This is a contradiction since it is said that both numbers were co-prime integers and we have shown that both are also even numbers. From this contradiction it follows that the root of two cannot be obtained from any two integers when implementing the ratio between them. This result is generalizable to any other non-quadratic value under the root and allows the understanding of the gap between Greek geometry (Pythagorean and Euclidean) and modern analytic geometry and algebra.
There are other notable commonly accepted cases of irrational numbers. For example, the ratio between the diameter and circumference of a circle is irrational and is represented by π ≈ 3.1415926535897932... Also, in The Vitruvian Man by Da Vinci, there is a ratio that is an irrational number, known as the Golden Ratio, Gorman (2002). The ratio between the total height of the Vitruvian Man and the height of his naval results on the irrational number: 
Wilkers (2010). This ratio is associated with the work of the Italian mathematician Fibonacci as the use of the sequence, which bears his name, is a way of obtaining the Golden Ratio by calculating limits of such a se- quence when the number of its terms tends to infinity, Spiegel (1963). This number has had different interpretations in art, ar- chitecture and for example in measuring female beauty, Tahan (1993).
A third case of irrational numbers corresponds to the napierian base of the exponential and logarithmic function, e ≈ 2.71828182846 ... used to model increasing behavior of different types and in different areas of knowledge (demography, economics, statistics, physics and others).
Contemporary notation, in ancient Greece would probably have hindered the compression of the demonstration of irratio- nal numbers as shown above. Indeed, the Indo-Arabic positional numbering of base 10 was developed several centuries (almost a millennium) after Hellenic ascension and was not introduced to Europe until the thirteenth century by the Italian mathematician Fibonacci. The standard algebraic notation for its part, crystallized recently thanks to René Descartes in the seventeenth century. These dates show the slow evolution of human thought and intellectual resistance to changes that occur in society.
Another example of the aforementioned is Zeno of Elea’s dichotomy paradox. The Greek thinker, with a biography as uncertain as that of Pythagoras, dedicated his life to defending the work of his teacher, Parmenides. There are two contributions to the evolution of thought by him and the Eleatic school. The first refers to the dialectical process of contrasting two op- posite versions of the same event and from them build a truth. This is 2300 years before the philosopher GWF Hegel would present his Thesis, Antithesis and Synthesis methodology. The second contribution of the school refers to the first known mention of the idea that would prevail in the coming millennia on infinitesimal calculus. Indeed, Zeno is the first philosopher to describe the physical world in terms of infinitely small time or space. Thus, intending to show it was impossible that physical reality was a reflection of the reality of the numbers, contradic- ting the Pythagorean sect.
Zeno of Elea, faithful to his style of dialect and intent on embarrassing Pythagoras’ followers, posed a series of paradoxes. One is that of Achilles, son of Peleus and the goddess Thetis, Trojan hero and traitor, murderer of Hector and the fastest of men, and a turtle. Zeno argues that not even Achilles could reach a turtle, the slowest animal in the world, if it were given a head start in a race. This was because when Achilles reached the position where the turtle was at the start of the hero’s race, it would have already advanced a certain distance. Then Achilles would reach the turtle’s new position, but it would have moved a further distance again obtaining the lead and thus extending the procedure infinitely, with increasingly smaller distances and Achilles always finding himself behind the turtle. This paradox is not very complex in the scheme of current thinking, but it was a puzzle to the Hellenic world. This paradox is an example of generating knowledge by direct observation.
The second of Zeno’s paradoxes is a variation of the aforementioned called the dichotomy paradox. If a stone is thrown from one point to another, it first must travel half of the distance, leaving the second half still to be travelled. Secondly, the stone must then travel half of the remaining distance or a quarter of the initial distance, still leaving half of the distance previously travelled. Then, the stone must again travel half of the remaining distance or an eighth of the initial distance, and there would still be remaining half of the new distance to be travelled. In this way the stone would never reach its destination thus making any type of movement impossible. With this, Zeno tried to show that the physical world does not exist as such and was just a projection from within each individual and not an abstract reality like Pythagorean numbers. There is also a higher incidence of empiricism, or observation of reality.
While Archimedes worked in the Third Century, BC with the idea of infinite sums similar to those which comprise modern calculus, Cajori (1909: 35), the mathematical language that en- ables efficient and thorough dissemination of the solution to Zeno’s paradox was not decided until the XVII century with the formal discovery of infinitesimal calculus. According to Cajori (Op.Cit.), Archimedes, in regards to infinitesimals, indicates that: “infinitesimals (infinitely small constants) were not used in rigorous demonstrations” Furthermore, it is stated that Archimedes “considered infinitesimals sufficiently scientific to suggest the truths of theorems, but not to furnish rigorous proofs”. One of the authors of this paper, working as a mathematics Teachers Assistant to freshman year engineering students, presented this exercise with excitement and enthusiasm, thus in just one line giving evidence of two millennia of mankind’s intellectual evolution. Unfortunately, none of the students were as interested or motivated as the Teacher Assistant.
The solution posited in continuation as the finite sum of a geometric progression with the first term 
and also with the ratio
. The distance traveled by the stone thrown by Zeno is represented by l. This is exhibited in the next expres- sion:
Taking the process to infinity, the stone would finally travel the desired distance l invalidating Zeno’s thesis. This simple exercise is a fairly elegant way of introducing the concept of limit and takes advantage of the formula for the sum of the first N terms of a geometric progression. However, from Zeno’s original approach, circa 460 BC, through Archimedes’ numerical approximation methods, until it was possible a formal approach, rigorous and universal solution to the paradox, after 1670 AD, it took at least 2130 years, which shows the long pe- riod of creation and consolidation of knowledge, for this particular case.
The slow pace with which societies recognize intellectual changes that lead to abstraction as the basis of thought not only ended Hippasus of Metapontum’s life 2500 years ago but interestingly also had consequences for James A. Garfield, the twentieth president of USA and amateur mathematician. Garfield was the second US president assassinated while in office after Abraham Lincoln. Prior to his tenure, he developed and published one of the many proofs of Pythagoras’ Theorem, available in basic math texts. This was done through the formation of a trapezoid, within three rectangle triangles inscribe and using an area calculus exercise and algebra to prove the theorem.
A proof of the theorem similar to that of President Garfield, is the following, Wilkers (2010): Consider the square in Figure 1 which consists of four right triangles with legs a and b and hypotenuse h joined at their vertices. The area of said square
corresponds to the square of one side, namely: A = (a + b)2. Moreover, the area of the square also corresponds to the sum of the area of each of the four right triangles that comprise the square of side h formed in the center of the figure. That is:
. Equating both expressions and reducing terms, we have the following:
, then: a2 + 2ab + b2 = 2ab + h2, then: a2 + b2 = h2.
With this, the validity of a theorem discovered more than two millennia ago is proved.
Square constructed with 4 triangles.
The cause of Garfield’s assassination goes beyond strictly political reasons and is related to the personal revenge of a North American lawyer whom was denied (unjustly in his opinion) a diplomatic post in Paris. As a product of his strange and incoherent reasoning as well as his erratic and aggressive behavior, it is speculated that Garfield’s murderer suffered from a major mental illness that led him to commit the crime. Paradoxically, it is possible that the two bullets that Garfield recei- ved could not have caused his death in the absence of assistance from multiple physicians. In the late nineteenth century, surgical equipment sterilization was not a common practice in medicine and was not fully accepted by the specialized society. Moreover, in the case of bullets, priority was given to extracting the projectile as soon as possible and at any cost. Therefore, during the seventy days that Garfield agonized, many doctors introduced their unsterilized fingers into the president’s wound trying to remove the foreign body. In this way, a relatively small wound became a fatal infection which finally killed the president on September 19, 1881.
Garfield’s death also shows the resistance to accept new ideas that occurs in society that break with established thought patterns, in this case in regards to medical ideas. In mathematics, this first occurred with irrational numbers, with zero (absurd in the Euclidean geometric design), with negative numbers and finally with imaginary and complex numbers. In other disciplines, such as the physics of Copernicus, Newton and Einstein, philosophical debates forced humanity to completely change its view of things. However, this resistance is often justified and grounded.
As shown above, the cited theorem is a truth in itself. Pythagoras’ Theorem, as any geometry freshman knows, is an irrefutable truth and universally accepted as such. Could there be any doubt? Respect to empiricism and its relation with Pythagoras’ theorem, Hessen’s work warrants mentioning. This indicates that Hume, who developed Locke’s empiricism, recognizes that mathematical knowledge is independent from experience and therefore universally valid. However the relationships between mathematical concepts are valid, independent of all experience. It is indicated that “the propositions that these relationships express, i.e., Pythagoras’ Theorem can be discovered purely through the activity of thought and do not depend on anything that exists in this world. Even if a triangle had never existed, the truths demonstrated by Euclid would al- ways conserve their certainty and proof”, Hessen (1993: pp. 64-65).
In the following, the theorem is subjected to an empirical experiment so that each observer obtains its conclusions according to their own perception of the facts.
An Experiment on Pythagoras’ Theorem
Pythagoras’ Theorem, from a theory of knowledge point of view, is primarily based on a strand of rationalism and therefore can be considered a universal law whose compliance is irrefutable by rational methods. This paper addresses the facts relating to the theorem through observation of reality, or rather, the use of empiricism where knowledge is obtained through direct observation of a phenomenon, captured by the sensory organs of the experimenter. Therefore, this is an experiment principally supported by fact.
The proof discussed below indicates that there are some points of the theorem that are difficult to understand when only the visual perception of the observer is relied upon. The aim of this experiment is not to create direct knowledge, but to present that an example of the fact that the truth about a given phenomenon depends on “the eye of the beholder” which is a typical situation in human behavioral sciences. The following exercise is intended to create research questions regarding the Theorem and also on how knowledge is generated based solely on direct process observation.
The proposed experiment consists of taking two rectangular sheets of paper of the same size (two sheets of printer paper). (see Figure 2).
The experiment involves the following steps:
Step 1. Take two rectangular sheets of paper of the same size (two sheets of printer paper) as shown in Figure 3, placing sheet No 1 onto sheet No 2.
Step 2. Verify, superimposing one page onto the other, that they are the same length (or width), as shown in Figure 4. This means that: 
on Figure 2.
Step 3. Join the corners, making them coincide so that the sheets of paper overlap exactly and that there seems to be single sheet of paper, as shown in Figure 5.
Step 4. With your fingers, slowly and carefully rotate the top sheet of paper, using corners 
as a pivot (see Figure 6). Figure 2 shows that with the displacement, an ABH or CDH right triangle is made.
Step 5. Measure with a ruler the segments 
y
.
Sheets No 1 and No 2 with their respective vertices.
Step 1 of the exercise.
Step 2 of the exercise.
Step 3 of the exercise.
Step 4 of the exercise.
Once familiar with the experiment, answer the following questions:
1. Are you familiar with the following mathematical expression: c2 = a2 + b2?
- Yes - No
2. The mathematical expression above represents:
1) Pythagoras’ Theorem, where “a” and “b” are the catheti of a right triangle and “c” is the hypotenuse
2) The square of the binomial
3) An equation whose unknown is “c”.
3. Have you ever doubted the compliance of the expression in question 1?
- Yes - No
4. In the proposed experiment, have you visually checked that the upper widths of the two sheets are exactly the same? That is:![]()
, where ![]()
is the width of the top sheet (Number 1) and ![]()
the width of the bottom sheet (Number 2), as shown in Figure 2, Or do you see a different result?
- Yes - No
5. In the proposed experiment with the sheets of printer paper, moving vertex B of the top sheet from D, downward from the lower sheet. Do you observe the following?
1) Vertex B remains precisely fixed to the vertex edge of the other sheet D to a point where vertex B inclines toward the left side (where B = H) and away from the edge of the second sheet (whose vertices are C and D).
2) Vertex B is never attached to the edge of the second sheet (with vertexs C and D).
3) Moving point B from point D of the bottom sheet, this (Vertex B) continues in the same direction as an inscribed circle whose radius is equal to the width of both sheets.
6. If in the previous answer (5), you accept the affirmation 1), according to your vision, the following should be present- ed:
1) Something absurd because, according to Pythagoras’ Theorem, this is an impossible situation.
2) That your vision is impaired and you cannot see the lines precisely. Should you go to an ophthalmologist?
3) You have visual doubts about Pythagoras’ Theorem
4) You should repeat the experiment several times because you have doubts about what you see.
7. If you repeat the experiment several times, do you see the same thing?
- Yes - No - Sometimes yes and sometimes no
8. If your answer was Number 7: “Sometimes yes and sometimes no,” What is the most predominant in your visual perception?
- Yes - No
Author’s Comments about the Experiment
Repeating Step 4 the following observations arise, depending on the point of view of the experiment. This exposes some ideas about the experiment and obviously invites the reader to analyze the discussion of knowledge generation. It is not intended, through these observations, to influence the reader’s vision and their conclusions.
It can be observed that a very acute ABH or CDH triangle is generated. The rectangle displayed in Figure 2 shows that from a point on the right side of the two sheets (Vertex H), the vertex of the top sheet starts to move away from point H on the right side of the sheet below and remains fixed. If this figure is observed, it would indicate that the upper side of both sheets have the same length![]()
, now moving the side of the upper
sheet, the side ![]()
becomes a hypotenuse of the triangle ABH formed by the displacement, it now appears as segment ![]()
however, it is the same segment![]()
. Thus, if this were the hypotenuse, it should not be the same as one side of the triangle when both sides were equal before displacement. That is, here the contradiction would be given that initially being equal to ![]()
now ![]()
is the hypotenuse of a rectangle triangle in which one of the cathetus is the same segment as the hypotenuse as initially it complied that![]()
.
If this is observed it gives rise to a central question: Is Pythagoras’ theorem only valid for some cases and would not be a general rule but a particular case when verified with a direct observation method based on empiricism? From a rationalism stand point, the test shown here does not have validity, however from an empiricism standpoint doubt arises for those who observe the presence of this triangle. In effect, visual observation would have a concrete answer through the concept of measurement error. The latter can be quantified using the theorem and the ratio between the height and the base of the triangle, allowing an empirical explanation as an observation of an imprecise measurement process.
Following the empirical method it would be necessary to repeat this case with many people and analyze the view of each one. This is the challenge that is posited in this paper. Obviously in order to respond to this empiricist concern the use of rationalist dogma is not necessary.
Some considerations could be given to the above explanations such as: imperfections in the sheets of printer paper, impressions caused by using a vertex as a pivot, measurement error, trigonometric errors when working with very small angles. Myopia, astigmatism and other variables that affect the observer’ senses could also be influences.
Conclusion
The answer to the proposed problem is not simple, because it is the basis of how knowledge is generated, if through what the observer brings incorporated in their mind and reasoning a pri- ori or processed in their brain what their sensory organs capture. This paper is not intended to generate an answer but to plant a doubt about the proposed problem as any observer could see a distinct reality.
The observations invite reflection on some scenarios. One position is that the Pythagorean result, from rationalism, is re- affirmed and another position could create a hypothesis in the sense that proof could be seen that physical reality does not necessarily behave in the same way as mathematical abstraction. Therefore, it could be contemplated. For some, the exercise po- sited, could also generate, among others, a doubt with respect to irrefutability of Pythagoras’ Theorem. From this the following questions arise: Is it reality perhaps that we can and want to observe? Does the empiricist method leave everything up to the perception of the researcher, the main problem of social and experimental sciences that complicate the validity of universal and irrefutable scientific law.
From a rationalist point of view, whether in mathematics or engineering, as in the aforementioned examples, what is proposed herein as a visual experiment of a theorem, does not pre- sent doubts because the differences can be satisfactorily explain- ed by measurement error. However, with behavioral sciences such as economics, sociology, psychology, such error is difficult to perceive quantitatively when the phenomenon is directly observed by the researcher, even when mathematical formality is used to describe the phenomenon. Thus, in these sciences the performance and perception of the subject studied, i.e., the observer, has a major influence on the creation of knowledge. In these latter sciences, the external perception of the phenome- non becomes crucial for the creation of knowledge.
References