then X and Y must have the same length. This would mean the length of the curves defined by the following functions are identical on [0,1]: f(x) = sin(Pi*x/2), g(x) = x, h(x) = x^2, j(x) = x^3, or even k(x) = x^n for any integer n>0.

Disproof #2

Assume that the Mathis’ “step method” can be used to calculate the length of a curve. In this situation our curve is the segment AC.
Let ABC be a right triangle with side lengths |AB| = 1, |AC| = 2, and |BC| unknown.
If we accept Mathis’ “step method” then we can approximate line segment AC by the red steps, and approximate the length |AC| by the lengths of the steps. As we make the steps smaller and smaller and pass to the limit, we have the length of the segment equal to the limit of the sum of the lengths of the steps, so as Mathis has demonstrated we have |AC| = |AB| + |BC|. Plugging in our numbers for |AC| and |AB| we get the equation 3 = 1 + |BC| which gives |BC| = 2.
From basic trigonometry we have sin(angle BAC) = opposite/hypotenuse = |BC|/|AC| = 2/2 = 1.

Now again from basic properties of triangles we can deduce that triangle ABC is “half” of an equilateral triangle. That is, if we reflected the triangle along the line segment BC and let A’ be the mirror image of A, then triangle ACA’ would be equilateral, as |AA’| = |AC| = |CA’|. This allows us to calculate angle BAC to be Pi/3 radians or 60 degrees.
Now given that we have found angle BAC to be Pi/3 radians we can calculate the sin of that angle in radians as sin(Pi/3) = sqrt(3)/2, or sin(60 degrees) = sqrt(3)/2)
So we have calculated sin(angle BAC) two different ways, and they should be the same.
Equating sin(BAC) = sin(BAC) and doing a few simple manipulations:

sin(BAC) = sin(BAC)
1=sqrt(3)/2
1^2 = [sqrt(3)/2]^2
1 = 3/4
4 = 3
4 – 3 = 3 – 3
1 = 0

Thus the assumption that the Mathis “step method” is an accurate way to calculate the length of a curve leads to the absurd conclusion that 1 = 0.
\Box QED

Afterthoughts

When I first set out to write this out I wasn’t entirely sure how to go about it. I found myself going on lengthy rants about Mathis’ ideas, methods, and conclusions. I could write a long piece attacking some of his ideas and explaining the folly in his way of thinking, but I had set out to discredit one specific paper. So I edited out my own rants and tried to make this as straight forward as possible and attack the method, not the man. I however must point out that the number Pi is so entrenched and intricately related to so many areas of mathematics and real world applications that if the value of Pi were to change, then everything in the following list would fall apart or at the very least need a significant rethink. So by Mathis attacking the value of Pi he is indirectly attacking so much more.
Here is a VERY short list of subjects and real world applications that Pi plays role in (it is more ‘important’ in some area than others):

  • complex analysis (contour integration in the complex plane)
  • probability, simulation, and testing
  • plane geometry
  • the GPS navigation system
  • signal processing (TV, radio, satellite)
  • physics (it shows up everywhere!)
  • real analysis (estimation see Stirling’s formula)

In closing I would like to quote Mathis from his long version of the Extinction of Pi, make a few short observations, and leave a question for Mathis and another for his supporters (I would really love to read a response!)

I show that in kinematic situations, Pi is 4. For all those going ballistic over my title, I repeat and stress that this paper applies to kinematic situations, not to static situations. I am analyzing an orbit, which is caused by motion and includes the time variable. In that situation, Pi becomes 4. When measuring your waistline, you are not creating an orbit, and you can keep Pi for that. So quit writing me nasty, uninformed letters.
-Miles Mathis

Pi is a constant so it never changes value based on the situation you are using it. I can only assume what he means is that in a kinematic situation he feels the constant Pi = 3.14159265… is an inappropriate constant to use and the number 4 should be used in its place. I promise you that that is not that case, but a proof of that will have to come at another time. I also acknowledge that I have not proved that Pi = 3.14159265, rather I have proved his method is flawed and any results derived from it are meaningless. There are many algorithms to calculate the exact value of Pi, one involves summation of the power series representation of the tangent function, some involve continued fractions, and some involve summation of other infinite series. Whatever the method used, the tools required are well outside the scope of this paper. To be completely honest, if you really believe that Pi equals anything other than the standard decimal representation, then the knowledge needed to understand methods of computing Pi is also well outside your grasp. Of the Mathis supporters I ask: what do you think is more likely; that in the history of scientific thought everyone else has been mistaken and the only person to truly understand the material is Mathis, or rather that Mathis misunderstands the very fundamentals and thus misunderstands anything built upon them? Take a minute to think about this and answer yourself honestly. My question to Mathis is if this notion Pi=4 only applies to kinematic situations, and if we assume by measuring your waistline you mean Pi = 3.14159265… for the purposes of plane geometry, then why did you bother with this little “proof” if you knew it was wrong?

- (by Dan)

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31 Responses to A reply to “The Extinction of Pi: The short version”

  1. sponsored says:

    Wow, I’m so glad you went through with the project to analyse his work.
    I’ve been eagerly checking back & this was a real treat. This is a great
    educational experience in that I chose to study the paper you picked very
    carefully and try to come to my own conclusions on the validity of the
    claims before reading your analysis. I hope to do this with them all! :D
    The new physics one is not my fortae as I haven’t studied SR but hopefully
    you all will continue with these posts, they help train the critical mind :p

    Yes I agree it’s a nice paper he wrote, but then all geometry reads like an
    elegant exposition, I admit it is a very smart idea of his, to break up AD + DC
    like that and take the limit. As for the C being a point on the graph & denoting
    the circumference, this to me was just an honest notational mistake, I didn’t
    even notice it until you mentioned it, unless there is some deeper meaning to it
    I think it’s nothing :p

    The simplest reasoning that he is wrong is just to integrate 1/8 of the arc of
    a circle and show that this distance is not equal to 1, no?
    I’m sure you could even use his modified version of the calculus to integrate
    1/8 of the arc of a circle & show it does not equal 1 :p
    To clarify, I assume OA = AB = 1 in my idea.
    It’s too simple what I’ve said to be enough I bet, but he is claiming
    that OA = AB = AD + DC & assuming we work on a unit length
    all we need to do is take an integral and show we do not have 1 for
    the arc.

    Thinking about this, it’s as if he’s claiming 1/8 of the circumference of
    a circle of radius 1 also has length 1, but that is a total contradiction in
    that the radius at one part is 1 while at another it simply can’t be 1 and
    also have the arc length to be 1/8.
    Im surprised that simple reasoning is not enough to convince him.
    I can mentally picture an ellipse having the properties he’s describing,
    i.e. the semi-minor axis has radius 1 and as you go clockwise from the
    top 1/8 of the curve could be set so that it’s arc length is equal to 1,
    but meh… This has nothing to do with pi…. If a circle was not equal
    to pi in any situation then it wouldn’t be a circle, it would be an ellipse
    or just a curve. Why bother with this in the first place?

    I think he just had a creative way to create a specific ellipse but
    went grandiose and claimed he’d revolutionized what those
    ‘gloriously negligent textbook authors’ haven’t mentioned in
    2000-odd years… :/

    • D says:

      Glad you enjoyed it!

      Initially I was going to appeal to the definition of arc length as an integral; however that runs into a couple problems. The first is that defining arc length that way is based on using chords to approximate the curve, which is exactly what Mathis claims is wrong with the method. The second problem is that polar coordinates are the most natural tool to use when looking at a circle, so consider the following:

      r = f(t) = 1 — equation of unit circle in polar coordinate t = theta.
      dr/dt = 0

      Length of arc from a to b = integral sqrt[r^2 + (dr/dt)^2] dt — from a to b
      = integral sqrt[1^2 + 0^2] dt — from a to b
      = integral 1 dt — from a to b
      = t — from a to b (I omitted the constant of integration, it’s not needed)
      = b – a

      So working in radians for one eighth of the circle we have a = 0 and b=Pi/4. Thus length = Pi/4.

      The problem here is that the answer involves Pi, so it could lead to circular logic. The same thing happens if we use Cartesian coordinates to find the arc length of one quarter circle, after integration we have to calculate arcsin(1) = Pi/2. So again the solution is dependent on the value of Pi. Any Mathis supporter could say “sure your method is correct and it yields the correct result, but our results agree since Pi = 4 to begin with”. So to prevent that sort of crap I decided to attack the method instead of a specific result of the method.

      As for SR, I know very little physics, but it is my understanding that the natural language to study relativity is tensor calculus. Mathis shares a view common among cranks, that all problems can be solved with high school algebra, a dash of watered down calculus, and some good ol’ fashion “common sense”. My guess is that this is as much math as he ever learned, and if all you have is a hammer then everything looks like a nail. Point being that Mathis lacks the tools needed to even properly analyze SR in a meaningful way. Einstein’s “though experiments” are an excellent aid for intuitive understanding, but any real analysis is done in a mathematical setting. I am not a physicist, so I could be wrong… but I doubt it.

      It can be very difficult to argue against him since he never offers any clear definition of terms or any real concrete analysis. Writing a proper analysis of his work requires you to clearly define his concepts for him, and to precisely formulate his theories for him. Then to argue against him you have to be precise when defining the language and concepts you will use. This can quickly turn into writing an introduction to the material at hand, something I don’t care to do. So if you found a Mathis “paper” with specific and well formulated claims you would like me to reply to, let me know. If I have time and can reply without writing an introductory textbook in the process I will.

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  3. sleestackvii says:

    sponsoredwalk recently challenged me to rebut Dan’s analysis above on Mathis’ Pi paper. I believe I have done just that.

    You can find it at:

    http://sagacityssentinel.wordpress.com/

    Look for further counter-criticism articles at this website in the near future.

  4. Dan says:

    You have rebutted nothing. Your arguments fail for the exact reasons mathis’ argument fail, they are demonstrably incorrect.

    You toss around words like vector analysis and complain that I did none, yet if I were to include even the simplest aspects of real analysis, specifically differential forms, it would be killing a mosquito with an atomic bomb. Do you even know what a differential 2-form or contour integral is?

    Quite honestly the mathis paper and you’re response are evidence that quite often a lack of formal education in mathematics precludes one from actually understanding anything but most trivial aspects of something even as basic as plane geometry.

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  6. Joe says:

    In reading your reply to the ‘The Extinction of Pi: The Short Version’ you seem to indicate that he is calling Pi or 3.1415… (4). In his intro to his long paper on Pi he says that 4 only applies to Pi in ‘Kinematic Situations’. He calls Pi 3.1415… ‘static Pi’ and says that is the correct value.

    In what he calls ‘Kinematic Situations’ dealing in Orbits where Time as a variable is involved as in Satellites he says ‘Pi’ is equal to (4). His abstract for his long paper on Pi is below and shows this distinction he is making between Pi and ‘Pi’, as it were.

    Here is quoting the abstract;

    “Abstract: I show that in kinematic situations, π is 4. For all those going ballistic over my title, I repeat and stress that this paper applies to kinematic situations, not to static situations. I am analyzing an orbit, which is caused by motion and includes the time variable. In that situation, π becomes 4. When measuring your waistline, you are not creating an orbit, and you can keep π for that. So quit writing me nasty, uninformed letters.”

    I am wondering if this changes your criticism in any way?

    I am also wondering if you have written a post on how he deals with the ‘Point’ in Calculus or Physics?

    The portion I am thinking about is below; (it’s a long one…sorry!)

    “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? I have exhaustively 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.”

    He deals with it extensively in his long paper on The Calculus;

    http://milesmathis.com/are.html

    Thanks Much!

    • Dan says:

      His disclaimer that Pi=4 only applies in kinematic situations in no way changes my assesment of his arguements. When one sits down and works with plane geometry, calculus, or general reltivity, to model some real world phenomenon you are bound by ‘rules’ of your chosen tool. Measuring a planets orbit is very hard, but modeling it with geometry is easy. So what the whole idea is is to discard the unimportant details (ie a planet is not a point, composition of the atmosphere, etc) and use the abstract framework of geometry to simplify the problem. Now that the problem is being modeled with geometry, one can use the tools of plane geometry, but must accept the rules of plane geometry. On of these rules is that Pi = 3.1415…, no matter what you are trying to model, the ratio of the circumference of a circle to its diameter will be 3.141…., it just cannot change because you say “curve f now represents an orbit, and line l represents a tangent…”, labelling the diagram cannot change the relationships among the objects in the diagram. I hope that made sense, I can elaborate more if that wasn’t clear.

      As for his nonsense on the calculus, I can write something up if you are still interested. I will tell you right now he is wrong, it is that simple. It is actually similar to what I wrote above, it seems he misunderstands the fact that “numbers and math” model physics, but are not physics, if that makes sense. The universe is not a set of equations, but much of it can be modeled with equations. About his entering and exiting an equation with the same number of dimensions, he is patently wrong. There is a very basic concept from linear algebra that any first year student will learn of, it’s called a linear tranform. A linear transform can “transform” one space into another, and their dimension need not be equal. All rants about solutions at instants, or solutions at points, are just wrong, completely wrong. Suppose I state you pour water into an empty glass for exactly 10 seconds at some constant rate. He would suggest that there is absolutely no way to tell how much water is in the glass after 5 seconds of pouring, because that would be a solution at an instant or a point.

      If you are still interested let me know, I can write a much clearer and more detailed analysis of this calculus business and ask the blog owners if they would post more from me.

  7. sponsoredwalk says:

    I want to recommend reading the amazon reviews for his book:

    There is a person on there doing his own “Mathis watch” of his latest articles.

    I remember writing about Mathis wikipedia comments and the hilarity of the
    arrogance contained within the article on K.E. when Sleestack tried to defend
    him on that thunderbolts forum, well this new critic has also picked up on
    that little quality of Mathis scientific writing & exploited it to hilarious consequences
    as well as showing his calculations & numerological refrains to be questionable :D

    So here is a sample of the amazon comments:

    “Throughout his essay, Mathis quotes the Wikipedia article on the Coriolis effect
    to argue that this “modern theory” doesn’t adequately explain water-draining
    behavior. Curiously, though, he doesn’t tell us what the article actually says about drains. Wikipedia clearly states that the Coriolis effect on a drain is vanishingly small;
    in controlled experiments, it is inevitably overwhelmed by the initial conditions of
    motion in the water. Mathis ignores this. That is what we call intellectual dishonesty”.

  8. Kris says:

    Miles is correct about pi – to an extent. There is a huge logical problem in maths. A little something we never take into consideration when even dealing with the hypotenuse of a triangle. A coordinate system can only plot lines with respect to that particular coordinate system. In a 2D rectilinear, we can only plot lines that are 0 or 90 degrees with respect to the system. No matter how much you divide the system, it is always rectilinear and can only ever plot those points that exist at the junctions of perpendicular lines. If I divide the system to infinity and zoom in with infinite expansion, I still have perpendicular intersections where points may be plotted. A hypotenuse cannot be drawn without first differentiating the system itself to a system at some angle with respect to the original system. So, with respect to the original un-differentiated coordinate system, the hypotenuse of a triangle is the sum of the two sides.

    The Pythagorean theorem is representative of smooth differentiation between two coordinate systems. A triangle with a true hypotenuse is 4 dimensional ( even though one of the dimensions is of zero size ).

    A true circle is representative of a continuously differentiated coordinate system. Pi with respect to the un-differentiated coordinate system is 4, as Miles finds. Pi with respect to a continuously differentiated system is as it stands, 3.14159… . A circle in an un-differentiated system is 2D. A circle in a continuously differentiated system, is infinite dimensioned – but only contains those dimensions that are respective of its plane.

    • Cee says:

      Dear Kris,

      What exactly do you mean when you say “differentiating the system”? If you could post a definition that gives a clear description of what to do to differentiate a system, that would be very helpful. (My guess is, however, that you can’t but instead have just an intuitive notion about that process.)

      Furthermore, I can’t see why I cannot draw a hypothenuse (basically a line between two given points) in a rectilinear coordinate system. Could I not just draw a line that connects the two points, independently of the coordinate system? What about drawing the line first and then assigning coordinates?
      Coordinates are just a tool that we use to describe the relationship between points – they are not identical to the points although we often think of them this way. You are confusing, it seems, the object we are describing and the description itself.

      Cheers,
      Cee

      • Kris says:

        Cee,

        I’ll give you the short version. To differentiate a coordinate system, we have to be able to break it down into its components- start from scratch and build the system. A single dimension breaks down into four components. The point, the normal, the magnitude, and a name.

        We can define this as: x ->>(p, n, t), and -x ->> (p, -n, -t)

        That is, a dimension, x, is characterized by some origin, p, with a given normal ( direction of expansion ), n, and a magnitude ( size of expansion ), t. It is important to note that these dimensional expansions are unidirectional from the origin, p. That is, the x axis as defined, does not contain any values other than is respective of its normal, which, respective of this are always positive. Negative values with respect to this normal require a second expansion that is mirror x, which is -x.

        All of the above values are arbitrary and have nothing to do with any other particular system, but declares that a dimension, x, exists and can be characterized by (p,n,t).

        Similarly, we can declare our y axis by stating y -> (p,!n->x,t). Which reads, y is characterized by the same origin as x, p, any normal that is not the normal of x, !n->x and the same magnitude of x, t.

        So, to define a complete system, s, we may say:

        p = 0
        t = oo

        x ->> (p, 0, t)
        y ->> (p, 90->x, t)
        -x ->> (p, -0->x, -t)
        -y ->> (p, -90->x, -t)

        s ->> (x,y,-x,-y) -> [+/- oo]

        The system, s, yields a rectilinear coordinate space and all values that exist in s must follow the rules of the space, s. That is, any value located in s must be characterized by points along the x,-x, y or -y dimensions. Since the dimensions y and -y have been defined as perpendicular to the dimensions x and -x respectively, there exists in this space, s, points that cannot be plotted. Everywhere there is a point in s that can be plotted, there exists an adjacent point that cannot be plotted – we have as much white space as there are values that can be plotted. Now that we have declared our space, we can differentiate by declaring there exists another space s1.

        x1 ->> (p, 45->x, t)
        y1 ->> (p, 90->x1, t)
        -x1 ->> (p, -45->x, -t)
        -y1 ->> (p, -90->x1, -t)

        s1 ->> (x1,y1,-x1,y1) -> [+/- oo]

        To give me a triangle, ABC with a true hypotenuse, I have to do something like this:

        AB = [(s->Ax,Ay)(s->Bx,By)]
        BC = [(s->Bx,By)(s->Cx,Cy)]

        CA = s1->>[(s->Cx,Cy)(s->Ax,Ay)]

        Essentially, this says – assign to s1 the values Cx, Cy, Ax and Ay from the space, s respectively. It is important to note that s1 stems from s by declaring x1 ->>(p,45->x,t). Without this definition, CA = s1->>[(s->Cx,Cy)(s->Ax,Ay)] would not be possible.

        You can’t draw a hypotenuse because it does not follow the rules of the respective space. You can, however, define a logical distance between two points. Your own words make my case: ” independently “. You can’t draw a line independent the system – unless it is arbitrary which then has nothing to do with the triangle, ABC. If you draw a line from C back to A – you’ve inherited the values of points A and C with respect to B. There is no independence that is possible here as the hypotenuse is dependent on AB and BC, which can reduce to dependence on B. If B is set to zero with respect to the original coordinate system, then B becomes p in a new system s2. A and C become dimensioned expansions from B.

        Kris

        • Cee says:

          Dear Kris,
          I am not convinced. In fact, I’m not sure I understand what you are trying to get at. Let’s try and clarify our misunderstandings, so here are several questions.

          (1) You write: “A single dimension breaks down into four components. ”
          It would have been nice if you had defined what a “dimension” is. I’m guessing that what you actually mean is what I would intuitively call an “axis”.
          I will think of it as an object with 4 components, according to your definition: “The point, the normal, the magnitude, and a name. ”
          This definition however begs several questions: What is the definition of a point? What is the definition of a normal? (I guess that the magnitude will be a real number, so I’m OK with that).
          Please realize that I know that you mean that it is defined by its origin, a line normal to it, its length and its label. However, this description does not make our understanding about what that object is any clearer. On the contrary, we now have even more undefined words that must be clarified! It is important to give clear definitions so that we can make clear predictions about the properties of objects.

          One last question on this point would be: Why do you think it is necessary to include the name as a component of that object? It seems rather, well, arbitrary as the name itself can be chosen arbitrarily and is not of much significance.

          (2) Why should this be true: “Negative values with respect to this normal require a second expansion”?
          Or this: “It is important to note that these dimensional expansions are unidirectional from the origin, p.”
          Methods currently used in linear algebra seem to work rather well without needing this restriction that ultimately leads you to conclude that somehow there are points that cannot be “plotted”.

          (3) You write: “Everywhere there is a point in s that can be plotted, there exists an adjacent point that cannot be plotted ”
          First of all, what does “adjactent” mean in this context? How do we calculate the distance between two points in your model? (On a second note, I’m not sure I understand what “plotting” is supposed to signify here. I think, very much like Mathis, you are confused about the difference between an actual picture of what we call a “graph” and its mathematical description).

          From your explanation about “differentiating” a system, I couldn’t figure out why this stops me from drawing a hypothenuse (or rather, why that should not be possible in current mathematics).
          (My example with independently drawing the triangle first was supposed to show that we can think of triangles as objects that are “there” independently of our descriptions of them. We try to model properties of these objects that we think are important in mathematics and then see what these properties imply.)

          I will not be at home over the weekend so it might take me until Tuesday to get back to you, should you choose to reply.

          Anyway, thank you for your patience and comment,
          Cee

      • Kris says:

        Cee,

        Actually, I was kind of surprised that someone responded so quickly.

        You are very particular as I suspected, and indeed was hoping you would ask these questions. Yes, you are a bit confused still. This will be long, and certainly carries the possibility of confusing you even more.

        1) You asked how I define what a “dimension” is. This is actually more important than one might think, because the classical definitions of dimension do not apply here. In my view, the mind is inescapable. There is no such notion of anything existing outside the mind. Remove all possible minds and there is no abstract notion of any mathematical constructs possible – only real, existential ones, which carry no meaning without the mind. Therefore, I place the mind as the primary origin, O. The mind exists at the top of the hierarchy in everything we understand. From this primary origin, we can work our way down and everything we create from here is subset it and relates to O in some way. In short, you cannot describe/define/understand a thing without a mind, which should be obvious, but apparently not since it is missing from almost all – if not all of the definitions in standard mathematics. Instead it is assumed that all minds will absolutely describe and understand the mathematical relationships the very same way. Again, it should be obvious that this clearly is not the case – all minds structure and understand differently, aside from, perhaps, a few anomalies.

        To define what a “dimension” is, I must first define what a “point” is. Since we’ve established that the mind is the primary origin, O, an existential quantity, we have provided a means by which these things can relate – in a hierarchical fashion. A “point”, p can then be defined as any other value or place that is not the same as or at the primary origin, O – either imagined or existential. A point may be given, arbitrarily, the same magnitude as the primary origin, O, but cannot be the origin O.

        O->p->>!O

        This says nothing about what a point actually is. It only says that a point, p exists at some place with some value that is not at the primary origin, O. I could call a planet, say Mars, a “point” if I really wanted to. It does not matter what existential qualities a “point” has – as long as it is not the primary origin, it can be a called point.

        I must also define what a “normal” is. A normal, n, is any direction with respect to the primary origin, O. The normal can be described as the entity that relates the point, p and the expansion, t to the primary origin, O.

        The magnitude, t is a simple value that expresses how much the primary origin, O must expand or extend to relate specific entities to O. Naming dimensions gives O a way to sort in a very selective manner, the quantities that exist underneath it – hierarchically. We need not limit the value of t to the set of reals.

        A “dimension” then can be defined as previously defined as a quantity that holds the components p, n and t. A “dimension” quite simply is a place holder in the hierarchical structure of the primary origin, O.

        x ->>(p, n, t): where “p” is the point, “n” is the normal, “t” is the magnitude of expansion and “x” is the name – all respective of O.

        You can call it an “axis” if you want to, but it doesn’t have to be.

        2)Methods used in linear algebra assume an undefined space. This undefined space is that of the particular mind examining the problem in question. It does not account for this in any of its definitions. There is a missing origin here which leads to a construct that removes the “normal” component. If you remove the “normal” component, then there is no way to actually bring the problem to the mind, as the mind inherently registers this normal component when examining the problem. We don’t seem to realize the steps we are actually taking to examine the problem. For example, we count a number of entities, n, but almost never realize that we actually started with ourselves. We give ourselves a value of zero and start counting. There may be a number of entities, n, with respect to us, but in total there exists at least a number of entities n+1. Negative values, as I had mentioned, are not really negative at all – they are only signed positives in relation to zero – which then relates to O. It is because there is a “normal” involved that two unique dimensional expansions must be made.

        3) Plotting refers to process of assigning value to a specific point existing underneath the spatial entity in the hierarchy of O. Adjacency then refers to values “right next to”, yet appears to hold the same value those values we have assigned to that point. Since it is set up in hierarchical form, two points may be assigned the same value, but given in respect to O, the two values are very different indeed. 1 != 1 may hold true, given certain respects to O. Calculating the distance between two points in my model depends on the circumstances – how these points relate to O.

        Other) I cannot fathom how objects can exist independent of their respective properties – real or abstract. Can you please explain to me how this is possible?

        I would agree that real objects may exist independent of our descriptions of them. However, I find that abstract mathematical objects simply cannot exist independent of the mind. If one forced these abstract mathematical objects into existence – say by drawing them in a book and defining them using some language, the abstractness of the object goes away as it then becomes a real object. Real objects must be analyzed by the mind to hold any meaning. The reading of the book, re-incarnates the abstractness of the object. Abstract objects always exist in the mind, which varies. A triangle, even when well defined, may not be understood by all minds in the same way. One may understand in his/her mind a triangle representative of three lines, while another may understand in his/her mind a triangle representative of three angles, and still another of three points. None are really any more accurate than the other two.

        It is extremely difficult to break habit. This mathematical knowledge has been drilled into our minds. To understand why you should not be able to draw a hypotenuse with the given standard definitions, one must be able to break down in detail the exact process in which he/she goes through to construct that hypotenuse.

        Thank you for your response.

        Kris

  9. crashloop says:

    Dear Kris,

    in your first post (read with a bit of charity) you argue that a cartesian coordinate system with only natural numbers for coordinates doesn’t allow for drawing “straight lines” in general, because it is something like a lattice (or pixel display).
    You go to great lengths to establish a new “coordinate system” (which is cleary ill defined) , but are totally ignorant of a “recent” development of mathematics which allows you to draw any line in any cartesian coordiante system, namely the real numbers. The property you are missing is that every subset of the reals has a supremum in the reals. This guarantees that between every two reals there is another one.

    • Kris says:

      Crashloop,

      “you argue that a cartesian coordinate system with only natural numbers for coordinates doesn’t allow for drawing “straight lines” in general, because it is something like a lattice (or pixel display).”

      Can you please quote me on that? Where did I make such a statement….?

      You seem to have misunderstood my statements. I did not say I could not draw “straight lines”. I said I could not draw a “hypotenuse” without first changing the system. I said that I could only draw lines that are respective of the defined coordinate system – that is, in a rectilinear Cartesian, 0 and 90 degrees, respectively. Sure – there is no reason for me to believe there doesn’t always exist another real that is immediately adjacent to any selected real in the set. If I did hold such a statement as true, I would be breaking continuity, and by gods that is precisely the principle I am building from. What I am saying is that there only exists those reals that are defined in the space – in which case, immediately adjacent points don’t exist there because they aren’t defined there. All values of the same magnitude are not created equal. A zero may be zero only with respect to the system in which it is being used. With respect to another system, a zero may or may not be zero.

      Since mathematics deals exclusively in comparisons, we can say that Relativity plays as big a role in mathematics as mathematics plays in Relativity. We must learn to look at it that way. While it is the absoluteness of mathematics that seems to be what makes it such an infallible language, ultimately it is the absoluteness of mathematics that breaks its back. There does not exist such a fundamental philosophical basis for mathematics that is infallible. Logical contradictions arise in unexpected places. This is simply accepted. Pick a school of thought on the philosophy of mathematics. It does not matter, They all share the common fate of being fallible when all is said and defined. Including my own.

      Personally, I am mostly in the camp of the Embodied Mind Theories when it comes to the Philosophy of Mathematics.

      Kris

  10. crashloop says:

    Kris,

    of course I can quote you on that. You write in your first post: “A coordinate system can only plot lines with respect to that particular coordinate system. In a 2D rectilinear, we can only plot lines that are 0 or 90 degrees with respect to the system.”. If I want to draw a “hypothenuse” (a straight line which is not orthogonal or collinear to the unit vectors defining the coordinate system) using this prescription this is equivalent of Mathis “step method” (see picture below Disproof #2 in the post).
    I strongly contest this notion. The axioms of affine spaces allow us to draw every line we want without changing the coordinate system. They are consistent and there is no “huge logical problem in maths”.

    • Kris says:

      You said “doesn’t allow for drawing “straight lines” in general”.

      What you quoted was very different from that.

      The axioms of ordered geometry, of which affine space are an extension of can be bashed by the very first 2 axioms which states: “There exist at least two points.” and “If A and B are distinct points, there exists a C such that [ABC]”

      There can’t exist “at least two points” without presuming there exists a space or set that allows these two points to be in logically different locations or quantities ( necessarily unique ) – thus making “A” and “B” thus there exists at least two points. In this case, the presumed set is linguistic – belonging to the set of letters, A-Z in the English Language. Or, perhaps, some other rather obscure symbolism that isolates A and B and unique entities. In other words, the logical definitions of the symbols, “A” and “B” as unique entities in some other set are being mapped to the axiomatic definitions here.

      Here is what happens when these sets are not presumed and mapped:

      1. There exists at least two points
      2. If A and A are distinct points, there exists an A such that [AAA]
      3. If [AAA], then A and A are distinct ( A != A )

      I think you get where this is going… The definitions blow up when already existing sets, such as the A-Z set in some other discipline are not presumed to exist, are accessible and mappable to the creation of new sets and/or spaces in mathematics. There is no axiom that tells me to select from an already existent set, or to create two unique stand in characters – without which the rest of the axioms completely fall apart because of the law of identity.

      The only reason many mathematical definitions, such as this one are successful and seem to be consistent is because we have a mind that is capable of mapping such sets that give definitions of unique entities over other sets from various disciplines without us realizing what we actually just did. That is, we have a mind that is capable of recognizing there is a difference between the symbols “A” and “B” without having to define there is a difference between “A” and “B”. Thus we use A and B presumptuously as primitive notions. This is a huge fundamental logical error because it assumes with a probability of 1 that something exists outside the mind. While something can be assumed to exist outside the mind with a probability close to 1, a probability of 1 cannot be held, in any case – no matter the circumstances.

      If I do not wish to hold such presumed sets or spaces, what I must do then is expand a set from the mind. Create it from a single unique entity – what I call the primary origin, O. I would say, axiomatically:

      1. There exists a mind, O.
      2. There exists from O a point, p, that is not O.
      3. If p != O, then there exists a point, t that is not p or O such that O->[p,t]
      4. If O->[p,t], then there exists a vector normal, n such that O->[p,n,t]

      Kris

      • Cee says:

        Dear Kris,

        I must admit I cannot see a significant difference between the axiom you take offense with:
        “There exist at least two points.”
        and the axioms you propose yourself:
        “1. There exists a mind, O.
        2. There exists from O a point, p, that is not O.”

        Your axioms, I feel, may be reformulated like this:
        1. There exists something that is O.
        2. There exists something else called p that is not O and has some relation to O.
        Or rather:
        1. There exist two things that are different, one called O and one called p.
        2. There is some relation between these two things.
        And even more simply:
        1. There exist two distinct things (we might as well call them points).
        2. There is some relation between these points.

        Making them sound very much alike to the axioms of geometry. Otherwise, I have to agree that the problem we are discussing is rather a philosophical one. Since however it is me that now brought up the issue with the axioms, I’m of course willing to discuss this further.

        Cheers,
        Cee

        • Kris says:

          The significant difference is in the law of identity.

          The axioms of geometry as they stand run into this issue, and so inherit primitive notions such that there is required an undefined set of symbols that are distinct ( appealing directly to the mind ), and mapped as stand in’s to create distinct objects in mathematics.

          “If [AAA], then A and A are distinct ( A != A )”

          The law of identity given the preceding axioms of geometry crashes this standard axiomatic approach.

          I need not use the symbols, p, n and t. I can re-use O over and over without running into a law of identity issue. I can say O != O and it can hold true, given certain preceding circumstances. I use p, n and t for ease of use, but it is not necessary – I just need an O.

          O->O, if O != O, then O->[O,O]

          The logical pointer as an extension of a hierarchy allows me to do this without crashing due to the law of identity and keep an ordered trace on it, though again, is difficult to keep track of which O you are talking about, thus p, n and t.

          My issues with all maths are entirely philosophical.

          Pi for instance is founded principally on the philosophical understanding that Zeno’s paradoxes were actually solved by the calculus and you can sum a series using Riemann sums and the like even though given such functions as:

          lim n->oo x^3/6(2+3/n+1/n^2)

          hold a logical contradiction.

          What happened? 3/n goes to Zero slower than does 1/n^2, yet they are *somehow* supposed to get there at the same time when n is the limit? Not possible even given an infinite number of iterations thus taking it to the limit.

          There is no real philosophical logical basis for infinite terms – summations to the limit of infinity does not work without holding a logical contradiction in it. Pi as a summation to the limit of infinity does not work without holding a logical contradiction in it. Zeno holds. My contention is that it is the mind that *makes* it work. That is, it is the mind that is capable of extracting finite values from infinite terms – forcing the infinite term into existence – to some arbitrarily specified degree of finiteness. Only a select portion of the infinite summation is selected. A true value is indeterminate because it will hold a logical contradiction.

          I hold the philosophy of mathematics above all else, because if there is a logical issue here, then guaranteed there will be a logical issue somewhere in the equation, no matter how simple.

          I do not disagree with the value of pi within a specified set of circumstances. I disagree that, not only pertaining to pi but all math in general, these things are not as absolute as mathematicians make them out to be.

          Kris

          • Cee says:

            Kris,

            You clearly do not know what a limit is in mathematics but hold only an intuitive notion of it. A limit is NOT (!) a process where you let some number increase and a function of that number approaches some value, its “limit”.

            If you knew the correct definition you would certainly not argue that the mind can extract the value from some sort of ininite process.

            I’m afraid you did not show a contradiction here.

            Cee

  11. crashloop says:

    Kris,

    your problem seems to be a philosophical one. But the question “are there things outside minds?” is not one I want to discuss in this blog (at least not at this very moment). It has been discussed in philosophy for a long time and I’ll recommend forums.philosophyforums.com as a platform for discussing such ideas.

    • Kris says:

      It has little to do with things outside the mind – actually. More so, it has to do with how the logical contradictions arise in the equations and definitions. Refer to last reply to Cee.

      Kris

      • crashloop says:

        I’ll give a definition of pi which doesn’t involve infinite series you loathe so much. Let pi denote the ratio of any circle’s circumference to its diameter. Pretty basic, isn’t it?

        Not understanding a concept doesn’t mean it contains a “logical contradiction” (What do you even mean by this term?).

  12. fgd says:

    Kris you are mathis. Now then, only post under your real name, so let’s get this straight: NO ASTROTURF CAMPAIGNING EVER.

    • Steve Urich says:

      Of course Kris is actually Mathis. Who else would defend such rubbish, except Mathis himself? Mathis is easy to spot; he’s always the one spouting misinformed nonsense.

  13. The fun thing is, this approach to calculate pi would work just fine if you take the area instead of the circumference.

  14. Rob says:

    The reason Mathis method doesn’t work is because he uses implicitly a different metric for the plane, he smuggles in a ‘manhattan’ metric, where distance between points is defined as:
    distance between point (x1,y1) and (x2, y2) = | y2 – y1 | + | x2 – x1 | instead of the usual metric defined as sqrt( (y2-y1) ² + (x2-x1)^2 ).
    And of course, that doesn’t converge to the actual distance over a curved line, it doesn’t even converge for any line which isn’t parallel to the x or y coördinate axis.

  15. David says:

    Sorry about the broken pingback link. Apparently WordPress automatically creates pingbacks whenever you link to another WordPress blog. There must be a bug in the WordPress software, since it created a faulty link. Anyway, the correct link is as follows:
    Miles Pantload Mathis (http://milespantloadmathis.wordpress.com/)

    Mod: I deleted the faulty link, I hope this is ok. I’ll have a look at your site. Cee

  16. Xavier says:

    Just a simple comment:
    Besides the round-about mathematical explanations I will like to point that Pi is an experimental number. Comes from dividing the perimeter by the radius. You can try that with any circumference and you will get 3,1 something (depending on the accuracy of measurements, but never 4, even if things are in motion.

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