Um… since a ray is bounded (has a starting point) and a line has another dimension by virtue of its not being bounded, following your “logic”, by extension *half* a plane ought to have three dimensions and a whole plane should have four, and a half volume should have five, and a full volume should have six.
And your linear algebra SUCKS if it considers negative numbers to be orthogonal to positive numbers. (Do you know what “orthogonal” means? And what it represents visually?) And do you realize your crazy multiplication example from three posts ago doesn’t make sense (as you said) because your premise that a negative number is orthogonal to a positive number is just flat-out wrong, and that it would make sense if you’d pitch that piece of idiocy overboard? If you are in fact justified, then why not consider things in the right half of the plane to be orthogonal to things in the left half of the plane? What justification is there for assuming a line is two dimensional?
I’ve had my fill of this; it’s nonsense on stilts.
It is your choice not to deal with me, or my ideas.
Just in case you might still want to deal, I’ll show you the formalism, because it is pretty simple.
I find linear algebra, and especially the notion of inner product spaces, to be conceptually unimpeachable as a framework for analyzing problems that can be mapped into a vectorial framework.
Orthogonal in linear algebra simply means that the projection of two vectors vanishes under the chosen inner product. The inner product is a matter of choice, but must be a proper inner product, i.e., it must satisfy the definitional axioms of an inner product.
Formally, then, you ought to have no trouble granting the following vector space construct:
Let P be the set of pairs of natural numbers.
Let R be the equivalence relation defined, for (s,t) and (u,v) elements of P, as:
(s,t)R(u,v) iff s+v==t+u
Let X be the set of equivalence classes of P under the relation R.
Define addition of elements x, y of X by reference to addition of their equivalence class representatives.
The additive identity element is the equivalence class of (0,0).
The equivalence class of (t,s) is the additive inverse of the class of (s,t), because (s+t,t+s)R(0,0) by the commutative nature of addition of natural numbers.
Under this addition, X is a proper vector space.
Note that the simplest, most reduced representative of each equivalence class has one component equal to 0. Thus, (N,0) is the prime representative of the class (N+K,K), and (0,N) for (N,N+K). In fact, all equivalence class prime representatives are of one of these two reduced forms. This is analogous to rational number formalism where 2/3==4/6 where 2/3 is the reduced, prime representation, but 4/6 is equivalent. (In fact, a similar process of considering pairs and forming equivalence classes is applicable when constructing the rational numbers – but the addition rule is defined differently)
X is seen to be isomorphic to the integers when the following correspondences are noted:
+1 → (1,0)
-1 → (0,1)
Thus it is a perfectly legitimate representation of the integers under addition; it highlights the vectorial nature of the concept “integer”, which is obfuscated in the current +x/-x formalism. The representations are isomorphic, so if one is clearly vectorial, the other must be, too.
So stop trying to multiply integers directly, you can’t do that with vectors!
What you can do is scale vectors and add them. The problem is that the current formalism encourages conflation of the concept “2” as a scalar multiple, with the concept “+2” as a vector displacement.
So, when one speaks like “-2 equals 2 times -1”, what one is really saying is “the vector (0,2) is equal to the vector (0,1) scaled by the scalar 2, i.e., (0,2) = 2*(0,1). My formalism makes it obvious what is going on here, it is a standard linear combination formalism.
Now one can proceed to choose and define an inner product on X, e.g., for the purpose of considering the projection of one vector onto the direction of another. What inner product to use depends on the context one is analyzing.
In the context of modeling spatial relationships, the key conceptual invariant is that spatial displacements are travelable, at least in principle. Since you can’t travel left by going right, I desire an inner product which respects the identification of displacements as travelable, including the fact that left and right are distinctly accessible directions.
The inner product I choose is simply the natural dot product between prime class representatives, so that (1,0) and (0,1) are orthogonal (in the sense of linear algebra, not Euclidean geometry). This fits with observations of physical reality, wherein you can’t ever go left by going right.
It seems strange only because you are used to thinking of lines as the basis of dimensionality.
To represent a plane takes three rays towards the vertexes of a triangle. Cartesian coordinates appear to use only two lines, but they use four rays – three is more efficient, especially so since only positives are necessary to represent the plane in my framework.
To represent a volume takes four rays towards the vertexes of a tetrahedron, not the redundant 6 rays found in Euclidean representations.
It’s a matter of conceptual efficiency, as I said.
If my sense of efficiency offends your sense of orthogonality, then one of us is wrong.
Cheers.