Computability



Above are a pair of textbooks on the theory of computation. Which is a set of foundational propositions and suppositions. That underpins the reasoning approaches used by modern computer scientists. Yet, this is only a theory. And, such is only like a Bill on Capitol Hill. Theories might change quite readily, even the theory of relativity..

Theories have not been universally accepted as laws among the sciences. As such, they do not carry as much weight. And like all laws among the nations, even these might be repealed and revised, after being ratified. Yet, most laws, especially natural and physical ones, rarely change. Such as the law of gravity. Anyone “fall up” lately. As far as, most of us know. We have not yet been raptured. And, this has not happenned.

Albeit, mathematics, which fundamentally is the art of computation, has been with us since the prehistoric times; electronic computing devices are relatively new. Their development started with the war effort during the late-1930s. And, it stems from the work of mathematicians during the late-nineteenth century. Who sought the creation of a machine that would simplify the tasks which they performed for government and industry.

So, do not treat someone like Kepler. If, they have a different view of the subject than the “typical or normal” mathematician. Some feel that all properly stated questions have an clear aand concise answer. This result simply might be unknown or forgotten by most, if not all, men. In terms of ability and intelligence, these who believe that all is computable might be among the highest or lowest sections of the normal curve that holds the population of modern mathematicians and computer scientists. This is a matter of debate.

And, this might be the heretic's supposition; however, everything most likely is computable. This belief mostly arises from the following premise:



After reading the textbooks above, one will see that most within in the field believe that some classes of problems do not have optimal solutions. Which one might find with a efficient procedure; in terms of computational resources. It might take weeks, months, years, decades, centuries, millennia, or eons computing the “perfect” and “best” result. And, it is accepted that other problems are intractable. Which being translated simply means difficult and not easily molded or shaped. Sadly, many modern students of computing falsely assume that "intractable" is a synonym for "impossible". Which it is not. This misunderstanding often arises when students derive the meanings of unfamiliar terms from their context of usage and do not look them up in a dictionary.

Often, it is the case that instructors tell their students that they should not waste time working on certain classes of problems, because they will reamin unsolvable for them. Question? What does the normal high school and college student do when an instructor says that they should not try certain vices because those things are more than they can handle. What do they do when that same instructor says the identical thing about a class of problems in a subject as he did about social vices and other bad habits.

Have you heard anyone say that life is a series of choices.

If, you are a science and engineering student with solid mathematics skills. Take the time and read Sipser. He writes in a rather narrative and conceptual style. That will give one pause. When, one hears the conceptual development surrounding the “intractable” problems in computing.

Those instantaneous intuitions, which make one question a theorem’s validity and that arise when reading or hearing the conceptual development, remain absent. When, one follows the formal mathematical development on a chalkboard, in a book, or on paper those never materialize. The mathematical “smoke and mirrors” prevent such.

If, you are a hardcore mathematics or computing student read the work of David, Sigal, and Weyuker, thoroughly. Then, enjoy Sipsers once again. Then, rewrite the theory. Maybe, start with disproving Goedel's Halting Theorem or show that accepting Cantor's Diagonalization Theorem means that one must undermines many of the very fundamental theorems in mathematics that is rest one. So, in ways, it is identically-false, a contradiction. Improvements are sorely needed in modern computing theory. For, a theory is simply a theory.

Does that compute.