On the Meaning of Mathematical Incommensurability.

This post is a summary I’ve written only for intelligent people. If you are not intelligent you are going to lose you time here but, because non-intelligent people love very much to lose their time, reading this post can be an enjoyable experience to you as well but, take care, you can find out that after all you’re smarter than you thought and maybe that won’t be so funny.

Think about how the human being could have started to measure linear lengths and areas. I guess to measure a linear length for the first time it was necessary to make an abstraction. We needed a referential segment based on a primary referential quantity; In that sense, we can suppose we created our referential segment of length 1 based on our primary number 1, one step, one thumb, or whatever other one. With that referential segment we can measure any linear distance. Our primary segment 1 is an abstraction, we did not measure how many points or lines are inside of that segment, we simply accepted it was right to consider that distance as our referential unity to measure linear distances; But that is not a total abstraction because of inside of that referential segment there is a central point which divides the segment in to two equal segments; that central point determines the inner symmetry of our referential segment 1.

Later we also could create a referential segment of length 2 based on our referential segment of length 1. The original inner symmetry of 1 is respected because we can divide the segment 2 into to equal segments of 1. The first problem arose when we wanted to combine the segments of 1 and 2 to measure a distance because the original symmetry is lost: the segment 2 is larger that the segment 1, we can not set our central point of symmetry in the middle of those segments. But that disproportion can be saved by creating a new referential segment based again on our referential segment 1. So we could create the referential segment 3: The symmetry is saved by setting the segment 1 in the middle of a right segment 1 and a left segment 1, calling the whole distance our new segment 3. That is the way I think we can imagine how prime numbers appeared in geometry.

Then, we also wanted to measure areas. And for that we needed to create a referential area based on our referential segment 1. So we took our referential segment 1 and built a square. And we agreed that the space inside of that square had the value 1. It is also an abstraction because we didn’t measure that inner space but the central point of that square 1 and its inner symmetry is not an abstract convention.

But the next problem arouse when we traced the diagonal inside of our referential square of area 1. And that problem could not be saved in same way we solved the disproportion when it came to measuring linear distances, because now the disproportion appears inside of our referential area 1 built with our referential segment 1. The disproportion appears inside of the original unity itself. So, we can not use the segment 1 or our square area 1 to fix that disproportion. And we see horrified that the hypotenuse of the equilateral triangle is irredeemably disproportionate with respect to the side 1 of our square area 1. The infinite decimals of incommensurable quantities appear, I think, because we are trying to compare two quantities and one of them is not referenced to our original reference of 1.

It occurs the same when we try to measure the area of the circle of radius 1 by using our referential square of side 1. We get the unsolvable disproportion between the perimeter and the diameter because they respond to different and disproportionate referential original (or primary) quantities, referential (primary) linear segments, referential (primary) square areas: the rational and the “irrational” ones.

I think the circumference of radius 1 is a complex area formed by two kind of different squares that carry different kind of disproportionate symmetries: the square of area 4 referenced to our “rational” referential square 1, and the square of area 2 related to an “irrational” square of area 1i. The square 2 is inside of the perimeter of the circumference touching with its four corners the circle, and the square 4 is outside of that perimeter touching with its four sides the circle.

It is clearer when we divide the exterior square in to four squares of area 1 and we divide the interior square in to four squares of area 0,50, and we set the central point of those squares. Measuring through any diagonal the distances from the central point of the circumference until the central point of those squares we see that we need necessarily two different kind of intervals or referential segments of measure, two different kind of referential metrics to reach the central points of symmetry of the squares of 0.25, 1 and 4, and the centers of symmetry of the squares 0,50 and 2.



I’ve used the term intervals because the resultant figures appear as musical scales. If we repeat those referential intervals, extending (or “projecting”) them in a consecutive way through the diagonal, we see that the disproportionate intervals concur at a specific point of the diagonal: at the 7th and 5th intervals the central points which represent the two kind of referential centers of the disproportionate symmetries converge.

So there are 7 blue intervals with 8 blue points and 5 red intervals with 6 points. I’m considering the fist and last points as complex zeros formed at the same time by blue and red points.

Following the diagonal we can see:

  • Zero blue an red points
  • 1 blue point (blue interval 1)
  • 1 red point (red interval 1)
  • 2 blue point (blue interval 2)
  • 2 red point (red interval 2)
  • 3 blue point (blue interval 3)
  • XXXXXX (no red point)
  • 4 blue point (blue interval 4)
  • 3 red point (red interval 3)
  • 5 blue point (blue interval 5)
  • 4 red point ( red interval 4)
  • 6 blue point (blue interval 6)
  • XXXXXX (no red point)
  • Zero blue and red points where the blue 7th interval and the red 6th interval converge at their respective end.

I think the intervals comprehended inside the two rows marked with XXXX are what is known in musical terms as the “tritone”. There, the periodical alternation between the two kind of intervals is altered and it is perceived by our senses as something unexpected. And until now, it has been considered as an irresolvable disharmony which seems to be born from the Nature itself.

I consider each “Do” note would be in that sense a complex musical note (formed by the convergence of the rational and irrational symmetries) repeated periodically through the same scale.

In this sense, taking the all the other notes of a diagonal I think they represent the only rational or the only irrational part of a complex note that only can be formed by combining different kind of scales, the 4 diagonal (Z) scales and the four XY scales interact to form the complex notes. The intervals of the XY scales are inverted with respect to the Z intervals. We only need to seek where and how the intervals that carry the irrational and rational symmetries converge in a projective way. For example, on the X coordinate will take place the projective convergence of two intervals of the upper right Z and the bottom right Z. The whole space would represent entangled musical scales that respond to the different kind of rational and irrational symmetries.

So, I think the music should be arranged and played in a circular “circular” way instead of only a linear one. The listeners of a musical orchestra should be the middle of the scene placed inside of two concentric squares; The music should be played by different orchestras placed on the different XY and Z coordinates of that flat space.

About music and irrationality I think you could be interested in reading the beautiful article of Peter Pesic mentioned above, published by the University of Chicago Press: “Hearing the Irrational. Music and the Development of the Modern Concept of Number” which first pargarapgh states – with capital letters – that “WE TEND TO TAKE FOR GRANTED the broad inclusiveness of our current concept of number (comprising integers along with rational and irrational quantities), as if it were inevitable and unalterable, though in fact it took its present shape only over the past few centuries.1 Indeed, the modern concept of “real numbers” differs profoundly from the concept of number as it was commonly understood in the West from antiquity until about 1600. Given this fundamental change, what is the relation between ancient and modern concepts of number? Why and how did the concept change? And what difference do these changes make? In the history of ideas, this shift of understanding is so consequential that it deserves much further study, not least because modern mathematics depends on it, hence also much of modern science.”

On the other hand, when it comes to measuring the area of the circle: If we project three times the square area 1 through the diagonal following one of the mentioned intervals, from the zero point until the point placed where the perimeter is touched by one of the corners of the square area 2, we get a resultant area of value 3. We know the area of the circle of radius 1 is not 3, it is an irrational quantity, 3,14… with infinite decimals. But I think we should be able to measure the area of the circle without considering irrational magnitudes by taking account of the two kind of mentioned symmetries, considering the circumference as a complex area.


So to measure the area of the circle by projecting our referential square 1 N consecutive times through the diagonal, we should project (displace) that square not only until the place where the perimeter is touched by the corner of the square 2 but also displacing it until the next point, which is related to the square 4 that touches the perimeter its side on the XY coordinates. I think we need to consider the rational and the irrational intervals. On the above left picture, the 0,14 (without infinite decimals) would be represented as the last blue stripe placed after the red colored squares. it is not possible to accept there the infinite decimals because at that point is where the irrational and rational referential intervals converge.


The consequence is that the use of a mathematical fact which is arithmetically correct as it is Pi, results conceptually and mathematically wrong. Irrational magnitudes appear then as misunderstood mathematical aberrations. The Gauss’ assertion about that arithmetic, and not geometry, is a priori and only its laws are necessary and true, appears then invalid. How could it be possible? If arithmetic gives us a conceptually inadmissible result, should not be more reasonable to think that our conceptual a priories are invalid instead of thinking that  an arithmetically correct operation could be wrong?

A similar question caused the break between geometry and arithmetic more than 20 centuries ago. That break was fixed later in an insufficient way, insufficient because it was at the expense of creating a break between our natural human reason – which now appears mathematically non-trustable – and arithmetic. The result is that we now use arithmetic in a blind and purely instrumental way. And you are not able to even to consider remotelly the possibility that your arithmetic operations could not be mathematically correct.

I think if arithmetic gives a conceptually unacceptable result we should review it. I think it should be necessary to reconsider why infinite decimals appear. Maybe it is because we are using our arithmetic in an incomplete way, or we are not taking account all the circumstances that concur on the reality we are working with, or we are not respecting the rules we have assumed, as it is that we can not compare directly things of different nature. What we cannot make is to scarify our reason for the sake of instrumentality. Rationality should be irrenunciable.  Mathematics is mainly to think and to understand, not to operate.

Lest’s consider irrationality in the Pythagorean theorem. We see that the length of the segments a and b (ando also a+b) is disproportionated with respect to the length of the segment c; If we compare a+b and c we get infinite decimals, an irrational magnitude. But the area resultant from a^2+b^2 is equal than the area of c^2. How is it be possible that those areas are equal if the sides of the squares can not be compared without getting an incommensurable magnitude? The area c^2 coincides with the areas a^2+ b^2 because the external elements of the squares a and b (the sides of those squares) are equivalent to the length of the internal elements of the square c^2 (the two hypothenuses placed inside of the square c^2; and their internal elements of the squares a^2 and b^2 (the two hypothenuses – four in total – of each square a^2 and b^2) are equivalent to external elements of the square c^2 (the sides of the square c^2). The sides of the squares and their hypothenuses carry the two different kind of originary symmetries, the rational and the irrational ones. I say “symmetries” but it could be said the two kind of referential metric magnitudes.


In this sense, a^2, b^2 and c^2 are complex areas that carry internally and externally the two kind of disproportionate referential metric symmetries. a^2+b^2 is not equal to c^2, those square areas are coincident or equivalent but their symmetries are different. Their symmetries are distributed in a different although equivalent way.

Mathematics have been developed in an abstract and instrumental way during the last centuries, I think, because just of the lack of comprehension of irrationality. This lack of comprehension also would explain why the discovering of “imaginary numbers” took place so late as mathematical objects suitable to be used arithmetically or algebraically without understanding why complex numbers appear and their conceptual meaning. Without having conceptual clarity the way gets slow and tortuous and the necessary efforts to advance become heroics.

I think understanding rationality in the way I explained above, clarifies conceptually what complex numbers are and lets to understand geometrically the problem of the insolubility of quintic and further equations with radicals. I think it is the same problem of the disharmony that appears in the musical scale with the named “tritone”.

I also think the complex zeros that appear when the mentioned intervals converge (saving the periodical disproportion) are the same thing than the Riemann’s non trivial Zeros or relevant Zeros when it comes to looking for the periodicity of the appearance of prime numbers.

I think the explanation I’ve made with a natural language is not far away from other attempts of explanation turning around similar ideas, the idea of an original magnitud, the referencial metric, that already appeared in the History of Mathematics since the Ancient Greece.

Already “Eudoxus gave a systematic development, done geometrically, of the ‘real numbers’, which the Greeks represented as geometric segments, called magnitudes, according to Wilder [25, p. 142]. Kline [14, Chap. 3] discusses Eudoxus’ development of a new theory of magnitude and proportion. Under his interpretation, a magnitude was not a number, but instead it represented an entity, such as a line segment or an area”. “Eudoxus developed the notions of a ratio of magnitudes and of a proportion (an equality) of ratios, which allowed for the treatments of both commensurable and incommensurable ratios”. (You can read the quoted paragraph in http://faculty.evansville.edu/ct55/Portfolio/WeierstrassArticle.pdf)

The Euclid V Book is about ratios and proportions and the book X explains incommensurability.

Aristotle distinguished between continue and discrete magnitudes in his theory of magnitudes.

Note that in “Greek mathematics, “One” was not considered a number (arithmos) because it was the “monad” (“Monas”) in terms of which all numbers were multiples”, from which all numbers are derived. The term “number” was restricted to integers greater than 1. (For this paragraph, see “Hearing the irrational. Music and the Development of the Modern Concept of Number” by Peter Pesic).

Later, Hermann Grassmann spoke about”primary units” and “extensive quantities” and defined mathematics as the “science of the connection of magnitudes”; Also Bernard Riemann considered mathematics as the “science of magnitudes”. And the Weierstrass’ theory of rational and irrational numbers was formulated as the theory of “numerical magnitudes”.

I think what I above called referential metric intervals, or referential originary kind symmetries are a similar idea than what Bernard Riemann considered as a “quanta” of varieties (or manifolds) (see also the notion of “multiply extended magnitudes”), in an abstract way, in his conference about the “Hypotheses which lie at the bases of Geometry”; and also they are a similar thing than the “gauges” considered by Hermann Weyl. I think both authors where speaking about different referential linear metrics carrying different kind immediately disproportionate symmetries. But each one of them made it from his own limited perspective and with their own new created language, and above all, without understanding irrationality.

In this sense I suspect that the named “non Euclidean geometries” are using – without being aware, in an abstract and instrumental way, through algebra and arithmetic – something similar to the different kind of referential magnitudes I mentioned above, and their connections and transformations, and that it is the root of its difference with the Euclidean geometries which only would use one kind of referential linear magnitude, the rational one. I think non euclidean geometries haven been developed by using the vertical plane, as “3D” volumes but not in an only exclusively flat space, as if the flat plane was already thoroughly known and already exhausted by the Euclidian postulates. (I remember in one occasion I saw a professor of maths saying that little kids were bored with maths because they first were thought the geometry of the flat space which was a very boring thing).

Maybe I have a wrong idea about non euclidean geometries but this is something I expect to clarify bey reading the book “Le problème mathématique de l’espace. Une quête de l’intelligible” by Luciano Boi. But I think that the circumference on a flat space is a non euclidean space because it is a complex area where the (at least) two kind of referential originary metrics-magnitudes – and the symmetries they carry – converge.

In this sense, the circumference and its perimeter traced on a flat plane are to me non euclidean geometries but not because of their relation with the Euclide’s postulate of the parallels but because they can not be sufficiently explained as complex entities by the Euclid’s book V where he explained the theory of ratios and the concept of magnitude. It is that book and the book X where Euclid should have presented a conceptual explanation about what irrationality is and how it is related in a successful and periodical way with rationality. Maybe he provided a solution in his geometrical point of view which has not been yet understood with the modern algebraical and arithmetical approaches but, without having read yet the Euclid’s books V and X, I think they did not find the solution because of the problem of squaring the circle; I think the circle and its area only can be measured in terms of a square by using two kind of referential metrics that converge periodically, two squares (the internal and the external) that carry or represent two kind of symmetries; and also because of the lack of explanation about why the squares areas are equal on the Pythagorean theorem while the related sides are not proportionate.

I will research about the Euclide books V and X. For that, a highly interesting and indispensable article to be read about the book X is “An invitation to read Book X of Euclid’s Elements” , by D.H Fowler.

Related to the bibliography on this wonderful article, you can read “The Croix des mathématiciens: The Euclidean theory of irrational numbers” by Wilbur Knorr, and also “Coloured Quadrangles: A Guide to the Tenth Book of Euclid’s Elements” by Christian Marinus Taisbak.

I hope I’ll have enough time to research through these treasures i just mentioned and some other references you can see on the Fowler paper.

David Herbert Fowler was a Historian of mathematics, he sadly died at 2004, you can read this obituary on the Independent to get an idea about his personality and his original and fresh ideas about ancient greek mathematics. He also wrote another highly interesting book to be read: “The Mathematics of Plato’s Academy: A New Reconstruction”, (look for the expanded second edition better than the first one).

On his article about the invitation to read the Euclide’s Book X, Fowler brings out the great difficulty that modern and current mathematicians – the Book X is considered “The cross of mathematicians” because of it – have for understanding the Euclide Book X because they tray to understand it from their arithmetic, abstract and algebraic point of view, when the Greek geometry was not arithmetized and they did not use algebra either; Actually Euclid uses words instead of numbers to state his definitions and propositions.

In this sense I’d recommend you to read the page 242 of the linked article, although I will transcript literally a paragraph where Fowler states that the:

“arithmetization of geometry starts by choosing, explicitly or implicitly, an assigned unit line; then all lines become endowed with a numerical length with respect to this line, with the essential feature that it is assumed that these lengths can be manipulated arithmetically. Similarly this unit line defines a unit square, whence all sufficiently simple plane regions (e.g., rectilinear, circular, etc.) are assigned numerical areas, and this definition is shown to be consistent with the definition of the area of a rectangle as the product of the lengths of its sides. Similarly for volumes. Lengths, areas, and volumes are then mixed and manipulated indiscriminately as dimensionless numbers, and we construct
and solve expressions involving these numbers; and these expressions are abstracted into algebraic formulae which are also manipulated irrespective of their geometric sense”.

The ideas of the mentioned page 242 and this particular paragraph are very similar to the ideas and the way I’m explaining them that I defend on this blog about the use almost exclusive of algebra and the arithmetization of geometry by modern mathematics. I think my approach – because I didn’t study mathematics – is purely geometric and visual but I think that geometry cannot exist independently of the idea of numbers, it derives and should be created starting from the idea of numbers. Its breaking with geometry likely took place with the discovering of incommensurable magnitudes and the impossibility to save it as it was made with numbers. But Geometry cannot be reduced to an abstract arithmetic because geometry is the visual representation of symmetries. This approach brings us to consider that numbers cannot be used in an abstract way either because they also represent symmetries, in a different way than lines, areas, and volumes, on the planes and spaces, but symmetries related to the distribution of quantities in any case. We can consider in an abstract way only the first originary unity that is or primary reference to measure the infinite space where we are.

It does not come to geometrizing arithmetic, in an opposite pendular movement as it always happens in the History of humanity, it comes to understanding what numbers are, to being aware that numbers also represent symmetry.

But Fowler, after all, also created a kind of particular algebraic or symbolic and so abstract representation to explain in that paper the ideas of the Euclid’s book X, which to me does not have much sense, although he recommended to people uncomfortable with his personal symbols to go to a translation of the Euclid book or even to the original Greek to read the text. That is what I’m going to do.

When I started to think about physics and made the models I explain in this blog, physicists told me that my ideas were not understandable by them because my pictures were not mathematics and mathematics has a subtlety that is not comparable to the natural language I used. My ideas were seen as artistic in the best case. But what I perceived discussing with them was that thy did not have a visual idea about the models they were working with and even worst, that they were unable to have a rational discussion about the ideas they have accepted, for example what is the vibration of the emptiness of the empty space in the Higgs mechanism, or what is a virtual particle when it comes to the atomic model; Actually they have many difficulties because their models have been developed in an abstract and instrumental way and are based on abstract mathematical developments also created in an instrumental way. When you show them the logical contradictions of their accepted models they say that things are in that way and end period. (The main argument that they use is the technological development that we have is just based on the currently accepted models, to me those models are the obstacle, the current ceiling for a true scientific revolution that let us cure our cellular illnesses and control gravity without destroying matter).

When I decided to start thinking about mathematics to be able to express my ideas in terms that were understandable by physicists, and I saw that the bases of the mathematics they are using have clear inconsistencies based on the lack of comprehension of irrationality, I found that the geometrical figures that I made were not understood – for my total surprise – by mathematicians. I thought, and I see now because of what Fowler comments about their difficulty to understand the geometry and language of the Eucluid’s book X I see I was very right, that because of the use of algebra and the abstraction and instrumentality of modern mathematics, they are not used to understanding pure geometry; And what is wors, that they are not used to thinking rationaly, they are only used to operate and in the best case to especulate in an abstract and imaginary way without reference to the simple and concrete reality. I even have been told for some people that geometry is not so accurate like arithmetic, as if geometry was not really trustable while only arithmetic and algebra were.


On the other hand, going back to the ideas about modern logic, thinking in terms of “classes”, if we think in primary-generatrix classes, as if they were sets, and the subclases derivate from them we would need to think about the kind of symmetries those subclases carry or represent externally (related to their perimeter) and internally (related to their areas), how those different symmetries are periodically disproportionate – can not be immediately compared, connected or related – how they can be connected locally and periodically (forming a “continuum”), and how the equivalence of those symmetries can occur by comparing the exterior with the interior symmetries of those subclases. The subclasses-square-areas are derivate extensions of the two primary classes (the referential square rational and irrational areas), which are derivate from the two primary linear classes (the rational and irrational referential segments, which are derivate from the number- quantity, the “monad” without extension, in terms used by Leibniz, number 1.

I’m very interested in and still reading about the Riemann lecture which is a key text in the development of the modern mathematics. If you are interested you can find about it in these books: I found the “Labyrinth of Thought. A History of Set Theory and its Role in Modern Mathematics” by José Ferreiros, “Bernard Riemann, On the hypotheses which Lie at the Bases of Geometry“ by Jurgen Jost. “A comprehensive introduction to differential geometry“, Vol 2, by Michael Spivak.

These ideas related to the History of maths are clues that I will try to developed in a clearer and deeper way if I have time. When people think about the fundaments of geometry or arithmetic or about what irrational numbers are or what the concept of number itself is, they and their ideas are automatically classified inside of the realm of “Philosophy” – in the better of the cases Philosophy of mathematics – or “logic”, or “linguistic”, or whatever thing that seems like outside of true “mathematics” because mathematics has been almost totally reduced to arithmetic and it seems that it only can be acceded from algebra. I also will try to clarify that misconception about the current arithmetised and “algebratised” mathematics in a more extensive way in further posts.

Another clue I’m going to developed when having time is related to a very interesting figure who is the german mathematician Gottlob Frege. Totally ignored during his live, he is currently considered the founder of the modern mathematical logic and the father of the analytical philosophy thanks to the works of Russell and Wittgenstein. Frege tried to derive the whole arithmetic from logic, that is to say to reduce arithmetic to logic. He was very aware that mathematics “was inadequately supported. This entire impressive construction, he claimed, rested on shaky foundations. Mathematicians did not really understand what they were about, even at the most basic level. The problem was not a lack of understanding of the true nature of imaginary numbers, or of irrational numbers, or of fractional numbers, or of negative integers; the lack of understanding began with the natural numbers such as 1, 2, 3. Mathematicians, in Frege’s view, could not explain the nature of the primary objects of their science”. And that’s because he tried to “setting out the logical and philosophical foundations of arithmetic” (The quoted phrases of this paragraph are taken from “Frege, an introduction to the Founder of Modern Analytic Philosophy”, by Sir Anthony Kenny)·

(Other books I’m really looking forward to reading are “The Foundations of Arithmetic” by Gottlob Frege, “Frege, a Critical Introduction” by Harold Noonan, and “Patterns of Change: Linguistic Innovations in the Development of Classical Mathematics” by Kvasz, Ladislav).

I think it is not possible to deduce arithmetic from Logic without previously understanding what irrational numbers are from a conceptual point of view. Frege reviewed the concept of number and spoke about classes and extensions of quantities as a key point of his logical system. But his whole system become untenable because of the objection that Bertrand Russell made about it. I’m very curios to read more about the Frege system and the Russell’s objection because I think if the system became inconsistent, Frege was very secure about his validity before getting the Russell’s critic, its inconsistency likely could be related to the lack of comprehension of irrational numbers, the existence of at least two different kind of referential magnitudes that create periodical asymmetries. Frege was not able to fix his logical system after the Russell’s objection (the Russell’s Paradox).

But I think maybe there are still some things to say about the named “logicism” considering it under the light of a different and clearer comprehension of irrationality and considering numbers or magnitudes, linear extensions, and areas and from the point of view of symmetry. To me is evident that irrationality has not been yet understood at all. The advances we have had in mathematics – and physics – during the last two centuries have been obtained through blind instrumental developments, but not because of conceptual advances. That lack of conceptual clarity, I want to remark, has difficult enormously and slowed down and limited the development of modern mathematics and physics. I think we can not think in abstract classes or in abstract sets; numbers, segments and areas are not abstract entities, they represent symmetries. And it is necessary to be aware that there is more than one originary symmetry and how and when the disproportionate symmetries converge periodically and we can pass in a “continued” way from one to another.

On the other hand, coming back to the different kind of symmetries and their relation to the problem of quintic equations considered from a geometric point of view, I think the pictures below show where and when we can and cannot pass through one kind of symmetry to the another one making “local transformations” of symmetry. (To understand the pictures below as equations of n degree it is necessary to consider the square areas built from the rotatory axis. When we speak in Geometry about 2^2 we are not speaking about an abstract entity which we use as an arithmetic instrument for performing arithmetic operations without a specific meaning, we are speaking about a square area built on a segment; here, that segments, the sides of the n squares are the segments traced from the zero point toward the different directions, which actually is the same segment displaced in a consecutive way toward the different directions for drawing the circle. I would recommend you to draw the pictures by yourself starting from the beginning and to observe them by yourself; By making that you will be making a geometric analysis. That is a concrete analysis. If you are not used to thinking in these geometrical terms without using abstract algebra and arithmetic , without having equations to clear, it won’t be easy to you to understand these pictures by only looking at them here).




I said above “the rotatory” axis because I think that we can not consider the Z diagonal as an independent axis on the flat space we are working on. The Z diagonal is the same Y or X axes displaced toward the right or left. We have no problem by displacing the XY coordinates if we displace the whole flat plane, but if we want to preserve our original XY referential coordinates and introducing a new coordinate Z on the plane space, we need to be aware that we have superpose a new plane – the one represented by Z – on our working space, and that Z plane has rotated with respect to the initial position of our original plane related to XY. The actual effect is the same as if we had expanded our physical plane-space when we are considering the 4 Z coordinates from the point of view of XY, or as if we had contracted the expanded plane-space when we are considering the XY coordinates from Z.

The same can be said when we think about the orthogonal plane. Considering X like a horizontal plane and Y as a vertical plane, perpendicular to X, we have no problem because we are working with a unique quadratic reference. The problem appears when we displace (without been aware about that displacement, thinking that it is an independent orthogonal coordinate) Y toward the right or the left.

Related to the idea of the periodical expansion and contraction, I also think that our physicists have tried to measure our Nature from their static and quadratic references which have made things more and more complicated. Think for example that the gravitational field we are orbiting expanded and contracted periodically. I know it has not been detected yet, but if it were it would have many implications because instead of considering the apparent but actually inexistent orbital ellipse (the orbital ellipse does not exist, it is only a figure we construct when we see the planetary motion) we should had been considering the circular and periodically expanding and contracting gravitational field, which we think is existent although it is invisible. That periodical expansion and contraction would cause the planetary motions and their periodical acceleration and deceleration. The “Big bang” of our universe would be only a moment of the periodical variation of two intersected fields varying with the same phase, contracting at the same time. When those fields expand at the same time it appears the opposite phenomenon, a “big silence”.

In this model, Gravity would be a force of pressure – idea already considered by Fatio and Le Sage, and even by Newton himself who was very aware that to speak about a force of attraction was not a mechanical explanation (See “Pushing Gravity: New perspectives on Le Sage’s theory of gravitation” – 2002, Matthew R. R. Edwards) which would create
the gravitational curvature. Expanding and contracting gravitational fields would be longitudinal waves.

The same could we said about about the structure of an atom. I think the atomic model could be represented and clarified by considering two intersecting fields that vary periodically, expanding and contracting. The fields created in and by their intersection would be the subatomic particles of the central and shared atomic nucleus. When the intersected fields vary with opposite phase the electron field will be the field moving toward the left side where the gravitational fields contracts; its Majorana antiparticle – the positron – will be the same field moving toward the right side where the right gravitational filed now contracts (while the left one expands). That explains that the positron is a “virtual particle” when the mention field acts as electron, and that the electron field is a “virtual particle” when that field acts as positron. In the pictures below you can see those different fields, how the neutron is the antiparticle of the proton, and how the mirror symmetry takes place in different successive times, and also how the mirror antisymmetry appears between the Neutron- antineutrino and Proton -Neutrino. When the intersected fields vary with the same phase, when they both expands the ascending motion creates an ascending pushing wave which is a photon which decay when both intersected fields expands at the same time. Here the force of pressure takes place up to down and follows a direction which is opposite by respect to the direction of the gravitational flux. When the decay of energy takes place, the anti gravitational force appears at the convex side of the intersected gravitational fields.




In this unconventional model subatomic particles apear as complex masses formed by the convergence on the the X or Y coordinates of the called “quarks”, forces of pressure coming from the Y or X coordinates and viceversa. It is the same idea expressed above when it comes to understanding the complex zeros.





I think the History of the whole mathematics of the last five centuries – and the physics they hold – has been totally conditioned and limited because of the lack of comprehension about what irrationality is and the aim of measuring and analizing all things in Nature from a static and unique referential square.

If you read about the History of mathematics you will see that most of the times true innovative ideas were at the beginning ignored or considered heretical by the prevalent orthodoxy of the moment, or remained totally misunderstood. Some times due to the lack of clarity of their developers, or because of their high abstraction, or because of the change of thinking they implied, or because of the challenge of intimately assumed worldviews of many people, etc. To change the deeply assumed ideas and believes, the paradigms deeply stablished each moment – maths and physics are full of aprioristic believes – needs time. It also requires people willing to make the effort of thinking by themselves.

More than thinking rationally and critically by themselves, human beings love to create myths and to admire the myths others created before, and to admire to their creators as if they were not persons but a kind of titans gifted with super human intelligences and supernatural powers. Maybe it is why so many times through the History it has been necessary to wait until the death of the “innovative” thinkers to see how their right or wrong – always limited in any case – ideas were considered and fructified in many also limited ways.

None of the mathematicians – note that almost exclusively they have been men, which indicates the limited way in which maths have been developed because women are so or even more able than men to understand Nature in all its aspects and from all kind of approaches – that thought the ideas that you where taught and you have internally assumed was not another thing than a simple and very limited person like you and me.

Myths do not exist. Genius either. There are no intelectual giants. There are only people thinking by themselves and people who never have thought by themselves.

Do not forgive, my dear, that you will always be able to think and to discover Nature by yourself.

Seek true instead of utility.

Be and stay free.

Alfonso De Miguel Bueno

Oct 8,2016. Madrid / Updated Oct 22, 2016.