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.”