Scalar Mathematics Research
Hestenes begins his book, New Foundations for Classical Physics, with the following observation:
There is a tendency among physicists to take mathematics for granted, to regard the development of mathematics as the business of mathematicians. However, history shows that most mathematics of use in physics has origins in successful attacks on physical problems…The task of improving the language of physics requires intimate knowledge of how the language is to be used and how it refers to the physical world, so it involves more than mathematics. It is one of the fundamental tasks of theoretical physics.
So it is that at the LRC, we are developing the mathematical language of use in scalar physics. The genesis of this work is found in the Reciprocal System of Mathematics (RSM), but the development of mathematical research conducted at the LRC is maintained in the Mathematics Research Division’s development record, called the Scalar Mathematics Document (SMD).
The SMD contains three sections:
The point of departure that distinguishes scalar mathematics from vector mathematics is the focus on the binomial expansion of the Greek tetraktys, where the difference between scalar dimensions and geometric dimensions is clearly manifest. The idea of scalar dimensions is a new, but fundamental, concept arising out of the operational interpretation (OI) of number.
However, the most important aspect of the algebraic difference that OI makes is the light it casts on the concept of n-dimensional multiplication. As, early as 1988, David Hestenes was pointing out to the mathematics community that the n-dimensional (geometric) product is essential in the definition of the concept of a vector. Indeed, he was writing that this fact is one of the neglected facts of mathematics, at the same time that Larson was writing his book, The Neglected Facts of Science!
While Larson wrote his book in the context of his vision of a universal motion, Hestenes wrote his paper, entitled “Universal Geometric Algebra,” in the context of his vision of a universal algebra of numbers. The problem with algebra that Hestenes was addressing centered on the definition of a vector and the dimensional space of vectors. He writes:
I like to distinguish between a linear space and a vector space. A linear space is defined as usual by the operations of addition and scalar multiplication, while a vector space is a linear space on which the geometric product is also defined. Thus, I regard the geometric product as one of the essential properties defining the concept of “vector.”
As the geometric product has not yet received the universal recognition I think it deserves, I must repeat its simple definition here. It is defined by the usual associative and distributive rules together with the special rule that the square of any vector is some scalar. The last rule implies that an n-dimensional vector space Vn is not closed under the geometric product. Rather, by addition and multiplication it generates a geometric (or Clifford) algebra G(V).
In other words, the multiplication of two, one-dimensional, numbers generates a two-dimensional algebra, as opposed to the single-dimensional algebra of scalars (counting numbers). This is because the squared number, of the one-dimensional vector product, is a zero-dimensional scalar number, not another one-dimensional number, or vector; that is, the vector space, Vn, is not closed under multiplication, it includes both the scalar and the vector space, or two dimensions, not one dimension (21 = 2).
If we agree with Hestenes that an n-dimensional vector space generates an n-dimensional (geometric) algebra, by addition and multiplication, then we need to understand how to add and multiply vectors, using the rules of the appropriate n-dimensional algebra, which is what Hestenes’ life’s work teaches us. However, if we find that an n-dimensional scalar space exists, then the need to understand the algebra of n-dimensional vector space, valuable though it might be, doesn’t necessarily apply in our case.
Fortunately, though, the n-dimensional scalar space has much in common with n-dimensional vector space. Therefore, a clear understanding of the issues and challenges facing Hestenes, in defining the rules for the n-dimensional addition and multiplication of the familiar vectors, is extremely helpful in defining the rules of these operations for the unfamiliar, n-dimensional, scalars.
The mission of the LRC’s Mathematics Research Division is to investigate this subject and develop the multi-dimensional mathematics document (MDMD). The MDMD is a hyperlinked document based on the logical structure of the mathematical development. It is located in the LRC Wiki.