Yes, at long last, here it is, the legendary (!?)
missing...


SOPHIST's CHOICEWhy 1=0 and how this proves Barrow should not



Way back in issue 008 (Nov 91), when we first began our series of Sophist's Choice, we promised that we would establish, once and for all, that Barrow should not have been voted out of the Football League in 1972. We weren't going to do this by just rehashing the usual arguments about Barrow not being bottom of the table, or by pointing out that the League breached its own rules when it held a new ballot after the initial vote was tied with Hereford United. No, though these points remain valid of course, we were going to take a fresh approach and demonstrate our case through the scientific application of the ineluctable logic of pure mathematics. This promise, however, was contingent on our former Editor either remembering or being able to work out an algebraic proof that 1=0 that he had first been shown as a mere whippersnapper back in Mr Punton's second year Maths class at the old Grammar School in 1970. Typically, in the end he did neither and got someone else to calculate it for him. But now that he has, we can at last substantiate, conclusively and beyond any shadow of a doubt, the previously preposterous proposition that one equals zero, and zero equals one. "Whaaat!? They've lost it this time!" I hear some of you saying, reaching for the back button already. Well, I shouldn't do that if I were you. For Old Ed also informs us that Mr Punton, in his other capacity as Physics master, informed his class on more than one occasion that there was going to be a total eclipse of the sun in Cornwall in August 1999. So as Old Ed quite rightly suggests, if his old tutor predicted that correctly, then why shouldn't he be right about one equals zero too? Ah, you're back. So, enough time wasted, here now is that arcane and esoteric formula as it was chalked on that blackboard so many years ago... 
Let: 
a = 
b 
Multiply both sides of the equation by a: 
a^{2} = 
ab 
Add (a^{2}  ab) to both sides: 
a^{2} + a^{2}  2ab = 
ab + a^{2}  2ab 
Factorise and simplify: 
2(a^{2}  ab) = 
a^{2}  ab 
Divide both sides by (a^{2}  ab): 
2 = 
1 
Subtract 1 from each side: 
1 = 
0 
But by now the implications for Barrow's predicament back in 1972 should be becoming clearer. In those days, of course, the bottom four teams in Division Four all had to apply to the League for reelection. Thus, the end of each season saw a mad scramble to finish at least 20th out of 24. In 1972 Barrow finished 22nd, two places shy of that crucial position, which was occupied by Chester City (see figure i) below). So if we are to prove that Barrow shouldn't have had to seek reelection then we must establish that they actually obtained more points than Chester. Now, bearing in mind the proof we determined above, then if one equals zero, then it must be valid to count Barrow's defeats as one point, and the Chester draws ("Oooh Arthur, where's me washboard?") as no points. In this case, while Chester would record only 20 points, Barrow finish with 59, well clear of the drop zone. Okay, no doubt you'll have spotted our somewhat dishonest
logical flaw in that one, but the fact remains that if both
clubs' draws are counted as no points, and both clubs'
defeats as one point, then in any of the three possible
permutations that allows (see figures ii)  iv) below),
Barrow finish above Chester every time. In fact Barrow
actually finish in nineteenth position, two places above the
bottom four, as these corrected tables show for each
calculation. The numbers in brackets are the historically
accepted positions and points, now definitively proven here
to be wildly, and perhaps wilfully, inaccurate. 



P 
W 
D 
L 
F 
A 
Pts 
17 Aldershot 
46 
9 
22 
15 
48 
54 
40 
18 Hartlepool United 
46 
17 
6 
23 
58 
69 
40 
19 Darlington 
46 
14 
11 
21 
64 
82 
39 
20 Chester City 
46 
10 
18 
18 
47 
56 
38 
21 Northampton 
46 
12 
13 
21 
66 
79 
37 
22 Barrow 
46 
13 
11 
22 
40 
71 
37 
23 Stockport County 
46 
9 
14 
23 
55 
87 
32 
24 Crewe Alexandra 
46 
10 
9 
27 
43 
69 
29 
Revised bottom eight based on one point for a draw and one point for a defeat 


P 
W 
D 
L 
F 
A 
Pts 
17 Exeter City (15th  43pts.) 
46 
16 
11 
19 
61 
68 
62 
18 Darlington (19  39) 
46 
14 
11 
21 
64 
82 
60 
19 Barrow (22  37) 
46 
13 
11 
22 
40 
71 
59 
20 Northampton (21  37) 
46 
12 
13 
21 
66 
79 
58 
21 Chester City (20  38) 
46 
10 
18 
18 
47 
56 
56 
22 Crewe Alexandra (24  29) 
46 
10 
9 
27 
43 
69 
56 
23 Aldershot (17  40) 
46 
9 
22 
15 
48 
54 
55 
24 Stockport County (23  32) 
46 
9 
14 
23 
55 
87 
55 
Revised bottom eight (no points for a draw and no points for a defeat) 


P 
W 
D 
L 
F 
A 
Pts 
17 Exeter City (15  43) 
46 
16 
11 
19 
61 
68 
32 
18 Darlington (19  39) 
46 
14 
11 
21 
64 
82 
28 
19 Barrow (22  37) 
46 
13 
11 
22 
40 
71 
26 
20 Northampton (21  37) 
46 
12 
13 
21 
66 
79 
24 
21 Chester City (20  38) 
46 
10 
18 
18 
47 
56 
20 
22 Crewe Alexandra (24  29) 
46 
10 
9 
27 
43 
69 
20 
23 Aldershot (17  40) 
46 
9 
22 
15 
48 
54 
18 
24 Stockport County (23  32) 
46 
9 
14 
23 
55 
87 
18 
Revised bottom eight (no points for a draw and one point for a defeat) 


P 
W 
D 
L 
F 
A 
Pts 
17 Exeter City (15  43) 
46 
16 
11 
19 
61 
68 
51 
18 Darlington (19  39) 
46 
14 
11 
21 
64 
82 
49 
19 Barrow (22  37) 
46 
13 
11 
22 
40 
71 
48 
20 Crewe Alexandra (24  29) 
46 
10 
9 
27 
43 
69 
47 
21 Northampton (21  37) 
46 
12 
13 
21 
66 
79 
45 
22 Stockport County (23  32) 
46 
9 
14 
23 
55 
87 
41 
23 Chester City (20  38) 
46 
10 
18 
18 
47 
56 
38 
24 Aldershot (17  40) 
46 
9 
22 
15 
48 
54 
33 