Shy a Doctoral Degree

The very reason why it was decided that earning a doctoral degree in computer science was impractical is the immaturity of the field. It seemed that as learning advanced certain basic aptitudes and understandings were lost.

One of the most glaring was the notion that infinity is not reachable.

This is contradicted by G. Cantor’s diagonalization argument. Which is fine for reasoning about the comparative growth of sets; however, it only truly shows that our positional numbering system has limitations. Which are not found among other numbering systems. Like, those based upon the grouping of quantity symbols. Such as, the system of Roman numerals.

For, it is personally believed that Cantor did not find nor fix infinity at a point. Consider, Cantor’s argument that the largest possible base ten number (999…) is along a diagonal in the table he created. Which is necessarily not square. As, it is often represented. When, this argument is “proven”. Since, the cardinality of a set of numbers in base ten outpaces its length significantly. Intuitively, how many numbers might be held is a set of numbers who are nine digits in length? One billion or ten with an exponent of nine.

Yet, adding any number in the set of all possible numbers representable by base ten and this largest number (999…) will yield a value smaller than the largest possible number found in base thirty-six (ZZZ…). With the alphabet serving as digits for numerals greater than ten.

So, Cantor did not locate a “true” infinity. His argument proves that quantities exist beyond the largest possible base ten value; however, that scalar is simply finite. Which is intuitive since ten is also. So, our base ten positional number system has limits.

This represents a limit of computation for the base but not computability in general.

It is highly possible that such a numbering system supporting a infinite base does not. Seeing that, infinity plus infinity is a new infinity. Yet, for mankind it is often most comfortable discussing computation in terms of base ten.

The popularity of this argument, Cantor’s diagonalization, and others that were seemingly as nonsensical are the very reason why doctoral studies were left. It was hoped that certain fundamental skills and aptitudes would never be lost by practicing very popular and community mandated dysfunctions as a doctoral-level education and career were sought. When, one might show that subtraction is not computable based upon Cantor’s argument. One has reached a sufficient and sound contradiction in his proof-work showing that Cantor’s argument is a fallacy; however, it seems that among modern computer scientists this is only fuel and motivation for more baffling proof-work establishing the hardness and intractability of some problems who were only meant as curriculum-based heuristics.

The words of a very well-respected and widely published professor in the Texas System will never be forgotten. He said, “If, you want your PhD. You will do it this way [and not question it]!” Then, he suggested a book by G. Polya on proper proof-work.

These ideas were shared with Co-Pilot from Microsoft and the system tanked. Seemingly, it feels that integers cannot have a infinite length; yet, if they cannot, then one cannot construct the diagonalization argument of Cantor.

And although, it has been many years since his name was first seen along with his diagonalization proof. It was thought that his first name was Gregor and not Georg. Yet, those are memories arising from the early-80s.

PHDP - Personal Hypertext Document Processor

Within the last three months, a general-purpose controller was produced along the lines of the CABOOSE project mentioned earlier in this web history. It was written in JAVA and might be combined with a JEE Servlet or a Helidon server using object composition.

As with CABOOSE, it places a partition between the static and dynamic view concerns. So, these might be created in parallel by software team members. This approach greatly simplifies application development.

Simply put, creating a PHDP application requires completing a storyboard of static concerns. One who has placeholders for any dynamic content. The placeholders for the dynamic "hotspots" are replaced in the view is rendered by the controller.

It is deemed general-purpose; because, one might build a vast array of different applications with the same controller mechanism. Which only requires parameterization. In fact, as a simple composable and inheritable object, it might be used among client-side applications.

This application was built using Microsoft Co-Pilot. Who did a wonderful job at suggesting various coding approaches. And, it was built in fewer than forty person-hours with less than a full KLOC during lulls in demand while working as a mathematics tutor at a local community college.

The history behind its development is a very long story. Who actually starts during the early-70s. When, a parent worked at the local medical center and was one of the first research technicians hired who was a person of color. 

She offered a suggestion one day during a research seminar, and she was immediately chastised by a doctoral level researcher. Who claimed that research ideas do not come easily. And that, they take a great deal of time developing. She claimed that her son, her only child at the time, could come up with a worthwhile research idea. 

At which, the researcher claimed that, if this author produced a fundable research idea before graduating secondary school, he would receive a full scholarship for a college education at the local state university. Which the author did during the Fall of '88. Albeit, he sought an education elsewhere for better or worse.

That confrontation in that weekly seminar, likely on a Friday, during the early-70s set the stage for a single person, this author, having a great influence on modern science and industry. Much, if not all, of the work developing those "suggestions" was done by others, who leaned upon those insights given, were credit them as the "originators" of those ideas. Which was not actually true. They incubated and hatched the products which arose from those ideas. Some of which came from a elementary school pupil.

The idea for a general-purpose controller arose while studying Pascal at Vanderbilt University during the Spring of '89. Seeing that, a starting document was sought for every program that one might write as an undergraduate. One sunny weekend afternoon was spent daydreaming and brainstorming in a dorm-room that was part of the freshman Quadrangle. Having personal contacts among the ranks of the international research community and in senior and executive leadership at SUN Microsystems, many of those ideas were developed. For one, the graduate assistant who taught that Pascal course asked if she could Zerox the notes drawn on pages of the author's research journal. After, some of those ideas were presented during classroom discussions. And, they proliferated through the ranks of computer science and software engineering professors among Vanderbilts departments at that time.

The author and his notes had been the subject of prying eyes. While, he attended secondary school. So, he had learnt that he should keep the best material hidden and non-obvious. Until, he found a serious collaborator. Which was the case for these notes.  Which had notions for the feature set of object-orientation, a series of related languages, including JAVA, the rudiments of frameworks and patterns, general-purpose controllers and other novel concepts who have had quite an impact since '90 listed in obvious terms.

In fact, modern computing, including the ACM, have worked diligently at covering over the actual origins of these innovations. For one, they claim that the first OOPSLA conference was in '86. Truth be told, it was during the early-90s. It was the renaming of a conference on software, languages, and programming which focused on structured development that had been occurring since the '70s or '80s. Objects, as abstract data types, did not land upon the computing scene until the Spring of '89. The construct called an object in Simula 67 was simply a dynamic type, like those found in Python 1.0. Which was not a object-oriented language. Objects as abstract data types came in a later version of Python. They have the same name but are entirely different things.  

And, many of the "best ideas" had that weekend afternoon were non-obvious concepts hidden behind the surface of the notes recorded. As such, IBM developed a "Domino" server. Who was not built upon a "domino" data structure. Which was seen as a configuration of interacting objects who relied upon dynamic invocation and started with the "initial 'tip' of the first one" yielding a cascade of computational activity. Which would settle upon a result.

Long story short, PHDP was a long-standing idea for a general-purpose controller. Who was a concept called PHP, Cold Fusion, and other "obvious" names in a personal research and idea journal kept during the '90s. These ideas were shared while looking for a collaborator; however, others took what "little information" that they were given and developed what they thought was intended. Which always was much more than needed. As, the source for PHDP and CABOOSE do show.

The source of PHDP might be found here: 

complete-web/phdp: HTML Document Preprocessor and General-Purpose Controller for Web Applications

Irrationals - Once Again

This account was temporarily lost. The password was forgotten, and the account could not be recovered via the traditional means. This was saddening; because, the readership had reach nearly 25K and it was gaining momentum. And, with the commotion that teaching college courses be, a concerted effort at recovering the account could not be achieved. By chance, Google sent a e-mail message which asked if the account could be validated and it used a smartphone number for recovery. So, the account was available for usage again.

Hopefully, that readership will resume. Although, it has been nearly a decade since the account was lost.

This forum was used for sharing some ideas which seemed valuable; however, they were not yet commonplace knowledge or were controversial. All of the work is "original" emanating from this sole author.

One topic and lesson that shall be taught again is that concerning "irrational" numbers. Which after this short mathematical development will seem quite possibly like a "irrational" concept.

During fourth grade, our class learnt the art of long division along with other much more advanced concepts often met in secondary school and college. By chance, one of the more advanced topics taught by our instructor, Mrs. Kaiser, was the development of the number hierarchy with its various sets: natural numbers, integers, rational numbers, irrational numbers, reals, and etcetera. It is believed that she also mentioned those who be complex, transcendental, and such; however, she said that we would learn more about those later. She was just introducing a few of the fundamentals.

Seeing that, we had just learnt about long division. We had discovered the power of the "vinculum". It is that bar who rest above the repeating part of a decimal number when one is determining the quotient of the division problem with its decimal expansion.

It dawned upon the author that the usage of the vinculum might yield a means for writing irrational numbers as ratios of integers. Which would make them rational and not irrational.

How might this be done? One asks!

The observation was made when the instructor described "pi" as a number with an infinite and never-ending decimal expansion. And, a comment was shared by the author. At which, the instructor exclaimed, "Do you want a Field Medal?"

That comment was this. [If, one places that bar used for repeating decimals above the zero in a ten. He would produce a "big honking ten" (it was fourth-grade), and he could multiply pi by a special form of one, the "big honking ten" divided by the "big honking ten" and produce a ratio of integers who each have an infinite length.]

So, seeing that, this editor will not produce a vinculum, we will describe our ten with a infinite number of zeros after it as 1(0)' . The ' signifies that the number or numbers in the parentheses repeats indefinitely. 

And,

[pi * 1(0)'] / 1(0)' = [pi * 1000...] / [1000...]

Which represents the ratio of an integer with a infinite number of digits divided by the same. So, although it might never be rendered by hand or mechanically, it is a ratio of integers. Unless, the number system does not allow for integers of infinite length. Which is does. 

At the age of nine, the author knew very little about Field Medals. And, life never permitted pursuing mathematics as a career field. Such that, a thesis, a dissertation, a research publication could be prepared that merited a Field Medal. If, such an idea is potentially sufficient for one.

So, this notion is shared once more; because, it seems that the concept of the irrational number is simply that, irrational. From, a simple-minded perspective that uses discrete mathematics concepts. An unending number of integers exist. When, these are composed with themselves once. The count of the total possible compositions is the square of infinity. Which is also infinity. So, the number of ratios must cover and map with all possible numbers. This is by the pigeon-hole principle.

Albeit, one might "prove" otherwise using a "diagonalization" argument which fixes the locus of infinity. Which by simple reasoning is unreachable and not fixable.

This is why Poincare said that Cantor [should cease with his insanity and arguing that infinity was reachable via an argument of diagonalization].

Arguments of diagonalization versus a very large fixed, finite value are great for comparing and contrasting the growth rate of mathematical objects like functions and sets; however, they place a hard-fixed and artificial cap on mathematical reasoning about the infinite. When, one claims that it is a fixed point with nothing beyond it. Such permits the "irrational" among mathematical reasoning.