I'll first exhibit data as a TABLE, then show the HOMOLOGY LISTING derived from it.
"Descriptive Variables & Laws of Motion for a Rigid Body" (adapted from New Foundations of Mechanics, D. Hestenes) TRANSLATIONAL MOTION ROTATIONAL MOTION MASS MOMENT OF INERTIA POSITION VECTOR ATTITUDE SPINOR LINEAR VELOCITY ROTATIONAL VELOCITY LINEAR MOMENTUM ANGULAR MOMENTUM KINETIC ENERGY ROTATIONAL KINETIC ENERGY FORCE TORQUE NEWTON'S LAW EULER'S LAW
AS A HOMOLGY LISTING
Above is a 2-column, 7-row Table (2x7 TABLE, c = 2, r = 7). It as a syntactic transform (meaning of this explained anon) as a HOMOLGY LISTING with 7 (= r) HOMOLOGIES, each making 2 (= c) COMPARISONS, or 4 (=2c) COMPONENTS:
- MASS: TRANS. MOTION:: MOMENT OF INERTIA: ROT. MOT.
POS. VECTOR: TRANS. MOT.:: ATTITUDE SPINOR: ROT. MOT. LIN. VELOCITY: TRANS. MOT.:: ROT. VELOCITY: ROT. MOT. LIN. MOMENTUM: TRANS. MOT.:: ANGULAR MOM.: ROT. MOT. KINETIC ENERGY: TRANS. MOT.::ROT. KIN. EN.: ROT. MOT. FORCE: TRANS. MOTION:: TORQUE: ROT. MOTION NEWTON'S LAW: TRANS. MOTION:: EULER'S LAW: ROT. MOT.
The above Homlist has the form [row 1st item]:[col. 1 item]::[row 2nd item]:[col. 2 item], for all rows of Table. That is, r[itm]i1:c[itm]1::ri2[itm]:c[itm]2, i=1,2,...,7. In general, the Homlist generated from a c X r TABLE has the form, r[itm]i1:c[itm]1:: r[itm]i2:c[itm]2::....::r[itm]i:c[itm]j, i = 1,2,...,r; j=1,2,...,c.
The above example clearly has semiotic content. Thus, if I know what "mass" means, and wish to know what "moment of inertia" means, the first homology tells me that "moment of inertia relates to rotational motion in the way that mass relates to translational motion", equivalently, "moment of inertia plays the role in rotational motion that mass palys in translational motion". By "same relation/role", I come to understand. This is why HOMOLOGY is perhaps our best EDUCATIONAL TOOL. It's easier understand this HOMOLOGY (HOMOMORPHIC MAPPING OF RELATION) that to DRAW INFERENCE FROM, say, mass being on the same row in the Table that moment of inertia, and then INFERING the significance of their different columns.
On the other hand, the following little table will show why this SYNTACTIC TRANSFORM carries over a FORM, but not necessarily MEANING (SEMIOTICS).
red::color::cubical:shape has the form of a HOMOLOGY, but tells us very little -- only that "RED (BLUE)" is A TOKEN OF THE TYPE "COLOR" & "CUBICAL (CYLINDRICAL)" IS A TOKEN OF THE TYPE "SHAPE". As Garfield would say, "Big hairy deal!"
BOXES COLOR SHAPE RED CUBICAL BLUE CYLINDRICAL So, choose your cases carefully, before you spend time TRANSFORMING A TABLE INTO A HOMLIST.