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The mathematical and philosophical concept of vector 1 
The mathematical and philosophical concept of vector 
by George Mpantes www.mpantes .gr 
Historical summary 
The vectors 
The transformation equations of coordinates 
the transformation equations of vectors 
Euclidean Geometry and Newtonian physics . 
Philosophical comments, Aristotle 
Historical summary 
“There is un unspoken hypothesis which underlies all the physical theories so far 
created, namely that behind physical phenomena lies a unique mathematical structure 
which is the purpose of theory to reveal. According to this hypothesis , the 
mathematical formulae of physics are discovered not invented, the Lorentz 
transformation , for example ,being as much a part of physical reality as a table or a 
chair”. ( RELATIVITY: THE SPECIAL THEORY J.L.Synge p.163) 
Indeed in our example the physical phenomenon is the force, and the 
underlying mathematical structure is vector analysis. But looking the historical 
process, mathematics create their truths independently, discover new entities, 
and their tendency for generalization goes ahead exceeding the initial physical 
presuppositions. The new discoveries of the mathematical process return to the 
physical theory where they create new unifications and generalizations, now for 
the phenomena of the actual world. Can we trust them? Can mathematics lead
The mathematical and philosophical concept of vector 2 
the physical theory? The answer seems to be positive, if the measurements 
agree with the mathematical conjectures (electromagnetic waves!). that is that 
mathematical structure extends the physical theory. So, for example, with the 
support of the use of vector methods , we had a development of theoretical 
physics and by the beginning of the twentieth century, vector analysis had 
become firmly entrenched as a tool for the development of geometry and 
theoretical physics. 
As we look back on the nineteenth century it is apparent that a 
mathematical theory in terms of which physical laws could be described and 
their universality checked was needed. Figuratively speaking two men stepped 
forward in this direction, Hamilton and Grassman. Hamilton was trying to find 
the appropriate mathematical tools with which he could apply Newtonian 
mechanics to various aspects of astronomy and physics. Grassman tried to 
develop an algebraic structure on which geometry of any number of dimension 
could be based. The quaternions of Hamilton and Grassmann’s calculus of 
extension proved to be too complicated for quick mastery and easy application , 
but from them emerged the much more easily learned and more easily applied 
subject of vector analysis. This work was due principally to the American 
physicist John Willard Gibbs (1839-1903) and is encountered by every student 
of elementary physics. 
The vectors . 
What is behind the physical phenomenon of the velocity; of the force; 
there is the mathematical concept of the vector. This is a new concept, since 
force has direction, sense, and magnitude, and we accept the physical principle 
that the forces exerted on a body can be added to the rule of the 
parallelogram. This is the first axiom of Newton. Newton essentially requires 
that the power is a " vectorial " size , without writing clearly , and Galileo that 
applies the principle of the independence of forces .
The mathematical and philosophical concept of vector 3 
These are the basic physical indications for the mathematical treatment, 
for “vector1 geometry”, where 
the term vector denotes a translation or a displacement a in the space.2 The 
statement that the displacement a transfers the point P to the point Q (“transforms” P 
into Q ) may also be expressed by saying that Q is the end-point of the vector a whose 
starting point is at P. if P and Q are ant two points then there is one and only one 
displacement a which transforms P to Q. We shall cal it the vector defined by P and Q 
 
. 
and indicate it by PQ 
There are two fundamental operations, which are subject to a system of 
laws, viz. addition of two vectors (the translation which arises through two 
successive translations (law of parallelogram), and multiplication of a vector by a 
number (is defined through the addition). These laws are 
A.Addition: a+b=c 
with the properties 
a+b=b+a 
(a+b)+c=a+(b+c) 
If a and c are any two vectors , then there is one and only one value of x 
for which the equation a+x=c holds 
B. Multiplication b=λ.a 
with the properties 
(λ+μ)a=(λa)+(μa) 
λ(μa)=(λμ)a 
1.a=a 
λ(a+b)=(λa)+(λb) 
In elementary physics , a vector is graphically regarded as a directed line 
segment , or arrow. This is the translation or the displacement described by 
Weyl. So in elementary physics , vector was something apparent, something 
concrete and intuitively simple. It was geometrical. In theoretical physics it 
became an idea, something cerebral, connected with algebra. The first was a 
1 The term vector was introduced from Hamilton 
2 This definition is from Weyl, (Space, time, matter)
The mathematical and philosophical concept of vector 4 
sketch of the second. This is the course of mathematics. The formula for 
algebraic vector was the old bold Cartesian binding of geometry with algebra viz 
this of a picture with the abstract and compact truth of numbers, a good 
combination between intuition and rigor, through concepts, in the center of 
which was the well-known coordinate system, one of the more significant 
generalization of mathematics. 
By means of a coordinate system, a set of ordered triples of real 
numbers can be put into one-to-one correspondence with the points of a three 
dimensional Euclidean space. However many aspects of modern-day science 
cannot be adequately described in terms of a three-dimensional Euclidean model. 
The ideas of vector analysis when expressed in a notational fashion are 
immediately extendable to n-dimensional space and their physical usage is amply 
demonstrated in the development of special and general relativity theory. 
With the change of the figurativeness of the points, change also the 
description of the vector. 
The set {A1, A2, A3} of all triples (A1, A2, A3), (A1΄, A2΄, A3΄) etc., 
determined by orthogonal projections of a common arrow representation on the 
axes of the associated rectangular Cartesian coordinate system is said to be a 
Cartesian vector. Many triples means many systems, but all these represent 
the same Cartesian vector, which has a family of arrows as its geometrical 
representative. The binding of orthogonal projections with the law of 
parallelogram is the base of all the formalism of vector analysis.3 
A Cartesian vector (A1, A2, A3), (3-tuple), can be represented 
graphically by an arrow, with it’s initial point at the origin and it’s terminal 
point at the position with coordinates (A1, A2, A3), but it is not the only possible 
arrow representation. An arrow with initial and terminal points (a,b,c) and 
(A,B,C) such that A1=A-a, A2=B-b, A3=C-c can be considered a representative 
of a 3-tuple. 
3 See my article “the mathema tical forms of nature, the tensors”
The mathematical and philosophical concept of vector 5 
A Cartesian vector with respect to a coordinate system, is 
characterized by a magnitude , a direction and a sense , and its components in 
any coordinate system satisfy the algebraic laws of the triples, viz the laws 1 
and 2 for the vectors, expressed algebraically , if we define a=(a1,a2,…..an) 
b=(b1,b2,….bn) 
i.e (a1,a2,…..an)+(b1,b2,….bn)=(a1+b1, a2+b2+……an,bn). 
λ.a=λ(a1,a2,…..an)= (λa1,λa2,…..λan) . 
Now an analytical treatment of vector geometry is possible, in which 
every vector is represented by it’s components and every point by its 
coordinates. 
How all these triples, (A1, A2, A3), (A1΄, A2΄, A3΄) etc., are related? 
The transformation equations of coordinates 
A fundamental problem of theoretical physics is that formulating 
universally valid laws relating natural phenomena. Because the transformation 
idea is of such importance, the development of vector geometry and later of 
vector analysis is build around this. 
A rectangular Cartesian coordinate system4 imposes a one-to-one 
correspondence between the points of Euclidean three-space and the set of all 
ordered triples of real numbers. A second rectangular Cartesian system brings 
about another correspondence of the same point . What is the nature of those 
transformations that relate such coordinate representations of the three-space? 
The specific transformations of coordinates for our example in the 
development of vector analysis, are called translations and rotations. They are 
linear transformations and they connect orthogonal Cartesian systems. All linear 
transformations have the characteristic that the fundamental relations (A) and 
(B) are not disturbed by the transformation viz they hold for the transformed 
points and vectors : 
4 We examine this particular case in our example.
The mathematical and philosophical concept of vector 6 
α΄+b΄=c΄ b΄=λ .a΄……… 
DEFINITION 1. The transformation equations that relate the 
coordinates ( , , ) and (x , , ) 1 2 3 1 2 3 x x x x x in rectangular coordinate systems , the 
axes of which are parallels are 
j j j x  x  x 
.......... .......... ....(1) 0 
where ( , , ) 3 
1 
0 x x x represent the unbarred coordinates of the origin of the 
0 
2 
0 
barred system O’. These are called equations of translation. The Cartesian 
vector concept is employed in obtaining them. 
DEFINITION 2. 
The transformation equations that relate the coordinates 
( , , ) and (x , , ) 1 2 3 1 2 3 x x x x x in rectangular coordinate systems, having a common 
origin and such that there is no change of unit distance along coordinate axes, 
are related by the transformation equations 
j x  c x 
.......... ..(2) j k 
k 
where the coefficients of 
transformations j 
k a are direction 
cosines satisfying the conditions 
p 
3 
 c j 
c j 
 j  
k p 
1 k
The mathematical and philosophical concept of vector 7 
These are called equations of rotation. 
The transformations of the coordinates (2) are a subset of the linear or 
affine transformations, with the general form 
 
 
.........( 3) 
1 
2 
3 
c c c 
3 
3 
c c c 
3 
2 
3 
1 
2 
3 
2 
2 
2 
1 
1 
3 
1 
2 
1 
1 
1 
2 
3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
x 
x 
x 
c c c 
x 
x 
x 
where apply the conditions of orthogonality, they are the orthogonal 
transformations that connect orthogonal Cartesian systems with common origin 
and are produced from the vectorial behavior of the vectorial units (bases) in 
the axes of the two systems 
. Physically they describe, as we have mention, the rotation of an 
orthogonal Cartesian system. The orthogonal transformations fulfill the first 
unification of geometry (the Euclidean metrical geometry in every orthogonal 
system) and as the geometry is a fundamental branch of physics, this unification 
will be the model of the unification of physical laws in all the 
systems.(universality) 
But what about the vectors? What is their deepest behavior in the scene 
of coordinate systems? 
the transformation equations of vectors 
We have seen that a Cartesian vector (A1, A2, A3) can be represented 
graphically by an arrow, but 
The components of this arrow, transform under rotation, as the coordinates . 
Proof: If the transformation (2) is applied to the coordinates of P0 and P1 , the 
coordinate differences { j j x x1 0  } satisfy 
j j x  x  c x  c x  c x  x 
j k k 
k 
j k 
k 
j k 
k 
( )......... .......... ..(4) 1 0 1 0 1 0 
that is the transformation (2). 
A corresponding verification of the statement holds for translations, where the 
vector components remain unaltered.
The mathematical and philosophical concept of vector 8 
So we have the definition of the Cartesian vector under the light of both 
transformations: 
A Cartesian vector (A1, A2, A3), is a collection of ordered triples , each 
associated with a rectangular Cartesian coordinate system and such that any 
two satisfy the transformation law 
.......... .......... .......... ......( 5) k 
j 
x 
 
j A 
k 
x 
A 
 
 
where the partial derivatives are the coefficients j 
i c of the linear 
transformation (3), of coordinates. 
We must notice that every component of the vector in the new system is 
a linear combination of the components in the initial variables. So if all 
components of the vector are zero in the initial system they will be also zero in 
the new variables. This is the more important property of vectors: a vectorial 
equation holds in every rectangular Cartesian system (for our paradigm), if it 
holds in one! This is the root of the universality of the physical or geometrical 
laws, as we see in the end of the article. Newton’s law is universal because it is 
written in vectorial form. It’s invariance in translation is the mathematical 
acceptance of the Newtonian principle of relativity. 
The scalars . 
A second concept which has evolved in the development of vector analysis is 
that of the scalar. The definition of scalar states that it is a quantity 
possessing magnitude but no direction. Such entities as mass, time, density and 
temperature are given as examples. But for mathematics, the prize example is 
the real number, as it does not have to be associated with magnitude. From a 
historical point of view scalar is a quantity invariant under all transformations 
of coordinates (Felix Klein). Whether a given algebraic form is invariant depends 
on the group of transformations under consideration. Again the scalars , as 
vectors, are associated with coordinate systems and transformations.
The mathematical and philosophical concept of vector 9 
Euclidean Geometry and Newtonian physics . 
The mathematical investigation showed that our Known geometrical 
vector (arrow) has hidden qualities which are raised by their correlation with 
coordinate systems: The laws of it’s transformation. The vector concept 
received much of its impetus from this fact, so it plays a fundamental role in 
many aspects of geometry and physics. This mathematical result underlies the 
principles of relativity of Newton and Einstein, that would be ungrounded 
without the mathematical discovery of the transformation theory of the vectors 
and (later) of tensors. 
Magnitude and angle are fundamental to the metric structure of 
Euclidean space. They are scalar invariants under the transformations of the 
orthogonal Cartesian set. 
The inner product transforms 
k 
P j Q j 
  c j P r 
c j Q s 
 c j 
c j P r Q s 
 P r Q s  
P k 
Q 
r 
s 
r 
s 
r 
j 
 j 
 j 
 k 
 
3 
1 
s 
3 
1 
3 
1 
3 
1 
( )( )  
1 0 , X j X (through 4) 
and the distance of the points j 
1 
1 (x  x )  (x  x )  (x  x )  (x  x )  (x  x )  (x  x ) 
3 2 
0 
3 
1 
2 2 
0 
2 
1 
1 2 
0 
1 
1 
3 2 
0 
3 
1 
2 
0 
2 
1 
1 2 
0 
These formulas carry out the first unification of metrical Euclidean geometry. 
An observer who measures a distance and an angle in a orthogonal Cartesian 
system uses the same formulas and finds the same results with somebody else 
who measures the same magnitudes in another orthogonal Cartesian system, 
which subsists a translation or a rotation of the first. The question of finding 
those entities (as distance and inner product) that have an absolute meaning 
transcending the coordinate system, is of prime significance. This gives us a 
direction as to which of the concepts considered in the framework of 
rectangular Cartesian systems should be generalized as well as how to bring 
about the generalizations. This is the criminal point of the universality of the 
physical laws. Moreover, the availability of the Cartesian systems of reference
The mathematical and philosophical concept of vector 10 
will be valuable when considering the special theory of relativity, in a later 
article. 
In vector formalism, we will now show the covariance of Newton's 
law in linear systems with given origin ( rotation) . 
In the system K we have 
d 
F   
( ) k k m 
Multiply by 
k 
r 
dt 
x 
 
x 
 
and summing with respect to k (from 5) we have 
x 
k 
m 
d 
F 
m 
 
( ) ( ) k r r 
x 
r 
F 
k 
k 
x 
x 
r 
    
dt 
d 
dt 
 
 
 
 
So the form of the equation remains the same in the new system ( 
covariant ) , and the mathematical formalism demonstrates that the laws of 
Newton have a universal application in Euclidean space, where we can adjust 
orthogonal Cartesian systems. The physical laws are invariant in form in all the 
orthogonal Cartesians systems (my article “covariance and invariance in physics”) 
Philosophical comments, Aristotle . 
The quotation of Synge about the discovery of mathematical structure 
of vector analysis , where the vector is as much a part of physical reality as a 
table, is very poetic, having construct a reality of visible and invisible objects, 
as the vector and the table. Vector exists as the table but it is invisible! 
Aristotle in his ideas of the theory of knowledge (Analytica posterioria) 
says that “the knowledge of a fact differs from the knowledge of a reasoned fact” . 
The theoretical foundations of the systems of this deductive reasoning, 
account of first principles where are the bases of every science. …. The 
scientific knowledge through reasoning is impossible if we do not know the first 
principles. ….it is clear that in science of nature as elsewhere we should try first to 
determine questions about the first principles …..αληθείς και πρωταρχικές και άμεσες και 
πρωθύστερες και αιτίες του συμπεράσματος,the first basis from which a thing is
The mathematical and philosophical concept of vector 11 
known……as regards their existence must be assumed for the principle ,(what a straight 
line is , what a triangle is ..)but proved for the rest of the system, by logical reasoning. 
How are these first principles to be established? ….they are arrived by 
the repeated visual sensations, which leave their marks in the memory. Then we reflect 
on these memories and arrive by a process of intuition (νους) at the first principles ….if 
there in not something intelligible behind the phenomena, there is not science for 
anything, science is not created from senses…. 
So vector is an intelligible creation, a first principle, that is a conviction, 
a support of the deductive reasoning. This reasoning constructs a mental logical 
reality in our brains, which is the human’s way of comprehension. Mathematics 
are neither discoveries nor inventions. Mathematics are creations , as poems, 
but logical creations, based on the rules of deductive reasoning. 
But in fact, their first principles are founded in nature. 
Sources 
Herman Weyl (space,time, matter,Dover) 
H.Eves (foundations and fundamental concepts of mathematics,Dover) 
J.L Synge (Relativity: the special theory, Noth Holland publising Company 
Amsterdam New York Oxford) 
Robert C.Wrede (introduction to vector and tensor analysis, Dover) 
Aristotle (Analytica posterioria, internet) 
George Mpantes mathematics teacher www.mpantes.gr

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The mathematical and philosophical concept of vector

  • 1. The mathematical and philosophical concept of vector 1 The mathematical and philosophical concept of vector by George Mpantes www.mpantes .gr Historical summary The vectors The transformation equations of coordinates the transformation equations of vectors Euclidean Geometry and Newtonian physics . Philosophical comments, Aristotle Historical summary “There is un unspoken hypothesis which underlies all the physical theories so far created, namely that behind physical phenomena lies a unique mathematical structure which is the purpose of theory to reveal. According to this hypothesis , the mathematical formulae of physics are discovered not invented, the Lorentz transformation , for example ,being as much a part of physical reality as a table or a chair”. ( RELATIVITY: THE SPECIAL THEORY J.L.Synge p.163) Indeed in our example the physical phenomenon is the force, and the underlying mathematical structure is vector analysis. But looking the historical process, mathematics create their truths independently, discover new entities, and their tendency for generalization goes ahead exceeding the initial physical presuppositions. The new discoveries of the mathematical process return to the physical theory where they create new unifications and generalizations, now for the phenomena of the actual world. Can we trust them? Can mathematics lead
  • 2. The mathematical and philosophical concept of vector 2 the physical theory? The answer seems to be positive, if the measurements agree with the mathematical conjectures (electromagnetic waves!). that is that mathematical structure extends the physical theory. So, for example, with the support of the use of vector methods , we had a development of theoretical physics and by the beginning of the twentieth century, vector analysis had become firmly entrenched as a tool for the development of geometry and theoretical physics. As we look back on the nineteenth century it is apparent that a mathematical theory in terms of which physical laws could be described and their universality checked was needed. Figuratively speaking two men stepped forward in this direction, Hamilton and Grassman. Hamilton was trying to find the appropriate mathematical tools with which he could apply Newtonian mechanics to various aspects of astronomy and physics. Grassman tried to develop an algebraic structure on which geometry of any number of dimension could be based. The quaternions of Hamilton and Grassmann’s calculus of extension proved to be too complicated for quick mastery and easy application , but from them emerged the much more easily learned and more easily applied subject of vector analysis. This work was due principally to the American physicist John Willard Gibbs (1839-1903) and is encountered by every student of elementary physics. The vectors . What is behind the physical phenomenon of the velocity; of the force; there is the mathematical concept of the vector. This is a new concept, since force has direction, sense, and magnitude, and we accept the physical principle that the forces exerted on a body can be added to the rule of the parallelogram. This is the first axiom of Newton. Newton essentially requires that the power is a " vectorial " size , without writing clearly , and Galileo that applies the principle of the independence of forces .
  • 3. The mathematical and philosophical concept of vector 3 These are the basic physical indications for the mathematical treatment, for “vector1 geometry”, where the term vector denotes a translation or a displacement a in the space.2 The statement that the displacement a transfers the point P to the point Q (“transforms” P into Q ) may also be expressed by saying that Q is the end-point of the vector a whose starting point is at P. if P and Q are ant two points then there is one and only one displacement a which transforms P to Q. We shall cal it the vector defined by P and Q  . and indicate it by PQ There are two fundamental operations, which are subject to a system of laws, viz. addition of two vectors (the translation which arises through two successive translations (law of parallelogram), and multiplication of a vector by a number (is defined through the addition). These laws are A.Addition: a+b=c with the properties a+b=b+a (a+b)+c=a+(b+c) If a and c are any two vectors , then there is one and only one value of x for which the equation a+x=c holds B. Multiplication b=λ.a with the properties (λ+μ)a=(λa)+(μa) λ(μa)=(λμ)a 1.a=a λ(a+b)=(λa)+(λb) In elementary physics , a vector is graphically regarded as a directed line segment , or arrow. This is the translation or the displacement described by Weyl. So in elementary physics , vector was something apparent, something concrete and intuitively simple. It was geometrical. In theoretical physics it became an idea, something cerebral, connected with algebra. The first was a 1 The term vector was introduced from Hamilton 2 This definition is from Weyl, (Space, time, matter)
  • 4. The mathematical and philosophical concept of vector 4 sketch of the second. This is the course of mathematics. The formula for algebraic vector was the old bold Cartesian binding of geometry with algebra viz this of a picture with the abstract and compact truth of numbers, a good combination between intuition and rigor, through concepts, in the center of which was the well-known coordinate system, one of the more significant generalization of mathematics. By means of a coordinate system, a set of ordered triples of real numbers can be put into one-to-one correspondence with the points of a three dimensional Euclidean space. However many aspects of modern-day science cannot be adequately described in terms of a three-dimensional Euclidean model. The ideas of vector analysis when expressed in a notational fashion are immediately extendable to n-dimensional space and their physical usage is amply demonstrated in the development of special and general relativity theory. With the change of the figurativeness of the points, change also the description of the vector. The set {A1, A2, A3} of all triples (A1, A2, A3), (A1΄, A2΄, A3΄) etc., determined by orthogonal projections of a common arrow representation on the axes of the associated rectangular Cartesian coordinate system is said to be a Cartesian vector. Many triples means many systems, but all these represent the same Cartesian vector, which has a family of arrows as its geometrical representative. The binding of orthogonal projections with the law of parallelogram is the base of all the formalism of vector analysis.3 A Cartesian vector (A1, A2, A3), (3-tuple), can be represented graphically by an arrow, with it’s initial point at the origin and it’s terminal point at the position with coordinates (A1, A2, A3), but it is not the only possible arrow representation. An arrow with initial and terminal points (a,b,c) and (A,B,C) such that A1=A-a, A2=B-b, A3=C-c can be considered a representative of a 3-tuple. 3 See my article “the mathema tical forms of nature, the tensors”
  • 5. The mathematical and philosophical concept of vector 5 A Cartesian vector with respect to a coordinate system, is characterized by a magnitude , a direction and a sense , and its components in any coordinate system satisfy the algebraic laws of the triples, viz the laws 1 and 2 for the vectors, expressed algebraically , if we define a=(a1,a2,…..an) b=(b1,b2,….bn) i.e (a1,a2,…..an)+(b1,b2,….bn)=(a1+b1, a2+b2+……an,bn). λ.a=λ(a1,a2,…..an)= (λa1,λa2,…..λan) . Now an analytical treatment of vector geometry is possible, in which every vector is represented by it’s components and every point by its coordinates. How all these triples, (A1, A2, A3), (A1΄, A2΄, A3΄) etc., are related? The transformation equations of coordinates A fundamental problem of theoretical physics is that formulating universally valid laws relating natural phenomena. Because the transformation idea is of such importance, the development of vector geometry and later of vector analysis is build around this. A rectangular Cartesian coordinate system4 imposes a one-to-one correspondence between the points of Euclidean three-space and the set of all ordered triples of real numbers. A second rectangular Cartesian system brings about another correspondence of the same point . What is the nature of those transformations that relate such coordinate representations of the three-space? The specific transformations of coordinates for our example in the development of vector analysis, are called translations and rotations. They are linear transformations and they connect orthogonal Cartesian systems. All linear transformations have the characteristic that the fundamental relations (A) and (B) are not disturbed by the transformation viz they hold for the transformed points and vectors : 4 We examine this particular case in our example.
  • 6. The mathematical and philosophical concept of vector 6 α΄+b΄=c΄ b΄=λ .a΄……… DEFINITION 1. The transformation equations that relate the coordinates ( , , ) and (x , , ) 1 2 3 1 2 3 x x x x x in rectangular coordinate systems , the axes of which are parallels are j j j x  x  x .......... .......... ....(1) 0 where ( , , ) 3 1 0 x x x represent the unbarred coordinates of the origin of the 0 2 0 barred system O’. These are called equations of translation. The Cartesian vector concept is employed in obtaining them. DEFINITION 2. The transformation equations that relate the coordinates ( , , ) and (x , , ) 1 2 3 1 2 3 x x x x x in rectangular coordinate systems, having a common origin and such that there is no change of unit distance along coordinate axes, are related by the transformation equations j x  c x .......... ..(2) j k k where the coefficients of transformations j k a are direction cosines satisfying the conditions p 3  c j c j  j  k p 1 k
  • 7. The mathematical and philosophical concept of vector 7 These are called equations of rotation. The transformations of the coordinates (2) are a subset of the linear or affine transformations, with the general form   .........( 3) 1 2 3 c c c 3 3 c c c 3 2 3 1 2 3 2 2 2 1 1 3 1 2 1 1 1 2 3                              x x x c c c x x x where apply the conditions of orthogonality, they are the orthogonal transformations that connect orthogonal Cartesian systems with common origin and are produced from the vectorial behavior of the vectorial units (bases) in the axes of the two systems . Physically they describe, as we have mention, the rotation of an orthogonal Cartesian system. The orthogonal transformations fulfill the first unification of geometry (the Euclidean metrical geometry in every orthogonal system) and as the geometry is a fundamental branch of physics, this unification will be the model of the unification of physical laws in all the systems.(universality) But what about the vectors? What is their deepest behavior in the scene of coordinate systems? the transformation equations of vectors We have seen that a Cartesian vector (A1, A2, A3) can be represented graphically by an arrow, but The components of this arrow, transform under rotation, as the coordinates . Proof: If the transformation (2) is applied to the coordinates of P0 and P1 , the coordinate differences { j j x x1 0  } satisfy j j x  x  c x  c x  c x  x j k k k j k k j k k ( )......... .......... ..(4) 1 0 1 0 1 0 that is the transformation (2). A corresponding verification of the statement holds for translations, where the vector components remain unaltered.
  • 8. The mathematical and philosophical concept of vector 8 So we have the definition of the Cartesian vector under the light of both transformations: A Cartesian vector (A1, A2, A3), is a collection of ordered triples , each associated with a rectangular Cartesian coordinate system and such that any two satisfy the transformation law .......... .......... .......... ......( 5) k j x  j A k x A   where the partial derivatives are the coefficients j i c of the linear transformation (3), of coordinates. We must notice that every component of the vector in the new system is a linear combination of the components in the initial variables. So if all components of the vector are zero in the initial system they will be also zero in the new variables. This is the more important property of vectors: a vectorial equation holds in every rectangular Cartesian system (for our paradigm), if it holds in one! This is the root of the universality of the physical or geometrical laws, as we see in the end of the article. Newton’s law is universal because it is written in vectorial form. It’s invariance in translation is the mathematical acceptance of the Newtonian principle of relativity. The scalars . A second concept which has evolved in the development of vector analysis is that of the scalar. The definition of scalar states that it is a quantity possessing magnitude but no direction. Such entities as mass, time, density and temperature are given as examples. But for mathematics, the prize example is the real number, as it does not have to be associated with magnitude. From a historical point of view scalar is a quantity invariant under all transformations of coordinates (Felix Klein). Whether a given algebraic form is invariant depends on the group of transformations under consideration. Again the scalars , as vectors, are associated with coordinate systems and transformations.
  • 9. The mathematical and philosophical concept of vector 9 Euclidean Geometry and Newtonian physics . The mathematical investigation showed that our Known geometrical vector (arrow) has hidden qualities which are raised by their correlation with coordinate systems: The laws of it’s transformation. The vector concept received much of its impetus from this fact, so it plays a fundamental role in many aspects of geometry and physics. This mathematical result underlies the principles of relativity of Newton and Einstein, that would be ungrounded without the mathematical discovery of the transformation theory of the vectors and (later) of tensors. Magnitude and angle are fundamental to the metric structure of Euclidean space. They are scalar invariants under the transformations of the orthogonal Cartesian set. The inner product transforms k P j Q j   c j P r c j Q s  c j c j P r Q s  P r Q s  P k Q r s r s r j  j  j  k  3 1 s 3 1 3 1 3 1 ( )( )  1 0 , X j X (through 4) and the distance of the points j 1 1 (x  x )  (x  x )  (x  x )  (x  x )  (x  x )  (x  x ) 3 2 0 3 1 2 2 0 2 1 1 2 0 1 1 3 2 0 3 1 2 0 2 1 1 2 0 These formulas carry out the first unification of metrical Euclidean geometry. An observer who measures a distance and an angle in a orthogonal Cartesian system uses the same formulas and finds the same results with somebody else who measures the same magnitudes in another orthogonal Cartesian system, which subsists a translation or a rotation of the first. The question of finding those entities (as distance and inner product) that have an absolute meaning transcending the coordinate system, is of prime significance. This gives us a direction as to which of the concepts considered in the framework of rectangular Cartesian systems should be generalized as well as how to bring about the generalizations. This is the criminal point of the universality of the physical laws. Moreover, the availability of the Cartesian systems of reference
  • 10. The mathematical and philosophical concept of vector 10 will be valuable when considering the special theory of relativity, in a later article. In vector formalism, we will now show the covariance of Newton's law in linear systems with given origin ( rotation) . In the system K we have d F   ( ) k k m Multiply by k r dt x  x  and summing with respect to k (from 5) we have x k m d F m  ( ) ( ) k r r x r F k k x x r     dt d dt     So the form of the equation remains the same in the new system ( covariant ) , and the mathematical formalism demonstrates that the laws of Newton have a universal application in Euclidean space, where we can adjust orthogonal Cartesian systems. The physical laws are invariant in form in all the orthogonal Cartesians systems (my article “covariance and invariance in physics”) Philosophical comments, Aristotle . The quotation of Synge about the discovery of mathematical structure of vector analysis , where the vector is as much a part of physical reality as a table, is very poetic, having construct a reality of visible and invisible objects, as the vector and the table. Vector exists as the table but it is invisible! Aristotle in his ideas of the theory of knowledge (Analytica posterioria) says that “the knowledge of a fact differs from the knowledge of a reasoned fact” . The theoretical foundations of the systems of this deductive reasoning, account of first principles where are the bases of every science. …. The scientific knowledge through reasoning is impossible if we do not know the first principles. ….it is clear that in science of nature as elsewhere we should try first to determine questions about the first principles …..αληθείς και πρωταρχικές και άμεσες και πρωθύστερες και αιτίες του συμπεράσματος,the first basis from which a thing is
  • 11. The mathematical and philosophical concept of vector 11 known……as regards their existence must be assumed for the principle ,(what a straight line is , what a triangle is ..)but proved for the rest of the system, by logical reasoning. How are these first principles to be established? ….they are arrived by the repeated visual sensations, which leave their marks in the memory. Then we reflect on these memories and arrive by a process of intuition (νους) at the first principles ….if there in not something intelligible behind the phenomena, there is not science for anything, science is not created from senses…. So vector is an intelligible creation, a first principle, that is a conviction, a support of the deductive reasoning. This reasoning constructs a mental logical reality in our brains, which is the human’s way of comprehension. Mathematics are neither discoveries nor inventions. Mathematics are creations , as poems, but logical creations, based on the rules of deductive reasoning. But in fact, their first principles are founded in nature. Sources Herman Weyl (space,time, matter,Dover) H.Eves (foundations and fundamental concepts of mathematics,Dover) J.L Synge (Relativity: the special theory, Noth Holland publising Company Amsterdam New York Oxford) Robert C.Wrede (introduction to vector and tensor analysis, Dover) Aristotle (Analytica posterioria, internet) George Mpantes mathematics teacher www.mpantes.gr