Thanks for a brief but power punched essay. You have put your finger on the malaise. But the patient should be willing to take your treatment.
Language is the transposition of one's/a system's thought/command in another person/system's mind/CPU. Mathematics does that for quantitative aspects of Nature. Hence, mathematics is a language of Nature. But it describes the quantitative aspect only - not all aspects. Like any language, it has its own grammar. It deals with numbers, which are scalar quantities. They return the same result under all substances. Hence undecidability is false.
Uncertainty is inherent in Nature due to our inability to know ALL factors affecting the outcome of any operation. This does not invalidate causality. Hence, given total input, the output is predictable and unpredictability is false.
But uncomputability stand on a different footing altogether.
One day I wanted tea. I prepared tea and used a strainer to separate leaves from the liquid. The other day, I prepared coffee using coffee powder, but again using the strainer. Nothing was left on the strainer. My friend, who was observing it, said, tea behaves like classical objects, but coffee is like quantum. If you apply this logic, then uncomputability is true. But is this logic itself valid? No. The essence of tea or coffee lies in its effect as a stimulant. Straining is linked to its physical volume and dissolvability. They describe independent aspects not linked to each other. Hence, uncomputability is a function of wrong or incomplete selection of parameters. Paradoxes arise only because of such wrong or incomplete selection of parameters.
As you say: "When a man processes data, the data is known and the logic of the man is known. With this information, a calculation (applying mathematics correctly) can be performed that would predict any decision the man might make". But if the data itself is incomplete or insufficient, the outcome will be GIGO - Garbage In, Garbage Out. You have rightly pointed out the malaise: "The theory here is that such a machine to do such a calculation cannot exist". When you choose reductionism and then extend the description of a part to describe the whole, you are sure to be misled.
Your statement: "From the perspective of the man" is important. That perspective is not the universal perspective. We have limited degrees of freedom, but that is within a broad domain of universal programming. Everything in the universe follows a pattern. If we use our little degree of freedom to create ripples in it, it will not last long. Only in the interim, we will be confused thinking it is uncomputable, forgetting that it is we, who have violated the natural sequence to create a temporary ripple.
We have violated these principles to land in uncomputability. This is what I have brought out in my paper.
