Postulates, Axioms, Theorems and Assumptions

I have suggested a method of dealing with problems in an earlier post. In a short reprise there are three steps: define the problem, develop possible solutions to the defined problem, and then select your favorite possible operational solution.

The crux of the approach is really based upon the problem definition. In doing so there are several types of information and data that will have to be considered.

The concept of axioms was probably first encountered in Plane Geometry. An axiom is a

claim which could be seen to be true without any need for proof. A postulate is an axiom, and so is an assumption. A theorem is something that can be proven by using previous theorems (which have been proven) and axioms. Much of this is mathematical jargon but it still is highly useful in considering problems of all types.

In defining a problem you must first consider what you know about the issue. What is fact? What do you know, absolutely know without any possible error about the problem? If you encounter a probable fact, but open to possible question, then that is equivalent to a theorem. It must be proven (in the mathematical sense) and rigidly defined in the world of humans. Is a fact true all the time? Then it is an axiom, or a postulate, or even an assumption. If a fact is true some of the time then what are the constraints which must be specified? Under what conditions is it true? You must know when something is true in order to build a problem definition. For a given set of circumstances a problem will have one set of facts; for a slightly different set of circumstances a problem will have another set of facts which may, or may not, be closely related to the first set of facts.

A simple example will illustrate. You build a system that allows you to see your enemy coming over a hill. Your system can see for 10 miles with clarity and precision. The hill is only 4 miles away. Will your system always work? The answer is no. If you are using your eyes, that will be true if it is a bright sunny day. But what if you are in a deep, opaque fog? The system will not let you see your enemy. So, the system works perfectly except for fog.

The problem is figuring out a system that will allow you to see your enemy when they come over the hill. It is not well defined. You have to also say under what weather conditions, day or night, etc. Once you have defined the problem (which I glossed over deliberately for the sake of the example) then your postulates (axioms and assumptions) will be understood.

This is not easy. It is not done in a blink of an eye. Yet grappling with the problem definition is critical for success in all fields of human endeavor. And failure to define the problem you are addressing with clarity and precision will almost certainly have unforeseen and potentially very unpleasant consequences.

Friday, February 26, 2010   ()

A Tipping Point

I am just starting to be involved in Blogging, so I had thought I would address certain issues which were not highly emotional or perhaps even controversial. But the events of the past few days have forced me to reconsider this posture.

A tipping point is the point beyond which you can no longer recover. From the Internet, “The tipping point is the critical point in an evolving situation that leads to a new and irreversible development. The term is said to have originated in the field of epidemiology when an infectious disease reaches a point beyond any local ability to control it from spreading more widely. A tipping point is often considered to be a turning point.”

I am addressing the issue of our political representatives. I am an old man, with substantial experience in many fields. For many years I have watched our political leaders insist that they will solve our problems, or at least address them. I have listened to these speeches for a long time and when I was younger I was enthusiastic and supportive. With time I have judged the efficacy of the leaders by comparing their campaign promises with their actions.

I am not a starry-eyed idealist. I recognize that politics of all kinds require a give-and-take effort on all sides. No one group or individual has the total answer for all people on any subject. In all types of human interactions when decisions have to be made for the good of any group with different constituencies it is necessary for all to recognize that half-a-loaf is better than no loaf. And, the brighter leaders will recognize and even publicly agree that a combination of different approaches, melded into a complex action might be the best solution. It is recognized that sometimes their joint decision leads does not lead to a good solution and therefore they must change what they agreed upon. This is the sort of political process that you recognize at the national level down to the local club decisions on any subject from gardening to street decoration at the holidays.

My observations, made over years, led me to believe that experience and understanding was a good thing to have in our elected representatives. After all, they were working for us, and as such, will at least be mildly responsive to our desires and will work the political give and take, as required, to get us half-a-loaf.

I have gone past my tipping point and now believe that I am wrong. I believe that the group of our elected representatives are worse than useless: they are a danger to the operation of the nation. I have reluctantly decided that the only action that might have an effect is to send a message to ALL the national Congressman (Senators and Representatives) by voting for and electing only non-incumbents.

To this sweeping statement many will object. I am not so happy myself. Perhaps there are Congressmen who are trying to solve the nation’s problems: Jobs, energy, economy, infrastructure, education, healthcare, two wars,etc. But I can’t find them. Here is the measure I use: Has there been ANY success in obtaining a solution in ANY of these areas? I think not. And the number of Congressmen who have actually helped in any solution are few and far between. I think it is time to send a message to all sitting office holders and future office seekers: we, the American voting populace have finally lost our patience. The penalty for their collective failure is for all of them, individually, to lose their seats. The good (hypothetically there are some) and the bad (inept where there are demonstrably many): all go down.

It is true that the new group who comes in will be less experienced than those they replace. But, look at this way: could they possibly be less effective than our current leaders? I am not talking about what they say: I am talking about what is actually done.

I will vote for ALL non-incumbents in the coming elections. I suggest that if we throw out (defeat for reelection) 50% (or so) of the current incumbents the message will be clear: we, the American electorate have had it.

Saturday, February 13, 2010 — 1 note   ()

Logic and Emotion in Problem Solving

I am frequently criticized as being too unemotional. In general people think that you should feel about things. Contrary to popular opinion, I do feel emotional about many things. But not about solving problems. My methodology in dealing with problems in the world around me is rank ordered and always works.

There are three steps that are required to address any problem which becomes part of your life. First, you have to define the problem. Second you have to consider all possible solutions to your problem. Third you have to pick one solution to implement. These are simple (in concept).

The definition of the problem is the first step. There are more errors made here than in any other part of the process. You put the problem into words, and frequently, not very carefully. All the people who are working with you nod, smile, and you assume that all of your colleagues (buddies, friends, etc) understand the words that you have used in the same way that you meant them to be understood. This assumption is frequently wrong. In order to be successful the problem definition must be understood in the same way by all the participants. They may use the same words, or not, but they must all understand the problem definition in precisely the same way. If not, disaster is just a step away.

Once the problem is defined you can proceed to the second step: possible solutions. These solutions should be based upon physics, administrative constraints, financial requirements, etc. It doesn’t matter what the constraints are, but the solutions must fit within a defined world which is feasible. So far we have not mentioned emotion. There is none. You may, or may not, like the possible solutions to the problem. But that issue comes later. The question you must address and answer is: is this particular proposed set of actions a solution to the well defined problem?

Now the rest is easy. Consider the set of allowable, feasible solutions. Some will be liked by you and not others. Some will be like by others but not you. Note the verb here defines an emotional response. There is really no method of selecting one solution over another except emotion. This is where your emotions come into play.

Now you have a proposed course of action. You have defined the problem. You have considered a set of possible solutions. And, you have selected one of the solutions for implementation. This approach, while considered by some to be a little cold-blooded, always works. It works for all problems: physical, managerial, and even emotional. I commend it to all of you.

Wednesday, February 10, 2010 — 1 note   ()

Introduction of Louis P. Solomon

My name is Louis P. Solomon. I am beginning this blog at the suggestion of my family. They are all inveterate bloggers with large followings. They know that I have written on many different subjects for many years and have strongly encouraged me to express myself in a way that essentially did not exist 10 years ago. I accept their challenge. Social networking is a new method of establishing networks for professional and social contacts and, while retired from previous career, I still am deeply involved in a large variety of different areas of interest to me.

My writings have focused for many years in several different areas. I wrote technical reports and technical articles which appeared in refereed technical journals. At the instigation of my Mother I began to write books. It was an amazing request, essentially a bolt-from-the-blue. With her usual tenacity she forced me to write my first novel (The Third Legacy). More books followed. These books are non-fiction and fiction. I was approached by the publisher of an Ezine who requested that I write a monthly column on different topics. I accepted her offer and have been writing columns for the past 5 years. These columns (called Viewpoints) are on (www.maryland20878.com).

While I write and enjoy it, I also run several different businesses. I am the founder of a very unusual consulting company (www.lpscolg.com) which focuses on solving unusual, non-standard problems using out-of-the-box approaches. My colleagues and I believe that there is no problem in the management, technical, economic, and administrative arenas which we cannot solve.

In addition to consulting, I also have founded a publishing company (www.pearlriverpublishing.net), give speeches to interested groups (www.lpsseminars.com) build web sites for special purposes (www.chelseatcondo.com), and operate ham radio (N3EXW): a hobby that I have engaged in since 1957. I use digital modes exclusively (operate mainly on 20 meters) using PSK31. This is an amazing technology which allows me to connect to people over most of the world, digitally, using roughly 30 watts of power. Think about that!

I will be posting comments on this blog frequently. I look forward to discussions on subjects which are of interest to me as well as responding to those of you who will make comments. Let me hear from you as the mood strikes you.

Sunday, February 7, 2010 — 3 notes   ()