Re: Peirce's 1870 “Logic Of Relatives ” • Comment 8.5

Thread:
JA:http://inquiryintoinquiry.com/2014/02/20/peirces-1870-logic-of-relatives-%E2%80%A2-comment-8-5/
CG:http://web.archive.org/web/20140221063252/http://permalink.gmane.org/gmane.science.philosophy.peirce/11855
JA:http://web.archive.org/web/20140222191841/http://permalink.gmane.org/gmane.science.philosophy.peirce/11868
JBD:http://web.archive.org/web/20140224152510/http://permalink.gmane.org/gmane.science.philosophy.peirce/11885
JA:http://web.archive.org/web/20140225042233/http://permalink.gmane.org/gmane.science.philosophy.peirce/11896
JLRC:http://web.archive.org/web/20140225042802/http://permalink.gmane.org/gmane.science.philosophy.peirce/11901
CL:http://web.archive.org/web/20140225043034/http://permalink.gmane.org/gmane.science.philosophy.peirce/11905
JA:http://web.archive.org/web/20140226060027/http://permalink.gmane.org/gmane.science.philosophy.peirce/11934

Clark, Jeff, Jerry, List ...

Here again is the definition of a graph that I'm using,
if for no better reason than it got burned into my brain
sometime in the late 70s, and so I'll be sticking to it:

<quote>

A ‘graph’ G consists of a finite nonempty set V = V(G) of p ‘points’ together with a prescribed set 
X of q unordered pairs of distinct points of V.  Each pair x = {u, v} of points in X is a ‘line’ of 
G, and x is said to ‘join’ u and v.  We write x = uv and say that u and v are ‘adjacent points’ 
(sometimes denoted ‘u adj v’);  point u and line x are ‘incident’ with each other, as are v and x. 
If two distinct lines x and y are incident with a common point, then they are ‘adjacent lines’.  A 
graph with p points and q lines is called a ‘(p, q) graph’.  The (1, 0) graph is ‘trivial’.

</quote> Harary, ‘Graph Theory’, p. 9.

Actually, the bit that defines a graph is the first sentence by itself:

| A ‘graph’ G consists of a finite nonempty set V = V(G) of p ‘points’ together
| with a prescribed set X of q unordered pairs of distinct points of V.

The rest of the paragraph goes toward anchoring a host of related nomenclature.

So if I give you a finite nonempty set of points V = {a, b, c, d, e}
together with a set X = {x1, x2, x3, x4, x5, x6, x7} of unordered pairs
of distinct points of V, say as prescribed below:

x1 = {a, b} = ab
x2 = {b, c} = bc
x3 = {c, d) = cd
x4 = {d, e} = de
x5 = {e, a} = ea
x6 = {a, d} = ad
x7 = {c, e} = ce

Then that is an example of a graph.

The very next thing that Harary writes after defining a graph is this:

“It is customary to represent a graph by means of a diagram and to refer to it as the graph.” (p. 9)

This is a very significant statement as it indicates how mathematicians — well, this school of graph 
theorists, anyway — view the relationship between a “graph”, a mathematical object, and a “diagram”, 
a type of sign that is drawn to represent the graph in whatever medium is at hand.

Regards,

Jon

-- 

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