Introducing Serendipity's Cousins - Leucippity/Aristopity - Validation/Rejection of a Pre-existing Significant Theoretical Insight

This feature is inspired by Professor Sheldon L. Glashow who coined the term Leucippity to describe a hypothesis/prediction that precedes the evidence required for its acceptance as a proper scientific theory.  A classic example of leucippity is Albert Einstein's 1915 prediction of gravitational waves which waited a full century before being experimentally observed in 2015.

In contrast to serendipity which refers to the occurrence of surprising & valuable discoveries that were not actually sought; leucippity refers to the deliberate, often slow, pursuit of experimental validation of an existing theoretical insight. 

The term leucippity is coined in honour of the ancient philosopher Leucippus (5th century BCE) who with his disciple Democritus is credited for founding Greek Atomism that all matter is composed of small indivisible particles which they called 'atoms'.  It took over two thousand years for their idea to become universally accepted!

Examples of leucippity are found in all fields of scientific endeavour - in physical sciences, medicine, geology, mathematics etc. Glashow (1, 2) gives some outstanding examples where the validation of a theory only came after a significant delay.  

The Slide lists a selection of leucippitous discoveries in physical sciences that were awarded a Nobel Prize:   

Leucippity mostly comes in play when a paradigm shift happens in a field of study; new ideas replace old explanations used to make sense of empirical evidence, and in turn generate a host of new predictions.  Such predictions await further empirical confirmation before the new ideas can be accepted as actual real theories (The Scientific Method). Einstein's Theories of Relativity (1905 & 1915) and Quantum Theory (1925) are two classic examples where the new theories explained a host of difficult-to-understand observations of preceding decades but the underlying assumptions in the theories were novel to the extent that few scientists were ready to accept these as real theories.  Einstein was not awarded the 1921 Nobel Prize for his theories of relativity but for his explanation of photoelectric effect - such was the reservations in the scientific community for his 'wild' ideas. Same is true for Quantum Theory.  Thankfully both theories made plentiful predictions and all predictions have been verified experimentally over the past 100 years.  The number of Nobel Prizes awarded for experimental work related to these theories is the proof that these are valid real theories with far-reaching significance for mankind's efforts to understand the laws that govern how nature works - and of course providing very valuable benefits to our every day life.  

We reserve the term leucippity for the verification of ideas/hypotheses that are of significant importance and might have waited for a reasonable time for it. Such ideas are deemed to be important and may substantially influence the progress/development of the field even while the idea is awaiting validation.  If a hypothesis is eventually proved to be invalid then it is likely that it would have done much harm to the scientific progress - particularly if the hypothesis was due to a scientist/philosopher of outstanding reputation. With these considerations, I introduce the word aristopity to describe such a phenomenon.

I have chosen the word aristopity to represent Aristotle (384-322 BCE).  Aristotle was a towering figure in ancient Greek philosophy, known for inventing formal logic and developing a comprehensive system of thought that influenced virtually every field, including ethics, politics, biology, natural philosophy (physics) and arts, essentially laying the groundwork for Western science and philosophy with his emphasis on observation and logical reasoning. 

However, Aristotle did not seek experimental verification of what his theories predicted - his theories were what observations and common sense supported.  The ideas of scientific method and tools (microscopes, telescopes etc.) came much later.  Aristotle's stature was such that his ideas went unchallenged for almost 2000 years and were accepted as eternal truth - to question them could result in getting burned at stake. It is not difficult to appreciate what harm such blind following to his erroneous theories might have done to the development of scientific knowledge. The next two slides list some of Aristotle's scientific ideas that have been discredited.

 

For aristopity, an erroneous hypothesis must survive over a long-enough period of time to affect thinking of a significant number of scientists.  This can be seriously harmful for proper development of scientific ideas. One indeed encounters many aristopitous practices that lasted a long time - for example, astrology, alchemy, phlogiston, Einstein's cosmological constant, Ptolemy's Almagest, Galen's Humourism, Lysenkoism, Phrenology, Miasma Theory and many more.

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Serendipity (Part 1) - Making Unplanned/Unsought Surprising Discoveries - Plays Major Role in All Aspects of Our Lives

 

Serendipity is the faculty of making fortunate and unexpected discoveries by accident                                 ---  Oxford English Dictionary (OED)

Sagacity is the quality of having or showing understanding and the ability to make good judgements.                       --- Cambridge Dictionary

The word serendipity was coined by Horace Walpole in 1754 in a letter to his friend where he recounted a Persian fairy tale 'The Three Princes of Serendip' in which the princes were always making discoveries by accident and sagacity.  Accidents become discoveries because of the sagacity.  In Appendix 1, I reproduce this charming story. 

Serendipity - smooth, pleasing and sweet sounding - has been called one of the most beautiful words of the English language.  Life is full of fortunate accidents and pleasant surprises - one only has to have  awareness/good judgement to own them when they come your way (have sagacity).  Serendipity is the occurrence of surprising & valuable events that were not being sought, often combining chance with sagacity (wisdom/insight).

There is no aspect of life that is untouched by serendipitous happenings - from finding your soulmate at the airport because you missed the flight, or being awarded Nobel Prize by noticing a completely unexpected flicker on a fluorescent screen metres away (discovery of X-rays).

The idea that chance plays an important role in the process of discovery is much older than the use of the word serendipity by Walpole to describe it.  In fact, Robert Hooke in 1679 wrote: " ......it will be much better to embrace the influence of Providence and to be diligent in the inquiry of everything we meet with.  For we shall quickly find that the number of considerable observations and inventions this way collected will a hundred-fold outstrip that are found by design." 

Curiocity, Perception, Cognitive Biases & Dogma:  

Louis Pasteur said 'In the field of observation, chance favours only the prepared mind'.  A chance accidental observation will remain that if the observer does not notice, question and follow through the unexpected finding (exercise perception).  A prepared mind is able to recognise the potential that others might ignore/discard. Curiosity drives the observer to investigate the event/result more in depth resulting in valuable discoveries. 

However, not every chance observation may be converted into valuable discovery nor  everybody is curious and prepared enough to question/investigate the chance offered (not sagacious enough). How can sagacity be encouraged and developed? First, we look into a couple of impediments - our cognitive biases and dogmatic thinking - it is important to enhance cognitive flexibility and actively challenge dogmas (rigid mindset, beliefs based on faith or authority).

We react to what is happening in the world outside according to our perceptions.  These perceptions are interpretation by our brain of the sensory signals mediated by numerous cognitive biases.  The result is that our perception of the world outside almost always represents a 'modified reality'.  

********** Have you read? 

Perceptions of the Brain - Why Our Perception of Reality is Almost Always Wrong?   A 12-hour course designed for adult education - for non-specialists.

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It is not surprising that in majority of situations, we fail to identify the importance of weak cues because they are overwhelmed by the cognitive biases that we all carry with us (cognitive biases are essential for survival in the complex world, and they are there for a reason).  The role of sagacity is of paramount importance and one needs to resist the  'obvious' interpretation because it is easier and less controversial.  Following Kahneman, in making decisions our brain ignores much of the incoming information to make quick decisions (system 1 thinking) in preference to the more energy intensive process of analysing the signals and then reaching a better/firmer conclusion (system 2 thinking) - this might happen in >90% of the decisions we make everyday. 

Overriding cognitive biases (creating cognitive flexibility) is not easy.  Such biases are deep rooted - I give an example (The Jastrow Illusion) in the following:  

(click on the slide for full page image)

Two identical sized rail-track pieces look very different in size when viewed as in the figure. In my lecture, I asked a student to measure the two tracks with a string - they are exactly the same size. (I encourage you to measure them yourself).  The interesting part is that even though our brain knows that the tracks are identical, it refuses to accept that because of the built-in way it interprets visual signals. The brain continues to ignore the factual information provided.   

It is interesting to explore how one can encourage behaviours leading to serendipity - Reference is a good place to start.   Cognitive biases help us navigate the world safely with ease and efficiently - while not perfect, they have helped human species to survive and indeed flourish. What one needs to enhance cognitive flexibility is a programme of 'improvement' in this sphere.  

One needs to venture out of their comfort zone - learn to accept ambiguities and contradictions that will help the brain to adapt to unexpected situations. 

A healthy brain is most desirable - brain training exercises, solving puzzles, exercise, good nourishing diet will help.

 Mindfulness/meditation improve focus and attention that improves flexible reactions to new situations (helps to be more open-minded).

Learn new skills - a new language for example to improve neuroplasticity.  

Besides improving the internal brain environment, much may be achieved by paying attention to external factors.  Interacting with  people/co-workers from different fields is always helpful in moving away from one's fixed biases and dogmas.   

I end this section by giving an example of how the input from an authority can influence one's decisions:  

End Note:  Many great discoveries have been made where serendipity has played a major role.  Many such discoveries have advanced our knowledge and been crucial to the welfare of the human race.  New research fields like nuclear physics, antibiotics etc owe their emergence to serendipitous discoveries. Indeed, every field has benefitted from serendipitous discoveries and almost a quarter of the Nobel Prizes have been awarded to such accidental discoveries  - and that is only since the year 1900.  Many ground breaking inventions and discoveries since historic times have contributed to the welfare and progress of our species.  In Part 2, I shall list some of the serendipitous discoveries in sciences and other fields of research.  

Of course, serendipity plays a pivotal role in our daily lives and many stories are told about lucky accidents that led to happy conclusions.  

In the end, I wish to point out that many a times, we fail to notice/appreciate the accidental break that is offered to us and we are worse off for that.  A fascinating example relates to Irene and Frederic Joliot-Curie who - not once but two times - failed to own the observations in their experiments with the result that they lost two opportunities of winning a Nobel Prize.  I shall discuss more examples of such 'negative serendipity'  in Part 2.  

Appendix 1:  The story is reproduced from Wiki

"In ancient times there existed in the country of Serendippo, in the Far East, a great and powerful king by the name of Giaffer. He had three sons who were very dear to him. And being a good father and very concerned about their education, he decided that he had to leave them endowed not only with great power, but also with all kinds of virtues of which princes are particularly in need."

The father searches out the best possible tutors. "And to them he entrusted the training of his sons, with the understanding that the best they could do for him was to teach them in such a way that they could be immediately recognized as his very own."

When the tutors are pleased with the excellent progress that the three princes make in the arts and sciences, they report it to the king. He, however, still doubts their training, and summoning each in turn, declares that he will retire to the contemplative life leaving them as king. Each politely declines, affirming the father's superior wisdom and fitness to rule.

The king is pleased, but fearing that his sons' education may have been too sheltered and privileged, feigns anger at them for refusing the throne and sends them away from the land.

The Lost Camel: No sooner do the three princes arrive abroad, they face clues to identify precisely a camel they had never seen.  They conclude that the camel is lame, blind in one eye, missing a tooth, carrying a pregnant woman, and wearing honey on one side and butter on the other.  When they later encounter the merchant who has lost the camel, they report their observations to him.  He accuses them of stealing the camel and takes them to emperor Beramo, where he demands punishment.

Beramo then asks how they are able to give such an accurate description of the camel if they have never seen it. It is clear from the princes' replies that they have used small clues to infer cleverly the nature of the camel.

Grass had been eaten from the side of the road where it was less green, so the princes had inferred that the camel was blind on the other side. Because there were lumps of chewed grass on the road that were the size of a camel's tooth, they inferred they had fallen through the gap left by a missing tooth. The tracks showed the prints of only three feet, the fourth being dragged, indicating that the animal was lame. That butter was carried on one side of the camel and honey on the other was evident because ants had been attracted to melted butter on one side of the road and flies to spilled honey on the other.

As for the woman, one of the princes said: "I guessed that the camel must have carried a woman, because I had noticed that near the tracks where the animal had knelt down the imprint of a foot was visible. Because some urine was nearby, I wet my fingers and as a reaction to its odour I felt a sort of carnal concupiscence, which convinced me that the imprint was of a woman's foot."

"I guessed that the same woman must have been pregnant", said another prince, "because I had noticed nearby handprints which were indicative that the woman, being pregnant, had helped herself up with her hands while urinating."

At this moment, a traveller enters the scene to say that he has just found a missing camel wandering in the desert. Beramo spares the lives of the three princes, lavishes rich rewards on them, and appoints them to be his advisors.

The three princes have many other adventures, where they continue to display their sagacity, stories-within-stories are told, and there is a happy ending.

A Tutorial on Alphamatic Puzzles - Wonderful Brain Training to Enhance Cognitive Function

Since my retirement 20 years ago, I have spent a lot of time in brain training activities - they say that exercising the brain is important for keeping age-related cognitive problems at bay.  I totally subscribe to this theory provided the activities are enjoyable, challenging and address different cognitive domains of the brain.  Playing Scrabble will sharpen a few of the skills relating to vocabulary, spellings, memory, spatial awareness but can only be one of the many different games and puzzles that one needs to indulge in.  I also like alphamatic puzzles as they help analytical reasoning and logic as well as boosting short term memory, attention and concentration - also they require knowledge of only school-level maths and everybody can enjoy them.

Alphamatic (aka cryptarithmetic) is a game where digits have been replaced by letters in an arithmetic operation - each letter represents a unique digit, with no two letters having the same value.  The goal is to find the digits - 0 to 9 - that the letters represent such that the resulting arithmetic operation is true. 

Slides A1 and A2 of the appendix explain the terms used in basic arithmetical operations with particular emphasis on digits carryover.

As a simple example, consider the following puzzle:

In this tutorial, I shall use the notation of numbering the columns from the right (rightmost column is Clm1 etc.) and identify the carry as cif where i is the number of column that generated it and f is the next column to the left of column i,  or f = i + 1) (see appendix slide A1 and A2 for more details) -  for example, the Units column is Clm1 and the digit carried over to the Tens column (Clm2) is c12.  Similarly,  the digit carried over from the Tens column (Clm2) to the Hundreds column (Clm3) is c23.

Also, by convention the number in the leftmost column is not a zero.

Carry plays an important role in solving the puzzle. It is also useful to look at the leftmost column as this can help to give a good idea of the range of values that the letters have in there.  If the sum row contains a higher value column not present in other rows, then the letter in the leftmost column will have a value equal to the carry that may be 1 or 2 (generally the value of a carry is 0, 1 or 2 , and rarely 3) - see slide 1 where A = 1.  If a higher value column is not present then in the leftmost column the sum of all the letters in rows + any carry from the previous column will be <10 - a useful piece of information.

Keeping track of all the details can get quite involved and one may lose direct awareness of the world outside - this might have a wonderful meditative effect that calms the system - particularly if you reach a successful result! 

Let us look at another example to practice the above information:  

Another example of a similar puzzle is as follows:

If the number ABCDEF is multiplied by 3 then it becomes BCDEFA.  Can you find the number?

The solution is given in appendix slide A3 - but try to solve the puzzle yourself first. Interestingly, the puzzle has two solutions showing two different numbers have this property.  Both answers are given in slide A3.

A puzzle that is slightly more difficult is described in slides 4 and 5.  Study the analysis to understand how to approach the solution.  Again, there is no unique way of solving a puzzle and you might wish to try your own method.

Now it is time to try some puzzles yourself!!

Enjoy! 

APPENDIX

 

Simple Templates to build 3X3 Sum and Product Magic Squares

 The history of the sum magic squares (SMS) goes back a long way.  There is something fascinating - almost magical - the way the numbers play out.  

In the following, I shall provide a template for building SMS and also introduce the product (aka multiplicative) magic squares (PMS).  PMS are not well known and few people are familiar with their properties and construction.  The methods described are rather straightforward and can also be enjoyed by those who find mathematics a difficult/confusing subject.  For clarity, I have restricted the discussion to 3X3 magic squares.

First we look at the much better known Sum Magic Squares.

3X3 SUM MAGIC SQUARES:  The SMS consists of 3 rows of numbers, each row having 3 numbers.  The square is an SMS if the sums of numbers in each row, each column, and both main diagonals are the same. 

An example is given in the following:  

                                2     7     6

                                9     5     1

                                4     3     8

The numbers in each row, each column and both diagonals sum to 15, called the magic sum.  What is not generally appreciated is that the sum of middle numbers (shown red below) is equal to the sum of the corner numbers (shown blue below) - it is 20 in this example (4 times the central number).           

                                2     7     6

                                9     5     1                        eq.0

                                4     3     8

Another property of an SMS is that the central number (5 in our example) is always one-third of the sum of rows or columns or diagonals (15 in our example).  Also the sum of all the numbers is 45 that is 9 times the central number (true for all 3X3 SMS)

Template to build an SMS:  A 3X3 SMS has nine elements.  Let us call them a, b, c, d, e, f, g, h and i.  Let the magic sum be equal to S. Then we have:  

                               a    b    c

                               d    e    f

                               g    h    i

where            S =  a + b + c  =  d + e + f  = g + h + i   

or                 3S =  a + b + c + d + e + f + g + h + i            eq.1

We can also write the sum of the two diagonals, the middle row and the middle column as follows

           4S = a + e + i + g + e + c + d + e + f + b + e + h

                = a + b + c + d + e + f + g + h + i + 3e

or       4S = 3S + 3e    (we have used eq.1 here)

Therefore,    S = 3e or e = S/3                                        eq.2

Eq.2 tells us that the centre number is always equal to one third of the the sum S (sum of rows, columns or diagonals).      Therefore, for integral numbers, S must be divisible by 3.

Now we can provide a template for building a 3X3 SMS. 

For this purpose, we chose e = 0, so that S is also equal to zero.  

A template is shown in the following:

                                  a      -a-b      b

                               -a+b      0       a-b                        eq.3

                                  -b     a+b      -a  

The way, I have built the template is by choosing centre number equal to zero and the top corner numbers as a and b.  Since the sum of the numbers in the top row is zero, the top middle number must be -a-b.  The rest follows. 

Now, if we wish to construct an SMS whose rows etc. sum to a number S = 3e then we simply add e to all the elements of the square - to obtain 

                               e+a      e-a-b      e+b

                               e-a+b      e       e+a-b                        eq.4

                                e-b     e+a+b     e-a  

This template also has the property that the sum of middle number of rows and columns is 4e - this is also true for the sum of numbers at the corners.

The SMS in eq.0 is obtained by choosing a = -3 and b = 1. 

3X3 Product Magic Squares (PMS): PMS are set of numbers arranged as a 3X3 square such that the product of numbers in any row, column or diagonal is the same.  An example is given below:

                                18    1    12

                                 4     6     9                          Eq.5

                                 3    36    2

It is easy to check that the product P of the numbers in any row, any column or any diagonal is 216 - this is equal to the central number raised to power 3. The product of the numbers at the middle of outer rows (shown red) and the outer columns (shown red) is 1296 - this is equal to the central number raised to the power 4.

Template to build an PMS: A 3X3 PMS has nine elements.  Let us call them                  a, b, c, d, e, f, g, h and i.  Let the magic product be equal to P. 

Then we have the PMS as:  

                               a    b    c

                               d    e    f

                               g    h    i

where the products of the numbers in the three rows are 

                    P =  a.b.c  =  d.e.f  = g.h.i  

or                P 3 =  a.b.c.d.e.f.g.h.i                     Eq.6

The product of numbers in each diagonal, middle row and middle column is P, therefore

                  P 4 =  a.e.i x c.e.g x d.e.f x b.e.h

or              P 4 =  a.b.c.d.e.f.g.h.i  x e 3  =   P 3 x e 3     

Hence,     P  =  e 3                                       Eq.7

Therefore, the product of the numbers in each row, each column or a diagonal is equal to the cube of the central number. This also implies that for integer numbers a to i in a PMS, the product P must be a whole cube - equal to the cube of the central number e.

Following the procedure for the construction of an SMS  template, we shall first choose the central number e = 1 and two other numbers a and b to give us the following template:  

For a=3 and b=2, eq.9 reproduces the PMS described in eq.5.  We also notice that the product of numbers in the middle of the top and bottom rows and the middle of the 1st and 3rd columns is  P 4 = a4b4  

Similarly, the product of numbers at the four corners is also a4b4      

Hope you have enjoyed this investigation into SMS and PMS.  

Now, it is possible to construct the magic squares so easily.

Have you read:  

  https://ektalks.blogspot.com/2018/10/additive-and-multiplicative-3x3-magic.html