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Vol.8, Issue 5; September - October 2023; 

ISSN: 2837-2964 

Impact Factor: 6.67 

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A CRITICAL EXAMINATION OF VECTOR CALCULUS: A LOGICAL 

APPROACH  
   

Akmal Khurshidov   

Home of Physical Problems, Pisatelskaya Tashkent, UZBEKISTAN   

 

ABSTRACT: This study addresses the foundational aspects of vector calculus, acknowledging its widespread 

use in the natural sciences. While vector calculus has proven highly effective in various scientific applications, 

its validity and foundational underpinnings warrant formal-logical scrutiny. The absence of a comprehensive 

formal-logical basis challenges the assertion of vector calculus as absolute truth. This paper endeavors to fill 

this gap by conducting a critical analysis of vector calculus, employing the unity of formal logic and rational 

dialectics as the methodological framework. By subjecting vector calculus to this rigorous analysis, we aim to 

establish a solid foundation for this mathematical formalism. 

 

Keywords: Vector calculus, formal logic, rational dialectics, foundational analysis, mathematical formalism. 

  

 

INTRODUCTION  

As is well known, the mathematical formalism of vector calculus is widely and successfully used in natural 

sciences [1-7].  However, this does not mean that the problem of validity of vector calculus is now completely 

solved, or that the foundations of vector calculus are not in need of formal-logical analysis. In my view, 

standard vector calculus cannot be considered as absolute truth if there is no formal-logical substantiation of 

this calculus. Recently, there has arisen a necessity for critical analysis of the foundations of vector calculus. 

But there are no works devoted the analysis of vector calculus within the framework of the unity of formal 

logic and of rational dialectics. The purpose of the present work is to propose the correct analysis of the 

foundations of vector calculus.  The analysis is carried out within the framework of the correct methodological 

basis: the unity of formal logic and of rational dialectics.    

1. ANALYSIS OF THE CONCEPTS OF “DIRECTION” AND “VECTOR"                           

As is well known, in mathematics, physics, and engineering, a vector (or Euclidean vector, or geometric 

vector, or spatial vector) is called quantitative characteristics which has not only a numerical value, but also 

the direction [7, 8]. In other words, vector is a line segment with a definite direction (or graphically is an 

arrow), connecting an initial point with a terminal point. I.e., vector is a geometric object that has magnitude 

(or length) and direction and can be added to other vectors according to vector algebra. Physical examples of 

vector quantities are material point displacement, velocity and acceleration of a material point, as well as a 

force. Therefore, analysis of the concept of “vector” is not possible without the definitions of concepts of 

“movement” and “direction”.  

1. Movement is a change in general, any interaction of material objects. Category of  

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“movement” is a scientific concept that reflects the most common and essential property of phenomena 

(processes), the most common and essential relations and connections in reality. Movement is an attribute of 

matter. In accordance with the dialectical principle of the unity of matter and movement, the movement does 

not exist without material objects. But the movement is not a material object. The movement is manifested as 

the unity of opposites: changeableness and stability, continuity and discontinuity. Concretization of the 

movement is the main forms of movement: mechanical, physical (thermal, electromagnetic, gravitational, 

atomic, and nuclear), chemical, biological, informational, and social ones.  

2. Change as a process can be of two types: a qualitative change and quantitative change. The qualitative 

change (i.e. a change of qualitative determinacy) is studied by dialectical logic and natural sciences. The 

quantitative change (i.e. a change of quantitative determinacy) within the limits of certain qualitative 

determinacy is studied by formal logic and mathematics. The quantitative change can be studied only within 

a reference system which contains a clock as component part.   

3. A clock (i.e., a device containing a working clock mechanism, moving the arrow and the fixed dial) 

determines the time and time characterizes the clock. Time is a concrete concept because it expresses the 

property of the clock mechanism (clock process). Time t is a universal variable (with the dimension of 

“second”), an information basis that is used to put in order of information about events and processes in the 

world. Time t is defined by the following mathematical expression [9]: tn n  where n 0, 1, 2,...;  is 

elementary (unit) duration which can be made as small as desired. Concrete numbers (denominate numbers) 

tn have one and the same qualitative determinacy (i.e., dimension of “time”). The set of numbers tn forms an 

ordered sequence. A member of the sequence is called a moment of time. Numerical values of quantity tn is 

changed due to clock mechanism which continuously changes numerical values of the quantity  n .  

4. The mechanical form of movement (in particular, the motion of a material point M ) is studied in a 

reference system which represents the unity of the system of coordinates and clock. The system of coordinates 

is a system of measuring devices which determines the position (i.e., the set of coordinates) of a material point 

M in space. (For example, the Cartesian coordinate system represents the system of three connected measuring 

scales (drawing scales): straight linesOx , Oy , Oz with  printed concrete numbers (denominate numbers) 

having the identical dimension of “meter”). The space of the object (for example, geometric space, and energy 

space) is the set of possible (available) states of the material object (in particular, the set of positions of the 

material point M ). Each state is characterized by a certain concrete number (denominate number) having a 

dimension. Movement of an object in space is a process of transition from some states to other states, i.e. the 

process of transition from some concrete (denominate) numbers to other concrete (denominate) numbers.  

5. A process has the beginning (i.e., the beginning of the changes) and the end (i.e., the end of the 

changes). The transition from the initial state to the final state represents the sum of elementary transitions 

and, therefore, is characterized by an increase of the changes. In other words, the total change is the sum of 

elementary changes. Since elementary change is characterized by the concrete (denominate) number having a 

dimension, the total change has dimension as well and is expressed by the following mathematical formula:   

sn n ,  

where n 0, 1, 2, 3, ...,  is elementary (single) change which has the dimension and is  

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assumed to be constant. The set of denominate numbers sn forms an ordered sequence. The numerical values 

of the denominate quantity sn are changed if the numerical values of the quantity n are changed. If the 

numerical values of the quantity n are not changed with time, the process is not realized.  

6. A process is characterized by the direction (directivity) of change, the rate of change, and acceleration 

of change. If the process is not realized, the direction (directivity), the rate, and acceleration do not exist. 

Explanation is that the direction (directivity), rate, and acceleration are the properties of the process and not 

the properties of the material object. Therefore, the direction of change determines the order of the number set 

but an ordered number set does not determine the direction. Neither pure mathematics nor applied mathematics 

(i.e., the mathematical formalism of the natural sciences) does not contain a mathematical (calculation) process 

because the mathematics does not represent a computer or some other material device that realizes the process 

of change of the values of the quantitysn .  Change of the valuesof the quantity n is carried out by an operator 

(person). Therefore, the correct mathematical formalism cannot contain the concepts of “direction 

(directivity)” and “vector”.  

7. If one assumes that the mathematical formalism contains the concepts “direction  

(directivity)” and “vector”, then the formula for the quantity sn  to be written in the following vector form:  

  

sn n   

 

where  is elementary (unit) vector. But since the numerical values of quantity n in this formula are not 

changed with time, the process of change of the numerical values of the quantity sn in mathematical formalism 

is not realized. Therefore, this formula does not describe direction, and the mathematical formalism does not 

contain the concepts of  

“direction (directivity)”, “vector”, and “unit vector”.  

Thus, the “direction (directivity)” and “vector” are not mathematical objects (concepts). The concepts of 

“direction” and “vector” do not correspond to any geometric object (for example, a line segment). Indication 

of the boundary points of the line segment and  designation of these points with the help of terms (words) 

”beginning” (“initial point”) and “end” (“terminal point”) do not define mathematically a geometric vector 

(because the order of points do not define the direction of movement). All points of the line segment have one 

and the same qualitative determinacy: concept of “initial point” and concept of “terminal point” are identical 

ones. Therefore, the terms “beginning” and “end” of the segment are not mathematical definitions of the 

concept of “direction (directivity)”. An arrow is a visual (graphic) image of course.  In other words, verbal, 

literal, symbolic, numerical, and graphical representations (display) of the beginning and the end of the 

segment are not a mathematical definition of the concept of “direction (directivity)”. Therefore, the correct 

mathematical formalism can not and must not contain the concepts of “direction (directivity)” and “vector”. 

The coordinate system represents a system of three connected drawing scales: straight lines 

Ox , Oy , Oz, which cannot be attributed to the direction. Also, straight linesOx , Oy , Oz cannot contain the 

unit vectors. From the point of view of formal logic, the terms “direction (directivity)” and “vector” in 

mathematics and theoretical physics mean representation, i.e.  

imaginary image of the process, which is depicted with the help of an arrow.  

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x 

 cos V  V x  
 

   

2. ANALYSIS OF THE STANDARD PROPOSITIONS OF VECTOR ALGEBRA  

The mathematical concept of “vector in generall” cannot be used in the natural sciences: this concept does not 

make sense in the natural sciences. The concept of “vector” as used in the natural sciences is characterized by 

the concepts of “denominate quantity” and “dimension of quantity”. Therefore, analysis of the standard 

propositions of vector algebra must be done from this viewpoint.  

1. As is known, the position of a vector in the Cartesian coordinate system Oxyz is determined by its 

projections. The projection of the vector on the axis is defined as follows.  

There are vector V whose module has the dimension (for example, the dimension of speed, acceleration 

dimension, the dimension of power) and the axis Ox with the denominate  

 

numbers which have dimension of length. The projection of the vector V on the axis Ox is  

  

called the quantity (module, length) V  of directed segment Vx  located (placed, putted) on the axis Ox :  

  

where  is the angle between the vector V and the axis Ox . In general case, the vector V  

is decomposed into components under the Cartesian orthonormal basis  i , j , k  as follows:  

    

V Vx   i Vy   j Vz   k  

    

where i , j, k are the unit vectors of the Cartesian coordinate system; Vx   , Vy   , Vz   are the projections of the 

vector on the corresponding axes. The principal importance of the basis  

 

i , j , k is that the linear operations on vectors under the given basis become the usual linear operations on 

numbers – the coordinates of these vectors. In my opinion, these standard expressions are not free from 

objection. The objection is that the standard expressions are contrary to the formal-logical laws.  

 

Really, the standard expressions assert that segment Vx lies on the axis Ox (i.e., segment   

 

Vx   coincides with the segment of axis Ox ). From the point of view of formal-logical law of identity, this 

implies that these segments have the same qualitative determinacy (i.e., the same sense, the same dimensions):   

(qualitative determinacy  of the segment Vx )  =  

(qualitative determinacy of the segment of the axis Ox ).  

  

But the segment Vx  cannot lie on the axis Ox (i.e., the segment Vx  cannot be coincided with a segment of the 

axisOx ) because these segments have different dimensions and, therefore, different qualitative determinacy 

(i.e., different senses). This statement is expressed by formal-logical law of absence of contradiction:   

(qualitative determinacy  of the segment Vx )    

(qualitative determinacy of the segment of the axis Ox ).  

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Consequently, the mathematical operation of finding the projection of the vector V  on the coordinate axes 

represents the formal-logical error: violation of the law of absence of contradiction.  

2. As is known, the rule of addition of vectors having the same qualitative determinacy is called the “triangle 

rule” or “parallelogram rule”. Standard operation of addition of two  

    

vectors is defined as follows: the sum V1 V2 of two vectors V1 and V2 is called the vector  

  

running from the beginning of the vector V1 to the end of the vector V2 under the condition   

that the vector V2 is applied to the end of the vector V1 . Under the addition of two vectors, their projections 

on an arbitrary axis are added, and under the multiplication of a vector by any number, its projection on an 

arbitrary axis is multiplied by this number. In my view, these standard assertions are not free from objection. 

The objection is that the standard assertions are contrary to the formal-logical laws. Really, segments of 

vectors and segments of arbitrary axis have different qualitative determinacy (i.e., different senses). This 

implies  

  

that the segments of the vectors V1 and V2 cannot lie on a segment of an arbitrary axis (i.e., the segments of 

the vectors cannot coincide with a segment of an arbitrary axis). From the point of view of formal-logical law 

of identity, these segments can be coincided if only they have identical qualitative determinacy (i.e., the same 

dimension, the same meaning).  

  

3. As is known, the scalar product of two vectors  V and F  is defined as follows: (a) one brings the initial 

points of vectors in coincidence with each other (i.e., the initial points are connected); (b) one postulates the 

relation   

  

V F V   F   cos   

where the point between symbols of vectors denotes the operation of scalar multiplication of  

 

vectors,    is angle between the vectors. The expression V cos  represents a denominate  

   

number: the projection of the vector V  on the vector F . Also, the expression F cos   

  

represents a denominate number: the projection of the vector F on the vector V . In my opinion, the standard 

definition of the scalar product of vectors is not free from objection. The objection is that the standard 

definition is contrary to the formal-logical laws.  

Really, the coincidence (connection) of initial points of vectors and the formation of the projections imply that 

the dimension of length (i.e., the qualitative determinacy) of the vector  

  

V is identical to the dimension of length (i.e., the qualitative determinacy) of the vector F :  

 

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(qualitative determinacy of the vector V ) =  

 

(qualitative determinacy of the vector F ).  

In general case, however, the dimensions of vector lengths are different. Therefore, these vectors cannot have 

a common point, and the multiplication can not be performed (i.e. the multiplication has no sense). This fact 

is expressed formal-logical law of absence of contradiction:  

 

(qualitative determinacy of the vectorV )   

(qualitative determinacy of the vector F ).  

Consequently, the mathematical operation of scalar product of two vectors represents a formal-logical error: 

a violation of the law of absence of contradiction.  

  

4. As is known, the cross-product of two vectors  V and F  is defined as follows: (a) one brings the initial 

points of vectors in coincidence with each other (i.e., the initial points are connected); (b) one postulates the 

relation   

    

H V F h V   F   sin    

where the cross between the symbols of vectors denotes the operation of vector multiplication  

 

of vectors,  is angle between the vectors,  H is  vector which is normal to the plane formed  

   

by the vectors V and F ;   h is unit vector which is normal to the plane. Under the  

  

established agreement, the direction of vectors H and h is determined by the “right-hand screw rule”. In my 

opinion, the standard definition of the cross-product of vectors is not free from objection. The objection is that 

the standard definition is contrary to the formal-logical laws. Really, the coincidence of the initial points of 

the three vectors means that the  

  dimensions of lengths (i.e., the qualitative determinacy) of the vectors V , F , and  H are  

identical:  

 

(qualitative determination of the vector V )  =  

 

(qualitative determination of the vector F ) =  

(qualitative determination of the vector  H ).  

In general case, however, the dimensions of the lengths of the vectors are different. Therefore, these vectors 

cannot have a common point, and the operation of vector multiplication cannot be performed (i.e., the 

operation of multiplication has no sense). This fact is expressed formal-logical law of absence of contradiction:  

 

(qualitative determinacy of the vectorV )   

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(qualitative determinacy of the vector F )   

(qualitative determinacy of the vector H ).  

Consequently, the mathematical operation of the cross-product of vectors is a formal-logical error: a violation 

of the law of absence of contradiction.   

DISCUSSION  

1. As is known, the confidence in the scientific method of research and in rational thinking replaced all 

other ways of cognition in the 20th century. Rational thinking represents the greatest achievement of mankind.  

Rationalization of thinking and of science is dialectical imperative of our time.  The development of rational 

thinking in the 21st century leads to critical analysis, reconsideration, and rationalization of the generally 

accepted theories created by the classics of science (for example, N. Bohr, E. Schrödinger, W. Heisenberg, A. 

Einstein, I. Newton, G. Leibniz, L. Euler, J. Lagrange, A. Cauchy, W.R. Hamilton,  J.W. Gibbs, O. Heaviside, 

etc.).  Rationalization and critical analysis of science are two side pieces (component factors) in progress of 

science.  Critical analysis and rationalization of theories are based on formal-logical analysis of scientific 

concepts, of the completeness of concepts, of the completeness of a system of concepts because “only the 

completeness leads to clarity” (Confucius). Recently, independent researchers give attention to critical 

analysis of theoretical physics, mathematics, biology, etc. (see, for example, www.gsjournal.net).  In the 

process of critical analysis and of interpretation of scientific theories, “...we can hardly rely on any of the old 

principles even if they are very common.  The only mandatory requirement is the absence of logical 

contradictions.” (N. Bohr).  Logical consistency of theories is achieved with use of the formal-logical laws. 

And a natural-scientific interpretation of theories is based on the use of rational dialectics. The system of 

universal (general-scientific) concepts and laws – i.e., science of the general laws of development of the 

Nature, human society, and correct thinking – is the unity of formal logic and rational dialectics. This unity is 

not only correct methodological basis of science but also the correct methodological basis for a critical analysis 

of theories.   

2. The origin of vector calculus is closely related to the needs of mechanics and physics: the idea of 

motion, the concepts of process, velocity, acceleration, displacement, force, and vector were introduced into 

mathematics in the 17-18th centuries. The modern meaning of the word “vector” represents generalization of 

its previous (out-of-date) meaning in astronomy, where, in 18th century, a vector is called an imaginary 

straight line segment connecting the planet to the center (focus) of the motion. At present, vector calculus is a 

branch of mathematics in which one studies the properties of operations on vectors. But the mathematical 

formalism does not contain motion, mathematical process. A mathematical process is carried out only in 

computers. (This is why continual mathematics must be replaced by discrete mathematics – computer 

mathematics).  

In specific scientific problem, one considers the quantities of the various natures. These quantities have 

different dimensions: length, area, volume, weight, temperature, speed, strength, etc.). If one selects a (define, 

explicit, appointed) determined unit, then each value of the quantity must be expressed by denominate number. 

But mathematics does not consider the specific quantities: the mathematical propositions and laws are 

formulated, abstracting from the specific nature of the quantities, taking into consideration only their 

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numerical values. In line with this, mathematics considers the quantity in general, the vector in general, and 

so on, neglecting the natural-scientific meaning of the quantity.   

Abstract mathematical propositions, theories, and models cannot be tested and used in the natural sciences. 

From the point of view of formal logic and of rational dialectics, in order to test and use mathematical 

propositions, theories, and models in practice, it is necessary to define the natural-scientific (practical) 

meaning of mathematical concepts (objects) and relations, i.e., to consider not a “quantity in general”, a 

“number in general”, “vector in general”,  but to consider the nature (i.e., dimensions) of quantities (length, 

area, volume, weight, temperature, speed, acceleration, displacement, force, etc.). From this point of view, the 

standard vector calculus does not have a natural-scientific meaning because the standard vector calculus is 

based on the concept of “vector in general”. Clarification of naturalscientific meaning of concept of “vector” 

and a logical analysis of operations on the “physical vectors” show that the standard propositions of vector 

calculus, relating to the “physical vectors”, are contrary to formal logic.  

3. There are two opinions about the existence of logical errors in generally accepted theories (for 

example, in physics and mathematics). The first opinion is that, although a theory (for example, the special 

theory of relativity) contains logical errors, “it works well” (Gerard’t Hooft). The second opinion is that the 

system of four fundamental formal-logical laws is incomplete and insufficient for a panchreston   (i.e., for 

complete explanation) and mathematical description of reality. In essence, these opinions are identical. 

However, in my opinion, these views are not free from objection. The objection is as follows. If one will 

discover additional formal logic laws, then the complete system of laws should not be contradictory: the four 

basic laws will retain its place and importance in a new, complete system (in other words, the four basic laws 

will not be refuted). In this case, the theories that are erroneous in “incomplete” logical system will also be 

erroneous in the “complete” logical system. And the theories that contain logical errors are false in essence. 

But the following questions will always remain open: Why devices that are based on false scientific theories 

(ideas) work? Why do the false scientific theories contribute to the development of mankind? Where is the 

limit of development based on false theories? What is the danger of development based on false theories? 

What are the essence and predestination of development?  

CONCLUSION   

Thus, the formal-logical and dialectical analysis of the foundations of vector calculus leads to the following 

main results: the standard vector calculus is incorrect theory because   

a. it is not based on the correct methodological basis: the unity of formal logic and of rational dialectics;   

b. it does not contain the correct definitions of concepts of “movement”, “direction”, and “vector”;   

c. it does not take into consideration the dimensions of physical quantities (i.e., number names, 

denominate numbers, concrete numbers), characterizing the concept of “physical vector”, and, therefore, it 

has no natural-scientific meaning;   

d. operations on “physical vectors” and the theoretical propositions of the standard vector calculus, 

relating to the “physical vectors”, are contrary to formal logic.  

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Kiyosi Ito. Encyclopedic Dictionary of Mathematics (2nd ed.), MIT Press, ISBN 978-0262-59020-4, (1993).  

A.B. Ivanov. "Vector, geometric", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 

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D. Pedoe. Geometry: A comprehensive course. Dover. ISBN 0-486-65812-0, (1988). [6] R. Aris. Vectors, 

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R. Feynman, R. Leighton, and M. Sands. "Chapter 11". The Feynman Lectures on Physics, Volume I (2nd ed 

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Mechanics. Berkeley physics course. V. 1. McGraw-Hill book company, (1964).  

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