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20 Things I learned about problem solving and theorem development — from Descartes to Lobachevsky

Men of Mathematics is a book by Eric Temple Bell published in 1937 — 81 years ago. It was nice to escape the social-proof cycle of and not read what the twittersphere is recommending. The book details the life of 34 mathematicians from the sixteenth century through the early twentieth century. It provides a somewhat idealized view of a pugnacious group of men from Pascal and Fermat to Lobachevsky and Boole. The men charted various paths to make their mark on mathematical fields such as elliptic functions, probability theory, geometry, arithmetic, analytic geometry, analysis, irrationals, and much more. It does have a glaring shortcoming: the book focuses exclusively on white men. No women. No people of color. No mention of folks from the Asia, the Middle East, or Muslims who all made great contributions to math. This is quite unfortunate. However, there are many, many women who have contributed to mathematics and Lynn Osen has documented some here.

  1. Math is the most exact science, due to the constructs on which the proofs are developed.

“Mathematics is the most exact science, and its conclusions are capable of absolute proof. But this is so only because mathematics does not attempt to draw absolute conclusions. All mathematical truths are relative, conditional.—CHARLES PROTEUS STEINMETZ (1923).”

The universe is broad – mathematics creates the conditions such that the proofs that are developed and tested – are just that, they are proofs. They stand as proofs to construct that is architected. However, they are not absolute, because new information is left to be uncovered, and most likely will disprove some conclusions.

  1. Modern mathematics really took off after the developments of analytical geometry and calculus.

“November 10, 1619, then, is the official birthday of analytic geometry and therefore also of modern mathematics.”

Modern mathematics really took shape from two important advances: analytic geometry circa 1637 via Descartes and calculus advancements circa 1666 via Newton.

  1. Mathematical theorems take hold, in part due to their broader applicability to related fields.

“It is difficult if not impossible to state why some theorems in arithmetic are considered “important” while others, equally difficult to prove, are dubbed trivial. One criterion, although not necessarily conclusive, is that the theorem shall be of use in other fields of mathematics. Another is that it shall suggest researches in arithmetic or in mathematics generally, and a third that it shall be in some respect universal.”

Some theorems, like concepts and ideas, can catch on much faster than others. I think this is in part due to human nature and how society accepts new ideas, and in part due to other fields ‘productizing’ the theorems. Almost as if the mathematicians are the engineers cooking up a new idea in the lab, and the related practitioners, say physicists, are the product managers validating the idea in the market.   

 

  1. Archimedes had an incredible ability to focus on the problem in front of him.

“Do not disturb my circles” – Archimedes.

Archimedes was so focused on his craft that when the Romans invaded his town, he kept working. The year was 212 BC. And the Second Punic War was in full swing. The Roman forces, led by Marcus Claudius Marcellus, captured Syracuse after a two year battle. Historical records (read: there is still some debate here) say Archimedes was contemplating a mathematical diagram in the sand, and the a Roman soldier stepped on part of the diagram and commanded him to meet the General Marcellus. Archimedes declined. He exclaimed directly at the soldier “Noli turbare circulos meos!” (in English: “Don’t disturb my circles!”). Those were his last words. He was beheaded on the spot. While a poor ending for Archimedes, his ability to focus on the task at hand and direct his efforts towards what he can control, is worth replicating.  

 

  1. Archimedes understood the laws of leverage.

“Give me a place to stand on and I will move the earth” – Archimedes.

While he did not invent the lever, his point was that with the right place to stand and a large enough lever, one could move an object no matter the size. Including the earth. And he developed the laws of levers and pulleys, still in use today.

 

  1. Archimedes was a prolific creator, and he solved problems with the resources around him.

“Archimedes composed not one masterpiece but many. How did he do it all? His severely economical, logical exposition gives no hint of the method by which he arrived at his wonderful results. But in 1906, J. L. Heiberg, the historian and scholar of Greek mathematics, made the dramatic discovery in Constantinople of a hitherto “lost” treatise of Archimedes addressed to his friend Eratosthenes: On Mechanical Theorems, Method. In it Archimedes explains how by weighing, in imagination, a figure or solid whose area or volume was unknown against a known one, he was led to the knowledge of the fact he sought; the fact being known it was then comparatively easy (for him) to prove it mathematically. In short he used his mechanics to advance his mathematics. This is one of his titles to a modern mind: he used anything and everything that suggested itself as a weapon to attack his problems.”

Archimedes was a Greek mathematician, engineer, physicist, inventor, and weapons-designer with a long list of creations to his name. He was not very well off. He was known to be frugal. Therefore he had to be creative with the resources around him. He solved problems with the resources available to him. He mentioned to his friend Eratosthenes that he used mechanics to advance his understanding of mathematics. It was a combination of using objects around him and thought experiments to further his thinking and discoveries. This led to many creations, including:

  • Hydrostatistics: how fluids perform when they are stable aka the ‘principle of Archimedes’
  • Infinitesimals: in The Method which was a letter to Eratosthenes, he wrote the term (the first explicit use)  infinitesimals, aka numbers that are really really small that there is no way to measure them.
  • Center of gravity used in physics
  • Archimedean Screw: used to extract water from under the ground.
  • The Odometer: built an apparatus which, when the wheel turned, it turned active gears and allowed one to measure distance traveled. This was important for military purposes of the day.
  • On his tomb, is a diagram of his favorite mathematical proof: that of a sphere and a cylinder. They are the same height and diameter. He proved that the volume and surface area of the sphere are exactly ⅔ that of the cylinder.
  1. Descartes was born (and lived) at a favorable time for his craft.

“Fermat and Pascal were his contemporaries in mathematics; Shakespeare died when Descartes was twenty; Descartes outlived Galileo by eight years, and Newton was eight when Descartes died; Descartes was twelve when Milton was born, and Harvey, the discoverer of the circulation of the blood, outlived Descartes by seven years, while Gilbert, who founded the science of electromagnetism, died when Descartes was seven.”

While he lived for only 53 years, (1596 to 1650), the French philosopher, scientist, and mathematician was surrounded by some pretty incredible talent to bounce his ideas off of.

  1. Descartes tackled his biggest problems in the morning.

“Descartes did not display his brilliance as a child. In fact, he had poor health, and was frail as a child which forced him to spend his mornings in bed. However, as he grew up he used this time to think and explore his intellectual curiosity.” Descartes had said:I desire only tranquillity and repose.”

In his adulthood, he continued to spend mornings in bed. It was his most productive time. Eventually it led him to develop Cartesian geometry. Descartes is credited with founding, in the sense of he is the one who’s name is frequently credited with (although there are probably other before his time who developed the same) describing analytical geometry which became known as Cartesian geometry. Cartesian geometry uses coordinate plane (x,y) to perform calculations in two and three dimensions. Many contributed to advancing the field, including the Greek Menaechmus, but Descartes and Fermat were credited with creating it. The former wrote La Geometrie in which he describes the methods and successes using those methods.

  1. Descartes fought dogma constantly. And questioned what we *actually* know.

“The authoritative dogmas of philosophy, ethics, and morals offered for his blind acceptance began to take on the aspect of baseless superstitions. Persisting in his childhood habit of accepting nothing on mere authority, Descartes began ostentatiously questioning the alleged demonstrations and the casuistical logic by which the good Jesuits sought to gain the assent of his reasoning faculties. From this he rapidly passed to the fundamental doubt which was to inspire his life-work: how do we know anything? And further, perhaps more importantly, if we cannot say definitely that we know anything, how are we ever to find out those things which we may be capable of knowing?”

Wrote Meditations on First Philosophy which is made up of six meditations. In it, Descartes disregards all things that are not absolutely certain, and then tries to establish what can be known with certainty. The default position of ‘how do we know anything’ and ‘ostentatiously questioning’ are two skills I seek to further develop.  

  1. Descartes reasoned independently based on the facts available. He was an early rational skeptic.

As never before the young soldier of twenty two now realized that if he was ever to find truth he must first reject absolutely all ideas acquired from others and rely upon the patient questioning of his own mortal mind to show him the way. All the knowledge he had received from authority must be cast aside; the whole fabric of his inherited moral and intellectual ideas must be destroyed, to be refashioned more enduringly by the primitive, earthy strength of human reason alone. His second conclusion was closely allied to his first: compared to the demonstrations of mathematics—to which he took like a bird to the air as soon as he found his wings—those of philosophy, ethics, and morals are tawdry shams and frauds. How then, he asked, shall we ever find out anything? By the scientific method, although Descartes did not call it that: by controlled experiment and the application of rigid mathematical reasoning to the results of such experiment. It may be asked what he got out of his rational skepticism. One fact, and only one: “I exist.” As he put it, “Cogito ergo sum” (I think, therefore I am).”

What Descartes was getting at was the very act of doubting one’s own existence served as proof that one’s own mind did exist.

  1. Descartes had many hobbies.

“During his long vagabondage in Holland Descartes occupied himself with a number of studies in addition to his philosophy and mathematics. Optics, chemistry, physics, anatomy, embryology, medicine, astronomical observations, and meteorology, including a study of the rainbow, all claimed their share of his restless activity.”

  1. Fermat solved problems that existed for thousands of years. The solution was proved 357 years after his death.

“I have discovered a truly remarkable proof of this theorem which this margin is too small to contain.” – Fermat.

He wrote this in the margins of the Arithmetica of Diophantus, translated by Claude-Gaspar Bachet, forming the proof for Fermat’s last theorem, which would be discovered 357 years later.   

  1. Fermat solved problems after his day job.

“In physics there are many similar functions, each of which sums up most of an extensive branch of mathematical physics in the simple requirement that the function in question must be an extremum; I Hilbert in 1916 found one for general relativity. So Fermat was not fooling away his time when he amused himself in the leisure left from a laborious legal job by attacking the problem of maxima and minima. He himself made one beautiful and astonishing application of his principles to optics. In passing it may be noted that this particular discovery has proved to be the germ of the newer quantum theory—in its mathematical aspect, that of “wave mechanics”—elaborated since 1926. Fermat discovered what is usually called “the principle of least time.” It would be more accurate to say “extreme” (least or greatest) instead of “least.””

He discovered a method to find the maxima and minima values of a curved line.

 

  1. Event Fermat had to fake it before he made it.

“Fermat’s life was quiet, laborious, and uneventful, but he got a tremendous lot out of it. Fermat was a born originator. He was also, in the strictest sense of the word, so far as his science and mathematics were concerned, an amateur. Without doubt he is one of the foremost amateurs in the history of science, if not the very first.”

  1. Fermat was a pure mathematician, he didn’t get drawn into philosophy or other fields.  

“Fermat seems never to have been tempted, as both Descartes and Pascal were, by the insidious seductiveness of philosophizing about God, man, and the universe as a whole; so, after having disposed of his part in the calculus and analytic geometry, and having lived a serene life of hard work all the while to earn his living, he still was free to devote his remaining energy to his favorite amusement—pure mathematics, and to accomplish his greatest work, the foundation of the theory of numbers, on which his undisputed and undivided claim to immortality rests.”

 

  1. Pascal was one of the most prolific mathematicians even though he suffered from health issues. 

“All of humanity’s problems stem from man’s inability to sit quietly in a room alone. ” – Pascal.

“All this brilliance was purchased at a price. From the age of seventeen to the end of his life at thirty nine, Pascal passed but few days without pain. Acute dyspepsia made his days a torment and chronic insomnia his nights half-waking nightmares. Yet he worked incessantly. At the age of eighteen he invented and made the first calculating machine in history—the ancestor of all the arithmetical machines that have displaced armies of clerks from their jobs in our own generation.”

 

 

His life is a running commentary on two of the stories or similes in that New Testament which was his constant companion and unfailing comfort: the parable of the talents, and the remark about new wine bursting old bottles (or skins). If ever a wonderfully gifted man buried his talent, Pascal did; and if ever a medieval mind was cracked and burst asunder by its attempt to hold the new wine of seventeenth-century science, Pascal’s was.

His lived for only 39 years (1623-1662) but his output was incredible: from Pascal’s theorem and Pascal’s triangle, to the theory of probability and Pascal’s units in hydrodynamics.

Pascal’s theorem

From Wikipedia: “In projective geometry, Pascal’s theorem (also known as the hexagrammum mysticum theorem) states that if six arbitrary points are chosen on a conic (i.e., ellipse, parabola or hyperbola) and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon (extended if necessary) meet in three points which lie on a straight line, called the Pascal line of the hexagon”

Theory of probabilities

 

  • “His theory of probabilities Pascal stated and solved a genuine problem, that of bringing the superficial lawlessness of pure chance under the domination of law, order, and regularity, and today this subtle theory appears to be at the very roots of human knowledge no less than at the foundation of physical science. Its ramifications are everywhere, from the quantum theory to epistemology.”

 

Pascal’s Units

  • From Wikipedia: The pascal (symbol: Pa) is the SI derived unit of pressure used to quantify internal pressure, stress, Young’s modulus and ultimate tensile strength. It is defined as one newton per square metre. It is named after the French polymath Blaise Pascal.

Combinatorial analysis which lead to Pascal’s triangle:

  • Triangle array of the binomial coefficients – while it is called Pascal’s Triangle in the Western World, mathematicians were doing something similar centuries earlier in Iran, China, India, and Germany.
  • From Wikipedia: The rows of Pascal’s triangle are conventionally enumerated starting with row n = 0 at the top (the 0th row). The entries in each row are numbered from the left beginning with k = 0 and are usually staggered relative to the numbers in the adjacent rows. The triangle may be constructed in the following manner: In row 0 (the topmost row), there is a unique nonzero entry 1. Each entry of each subsequent row is constructed by adding the number above and to the left with the number above and to the right, treating blank entries as 0. For example, the initial number in the first (or any other) row is 1 (the sum of 0 and 1), whereas the numbers 1 and 3 in the third row are added to produce the number 4 in the fourth row.

  1. Newton could see claims that were falsifiable and those that weren’t at the very first glance.

“Hypotheses (non) fingo.” – Newton.

Translation: “I frame no hypotheses”. This quote, mentioned in Principia, basically states that Newton saw a difference between claims about the world – such as hypotheses that can’t be tested and should be avoided – and inductions, which are ground in experiment, can be tested, and should have hypotheses. But the former, definitely not. The mathematician and scientist lived from 1643 – 1727.

  1. Newton’s ability to utilize subconscious assimilation help him rise to become one of the most influential scientist and mathematician of all time
  • Newton was the architect of dynamics and celestial mechanics. And he rounded out the proof developed by those before him for the Fundamental theorem of calculus: Newton made a capital discovery: he saw that the differential calculus and the integral calculus are intimately and reciprocally related by what is today called “the fundamental theorem of the calculus”
  • “Newton’s three laws of motion:
    • I. Every body will continue in its state of rest or of uniform motion in a straight line except in so far as it is compelled to change that state by impressed force.
    • II. Rate of change of momentum [“mass times velocity,” mass and velocity being measured in appropriate units] is proportional to the impressed force and takes place in the line in which the force acts.
    • III. Action and reaction [as in the collision on a frictionless table of perfectly elastic billiard balls] are equal and opposite [the momentum one ball loses is gained by the other].”

 

 

 

 

 

  • Principia – published Philosophiæ Naturalis Principia Mathematica (English: Mathematical Principles of Natural Philosophy). The term ‘Natural Philosophy’ was used in his day, but is what we now call ‘Science’. However, this term wasn’t coined until 1833. From wikipedia: The Principia states Newton’s laws of motion, forming the foundation of classical mechanics; Newton’s law of universal gravitation; and a derivation of Kepler’s laws of planetary motion (which Kepler first obtained empirically).

Newton solved problems through subconscious assimilation: eg returning to the problem and better able to solve it.

 

  1. Lobatchewsky essentially created non-euclidean geometry and doesn’t receive the credit he deserves.

“Euclid in some sense was believed for 2200 years to have discovered an absolute truth or a necessary mode of human perception in his system of geometry. Lobatchewsky’s creation was a pragmatic demonstration of the error of this belief….If Euclid did not, his predecessors did, and by the time the theory of “space,” or geometry, reached him the bald assumptions which he embodied in his postulates had already taken on the aspect of hoary and immutable necessary truths, revealed to mankind by a higher intelligence as the veritable essence of all material things. It took over two thousand years to knock the eternal truth out of geometry, and Lobatchewsky did it.”

He lived from 1792 – 1856. For two thousand years, mathematicians were trying to deduce Euclid’s fifth axiom (for any given line and point not on the line, there is only one line through the point not intersecting the given line) from other axioms, but Lobachevsky instead challenged the whole line of thinking from the foundation. And he developed geometry on which the fifth postulate was not true. He published these findings in A concise outline of the foundations of geometry in 1830. “Bolyai–Lobachevskian geometry” aka Hyperbolic geometry is what he founded.

William Kingdon Clifford gave him the nickname “Copernicus of Geometry” because of his revolutionary approach.

 

  1. Lobatchewsky was inspired by solving problems that society thought were not solvable.

“The boldness of his challenge and its successful outcome have inspired mathematicians and scientists in general to challenge other ‘axioms’ or accepted ‘truths’, for example the ‘law’ of causality which, for centuries, have seemed as necessary to straight thinking as Euclid’s postulate appeared until Lobachevsky discarded it. The full impact of the Lobachevskian method of challenging axioms has probably yet to be felt. It is no exaggeration to call Lobachevsky the Copernicus of Geometry, for geometry is only a part of the vaster domain which he renovated; it might even be just to designate him as a Copernicus of all thought.”

 

  1. Lobatchewsky was a first principles thinker.

“Lobatchewsky was a strong believer in the philosophy that in order to get a thing done to your own liking you must either do it yourself or understand enough about its execution to be able to criticize the work of another intelligently and constructively.”

Two things I learned about active open-mindedness and asking questions from Richard Feynman

On February 6, 1975, Feynman gave a talk at the First Annual Santa Barbara Lectures on Science and Society, at the University of California at Santa Barbara.

His talk was one of nine lectures in the series “Reminiscences of Los Alamos, 1943-1945” based on the work of the Manhattan Project.

Two takeaways:

First, Feyman demonstrates the ‘greatness’ and active-open mindedness of the members on the committee – and their ability to retain information, and change their point of view when new information was presented:

One of the first interesting experiences I had in this project at Princeton was meeting great men. I had never met very many great men before. But there was an evaluation committee that had to try to help us along, and help us ultimately decide which way we were going to separate the uranium. This committee had men like Compton and Tolman and Smyth and Urey and Rabi and Oppenheimer on it. I would sit in because I understood the theory of the process of what we were doing, and so they’d ask me questions and talk about it. In these discussions one man would make a point. Then Compton, for example, would explain a different point of view. He would say it should be this way, and he would be perfectly right. Another guy would say, well, maybe, but there’s this other possibility we have to consider against it.

I’m jumping! Compton should say it again! So everybody is disagreeing, all around the table. Finally, at the end, Tolman, who’s the chairman, would say, “Well, having heard all these arguments, I guess it’s true that Compton’s argument is the best of all, and now we have to go ahead.”

It was such a shock to me to see that a committee of men could present a whole lot of ideas, each one thinking of a new facet, while remembering what the other fellow said, so that, at the end, the decision is made as to which idea was the best – summing it all up without having to say it three times. So that was a shock. These were very great men indeed.

Second, he asked questions to challenge ideas. It advanced the thinking of those around him.

Every day I would study and read, study and read. It was a very hectic time. But I had some luck. All the big shots except for Hans Bethe happened to be away at the time, and what Bethe needed was someone to talk to, to push his ideas against. Well, he comes in to this little squirt in an office and starts to argue, explaining his idea. I say, “No, no, you’re crazy. It’ll go like this.” And he says, “Just a moment, “ and explains how he’s not crazy, I’m crazy. And we keep on going like this. You see, when I hear about physics, I just think about physics, and I don’t know who I’m talking to, so I say dopey things like, “No, no, you’re wrong, “ or “You’re crazy.” But it turned out that’s exactly what he needed. I got a notch up on account of that, and I ended up as a group leader under Bethe with four guys under me.

Full transcript at Caltech Library. 

6 Things I learned about bounded rationality and conscious filtering from Herbert A. Simon

models of my lifeHerbert A Simon was one of the most influential thinkers of the 20th century. Simon was born in Milwaukee, WI in 1916.  His father, Arthur, was an Electrical Engineer of Jewish origin who emigrated from Germany in 1903 and his mother, Edna, was a pianist who was a third generation American.

In 1975, Simon was awarded the Turing Award for his contributions to human cognition and artificial intelligence. In 1978, he followed that up by collecting the Nobel Prize in Economics for developing the theory of bounded rationality.

He was a polymath’s polymath. Simon graduated with a PhD in Political Science, did research in Psychology and Organizational Behavior, was a professor in Computer Science and Artificial Intelligence and won a Nobel in Economics. The unifying question that drove his life’s work over a 40 year span was to understand decision making and problem solving. His curiosity led him to pursue a doctorate, write dozens of papers and numerous books, and develop new theories that challenged our understanding of decision making. As of 2016, he is the most referenced author about artificial intelligence and cognitive psychology on Google Scholar [i].

Here are six things I learned Herbert A. Simon’s autobiography, Models of My Life.

1. Our ability to make rational decisions are bounded by the situation.
Simon was awarded the Nobel Prize in Economics in 1978 for his research on decision making within organizations. Much of his research focused on improving our understanding of how people make decisions. Traditional economic theory said that managers make rational, profit-maximizing decisions. What economists call rational choice. Always logical. Always prudent. Always utility maximizing.

Simon found that rational choice was an incomplete view. He described the firm as a complex, dynamic system. And within this system, people attempt to make rational decisions, but their capacity to do so is limited because of social connections, time constraints, and limited knowledge about the consequence of their decision. Put another way — the world is really complex, we don’t want to let others down and we have a lot going on — so we don’t fully think through our decisions. And, it’s pretty much impossible to consider all the information when making a decision and all the potential outcomes.

This leads people to make decisions that are satisficing — combining ‘satisfying’ and ‘sufficing’ — rather than maximizing.

This is what is known as bounded rationality [ii].

Bounded rationality states that people would not decide differently if they had more information or more time. Rather, it’s that people can’t process all the information even if it was available. Therefore if we want to anticipate people’s actions, knowing the amount or quality of information is not enough. Rather we should understand the cognitive process used to select from the information available. Hence people often seek solutions to decisions that are “good enough” using “rules of thumb”. These are known as heuristics.

As Simon notes:

We act out our lives within the mazes in which Nature and society place us.

Bounded rationality applies to individuals and organizations. Say you want to buy a TV, so you read a few online reviews and talk to a couple friends. When the sales person at the electronics store offers you a better deal on a comparable TV, you turn it down. This is because your reality has already been shaped by your friends and you are not willing to consider the other options.

In an organizational setting, let’s say a decision needs to be made. Rational decision making means that you will follow a logical, step-by-step approach analyzing the situation, considering facts, and make a decision that is made to produce the optimal outcome. However, the decision has to be made within a time constraint. Therefore bounded rationality is often manager – using a shorthand ‘heuristic’ to make the decision. It may be that your boss has made a similar decision or others won’t question the decision. It may not be the optimal decision for the company or the individual – but the manager will make a decision while not considering all the alternatives and select the first alternative that is satisfactory.

Finally, let’s say you are completing a survey. It’s been found that respondents choose answers that are satisfactory rather than optimal. This is because the respondents care more about what their answers signal to research more than they care about the accuracy of their response. This leads researchers to false conclusions.

A theory I would like to be explored is a rationality spectrum. Why doesn’t something like this exist? On one end, is complete rationality. On the other, bounded rationality. In each situation, there are certain variables which affect how rational the decision maker can be. Said another way, certain situations are more ‘bounded’ than others.

2. Our attention is finite. We must consciously filter information to make sound decisions.

Simons_3_stages_in_Decision_Making
Reference [iii]

Simon’s most famous quote is:

“a wealth of information creates a poverty of attention.”

The broader context is important to understand, his full comment was:

“In an information-rich world, the wealth of information means a dearth of something else: a scarcity of whatever it is that information consumes. What information consumes is rather obvious: it consumes the attention of its recipients. Hence a wealth of information creates a poverty of attention and a need to allocate that attention efficiently among the overabundance of information sources that might consume it.”

This is especially true in 2017. Technology brings more and more information to us in every conceivable moment. Simon developed the Model of Decision to help us filter information. The model has three steps: intelligence, design, and choice.

In the intelligence phase, the problem is identified as the decision maker ‘decides what to decide’. Problem searching involves comparing the actual state to the standard and problem formulation of the why. The output is a decision statement. The second phase is the design phase, which involves stating the objectives, research alternatives and analyzing the pros and cons.

In the choice phase, the alternatives are evaluated and the decision is made. Tools such as decision tree analysis are often implemented during this stage. The output is a decision to move forward with.

3. The General Problem Solver paved the way for artificial intelligence.
In 1939, when Simon was 33 he headed to Carnegie Institute of Technology, which would become Carnegie Mellon University.

He teamed up with Allen Newell and Cliff Shaw to develop a computer program called Logic Theorist in 1955, which was designed to imitate human problem solving. They were at the forefront of this field. In fact, the term ‘artificial intelligence’ was not coined until the following year.

The Logic Theory Machine was essentially the beta of the General Problem Solver. In 1957, Newell and Simon created the General Problem Solver (GPS) which had rules for solving problems based on the information processed.

GPS used means-end analysis as the central tool for problem solving. As Simon notes:

“…both Al and I, apparently independently, found in a particular thinking-aloud protocol clear evidence that means-ends analysis was the subject’s principal problem-solving tool. Means-ends analysis is accomplished by comparing the problem goal with the present situation, and noticing one or more differences between them—for example: I am here, I want to be there; I am five miles from my goal. The observed difference jogs memory for an action that might reduce or eliminate it (take a bike or an automobile; walk). The action is taken, a new situation is observed, and, if the goal has still not been reached, the whole process is repeated.”

4. We view the world differently because we employ different mental representations. These models encode information differently leading to different solutions for the same problem.

The representations one holds in their mind will shape how they approach problems. What representations do mathematicians use when researching problems? What about scientists, teachers, or artists? And where do these representations come from? While some think in words, others think in “mental pictures”.

Jacques Hadamard, the French mathematician, mentioned in his book, The Psychology of Invention in the Mathematical Field that mathematicians (specifically, American mathematicians) think in images. Hadamard noted in this letters to Einstein:

“[They] avoid not only the use of mental words but also, just as I do, the mental use of algebraic or any other precise signs; also as in my case, they use vague images.” [iii]

He followed up by saying:

“The words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be “voluntarily” reproduced or combined… The above mentioned elements are, in my case, of visual and some of muscular type.” [iv]

Simon believed that heuristic search was the right mental representation:

“One segment, under the banner “Let language lead the way,” takes verbal reasoning as its metaphor for the problem-solving process, and thinks of reasoning as some kind of theorem-proving structure. The second segment of the cognitive science community uses heuristic search through a problem space (a mental model of the task domain) as its metaphor for problem solving. Human Problem Solving (Newell and Simon 1972) adheres strictly to this viewpoint.”

5. The Cowles Commission created Econometrics and the General Equilibrium Theory and produced nine Nobel Laureates.

The Institute for Advanced Study in Princeton, NJ is one of the most notable independent research centers in the U.S., with over 33 Nobel Laureates who have passed through its doors including Einstein and Godel. While lesser known, the Cowles Commission was a formidable challenger. The Commission was formed at the University of Chicago in 1932 and had a twenty three year run till 1955. They had quite the run in 23 years — changing the direction of economics and producing 9 future Nobel Laureates.

How do they do it? Simon hints at the diversity and debate:

There was a mix of backgrounds and languages and strong accents spoken…The accents may have been more of a help than a hinderance to understanding. When several speakers tried to proceed simultaneously, by holding tight to the fact that you were trying to listen to, say, the Austrian accent, you could sometimes single it out from the Polish, Italian, Norwegian, Ukrainian, Greek, Dutch, or middle American. As impressive as the cacophony was the intellectual level of the discussion, and most impressive of all was the fact that everyone, in the midst of the sharpest disagreements, remained warm friends.”

The charter of the Commission was to link economic theory to statistics and mathematics [v]. Basically they wanted to utilize math proofs to validate economic theories. They held weekly seminars where a bunch of future Nobels got in a room and debated. It must have worked. The group is credited with creating two fields: General Equilibrium theory and Econometrics, as well as advancing linear programming, identifiability, and the simplex method.

During those twenty three years, the Commission output and influence on the field of economics is profound. Here are a few:

  • Tjalling C. Koopman: Nobel 1975 for his theory of optimal allocation of resources.
  • Ragnar Frisch: Nobel 1969 for founding econometrics.
  • Kenneth Arrow: Nobel 1972 for his contribution to General equilibrium theory, also Arrow’s Impossibility Theorem. Also, he is the uncle of Larry Summers.
  • Lawrence Klein: Nobel 1980 for his statistical models for economics.
  • Leonid Hurwicz: Nobel 2007 for his mechanism design.
  • Don Patinkin: monetary economics and money demand.
  • Gerard Debreu: Nobel 1983 for his contributions to General equilibrium theory
  • George Stigler: Nobel 1982 for his theory on the effects of public regulation on markets.
  • Andreas Papandreou: became Prime Minister of Greece.
  • Milton Friedman: Nobel 1976 for his consumption analysis and monetary theory
  • Oskar Lange: market pricing in a socialist system.
  • Trygve Haavelmo: Nobel 1989 for his contributions to econometrics and probability theory.
  • Herbert A. Simon: Nobel 1978 for his theory on bounded rationality.

6. Simon’s thinking was influenced by farmers, authors, economists, political scientist and engineers
Simon’s work spanned multiple industries. His ability to combine art and science led him to develop theories and insights that many had passed over. Below are the people and experiences that had an outsized impact on Simon’s world view:

  • Arthur Simon: Simon mentions the following about engineers, and his father by extension as he was an electrical engineer, “engineers believe in real things like machines and bridges and land. They are less confident that intangibles like money and organizations really exist, and the Great Depression enforced their skepticism.” This is part of what shaped his early beliefs.
  • Harold Guetzkow: college classmate, friend, and shaped Simon’s thinking for over 25 years
  • Maurice Davis: While Simon traveled the world, he says the real adventure of his life was Rockmarsh, a cattle farm 40 miles northwest of Milwaukee that covered nearly 3 square miles. Davis was the founder of the farm who was a manic depressive from his experience in World War I. Simon lived with Davis and two other young men on the farm. Simon credits that down-home experience outside academia as the single most important experience to develop his interpersonal skills. The farm eventually failed – from Davis’ suicide, to a widespread fire and a bout of pinkeye amongst the cattle – but this failure shaped was a wakeup call for what is controllable for Simon: “while any theories, however plausible and valid, can be destroyed totally by the obstinate facts of the real world.”
  • Dick Cyert & Jim March: published the The Behavioral Theory of the Firm in 1963, which informed Simon’s belief that the decision-making process within the firm was essentially a problem-solving process.
  • Jacob Marschak: head of Cowles Commission from 1943-1948, known as the founder of Econometrics, introduced independence axiom, and laid the groundwork for portfolio theory
  • Allen Newell: computer scientist and cognitive psychologist at Carnegie Mellon and the RAND Institute. Worked with Simon to develop Logic Theorist and General Problem Solver.
  • Yuji Ijiri: Cco-authored the Skew Distributions and the Sizes of Business Firms paper.
  • Tjalling Koopmans: while Simon was teaching at the Illinois Institute of Technology he started attending the Cowles Commission seminars. Koopmans was one of the leaders who organized these sessions, which Simon said were his ‘second education in economics’.

References:
[i] Herbert A. Simon, Wikipedia https://en.wikipedia.org/wiki/Herbert_A._Simon
[ii] Simon coined this term in 1947 in his book, Models of Man, Social and Rational- Mathematical Essays on Rational Human Behavior in a Social Setting
[iii] MrunaltPatel. Herbert A. Simon, Wikipedia: https://en.wikipedia.org/wiki/Herbert_A._Simon#/media/File:Simons_3_stages_in_Decision_Making.gif
[iii] Jacques Hadamard, The Mathematician’s Mind: The Psychology of Invention in the Mathematical Field. Princeton University Press, Princeton NJ. 1945. Pages cited: 84, 143, 104-106.
[iv] Albert Einstein, Ideas and Opinions, Broadway Books; Reprint edition (June 6, 1995).
[v] The History of Economic Thought. http://www.hetwebsite.net/het/schools/cowles.htm

3 reasons why e-commerce sites are opening brick and mortar stores

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For retailers, as for any company today, a strong digital presence is a mandate.

Millennials interact via social media, email, and mobile devices. To reach that demographic, brands need a robust digital marketing strategy. However, there is a limit to the strength of a digital relationship. For one, building trust is hard. Then, there is the is the lack of a tactile relational experience with the product or service. The entire experience, while convenient, is transactional.

By design e-commerce experiences limit the level of trust that can be established with a shopper. However, in-person experiences offer greater depth, and much greater opportunity to build trust. Brands that have a personal, physical relationship with shoppers have an opportunity to build a social connection.

Anthropologist Robin Dunbar’s research helps explain why. On average, Dunbar’s Number suggests, humans can have only 150 meaningful social connections. Part of the reasoning behind this is that the sensation of touch triggers endorphin release in the brain. The evidence is clear — it shows that humans are not yet able to develop robust social relationships via digital channels alone. To enter customers’ 150-connection circle, retailers must develop authentic relationships through offline channels.

Some companies have already begun rolling out their Dunbar’s Number strategy to relationship-building. These strategies are geared toward providing an authentic customer experience by building face-to-face conversations, enabling a complete sensory experience and meeting customers where they live.

1. Face-to-face interactions matter, a lot

In March, Apple updated its Genius Bar service. In the past, customers had to sign up for appointments online, or head to Apple stores for a walk-in appointment. But the walk-in process required customers to wait in the store and watch a scrolling display of names, lest they miss their time slot.

Customers now describe their issue to a store representative, who adds that information to the database of waiting customers. An algorithm determines wait time and queue priority. In the meantime, customers are free to go about their business. Apple’s new system alerts customers via text when it’s time to return to the store. The Genius bar experience has solidified Apple’s gold standard in delivering customer service.

The defining reason why the Genius bar is remarkable is because it facilitates trust, interaction and service between Apple and its customers. For its efforts, consumers voted the electronics giant as the number one company in Brand Keys’ 2015 Customer Loyalty Engagement Index.

Robert Passikoff, Brand Keys’ founder and president, writes:

“In a marketplace where brands struggle to create meaningful differentiation and engagement, those who [are] able to identify customers’ expectations and address them via authentic emotional values, will see tangible bottom-line results.”

Apple isn’t the only company to take the idea of authentic customer relationships to heart. Bonobos, the online clothing retailer, operates Guideshops, pop-ups that allow customers to try on items and speak with salespeople. E-commerce platform Shopify is in the midst of opening week-long pop-ups in a number of North American cities. The stores don’t sell anything; they house workshops where Shopify clients can learn about the platform and how it can help their businesses.

The goal of the Genius Bar, of Guideshops, of Shopify workshops, is the same. Provide customers an authentic, in person experience with the brand. This connection helps fuel trust — the second-most important factor behind Millennials’ brand loyalty, according to this NewsCred report. Trust breeds loyalty, and brand loyalty pays off, a BIAKelsey and Manta report finds. Repeat customers, it says, spend 67% more than new customers. Create trust-building authentic relationships, engender brand loyalty, realize stronger bottom line.

2. Brick and mortar enables a complete sensory experience

On Psychology Today, Kit Yarrow writes about sociocultural shifts that affect shoppers’ psychology. She references Edelman’s Trust Barometer, and notes that what people trust most are other people. It’s why, she writes, “‘crowd cred’ is more important than ever.” This crowd cred, she explains, is how “retailers that showcase the opinions of other shoppers — through things like ‘most pinned’ tags in stores or product ratings online — simply look more trustworthy.”

Part of what advances the perception of trustworthiness in shoppers’ minds is their sensory experience. “Sensory components of shopping are more influential than ever,” says Yarrow. “Today’s shopper… form[s] opinions through visual and symbolic information such as colors, assortments, and organization much more readily than words.” Customers want information. Information helps people develop opinions and, eventually, trust. Online product information helps, but nothing can replace the emotional connection shoppers feel when they can walk into a store, and compare and contrast physical items.

3. Meet customers where they live

Another company leveraging tactile experiences with consumers is Tesla. Tesla is showcasing their cars in pop-up on the back of a flatbed truck. And by doing this they are fundamentally shifting the paradigm. They’ll bring the dealership to you. Customers then have a chance to touch and examine the cars in person. As a result, Tesla will likely widen its potential customer base. Most importantly, these interactions will help Tesla develop authentic relationships with customers they wouldn’t otherwise reach.

Authenticity, trust, and loyalty

Results from a 2006 Innovative Marketing study suggest, “brand trust is an important antecedent of both attitudinal and purchase loyalty as two different types of brand loyalty.” And if repeat — loyal — customers spend 67% more than new customers, well, you get the picture. Building trust to develop brand loyalty isn’t a new idea.

They key is to building trust is through authentic customer experiences. A strong digital presence is necessary, but robust in-person relationships are critical. As a brand, you are competing with hundreds of other social connections for a customer’s attention, trust and loyalty. And the the only way forward is through authentic, tactile experiences.

The Maine Hunting Boot That Started The Omni-Channel Movement

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Field Biologist George Schaller’s 1956 Sheenjek Expedition in Alaska. Credit: The Murie Family Collection

In 1911 Leon Leonwood Bean became an early pioneer of multi-channel retail. Bean, who was orphaned at age 12, spent most of his early adulthood doing door-to-door sales and other odd jobs, and didn’t become the name we know today until he was 40. To sell his Maine Hunting Boot, Bean turned to a mail-away flyer, which later became his famous catalogue. By focusing on multiple transaction channels, Bean turned his rural Maine hunting equipment store into the first semblance of the internationally recognized L.L. Bean. Between 1967 and 2001 the company leveraged multi-channel marketing and retail methods to grow from a $4.75 million establishment to a billion dollar enterprise. Today L.L. Bean combines its online store with co-branded credit cards, and brick and mortar shops, among other channels, to entice customers.

Seamless Shopping


Mail-order catalogs were the original, albeit slower, on-demand model. Credit: Fish in A Barrel

Mail-order catalogues may be a bit outdated, but the practice of driving retail sales via a number of different channels continues to grow in prevalence. United States Census data suggests e-commerce accounted for 6.5% of all retail sales in 2014. But there is a clear upward trend. Today’s retailers would do well to leverage their omni-channel presence to drive sales.

To be clear, there is a distinction between L.L. Bean’s multi-channel shopping experience and today’s omni-channel strategies. Multi-channel retailers allowed customers to purchase goods through a variety of channels. Those channels, however, weren’t always seamlessly connected. A catalogue buy or e-purchase might not be returnable to brick and mortar stores, for instance. Omni-channel retail strives to unify the myriad channels available to shoppers and retailers. Omni-channel retailers don’t just offer two or three purchase methods. They use the channels available to them to develop a holistic view of customer preferences, and turn that information into a seamless shopping experience.

Mobile is the New Mail-Away Flyer

It’s a trend that is poised to continue to grow. Retail Online Integration found in 2013 the top 100 omni-channel brands improved their year-over-year sales anywhere from 2.3% to nearly 70%. Much of that growth comes from digital channels, particularly mobile. According to Deloitte research, in 2012 mobile influenced 5.1% of all retail store sales in the United States. That figure represents about $159 billion in total sales. Deloitte forecasts that mobile will eventually influence 17–21% of all retail store sales — up to $752 billion — by 2016. The firm also believes “smartphone shoppers are 14 percent more likely than non-smartphone shoppers to convert in store.”

An L2 Business Intelligence report found that digitally influenced retail sales have risen from 14% in 2012 to nearly 50% today. Not surprising given 72% of customers “showroom,” or buy online after browsing in-store, while 78% of shoppers “webroom,” that is, they buy in-store after browsing online. The message? Digital and physical purchase channels are becoming increasingly intertwined. This means retailers must create a seamless shopping experience between all of their channels.

The 2 F’s of Omni-Channel: Fast Fulfillment

Research suggests the omni-channel experience goes beyond a unified return policy. A Retail Systems Research report, commissioned with SPS Commerce (a supply chain management solutions company), points to something called the “Amazon effect.” Because of Amazon’s quick fulfillment capabilities, customers are increasingly concerned with how quickly their retail purchases arrive. The retailers surveyed believe three of the top four actors on the retail supply chain in 2014 were: rising consumer expectations for item information, inventory availability, and rapid fulfillment. To drive sales, omni-channel strategies will need to address those concerns.

Big box retailers and smaller companies alike are turning to omni-channel methods. Crate & Barrel leveraged a hybrid experience to address customer concerns. The company first updated their online product descriptions — thereby addressing consumer expectations for item information. Then, they implemented a seamless purchasing program by which customers may buy products online and pick them up at a nearby store. In two maneuvers, the furniture and home goods giant addressed the three main customer concerns. Crate & Barrel’s total year-over-year web sales grew 5.1% in 2013, thanks in part to their omni-channel focus.

Warby Parker Goes Omni-Channel


Warby Parker’s Winter 2014 Collection.

Warby Parker made their omni-channel name by addressing customer expectations for item information. Especially with retail verticals that require a touch-and-feel component to collecting information, a list of product details on a website often isn’t enough. Warby Parker offers a program called “Home Try-on.” With this program they ship customers five different frames free of charge. After five days, the customer must return the frames, with free shipping. This allows potential customers to generate the tactile product information they desire, and then easily complete the purchase online.

The eyeglasses retailer took their omni-channel strategy one step further. Warby Parker also opened a number of showrooms. These showrooms give potential customers a view of the company’s entire inventory. Customers may order the pair of their choice, and Warby Parker then ships the selected frames, as if the customer completed the order online. In this manner, Warby Parker is also able to address consumer concerns about inventory availability.


Warby Parker’s flagship retail store in Soho. Credit: Warby Parker Blog

MIT Sloan studied Warby Parker’s omni-channel initiatives. They found a 9% increase in total sales in areas near Warby Parker showrooms. Online sales to customers in zip codes surrounding showrooms also grew by 3.5%. It’s why sales per square foot is the wrong metric. MIT found that while Warby Parker’s Home Try-on program decreased in overall sales, the initiative’s try-on-to-purchase conversion efficiency increased. (Full disclosure: Storefront is mentioned in one of MIT’s reference links).

Easy on the Wallet


The showroom comes to you. Credit: Warby Parker

Renting a showroom isn’t exactly within the budget of smaller operations. Luckily brands like Warby Parker have shown expansive showrooms aren’t the only option for retailers looking to embark on an omni-channel strategy. Warby Parker’s Class Trips utilize a bus in which representatives drive to a designated location with a vehicle full of frames. Bonobos, an online fashion retailer, uses what they call “guideshops.” Guideshops offer inventory shoppers may try on, but must order for delivery later. These options provide a glimpse at ways companies have found creative methods to expand their offerings, without breaking the bank. And, they allow Bonobos to test a hypothesis.

Even if retailers opt to go after a more traditional model, many are discovering the costs are not as prohibitive as they thought. The current real estate climate is ripe for retailers looking for shorter-term fixes. Jones Lang LaSalle found that in 2014, 55% of all retail building projects were “single-tenant, freestanding general purpose commercial buildings.” Meanwhile projects like shopping malls and retail centers built around massive anchor tenants made up just 41.7% of all retail construction projects in 2014, down from 67.6% in 2008. And tenants are finding that short-term rental rates for those smaller buildings are far lower than traditional long-term commercial leases. More commercial real estate is available to retailers with shorter lease terms and lower rates. For example, the Westfield mall in San Francisco recently opened a space dedicated to making it easy for emerging brands to pop-in for a short term.

It’s important to smaller retailers that they are able to find ways to develop a physical showroom-type space. Despite the overwhelming presence of e-commerce plays like Amazon and Overstock, the majority of consumers still make their retail purchases in physical stores. While the e-commerce slice of the retail sales pie is growing, it is clear consumers still value the ability to touch and explore merchandise in person.

L.L. Bean and other large retailers — think Macy’s, Crate & Barrel, Starbucks, even — began as brick-and-mortar shops that expanded into the world of e-commerce. Warby Parker took the opposite route, and turned their online shop into an experience with a physical presence. In either case, the company was able to address customer demands by melding two once-independent methods of purchasing retail goods.

There are more pieces to the omni-channel puzzle than simply establishing a physical store and complementary online presence. Drop shipping and supply chain rearrangements, updating back-end tools like CRM software, and mobile application and website development, are also important devices needed to drive omni-channel retail sales. If L.L. Bean’s Maine Hunting Boots are any indication, sometimes a little creative thinking goes a long way.

Commerce as Creativity


“We guarantee them to give perfect satisfaction in every way.” — Leon Leonwood Bean Credit: L.L. Bean

Leon Leonwood was farther ahead of his time than even he could realize. His mail-order, brick-and-mortar hybrid addressed the three major concerns today’s retail customers have: transparency into product information, inventory, and rapid fulfillment. There is no quick fix to developing an omni-channel strategy. The barrier to entry is not as low as a snail mail catalogue. Yet the cost to establish a physical location in tandem with an e-commerce site isn’t as high as it has been in the past. Small retailers looking to make a big footprint need just a dash of creativity, and to remember the story of the Maine Hunting Boot.