Relativity of the mathematical exactitude, philosophy of science

“The geometric analysis of the ancients and the algebra of the moderns, besides that they embrace only matters highly abstract, and, to appearance, of no use, the former is so exclusively restricted to the consideration of figures, that it can exercise the understanding only on condition of greatly fatiguing the imagination; and, in the latter, there is so complete a subjection to certain rules and formulas, that there results an art full of confusion and obscurity calculated to embarrass, instead of a science fitted to cultivate the mind. By these considerations I was induced to seek some other method which would comprise the advantages of the three and be exempt from their defects.”
Rene Descartes. Discourse on the Method of Rightly Conducting the Reason and Seeking Truth in the Science, 1637, Chapter 2
Index
1. Scope of the thesis
2. Essence of the thesis
3. Importance of Rene Descartes
4. Opening new doors
5. Conclusions
1. Scope of the thesis
The objective of the thesis is to demonstrate that mathematics is relative and not an exact science. This fact opens the door to new point of views, new approaches in the philosophy of science. If mathematics is not an exact science, but relative, their results and the results of the other sciences based in the mathematics must be also relatives.
2. Essence of the thesis
What things are the numbers?
Do they exist in absolute and tangible form?
What things are the geometric figures?
Do they exist?
If they exist, then how, where, why and when can be appreciated by our senses?
A number is an abstract entity, a geometric figure too; they are creations of our minds that do not exist in material form.
The exactitude of mathematics is relative.
A number is a mental representation of one unit. Starting from that representation of the unit has been created all the numbers, which are additional mental representations of one.
One is the unique and primary amount. God is one; the point from which begins all the things that exists in the universe. One represents the same in mathematics. One is the point from where begin all the other measures.
All the numbers that you know come from one. 2, 3, 4, 5, 6, 7, 8, 9, are simple derivations, additions, to one. They do not have own entity. This idea denies the infinitude of the natural numbers. The unique real unit of measure is one. One adopts all the forms that exist in the objective world, as the water takes the form of the container that keeps her. Cero, alone, is nothing. Cero is the contrary expression of one, which is all. 1,000 is one one thousands times. Cero, when is associated to a number, is only a mental representation of the number of times that one has been repeated in that amount.
The primary amount is one; the other amounts are only the result of the increase of that primary amount. Observe that I have employed only the word increase. You can sum, rest, multiply or divide an amount but the result, ever, will be an amount that represents one. All the numbers are simple derivations of one, because of a rational, logic and simple reason: all the mathematical expressions are units, this mean one. The fractions are, per se, units. If you cut a bread by its middle, ½, this new middle is a new unit, a new one itself. ½ is regarding the original bread that already does not exist when you cut the bread; then, ½ is relative; what exist are two new unities integrated by two half, and each one of them is one unity, this mean one. This reveals that the numbers different to one are only names, appearance. One electron is one unit that integrates a major unit, the atom, but the electron, per se, is an independent unit, since the mathematical point of view. The electron is one itself.
One represents the force of God, the primary creation. In that primary creation are the germs of life, the essential elements of all the existent things. One is the atom, which contains the primary source, the primary energy of the universe. The atom encompasses one force ---not two, three, four, etc--- that, when is associated to other one, can develop millions of new ones; that is the principle of the chain reaction. I think that Galileo Galilei understood in a broad sense all this issues and for that reason he said that mathematics is the language of the universe.
The Holy Scriptures says that God created man ---one man--- and from that man God created later woman. From this first one was created a new life. That is the first mathematical expression of the numbers. It means that from one was created other one and was formed the number two that is the sum of one plus one. Man is, therefore, one at the image and similarity of God. Man was one; woman was two, the addition of one to the first one.
The geometric figures are also derivations of one. A triangle is one three times in a mental creation in the space. A circle is the prolongation of one since one point and its return to the same point. A straight line is one mental creation of one dimension between to extremes. Triangles, circle, straight line are simple names that our mind associate to determined figures created also by our own minds. But none of those figures exists per se, with own corporeity, in the objective world. The reason is able of create logic explanations. The problem is that sometimes the logic explanations are not true.
I remember the concept of my first professor of philosophy regarding mathematics. He taught: mathematics is the study of the exact dimensions in the time and the space: in the time is the arithmetic and in the space is the geometry. Some time later of studding that concept I asked myself: exact dimensions regarding what?
There are matters about which the people do not give its opinion because consider that they are truths that can not be set in doubt. And, on the contrary, there are subjects about which the people express their opinions because are not proven truths, or not enough proven truth.
The people consider that mathematics is an exact science; this concept is not enough discussed or not discussed. The exactitude of mathematics is an axiom. The people say yes, mathematics is an exact science and you commit a sin if you express any doubt about it.
But mathematics cannot be an exact science because of a simple reason: mathematics is a creation of man mind and, in consequence, it does not exist in the material reality per se, with own life. You cannot touch, smell, taste or see a number or a geometric figure but like representations of the real objects that exist in the reality.
The numbers and the geometric figures are only mental representations of the things that exist in the objective world.
Mathematics is a set of conventions built by the human being mind. For example, two centuries ago, man established that one meter was a portion of the meridiem that crossed Paris; currently, in the 21st century, man changed that concept and now he says that one meter is a fraction of the speed of light. This new concept was approved recently by the International Agency that controls the measures in the world. This simple example confirms that the exactitude of mathematics is relative. Man changes the concepts and the mathematical measures at his convenience; this demonstrates that mathematics is not an exact science.
When I was a child, a small friend told me: Pablito, (little Pablo), broke the toy to see what does has it inside; I remember that I said to my small friend: I want to play with the toy and to know how it work but I do not desire to destroy it. Now, I would like to know some things but without destroying them.
There is much kind of minds; ones accept the things like they are in appearance while others ask why.
I follow my own way, the dictates of my conscience, hence I write my opinions, it does not matter if they coincide or not with the generalized opinion of the people. There is not a unique truth. I think that the exact sciences are the sciences of nature that can be proven in the reality by mean of the senses. However, there is a hide reality that we cannot perceive with our own senses. We cannot see the atoms but we can experiment their consequences through the experience of their victims (Hiroshima and most recently Chernobyl) for example. One of the objectives of science must be to discover the hide reality of the things, the not apparent.
It is the sensible experience ---direct and indirect--- the main source of the knowledge. The direct experience is our own experience and the indirect the acquired by mean of the experience of other people.
The reason is the other source of knowledge. Our mind has the capacity to create and discover new things. But the creations and discoveries of the mind must be proven in the practice, in the reality of the life. Mathematics is a creation of our mind developed to make the real world more accessible. The mathematics is a form of expression, a language, in the same sense that is the oral and written language. But, while the oral and written language employ words, the mathematics use numbers and geometric figures.
The words, the same as the numbers and geometric figures, are abstract creations of our mind to name the things of the reality.
3. Importance of Rene Descartes
We must mention the work of Rene Descartes (1596-1650), because he is considered the father of the modern Rationalism, philosophical conception that assigns to the intellect the cause of the knowledge. The opposed thesis to the Rationalism, is the Empiricism, which principal exponent was the British philosopher Francis Bacon (1561-1626), creator of the scientific method of research.
Mathematics is a result of the reason; it is, then, a rationalist discipline, different to the empiricist sciences; they are the sciences which results can be proven in a tangible and practice form, following the scientific method of research. You can prove the existence of a tree in any place, but you cannot prove the absolute or corporeal existence of a number or a geometric figure. They belong to the intangible world, to the world of the ideas. The ideas can be transformed in real things or they can represent real things, but the ideas, itself, in its essence, do not have corporeal existence. So that mathematics is an abstraction that does not have corporeal, real existence.
Numbers and figures do not exist; what exist are the material things of the world; we only give name and measures to the things that exist in the objective reality. Man has conceived the figure of a triangle, but the triangles exist only in his imagination. The forms built by our minds sometimes coincide with the objects of the reality and, therefore, the human beings have developed theories and practice about the numbers and figures.
Rene Descartes is one of the most influential figures of knowledge; moreover of philosopher of clear expression, was a relevant mathematician and is considerer the father of the analytic geometry. But the clear thought of Descartes led him to see the knowledge and the science from an objective perspective. For example, in the second chapter of the Discourse on the Method of Rightly Conducting the Reason and Seeking Truth in the Science, 1637, he wrote the following:
“The geometric analysis of the ancients and the algebra of the moderns, besides that they embrace only matters highly abstract, and, to appearance, of no use, the former is so exclusively restricted to the consideration of figures, that it can exercise the understanding only on condition of greatly fatiguing the imagination; and, in the latter, there is so complete a subjection to certain rules and formulas, that there results an art full of confusion and obscurity calculated to embarrass, instead of a science fitted to cultivate the mind. By these considerations I was induced to seek some other method which would comprise the advantages of the three and be exempt from their defects.” End of the quote.
The logic question that arises after reading those ideas is: how the father of the analytic geometry could express those opinions about the geometry and the algebra?
The words of Descartes do not require more explanations. Obviously, he had a deep rejection for the algebra and for the forms employed by the ancient geometers; however, he worked in the geometry and developed his own system, named after his death the Cartesian coordinate system.
In his Discourse on the Method, Descartes left for the posterity his four rules to get the truth in the science; the first rule, and most important, is the methodical doubt.
Descartes wrote:
The first rule was never to accept anything for true which I did not clearly know to be such; that is to say, carefully to avoid precipitancy and prejudice, and to comprise nothing more in my judgement than what was presented to my mind so clearly and distinctly as to exclude all ground of doubt.
The second, to divide each of the difficulties under examination into as many parts as possible, and as might be necessary for its adequate solution.
The third, to conduct my thoughts in such order that, by commencing with objects the simplest and easiest to know, I might ascend by little and little, and, as it were, step by step, to the knowledge of the more complex; assigning in thought a certain order even to those objects which in their own nature do not stand in a relation of antecedence and sequence.
And the last, in every case to make enumerations so complete, and reviews so general, that I might be assured that nothing was omitted.
The long chains of simple and easy reasonings by means of which geometers are accustomed to reach the conclusions of their most difficult demonstrations, had led me to imagine that all things, to the knowledge of which man is competent, are mutually connected in the same way, and that there is nothing so far removed from us as to be beyond our reach, or so hidden that we cannot discover it, provided only we abstain from accepting the false for the true, and always preserve in our thoughts the order necessary for the deduction of one truth from another.” End of the quotes.
4. Opening new doors
The acceptance of the first Descartes’ rule opens innumerable questions in many fields of the science. He warned that it not must be accepted like truth something that has not been proven without any doubt. I do not want make explicit mention of doubts, but only remember that the intellectual formulations and speculations are not enough. Something that has not been proven in the practice, in the reality, do not deserve to be considered like a scientific truth, an irrefutable truth; some mathematical assumptions cannot be an exception.
Mathematics is the system of measures employed by the sciences and, in consequence, if some mathematical conceptions can be in doubt, it means that the results in other sciences might be also put in doubt.
There are logic and rational questions linked to the mathematical exactitude, like how, with which level of certainty have been measured the speed of light, the motion of the heavenly bodies, the distance between them? How can be confirmed that those appreciations are right? What is right? What is true? How can be proven that the Einstein’s paradox of the twins is true? How can be assured that some of his appreciations ---that represented a deep change for the sciences of the 20th century--- were not relatives?
5. Conclusions
- Mathematics must be taught with a practice and useful sense and not like a sacred science that do not admit any kind of doubts about their postulates.
- Mathematics should be reduced to the simplest formulas for the general comprehension of everybody.
- The complexity of mathematics comes from the complex minds of men that do not have the clarity to expose their thoughts with clarity; therefore, its ideas ---theorems, formulas, calculus and others--- are incomprehensible.
- Many of those complex mathematical assumptions are good for nothing, like Rene Descartes described the work of the ancient geometers and the algebra.
- An important intellectual task of the sciences must be to review the thesis that has been built in base of mathematical assumptions and, in consequence, ideas not enough proven in the objective reality, but based only in the imagination and the reason. This would be a way to conciliate the two extremes between Rationalism and Empiricism.