Information Theory: How the Genius of Claude Shannon Changed Our Lives By Thinking “Outside the Box”

Claude_Elwood_Shannon_(1916-2001)[1]Claude Shannon: have you ever heard the name? How about Isaac Newton, Albert Einstein, and Charles Darwin? Those three names are universally familiar to the general public even though all but a small segment of the population would find it difficult to elaborate significant details of the work that made them immortal in scientific history. In Shannon’s case, his name, his face, his genius, and his immense impact on our world are all virtually unknown, thus unappreciated, outside the realms of mathematics and electrical engineering. Claude Shannon is the “father of information/communication theory” and primarily responsible for the vast networks of computers, data processing, and mass communication that power modern society. It is my intention, here, to at least do minimal justice to his rightful legacy among the great minds of mathematics, science, and engineering.

Shannon’s contributions are numerous and varied, but a closer look reveals that the central theme of most of them are well characterized by his most famous of many publications over the years, The Mathematical Theory of Communication, which appeared in 1948. Most of Shannon’s published papers were issued under the imprimatur of the Bell (Telephone) Labs Technical Journal. Bell Labs had a long and illustrious run as an incredible incubator for many of the most important math, science, and engineering advancements in America during the twentieth century. Accordingly, many of the country’s top minds were associated with the Lab and its activities. Claude Shannon was one of them.

All Information Can Be Represented By Data 1’s and 0’s!

3653[1]Have you ever marveled at the fact that modern computers can store and reproduce any-and-all information – text, audio, color pictures, and movies – using only organized collections of data 1’s and 0’s? Think of it! A modern computer is little more than a collection of millions of microscopic electronic switches (think light switches) which reside either in an “on” state (a data 1) or an “off” state (a data 0). If that reality has never occurred to you, pause for a few moments and reflect on the enormity of the fact that anything and everything called “media” can be displayed on-command by calling-up organized collections of data 1’s and 0’s which reside in the bowels of your personal computer! In addition, the computer’s “logical intelligence” – its ability to respond to your commands – also resides in the machine’s memory bank in the form of data 1’s and 0’s. In the nineteen-twenties and thirties, Claude Shannon was among other computing pioneers who understood the possibilities emerging from the burgeoning progress of electronics. The notion of a binary (or two-state) number system in a computing device was evident as far back as the eighteen-thirties when Charles Babbage designed and built his first bulky, mechanical computing machines.

Today, in our everyday lives, we use the decimal number system which is inherently unsuited to computers because that number system requires each digit in a number representation to assume one of ten states, 0 thru 9. Modern computers are designed around the binary (or two-state) system in which each digit in a binary number assumes a value, or weight, of either one or zero. A simple light switch or an electronic relay (open or closed) are examples of simple, two-state devices which can be used to represent any single digit in a binary number. In actuality, the two-state devices in modern computers are implemented utilizing millions of microscopic, individual solid-state transistors which can be switched either “on” or “off.” The binary number system, requiring only simple two-state devices (or switches), is the optimal choice.

Shannon would be the first to admit that he was never motivated to change the world by the work he pursued. Nor was he motivated by any prospects of fame and fortune for his efforts. Rather, he was endlessly fascinated by the challenges inherent in pursuing theoretical possibilities, regardless of any possible practicality or profit stemming from his efforts. Claude Shannon’s persona had multiple facets: a genius, out-of-the-box thinker, an inveterate tinkerer and inventor of gadgets, a juggler (circus-type), and a devotee of the unicycle – a conveyance he both rode, designed, and built himself! This most unusual personality forged much of the “quiet legend” which surrounded the reclusive, mysterious Mr. Shannon. Even though he was a tinkerer and builder of “toys and gadgets,” he lived for and thrived on elevated ideas – creations of the mind. In many respects, he was much like Albert Einstein in his outlooks, his rampant curiosity, and his dogged persistence, all of which were on full display as Einstein tackled the mysteries of both special and general relativity.

The Most Important Master’s Degree Thesis Ever Submitted!

In 1937, Claude Shannon submitted a thesis for his master’s degree in electrical engineering at MIT. Normally, a master’s thesis proves to be significantly less impressive in terms of originality and impact than that required for a Phd. Shannon’s master’s thesis proved to be a startling exception to the rule – the first of many unorthodoxies that characterized his unusual career. As an undergraduate at the University of Michigan, he had earned dual degrees in mathematics and electrical engineering. It was at Michigan that he learned the “new math” developed by the English mathematician George Boole and introduced to the scholarly community in 1854 under the title, An Investigation into the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities. This work was the most important contribution to emerge from the genius of Boole who died much too young from pneumonia at the age of forty-nine years.

Shannon's ThesisShannon was prescient enough to recognize that Boole’s algebraic treatment of the binary number system uncannily lent itself to the development of real-life logical systems (computers) which could be simply implemented using electrical relays – binary (two-state), on/off devices which had cost, space, power, and reliability issues, but which could nevertheless demonstrate computing principles in the nineteen-thirties and forties. In simplest terms, Shannon demonstrated in his master’s thesis that, using Boolean algebra and simple two-state electrical devices, a computer could be designed to “think logically” while processing and displaying stored information.

Shannon’s prescient recognition led to the characterization of his thesis as “The most important Master’s thesis ever written.” Indeed, Shannon opened the doors to a new and exciting vista, one that he vigorously explored while working at AT&T’s Bell Laboratories, and later, at MIT.

Shannon Sets These Major Goals for Himself – No Small Tasks!

How do we define “information,” how do we quantify information, and how can we transmit information most efficiently and reliably through communication channels?

I suggest that the reader pause a moment and ponder the thin air in which Claude Shannon pursued his goals. How in the world does one define and quantify such an “airy” concept as “information?” 

Here are some examples, the easiest entry-point into Shannon’s methodology regarding the definition and quantification of information:

When we flip a coin, we receive one data-bit of information from the outcome, according to Shannon’s math! In this case, there are only two outcome possibilities, heads or tails – two “message” possibilities, if you will. Were we to represent “heads” as a binary data “1” and tails as a binary data “0”, we can visualize and quantify the outcome of the coin flip as the resulting state (“1” or “0”) of a single “binary digit” (or “bit”) of information gained in the process of flipping the coin. In Shannon’s world, the amount of information received would equal precisely one-bit of information in either case – heads or tails – because each case is equally probable, statistically. The final comment concerning probabilities is important.

Here is how probability/statistics enters into Shannon’s treatment of information: What would be the case if I had a bona-fide, accurate/true crystal ball at my disposal and I queried it, “Will I still be alive on my upcoming eighty-second birthday – yes or no?” There are only two possible predictions (or messages), but, in this case, the information content of the message conveyed is dependent on which outcome is provided. If the answer is yes, I will make it to my 82nd birthday, I receive (happily) lessthan one bit of information content because actuarial tables of longevity indicate that, statistically speaking, the odds are in my favor. If the answer is no, I (unhappily) receive more than one bit of information due to the probability that not reaching my next birthday is statistically less than 50/50. A message whose content reveals less likely outcomes conveys more information than a message affirming the more likely, predictable outcomes in Shannon’s mathematical model of information.

Here is a third example of Shannon’s system: Consider the case of rolling a single die with six different faces identified as “1” through “6.” There are six possible outcomes, each one having the equal probability of 1/6. According to Shannon’s mathematical model, the amount of information gained from a single roll of the die is 2.59 binary bits. The outcome of a single roll of a die carries 2.59 bits of information vs. only one bit of information from the single flip of a coin. Why is that? It is because any one of six equally likely possible outcomes is less likely to occur than either outcome of a coin flip which presents only two equally likely outcomes!

Lest you think that quantifying the information content of messages strictly on a statistical basis with no regard for the content of the message itself seems a silly bit of elite hair-splitting on the part of math/engineering crackpots, I can assure you that you are dreadfully mistaken for these and numerous other derivations and conclusions that sprang from the curious mind of Claude Shannon form the backbone of today’s trillion dollar computer and communication industries! Shannon and his information/communication theories, like Einstein and his relativity theories, has been proven correct by both time and actual practice. Because of both men, our world has been immensely altered.

A Good Stopping Point for This Journey into                                  Information/Communication Theory

At this point in the story of Claude Shannon and his information /communication theories, we approach the edge of a technical jungle, replete with a formidable underbrush of advanced mathematics, and this is as far as we should go, here. For those well-versed in mathematics and engineering, that jungle path is clearly marked with signposts signifying that “Shannon has passed this way and cleared the pathway of formidable obstacles: proceed…and marvel.” The pathway that Shannon forged guides fortunate, well-equipped adventurers through some deep and beautiful enclaves of human thought and accomplishment.

Claude Shannon was a remarkable original, an imaginative thinker and doer. Inevitably, great milestones in math, engineering, and science are not without some degree of precedence. In Shannon’s case, there was not much to build from, but there was some. Certainly, the Boolean algebra of George Boole was a gift. As mentioned earlier, Shannon’s first publication of his own findings, titled The Mathematical Theory of Communication, appeared in the Bell System Technical Journal of !948.

IMG_2500Hartley BSTJ 1928 Enhanced 21928 Nyquist Sampling_1

His paper was quickly published in book form in 1949 by the University of Illinois Press. In his paper, Shannon mentioned the earlier work of Ralph V. L. Hartley and Harry Nyquist, both earlier Bell Laboratory employees, like Shannon. Hartley published his prescient views on the nature of information in the Bell System Technical Journal, dated July 1928. His paper was titled, The Transmission of Information. Although rudimentary, the paper was original and set in motion ideas that led Shannon to his triumphant 1948 publication in the Bell Journal. In Nyquist’s case, in addition to discussions re: the importance of optimal coding for the efficient transmission of information in an earlier, 1924 issue, Nyquist published, in the August 1928 Bell Journal, his ground-breaking analysis of the minimum waveform sampling rate of an analog (continuous) signal necessary to accurately reconstruct the original waveform from stored digital data samples – as is routinely done, today. Nyquist’s famous sampling theorem provided the necessary “bridge” between the world of analog information and digital representations of analog data that was so necessary to make Shannon’s theories applicable to all formats containing information.

Two Crucially Important, Parallel Technology Upheavals Which Enabled Shannon’sTheories in the Real World

The first of these upheavals began with the announcement from Bell Labs of the solid-state transistor in 1948, ironically the same year that Bell Labs published Shannon’s The Mathematical Theory of Communication. Three Bell Labs researchers led by William Shockley won the 1956 Nobel Prize in physics for their work. The transistor was a remarkable achievement which signaled the end of the cumbersome, power-hungry vacuum tubes which powered electrical engineering since their introduction in 1904 by Lee de Forest. By1955, the ultimate promise of the tiny and energy-efficient transistor came into full view.

The second major technology upheaval began in 1958/59 when the integrated circuit was introduced by Jack Kilby of Texas Instruments and, independently, by a team under Robert Noyce at newly founded Fairchild Semiconductor, right here in adjacent Mountain View – part of today’s Silicon Valley. The Fairchild planar process of semiconductor fabrication signaled the unprecedented progress which quickly powered the computer revolution. Today, we have millions of microscopic transistors fabricated on one small silicon “chip” less than one inch square. The versatile transistor can act as an amplifier of analog signals and/or a very effective high-speed and reliable binary switch.

These two parallel revolutions complete the trilogy of events begun by Shannon which determined our path to this present age of mass computation and communication.

A Final Summation

My goal was to make you, the reader, cognizant of Claude Shannon and his impact on our world, a world often taken for granted by many who daily benefit immensely from his legacy. We have come a very long way from the worlds of the telegraph – Morse and Vail, and the telephone – Alexander Graham Bell, and radio – Marconi, and Armstrong. The mathematical theories and characterizations proposed by Claude Shannon have essentially all been proven sound; his conclusions regarding the mathematical theory of communication are amazingly applicable to all modes of communication – from the simple telegraph, to radio, to our vast cellular networks, and to deep-space satellite communication.

I respectfully suggest you keep a few things in mind, going forward:

-Your computer is what it is and does what is does in no small part thanks to Claude Shannon’s insightful genius.

-Your cell phone can connect you anywhere in the world thanks largely to Claude Shannon.

-The abiliity to store a two-hour movie in high-definition and full, living color on a digital compact disc called a DVD is directly due to Claude Shannon.

-The error-correction capability digitally encoded on CD’s and DVD’s which insure playback with no detectable effects even from a badly scratched disc is absolutely the result of Claude Shannon’s ground-breaking work on error-correcting digital codes.

-Your ability to encrypt the data on your computer hard drive so that it is impenetrable to anyone (even experts) who do not possess the decoding key is, yet again, a direct result of Claude Shannon’s cryptography efforts.

And, finally, we arrive at the most surprising fact of them all: how is it that virtually 90 per-cent of the world’s population has benefitted so immensely from the legacy of Claude Shannon, yet so few have even heard of him? Perhaps there are some lessons, here?

Kudos to Claude Shannon and all the other visionaries who made it happen.

Isaac Newton and the Plague of 1665/66: Perhaps the Greatest Year in Science!

Today, we have the Covid-19 virus pandemic which threatens America – indeed the entire globe. Many of us are just now emerging from weeks of “sheltering-in-place” while avoiding the virus and its risks. Virtually overnight, we found ourselves confined to home with copious spare time on our hands, time to do all those “other things” which prove to be so elusive in normal times. Many are the voices which have expressed this as a surprise blessing! Indeed, what have you been able to accomplish using this unexpected windfall of extra time at home?

Woolsthorpe Manor, Lincolnshire

The all-time poster-child for shelter-in-place achievers happens also to be the greatest scientist who ever lived, Isaac Newton (yes, even greater than Albert Einstein who holds second position – in my humble estimation!).

As a young, unknown student, Newton had just completed his undergraduate work at Cambridge University in the year 1665 when the fearsome bubonic plague, the “black death” as it was called, swept through London and regions of England. Armed only with the most rudimentary medical knowledge, Londoners and folks in the countryside resorted to the only option available to them: sheltering-in-place to avoid exposure. Sounds familiar, does it not?

 In 1665, despite centuries of recorded plagues and millions of deaths, the origin and transmission of such deadly pandemics were to remain unknown for a surprisingly long time. It was not until 1894 that Alexandre Yersin identified the bacterium responsible for such a horrible affliction. In 1898, Jean-Paul Simond revealed that the bacterium was spread through flea bites. Rodents were identified as the principal hosts and transmission vehicle for these fleas. Although largely treatable and well-controlled, today, “the black death” surprisingly still stalks the earth and its human populations!

The year 1666 is known as Newton’s “annus mirabilis,” the “miracle year” in science due to thought processes and experiments that took place in a tiny manor house in Woolsthorpe, Lincolnshire, near Cambridge. It was there, in his mother’s rustic farm-house, the place where he was born, that young Newton secluded himself from the plague for more than a year of intense contemplation, investigation, and writing.

At Woolsthorpe, Newton formulated three fundamental cornerstones of science and mathematics: first, the foundation of modern calculus, known then as Newton’s theory of fluxions; second, experiments with prisms and light which led to his second masterwork book in 1704, the Optics; and finally, his thoughts on the strange nature of gravitational attraction which led to his ultimate masterwork of 1687, Philosophie Naturalis Principia Mathematica which translates from Latin as: Mathematical Principles of Natural Philosophy.

The Principia is universally regarded as the greatest scientific book ever published, being the product of perhaps the most fertile mind in the recorded history of mankind. In the book, Newton combined his prodigious knowledge of Euclidean geometry with fledgling elements of his new calculus to describe mathematically, for the first time, no less than the motion of the planets through the heavens. Also revealed are Newton’s three laws of motion, the basis of modern physics/mechanics, and his notion of universal gravitational attraction.

Newton’s prodigious output during that year-plus of sheltering-in-place at Woolsthorpe is legendary because his investigative conclusions at that time led directly to his later, refined publications and their great advancement of scientific knowledge and method.

In stark contrast to Newton, this writer will be happy to further organize his den, write a few blog posts (such as this one), and clean-out the garage over the next several months. Oh…and I hope to give myself a much-needed haircut, soon! Like Newton, we can all strive, in our own way, to make the best of a terrible situation.

Hermann Minkowski, Albert Einstein and Four-dimensional Space-time

Is the concept of free-will valid as it relates to humans? A mathematics lecture presented in September of 1908 in Cologne, Germany by Hermann Minkowski not only paved the way for the successful formulation of Albert Einstein’s general theory of relativity in 1916, it also forced us to completely revamp our intuitions regarding the notion of time and space while calling into question the concept of human free-will! Some brief and simplified background is in order.

Prior to Minkowski’s famous lecture concerning Raum Und Zeit (Space and Time), the fabric of our universe was characterized by three-dimensional space accompanied by the inexorable forward flow of time. The concept of time has long been a stubbornly elusive notion, both in philosophy and in physics. From the mid-nineteenth century onward, there had increasingly been problems with our conception of “time.” The difficulties surfaced with the work of James Clerk Maxwell and his mathematical characterization of electromagnetic waves (which include radio waves and even light) and their propagation through space. Maxwell revealed his milestone “Maxwell’s equations” to the world in 1865. His equations have stood the test of time and remain the technical basis for today’s vast communication networks. But there was a significant problem stemming from Maxwell’s work, and that was his prediction that the speed of light propagation (and that of all electromagnetic waves) is constant for all observers in the universe. Logically, that prediction appeared to be implausible when carefully examined. In fact, notice of that implausibility stirred a major crisis in physics during the final decades of the nineteenth century. Einstein, Poincare, Lorentz and many other eminent physicists and mathematicians devoted much of their time and attention to the seeming impasse during those years.

Enter Einstein’s special theory of relativity in 1906

In order to resolve the dilemma posed by Maxwell’s assertion of a constant propagation speed for light and all related electromagnetic phenomena, Albert Einstein formulated his special theory of relativity which he published in 1906. Special relativity resolved the impasse created by Maxwell by introducing one of the great upheavals in the history of science. Einstein posited three key stipulations for the new physics:

A new law of physics: The speed of light is constant as determined by all “observers” in the universe, no matter what their relative motion may be with respect to a light source. This, in concert with the theoretically-based dictate from Maxwell that the speed of light is constant for all observers. Einstein decreed this as a new fundamental law of physics. In order for this new law to reign supreme in physics, two radical concessions regarding space and time proved necessary.
Concession #1: There exists no absolute measure of position and distance in the universe. Stated another way, there exists no reference point in space and no absolute framework for determining distance coordinates. One result of this: consider two observers, each with his own yardstick, whose platforms (habitats, or “frames of reference,” as it were) are moving relatively to one another. At rest with respect to one another, each observer sees the other’s yardstick as identical in length to their own. As the relative velocity (speed) between the two observers and their platforms increases and approaches the constant speed of light (roughly 186,000 miles per second), the other observer’s “yardstick” will increasingly appear shorter to each observer, even though, when at relative rest, the two yardsticks appear identical in length.
Concession #2: There is no absolute time-keeper in the universe. The passage of time depends on one observer’s velocity with respect to another observer. One result of this: consider our same two observers, each with their own identical clocks. At rest with respect to one another, each observer sees the other’s clock as keeping perfect time with their own. As the relative velocity (speed) between the two observers and their platforms increases and approaches the constant speed of light, the other observer’s clock appears increasingly to slow down relative to their own clock which ticks merrily along at its constant rate.

Needless to say, the appearance in 1906 of Einstein’s paper on special relativity overturned many long-held assumptions regarding time and space. Einstein dissolved Isaac Newton’s assumptions of absolute space and absolute time.The new relativity physics of Einstein introduced a universe of shrinking yardsticks and slowing clocks. It took several years for Einstein’s new theory to gain acceptance. Even with all these upheavals, the resulting relativistic physics maintained the notion of (newly-relative) spatial frames defined by traditional coordinates in three mutually perpendicular directions: forward/backward, left/right, and up/down.

Also still remaining was the notion of time as a (newly-relative) measure which still flows inexorably forward in a continuous manner. As a result of the special theory, relativistic “correction factors” were required for space and time for observers and their frames of reference experiencing significant relative, velocities.

This framework of mathematical physics worked splendidly for platforms or “frames of reference” (and their resident observers) experiencing uniform relative motion (constant velocity) with respect to each other.

The added complications to the picture which result from including accelerated relative motions (the effect of gravity included) complicated Einstein’s task enormously and set the great man on the quest for a general theory of relativity which could also accommodate accelerated motion and gravity.

Einstein labored mightily on this new quest for almost ten years. By 1913, he had approached the central ideas necessary for general relativity, but the difficulties inherent in elegantly completing the task were seriously beginning to affect his health. In fact, the exertion nearly killed Einstein. The mathematics necessary for success was staggering, involving a complex “tensor calculus” which Einstein was insufficiently prepared to deal with. In desperation, he called his old friend from university days, Marcel Grossman, for help. Grossman was a mathematics major at the Zurich Polytechnic, and it was his set of class notes that saved the day for young Einstein on the frequent occasions when Einstein forsook mathematics lectures in favor of physics discussions at the local coffee houses. Grossman’s later assistance with the requisite mathematics provided a key turning point for Einstein’s general theory of relativity.

Enter Hermann Minkowski with Raum Und Zeit

The initial 1909 publication of Raum Und Zeit

On September 8, 1908 in Cologne, Germany, the rising mathematics star, Hermann Minkowski, gave a symposium lecture which provided the elusive concepts and mathematics needed by Einstein to elegantly complete his general theory of relativity. Similar to Einstein’s 1906 special theory of relativity, the essence of Minkowski’s contribution involved yet another radical proposal regarding space and time. Minkowski took the notion of continuously flowing time and melded it together with the three-dimensional coordinates defining space to create a new continuum: four-dimensional space-time which relegated the time parameter to a fourth coordinate point in his newly proposed four-dimensional space-time.

Now, just as three coordinate points in space specify precisely one’s physical location, the four-dimensional space-time continuum is an infinite collection of all combinations of place and time expressed in four coordinates. Every personal memory we have of a specific place and time – each event-instant in our lives – is defined by a “point” in four-dimensional space-time. We can say we were present, in times past, at a particular event-instant because we “traversed-through” or “experienced” a specific four-dimensional coordinate point in space-time which characterizes that particular event-instant. That is very different from saying we were positioned in a specific three-dimensional location at a specific instant of time which flows irresistibly only forward.

What do Minkowski’s mathematics imply about human free-will?

By implication, the continuum of four-dimensional space-time includes not only sets of four coordinate points representing specific events in our past (place and “time”), the continuum must include points specifying the place and “time” for all future events. This subtly suggests a pre-determined universe, where places and “times” are already on record for each of us, and this implies the absence of free-will, the ability to make conscious decisions such as where we will be and when in the future. This is a very controversial aspect of Minkowki’s four-dimensional space-time with distinctly philosophical arguments.

For certain, however, is the great success Minkowski’s mathematics of space-time has enjoyed as a basis for Einstein’s general theory of relativity. Most, if not all, aspects of Einstein’s special and general theories of relativity have been subjected to extensive experimental verification over many decades. There is no instance of any validly conducted experiment ever registering disagreement with Einstein’s special or general theories. That is good news for Hermann Minkowski, as well.

Minkowski’s new reality takes us beyond the two-dimensional world of a flat piece of paper, through the recent universe of three-dimensional space plus time, and into the brave new world of not only four-dimensional space-time, but curved four-dimensional space-time. The nature of curved space-time serves to replace the Newtonian notion of a gravitational force of attraction which enables the celestial ballet of the heavens. For instance, the orbit of earth around the sun is now regarded as the “natural path” of the earth through the curvature of four-dimensional space-time and not due to any force of attraction the sun exerts on the earth. According to the general theory of relativity, the mass of the sun imposes a curvature on the four-dimensional space-time around it, and it is that curvature which determines the natural path of the earth around the sun. Minkowski and his mathematics provided the final, crucial insight Einstein needed to not only radically redefine the nature of gravity, but to also successfully complete his general theory of relativity in 1916. Einstein’s theory and its revelations are generally regarded as the most significant and sublime product ever to emanate from the human intellect. Take a bow, Albert and Hermann.

My eulogy to Hermann Minkowski

Albert Einstein is assuredly the most recognized individual in human history – both the name and the image, and that is very understandable and appropriate. Very few in the public realm not involved with mathematics and physics have ever even heard the name, “Hermann Minkowski,” and that is a shame, for he was a full participant in Einstein’s milestone achievement, general relativity. Minkowski’s initial 1907 work on Raum Und Zeit came to Einstein’s attention early-on, but its mathematics were well beyond Einstein’s comprehension in that earlier time frame. It was not until several years later, that Einstein and Marcel Grossman began to recognize Minkowski’s gift to general relativity in the form of his mathematics of four-dimensional curved space-time.

Hermann Minkowski delivered his by-then polished lecture on space-time at Cologne, Germany, in September, 1908. Tragically, he died suddenly in January, 1909, at the young age of forty-four – from a ruptured appendix. His latest findings as presented in the Cologne lecture were published in January, 1909, days after his death, sadly.

The “lazy dog” has the last bark

Albert Einstein and Hermann Minkowski first crossed paths during Einstein’s student days at the Zurich Polytechnic, where Minkowski was teaching mathematics to young Einstein. Noting Einstein’s afore-mentioned irregular attendance at lectures in mathematics, the professor reportedly labeled the student Einstein as, “a lazy dog.” Rarely in the annals of human history has such an unpromising prospect turned out so well! I noted with great interest while researching this post that Einstein long regarded mathematics as merely a necessary tool for the advancement of physics, whereas Minkowski and other fine mathematicians of the past tended to consider mathematics as a prime mover in the acquisition and advancement of knowledge, both theoretical and practical; they viewed physics as the fortunate beneficiary of insights that mathematics revealed.

In the late years, Einstein came to appreciate the supremely important role that mathematics plays in the general advancement of science. As proof, I will only add that the great physicist realized his dependence on the mathematicians Grossman and Minkowski in the nick of time to prevent his theory of general relativity from going off the rails, ending on the scrap heap, and leaving Albert Einstein a completely spent physicist.

Note: For a detailed tour and layperson’s explanation of Einstein’s relativity theories, click on the image of my book: The Elusive Notion of Motion – The Genius of Kepler, Galileo, Newton, and Einstein – available on Amazon