In this communication I wish first to show in the simplest case of the hydrogen atom (nonrelativistic and undistorted) that the usual rates for quantization can be replaced by another requirement, in which mention of "whole numbers" no longer occurs. Instead the integers occur in the same natural way as the integers specifying the number of nodes in a vibrating string. The new conception can be generalized, and I believe it touches the deepest meaning of the quantum rules.
5.4 The question of accumulation. If life is a wager, what form does it take? At the racetrack, an accumulator is a bet which rolls on profits from the success of one of the horse to engross the stake on the next one. 5.5 So a) To what extent might human relationships be expressed in a mathematical or logical formula? And b) If so, what signs might be placed between the integers?Plus and minus, self-evidently; sometimes multiplication, and yes, division. But these sings are limited. Thus an entirely failed relationship might be expressed in terms of both loss/minus and division/ reduction, showing a total of zero; whereas an entirely successful one can be represented by both addition and multiplication. But what of most relationships? Do they not require to be expressed in notations which are logically improbable and mathematically insoluble? 5.6 Thus how might you express an accumulation containing the integers b, b, a (to the first), a (to the second), s, v? B = s - v (/+) a (to the first) Or a (to the second) + v + a (to the first) x s = b 5.7 Or is that the wrong way to put the question and express the accumulation? Is the application of logic to the human condition in and of itself self-defeating? What becomes of a chain of argument when the links are made of different metals, each with a separate frangibility? 5.8 Or is "link" a false metaphor? 5.9 But allowing that is not, if a link breaks, wherein lies the responsibility for such breaking? On the links immediately on the other side, or on the whole chain? But what do you mean by "the whole chain"? How far do the limits of responsibility extend? 6.0 Or we might try to draw the responsibility more narrowly and apportion it more exactly. And not use equations and integers but instead express matters in the traditional narrative terminology. So, for instance, if... " - Adrian Finn
The page of my notebook was filled with many messy integrals, but all of a sudden I saw emerge a formula for counting. I had begun to calculate a quantity on the assumption that the result was a real number, but found instead that, in certain units, all the possible answers would be integers. This meant that areas and volumes cannot take any value, but come in multiples of fixed units.
The trouble with integers is that we have examined only the very small ones. Maybe all the exciting stuff happens at really big numbers, ones we can't even begin to think about in any very definite way. Our brains have evolved to get us out of the rain, find where the berries are, and keep us from getting killed. Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions.
I have tried, with little success, to get some of my friends to understand my amazement that the abstraction of integers for counting is both possible and useful. Is it not remarkable that 6 sheep plus 7 sheep makes 13 sheep; that 6 stones plus 7 stones make 13 stones? Is it not a miracle that the universe is so constructed that such a simple abstraction as a number is possible? To me this is one of the strongest examples of the unreasonable effectiveness of mathematics. Indeed, I find it both strange and unexplainable.
Arithmetic starts with the integers and proceeds by successively enlarging the number system by rational and negative numbers, irrational numbers, etc... But the next quite logical step after the reals, namely the introduction of infinitesimals, has simply been omitted. I think, in coming centuries it will be considered a great oddity in the history of mathematics that the first exact theory of infinitesimals was developed 300 years after the invention of the differential calculus.
The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules... One might therefore conjecture that these axioms and rules of inference are sufficient to decide any mathematical question that can at all be formally expressed in these systems. It will be shown below that this is not the case, that on the contrary there are in the two systems mentioned relatively simple problems in the theory of integers that cannot be decided on the basis of the axioms.