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