This actually isn't true if one formula given requires initial conditions given by another formula, i.e. if the formula is somehow stated incompletely. The situation worsens considerably if we equate analytical formulas with approximative numerical formulas, without carrying out the approximation numerics correctly.That is a contradiction. We get two unambiguous formulas, one to calculate "ch" and one to calculate the fine-structure constant based on "ch". Why would you need anything else if the formulas were correct?
If I ask you to find x, and tell you that 2+5=x, do you need to read section 8 of my post to find x? Section 8 might have a different way to do so (in this case it is unclear what section 8 actually suggests to do), but surely it should give the same result.
This is exactly what is stated here, in the bolded parts:
A computer will only give the correct results given a sufficient speed of convergence.pg 8 said:To use (7.1) for computation, we need to specify the initial data, something which will be done in section 8. The numerical verification that Ж agrees with 1/α to all decimal places, so far calculated, follows from the numerics of section 8. This comes in three steps, the first involving the sum and integral of the formulae (1.1) and (7.1) as with γ. But, as Euler discovered, the convergence in this process is too slow for effective computation.
Now the situation even gets hairier in section 8 when he starts using iterated maps of exponentials. I think the way the section is written in that multiple backtracks are necessary is what creates much of the ambiguity and ensuing confusion. Especially the statement regarding ignoring he first term in 8.5, when comparing 8.5 and 8.6 seems to trigger distrust by most readers that something must be amiss.; mathematically speaking however, the argument is clear using induction.
Further confusion then seems to arise again for 8.7 and 8.8 (especially for those not very
familiar with using the monotone convergence theorem and/or limit comparison test) because he backtracks and then talks about, I'm presuming, an unfamiliar technique to most readers. This gets worse because he then uses the Mars rocket analogy instead of a mathematical argument and I suppose most people, definitely pure mathematicians, just give up reading even though he gives a reference to Hirzebruch’s proper demonstration in the very next section even explaining how ##t## needs to be interpreted differently, how the stopping rule is justified because of monotone convergence and how to formalize this using Bernoulli numbers.
Another historical backtrack to Eddington throws the reader off again before he finally ends by giving an explicit prediction in 8.9 based on his usage of the formula in section 8.
It seems to me that this backtracking and throwing in of historical sidenotes in the main text is the main problem with his paper for most readers, especially his throwing in of theological metaphors and the word 'magic': his style of writing is blatantly non-Bourbakian and therefore suggestive of being "not proper formal mathematics"; it instead reeks of popular science writing. Many others and myself will agree that his writing style is definitely non-Bourbakian, but this has no bearing whatsoever on his mathematical argument itself; just dismissing some argument because you don't like how it is written is definitely a case of throwing out the baby with the bathwater.
For the younger people who do not know this: the Bourbakian writing style characteristic of contemporary academic mathematics is a very novel invention, which only became universally standard in the mathematical community long after the generation of Atiyah were already working mathematicians.
Important to note is that physicists don't use it, and many old mathematicians, especially those that also do physics, actually chose not to adopt the Bourbakian writing style, because it is that and that only: a writing style. Of course, used correctly, it can be much more clear than regular writing but that is only because it is overly pedantic, while being simultaneously absolutely sterile in a literary sense.