dkspost_2021 Parts 2.61 to 2.70

4/19/2021 Part 2.61

Syntactic learning vs semantic learning.

Mr. Bala Ajjampur has a comment on mathematics learning:

“Learning happened differently at different times. So, let me paraphrase one learning proverb from many cultures – ignorance is not a sin, but refusing to learn is. There is nothing to be afraid of learning mathematics. But it is difficult to convince many just because of what you said in the beginning of your post. As jargons play a major role in judging someone’s knowledge, the jargon of mathematics once understood, makes you an expert. But teaching Mathematics has always been tough, because of abstractions and lack of real-life problems and failing to see that the whole world is mathematically aligned. The opposing view would be that in trying to understand the patterns we created mathematics. So, is this a chicken and egg problem that will continue to haunt human kind forever?”

My response

Thanks, Mr. Bala. This gives me an opportunity to go into syntactic vs semantic / contextual learning. It is not a chicken and egg problem. It is an evolution/ intuitive activity – new knowledge coming from old ones. The only thing is analysis and explanations came later in most cases.

Let me ask a question – does anyone learn a language by learning grammar alone?

The answer to this question will lead us to light with what I have been calling semantic learning and contextual learning. I gave an example from Engineering before I looked at mathematics. Before teaching Young’s modulus can I teach where it is applied or needed. I then provide clarity to a student and bring in his attention.

What we are doing with teaching mathematics is teaching grammar. That is why you call it tough. Definitely there are many who understand abstractions and become teachers and researchers in mathematics and allied areas. But there are a large number of people who want to use mathematics. They don’t understand the abstractions.

In the early days, many people attended Gurukula. But slowly this dropped to a very small number of people. This is due to abstractions and non-relevance. It drove away most of the people from formal education. Their education was at home and neighborhood.

So, most people need a semantic frame work. It is definitely possible to build this framework for most topics. In other cases, like abstract algebra need setting can be difficult. Even mathematics researchers will appreciate the semantics and invent more. I taught mathematics to biologists and reoriented my teaching to bring them into attention. Not easy, but doable. Actually, the context of abstract mathematics like number systems, geometry in early India came up because of applications in Astronomy, Astrology, construction of temples, yaga shalas etc. Bhaskara’s Lilavathy has many real life problems.

As I said this semantic framework is needed in many subjects like fluid mechanics, electric fields, automata theory, systems theory etc.

Let us look at civil engineering. The curriculum does not have many aspects of real life and it is having many discrete topics. I think we should start with a guided tour of civil engineering first may be a guided tour of engineering and technology first. Then choose those aspects like buildings etc. which will be part of the curriculum. Then put down 30 to 40 major problems in construction of buildings, dams, bridges, roads etc. Define these problems in detail with sub problems. Then discuss problem solving techniques in general like what Polya has done or Drume has talked about. Then move to discuss in some detail about concepts, principles, techniques etc. Then discuss principles and theories in depth. Then show applications. Explainability and understandability improves considerably by this approach of semantic teaching. Recall will be better. This is the important step to cognitive learning. Otherwise, it will be a dream or sham.

4/21/2021 Part 2.62

Engineering education or training?

Mr Aravind TCS with phenomenal interest in education has another question. Surprise questions come from non-teachers.

“Sir-another related doubt- In Higher Education (e,g, Engineering), when the student is generally 17 years old is it that we impart Education or Training. Education is generally given in the younger years of the student which happens both at school and home through friends and people at home especially grandparents. The question is how difficult is for a student to get educated during Higher Education, since he/she would be influenced in the earlier years to develop thinking and get educated? In simple words how much do we educate and train a student in Engineering as a broad proportion?”

My response.

We see education as building a person holistically. It has multiple dimensions. At school level, the focus is supposed to be character building, social aspects and basic sciences in a broad manner of looking. Education at engineering level is still holistic and so not skills oriented and hence not training. The focus is on engineering principles, practices, performance measures, impact factors, reliability and fault tolerance, experimentation, analysis, design, product development and highly required areas of applications. Engineering is complex and fast changing. Automation is all encompassing and dominating engineering design, production and operations. So learning needs to be cognitive and holistic. So engineering is definitely an educational activity and not just skills development. Plumbers, carpenters, smiths, linemen are skilled persons but not engineers.

Engineering and engineering education have been evolving over centuries and moving and changing fast in last two decades. Early engineering till the beginning of twentieth century focused on materials, devices operations and practices. Applications were a few and fixed. Civil engineering grew quite a lot. Mechanical engineering focused on machines power plants and practices. Only when electrical engineering started, focus was on circuits and the idea of putting together many components into a system – generators, transformers, lines and circuit breakers, relays etc. – happened and we got the concept of power systems. IISc is the first institution to start electrical technology in 1909 and ECE in 1948 almost in line with major universities in west.

Till 1950s, we concentrated on principles and practices mostly in our engineering education. There was very little science except in electrical branches. Electrical engineering used electricity and magnetism heavily. But engineering took a turn by 1970s. Focus was on mathematical models and solutions. There were doubts whether we are producing engineers or applied mathematicians. By 1980 s, computers played a major role in engineering. It is quite dominant now; with the result we are now producing programmers and the core areas are becoming of secondary importance. There is a talk of biology playing a major role. Also engineering will expand beyond the known disciplines by including health, all utilities including smart City, security, – see Marc Goodman’s book on future crimes -, retail, etc. Focus is now on integration and cyber physical systems. We now deal with highly complex, interconnected, networked self-repairing and self-changing engineering systems with millions of subsystems and users. We need to build an education system to handle these. Are we prepared?

4/23/2021 Part 2.63

Examinations – are they needed?

Mr Aravind has come up with a very touchy, very important and favorite topic of mine – examinations.

His comments:

“Another doubt regarding examinations during high school. While we have continuous evaluation, reasonable teacher-student ratio, can we reduce the final exam especially in the 10th standard? Also, there is no way presently to remove the teacher/school bias if continuous evaluation is done, when a million students take the 10th standard exam. How can we evaluate the process of Education (thinking process, understanding the world etc.,) during the high school especially in 10th standard final exam? A student may score very high marks currently in the state level exam of 10th standard, but may not have attained the ‘basic minimum’ of Education”

My response:

Examinations have been playing the central role in our education system. From 1900 Lord Curzon time onwards, there were recommendations to remove examinations. This was and is being resisted. Most teachers believe exams measure quality of learning by each student. Merit is based on marks. Admissions at Universities are based on marks. Jobs are based on marks. Fortunately, the large diversity in marks by several boards and universities led to the creation of entrance examinations for higher education in many places and even for jobs in public sector and private sector. UPSC always conducted its own examinations for civil services and engineering and other services and topped it with interviews. Now we have two exams. Same is true with IITs and PG programs, medical admissions etc. So, the society when it comes to serious selections gives no respect to exams and marks. They play a threshold role only. This has resulted in a student taking tens of exams and tuition is a multibillion business. The exams in education – schools and colleges-play a marginal role only.

Further what do you examine in an exam? Memorizing and doing some arithmetic. That’s all. Most papers are not challenging. There is no justice of evaluation of cognitive understanding of concepts and principles and their applications. Why we don’t want open book exams? You cannot get into the detailed assessment of a student’s understanding of knowledge and it’s applications to large problems and capabilities for design, critique and analysis. In addition, it creates tension and fear to most students. So, we do not get a good reliable indication of the capabilities of a student. Mostly we get a biased and wrong idea about a student.

Does a teacher get time to evaluate an exam assuming one can ask challenging questions? There are a large number of papers and a smaller number of evaluators. So, papers are not evaluated and assessed but corrected. Time available is very little. So, there is no opportunity to think and analyze on evaluation but look for errors and give marks on that basis. I want to give two personally involved examples. A student at school level was studying analogies in English like

As white as snow and as brave as a warrior. We suggested a change

As white as milk – the child has seen milk but not snow.

As brave as a tiger.

The child got zero marks. The teacher went by what is in the book and did not apply her mind.

The second example is with respect to a programming exam for first year in engineering. He came to me and said he wrote an algorithm differently. He wanted me to check it. I found it correct and appreciated him. He got zero marks because the teacher knows the program in the book and does not have time or inclination to verify the program submitted by students. I have more horror stories. That is why I call examinations are capital punishments or worse. The evaluator declares a student fail. He or she is a failure through life. Is this correct. A few minutes to correct a paper to decide the fate of a person? Can we accept this. Many schools and institutions have changed. Why not we?

When autonomy was to be given to engineering colleges, a lot of people talked about inflation of marks, biases and dilution of quality. now it is more than a decade. Autonomous colleges have done better. Definitely more students pass. It may not always be due to influences and biases. Motivation and time for assessment may be other factors.

But I see your concern about bias and vindictiveness. We have an ombudsman for banks who look at customer grievances. Why not one for students’ performances. Universities were hiding their inefficiencies by saying they will not give answer books. They will only re-total and not reevaluate. Are these justifiable? They are changing. But remove all school regulators and inspectors. Create a robust ombudsman structure to check quality of education and students’ performances.

Hope there is some light and fondly hope people support the continuous evaluation process. It has been proven beyond doubts as the best way of evaluating one’s knowledge. There several ways of assessment to be done on a weekly basis based on understanding, application, analysis, problem solving, design , experimentation and reading books and writing blogs ,abstractions, and critiques. The shape of marks sheet should change to include the merits of a student.

4/26/2021 Part 2.64

How to change mindset on examinations?

There are many comments welcoming a change. But some raised a concern about implementation. There is a feeling amongst many faculty that the change to continuous evaluation is necessary and desirable. But most feel that the senior faculty who are in positions of power do not agree for change. The seniors are convinced that the existing exam system is good and it gives quality and measurements to education and so they strongly resist changing to continuous evaluation. Some feel changing will lead to trouble. Many seniors feel old is gold. They came through the exam system and are happy and comfortable with it. They do not allow for change. This reminds me of the situation where a vice chancellor opposed giving autonomy to colleges.

I can appreciate this concern. It is serious and needs to be addressed. I encountered serious objections to doing research as an academic activity from senior professors at NITs. They were happy teaching the same subject every year. They don’t want to do research. It is the guidelines of UGC and aspirations of younger faculties that drove the research wagon. Similarly, the seniors resisted changes in curriculum. While computer science will not survive on an old curriculum with changes happening every year, that was not the story with civil and mechanical engineering. Electrical engineering evolved continuously from seventies. When I discussed the need for regular revision of curriculum, one senior professor at a good NIT said that the curriculum passed the test of time. So it is good. Professors mostly are inward looking on their subjects and may not be aware of transformations happening in their fields.

So it is a tough problem which has to be tackled. You cannot confront them. My advice to many directors and principals is to put younger faculty on policy making bodies, involve them in curriculum design and exam reforms and in TEQIP committees. This worked well.

We need to define a structure for continuous evaluation. It should provide flexibility to teachers. It should not have any rigid features. It should have multiple options. Each teacher can come with a plan in advance and get it approved. There should be a range of weightages for each component of the scheme. This should be detailed from implementation perspective. All faculty members should be given a questionnaire on the scheme. The questionnaire should be given to students, alumni and industry experts. Create a committee to moderate and advice teachers in the beginning so difficulties are ironed out. Talk to diehard opposers individually. Management should see the importance of continuous evaluation for the survival of the institution particularly if NEP gets implemented. You have no escape then.

4/28/2021 Part 2.65

4/30/2021 Part 2.66

Prof Vasudev Parvati from SDMCET, Dharwad comments on exams and continuous evaluation.

“It is indeed heartening to hear the above comments from you.

I have always felt that the ‘decide all’ examinations at the end of semester/year are a big impediment to a student’s academic progress since it requires him to do rote learning and evaluation done is only perfunctory.

The major issue with continuous evaluation in large systems is ensuring objectivity and uniformity. I believe there are techniques to make this possible, in spite of inherent biases.

I request you to kindly suggest how to make formative or cumulative assessment foolproof in large systems such as state SSC or PUC boards.”

My response

Our primary objective in education is learning, understanding and applying learnt concepts. It is not passing exams. Exams are corrective processes to improve learning and so should be continuous. Single exams do not correct a student’s learning but punishes him or her.

A multistep look is needed to arrive at a scheme for continuous evaluation. We are clear that there is no single set of components. They can vary with the type of courses – theory, experiments, descriptive, design, analysis, principles etc.

So, identify what is the objective or focus of a course. This can deal with may be ten or more subobjectives. Our job is to look for competencies in each subobjective. A final marks sheet may contain these sub objectives and the performance. So, it is not a single test or exam. We can decide on an evaluation method suitable for each sub objective. So, there is uniformity in defining focus and objectives for a course like physics. Please be transparent. Currently secrecy rules our exam system. Publish focus, sub objectives, reference materials, problem sheets, applications, experiments, design problems etc. The job of board is not to conduct exams. It should change its mindset. Remove administrator mindset and control. Create learning mindset. Publish links to information . Put information into CDs and send to all schools.

Remember exams do not make or break a person’s life. It does not make a person a failure. It should not drive a student to drop out. Make learning and evaluation a pleasant, tension free process.

Now divide it into two components one evaluation done by someone other than teacher and another by teacher. Create an ombudsman for justice. This means you may need to evaluate multiple times in a year or semester.

The components for continuous evaluation may consist of:

1.Abstraction of what a student learnt in a course every fortnight.

2. Quizzes may be one per sub objective minimum.

3. Problem sheets

4. Critique on a topic or a paper/ article/ book chapter – what different people say, what are the conclusions, correctness of observations etc. Board and schools can publish a big list and it can be put up in their notice board or web site and materials available in the library. This can be evaluated not necessarily by the teacher.

5. Discussions on some topics – school can have a panel to evaluate this.

6. Monthly tests may be centrally evaluated, if necessary. While quizzes are short answer questions. The tests can test depth of knowledge and application.

7. Solving a large problem or doing a project. This should go to the score card also.

So you have a number of options. It keeps the teacher on his or her toes with more evaluation and move away from tuitions. Bar tuitions.

Finally remember it is only a stepping stone. We don’t select candidates for higher education using these marks. We have another exam like CET for that. Do people believe in exam results. Looks like no is the answer.

Objectives need to be defined carefully. Don’t repeat syllabus as objectives. I have seen this happening. For example, if you teach a programming course, the objective is not learning syntax but writing programs to solve problems.

While the concern of fairness is reasonable and needs a checks and balances system, this means the entire massive education departments in states should change their outlook. They should focus on students and learning and not on rules and transfers. I am sure that teachers with training will accept and adjust to this scheme of continuous evaluation.

I gave one way of making evaluation objective and uniform. There are other ways of doing this also. I didn’t discuss evaluation for engineering subjects.

5/02/2021 Part 2.67

Cognitive learning – thinking

Let us go back to cognitive learning. We saw a learning map. But there some important things needed to achieve cognition. These are not taught in schools or colleges. Many were part of early Indian philosophy and these are part of philosophy now but not of other disciplines. I think every discipline should have a philosophy component dealing with the basic foundations for cognitive learning. Most of us pick up some traits but a formal approach and application in each course is essential for effective cognitive learning. What is the first one – simple – thinking? When we encounter big problems, we may think or seek advice. But mostly we don’t think. We become mechanical. A good example to start this discussion concerns Newton, I think. We cannot question his intellect or his deep analysis. So, he was a thinker. But he had a problem to solve. The cats one big and one small need to get into the house. He told the carpenter to make two holes in the door for this. Carpenter asked why two. One big hole is enough. Newton didn’t think this solution. If he spent an instant, he would have got it. It is not important enough to waste his time.

We have to make selections and decisions several times in a day. Some trivial, some major ones; some in family life, some in professional life. Doctors need to think in most cases.

What is thinking?

thinking according to dictionary is

noun

the process of considering or reasoning about something.

Example “the selectors have some thinking to do before the match”/ doctor has to prescribe a treatment for a patient.

adjective

using thought or rational judgement; intelligent.

Example -“he seemed a thinking man”

Let us various ways of looking at thinking

1.Remembering and Recalling. Retrieving or repeating information or ideas from memory. … This is first step to absorb what was learnt. So, recall is easy. After each lecture, if a student spends five minutes to think about what he learnt and write down essential points, doubts and questions.

2.Understanding. explaining, Interpreting, constructing meaning, inferring, or explaining material from written, spoken, or graphic sources. Looking for the reason why it was developed, why it was needed, how it affects other social, environmental, and economic aspects. This is second step in learning. It needs planning, organizing activities, Q and A sessions, group discussions etc. Every lecture should be based on a set of questions or problems. Not just telling what is in the book. Rework is needed here. Reduce lectures increase activities.

3.Applying. Using learned material or implementing material in new situations. Define a number of problems at least a few should be big ones and a few depend on not just mathematics but logic and reasoning. Experimentation and problem generation and solving are essential parts of a course.

4.Analyzing – modeling, solving, working mechanisms, failures and diagnostics, effects, impacts, improvements, transformations etc.

5. Evaluating – performance measures and measurements, parameters for selection, comparisons not just cost but other aspects like maturity, usability, maintainability, ruggedness, reliability and stability of the products for five years etc., selections, justifications,

6. Creating. – new things out of existing ones, innovations, improvements, integrations, disruptions etc.

The Skills We Need for Critical Thinking

The skills that we need in order to be able to think critically are varied and include observation, analysis, modelling and solutions techniques, interpretation, reflection, evaluation, inference, logical reasoning, explanation, guessing and imagination, problem generation and solving, disruptive thinking, common sense optimization and decision making. G Polya the great mathematician who wrote” how to solve it ” says ” it is important to prove results but more important is guessing”. Einstein feels imagination gives us very large number of options. Your known theories restrict you.

So we are clear thinking is not autonomous. It needs to be acquired and practiced regularly and frequently. Calmness of mind, focusing and concentration as well as good health are accelerators for thinking.

Many of these can be expanded. Each aspect needs further study.

5/04/2021 Part 2.68

Thinking

Why do we need to think?

To explain various activities, behaviors, happenings

To decide on various matters occurring daily

To know what/ why/ how of various things

To be happy and helpful

To face and handle difficulties;

We saw thinking needs effort. We think based on our beliefs. We accept statements and assertions from others easily without thinking. This led to binary thinking. We also may have positive thinking or negative thinking. Thinking needs to be cultivated as a good habit. Balanced unbiased thinking needs analytical mind and no assumptions and only minimal beliefs. Nothing wrong with beliefs. Even scientists have axioms and conjectures. We may start with guesses and hypotheses also. Most iterative problem-solving approaches start with a guess. But the guess should be intelligent and realistic Let us say that we have an electrical network in a house. We need to calculate voltages at various points. The circuit is nonlinear, say. We need to get solution by iterative methods. We don’t go with a guess of voltage to be say 500 v. That will be impractical. You may not get a converged final solution. So, it is desirable to guess a value closer to say 220 volts. You may get a solution to your iterations. But the focus of thinking is proving your guesses and hypotheses, not blindly going by them. The confused situations of trying to find answers to corona virus is very evident. Some believe in data. Some look at past experiences with pandemics, some look at observing and treating many patients, some look at it from the scientific views and experimentation and analysis. Beliefs are quite strong here. If you take treatments, million solutions pop up, most are fakes not explained by science. I just want to look at one example. Autopsy of a dead covid patient showed bacteria and so they concluded that it is not a virus but a bacterium – a flawed conclusion one can easily see through the fallacy here. Viruses are thousand times smaller than bacteria. So we need very high power mass spec or atomic focus scopes to see them. A difference of 1000 times is missed. A flexible and open approach is needed. Good thinking makes life happy.

We think to find why something is happening or to decide on many day-to-day actions like what to eat, or bigger ones like what and where to study and what jobs to look for. Most find some answer for the question whom to vote for in an election.

We saw several approaches to thinking. We saw design thinking, systems thinking and holistic thinking. So thinking is the base for cognitive learning. Thinking may not give the correct answer always particularly in complex problems. Many problems in life are made complex by us.

Reasoning goes in tandem with thinking. While thinking may not give correct explanations or answers, reasoning will provide logical and mathematical or analytical or linguistic approaches to find reasons – why something’s happen? What is a good decision? Etc.

A simple definition from dictionary is

reasoning is the action of thinking about something in a logical, sensible way.

For example, we have several experts explaining possible or plausible causes for covid illnesses, but there is no clarity. Deccan herald on Sunday has almost a two-page reasoning to use data for decisions.

5/05/2021 Part 2.69

Beliefs

Prof Vinod Vyasulu has a suggestion:

“You should elaborate on the point about belief and axiom. All of us have them. Where do they come from? When can they mislead us?”

So let us look at belief first. As he says all of us have beliefs – including science-. Some have a few, some have many; some strong, some changeable; some mostly harmless some affects others. Philosophy started with beliefs.

Let us see the definition first

“Belief is a state or habit of mind in which trust or confidence is placed in some person or thing”. her belief in God, a belief in democracy. I bought the table in the belief that it was an antique.

Dictionary definition

1.belief is an acceptance that something exists or is true, especially one without proof.

“his belief in extraterrestrial life”

2. It is trust, faith, or confidence in (someone or something).

Beliefs may be forced by authorities like leaders or kings. Or it can be forced by religion like “sun goes round the earth” or there is a god or follow the religion or you will go to hell. Heaven and hell are also beliefs from religions. Majority of beliefs came from religions. Religions are built with the belief that people need to be forced into beliefs. So, preachers are focused on that and not rational. This unfortunately is true of even Buddhism. Rig Veda has a lot of beliefs – many gods and goddesses controlling various aspects of nature. It also has dictums which cannot be called beliefs- may be values- but are desirable. Examples are knowledge is free; accept knowledge from wherever it comes. It talks about governance which is not belief oriented. It talks about cleanliness and rituals. Upanishads also talk about various aspects. We are seeing a slow transformation of beliefs to rules. Society also introduced many beliefs. It dictated rights and wrongs to people. A lot of beliefs may also come from parents, elders and society. These consider good behaviors, values, obedience, respect, discipline, etc. Some people have hundreds of beliefs. Many beliefs lead to attitudes and behaviors. Belief can be strong or flexible.

Beliefs started when humans organized into groups. Order was needed. So, beliefs were built/ created and imbibed in people. It looks like everyone has some beliefs. Atheism is also a belief in non-god. Philosophers built many beliefs, -I am using the word belief, because the assertions made by early philosophers were not provable, most are conjectures.

When you find a belief which seems to be a permanent fact and it gives birth to many other facts, that means there is a stronger basis for that assertion and Greek philosophers introduced the concept of principles. So guessing and hypotheses played a dominant role in this scheme which builds science. Indian philosophy is multifaceted. It allows flexibility – multiple paths for life – jnana, bhakti, karma etc. It includes yoga also as an integral part of philosophy. Samkhya, numbers based is also a philosophy component. Indian philosophy did not concentrate on beliefs but on proving assertions and beliefs. People argued on these and Dharma and rituals and living for long and used logical approaches, inference, mathematics to prove their beliefs. Various methods of proving and finding truth can be seen in several works. There are many who believe strongly that mathematics – now data- will solve every problem including covid 19.

There are four Kinds of Beliefs

1.Meta: beliefs about beliefs.

2.Perceptions: beliefs about how the world seems to be, based on the evidence I have.

It also includes various aspects of life, living, nature, and their inter relationships.

3.Opinions: beliefs about how I should interpret reality. …

4.Predictions: beliefs about how I think things will end up in the future based on what I know now.

Sometimes beliefs can be wishes and hopes. Remember Vasudeiva Kutumbakam. It is a hope for being helpful to others. Ahimsa is the basic hope of Jain’s. They had several practices to follow ahimsa principles.

Scientists also believed in certain facts. They call them conjectures. Once it works and it is proved through experiments and analysis, it becomes a law.

Fundamental principles on which laws are built are axioms. Axioms have almost universal applications. They are true under a large number and types of situations and conditions. Space is very large and expanding is an axiom. Chandrasekhar limit is close to a law. We are familiar with Newton s laws, and many others.

Sometimes fact and fiction get mixed. Example is Astronomy and Astrology particularly in India.

We need to have an open mind to accept or reject a belief. It is one’s choice. But we can’t tread on others toes.

There is lot more one need to talk about belief because of its power, it’s tendency to make you a slave, it’s interrelationships with ethics, values, concepts of life, nature and cosmos.

One’s belief should not destroy others or nature or disturb the harmony and peace of a place or should not lead to violence. So long as it is bounded, does not interfere, benevolent and flexible, beliefs are fine.

Thanks Vinod

5/07/2021 Part 2.70

Let us see axioms.

Definition

Axioms: a statement or proposition which is regarded as being established, accepted, or self-evidently true.

“The axiom that sport builds character”

MATHEMATICS

a statement or proposition on which an abstractly defined structure is based.

Axioms are not provable. They form the first principles.

When an axiom was found not true, it is discarded or modified. They are general truths applicable in a wide variety of situations.

Some examples are

1.Two Parallel Lines Never Intersect Each Other.

2.India is a Part of Asia.

3.Probability lies between 0 to 1.

4.The Earth turns 360 Degrees Every day.

5.All planets Revolve around the Sun.

Another way of looking at axiom is: An axiom, postulate or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek axíōma (ἀξίωμα) ‘that which is thought worthy or fit’ or ‘that which commends itself as evident” the first term used was first principles and later it became axioms.

There are some differences in meaning.

The term has subtle differences in definition when used in the context of different fields of study. As defined in classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. As used in modern logic, an axiom is a premise or starting point for reasoning.

Deductions were developed by Greeks to start from first principles and reach by deductive reasoning to a new fact. Mathematics, Geometry and Logic use axioms extensively. Mathematics defines an algebra and develop operators and theorems. Logic starts with a proposition that is true and deduce a new fact. It was qualitative in the beginning and after Bertrand Russell introduced predicates with variables A B C etc . Boolean logic starts with Huntington’s postulates.

Aristotle talked about geometry and many Indian mathematicians also proposed and used geometry.

It was Euclid who defined axioms and used it to create proofs. He wrote the book titled elements. He categorized axioms into two parts.

Some examples of Euclidean postulates are

1.It is possible to draw a straight line from any point to any other point.

2.It is possible to extend a line segment continuously in both directions.

3.It is possible to describe a circle with any center and any radius.

4.It is true that all right angles are equal to one another.

5.(“Parallel postulate”) It is true that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, intersect on that side on which are the angles less than the two right angles.

Common notions are

1.Things which are equal to the same thing are also equal to one another.

2.If equals are added to equals, the wholes are equal.

3.If equals are subtracted from equals, the remainders are equal.

4.Things which coincide with one another are equal to one another.

5.The whole is greater than the part.

Beliefs are for individuals and they are not general. They can be contradictory. For example, one believes coffee drinking is good for heart. Another believes coffee is bad for heart. One believes in God. Another says there is no god. So many beliefs may not be rational. Even professionals have beliefs. They call them rules. Even rules may change. For example, in the sixties, doctors advised heart patients to stay in bed and not to exercise. It has changed now.

Similarly, nurses used to remove bandages with a belief that doing it- pulling bandages from skin- fast is better than pulling it slowly.

One performs a lot of tasks in a job and devices approaches based on experience. Another example: consider the newspaper’s Page maker and the ways by which he puts together various news items into a page. It is not a simple job. When Dantzig developed linear programming’s simplex algorithm, this was a problem he solved. But when he checked with the Page maker, he found the solution by Page maker is the same as his. I call this common-sense optimization. Even fruits are shaped correctly as spheres based on optimization. So, beliefs can evolve.

But axioms deal with general principles and applicable in many situations. After all these decades and centuries most or shall I say all, mathematical axioms are fine and true. They stood the test of time.

There is an interesting book

“Predictably irrational” by

Dan Ariely.

He says humans are not rational and their irrationality say in options and selections can even be predicted. He talks about behavioral economics. Interesting analysis of our selecting approaches.