When the quintessential genius Sir Isaac Newton was praised for his brilliance in deriving Kepler’s mathematical laws, mathematizing gravitational force and inventing calculus, he expressed with remarkable insight the vast difference between the magnitudes of solved and unsolved problems:
I do not know what I may appear to the world; but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smother pebble or prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.
In this comment, Newton expressed his recognition that what we humans know and understand is only a minuscule fraction of what there is to know and understand. In fact, many of the “truths” in Nature might be beyond the range of human understanding. This recognition has increased in subsequent centuries as we probe more deeply into Nature’s secrets.
Many great mathematicians have acknowledged that the solution of certain problems are beyond their reach. Karl Friedrich Gauss, considered by many to be one of the greatest mathematical geniuses of all time, was once asked whether he could prove Fermat’s Last Theorem. He responded that he did want to invest the necessary time on what would be a probable failure. This theorem was ultimately proved in the late 1990’s by Sir Andrew Wiles who dedicated almost his entire lifetime to its solution.
In 1900, David Hilbert proposed an agenda for 20th-century mathematics, announcing 23 outstanding problems that mathematicians should investigate. In the years that followed, many of these problems were solved, but others remain unsolved by today’s greatest mathematicians. A century later, the Clay Mathematics Institute of Cambridge, Massachusetts (CMI) celebrated the Millennium by offering a reward of $1,000,000 for the solution of any of seven posted unsolved mathematical problems. In the quarter century following this offer, only one of these seven problems, the Poincaré Conjecture, was solved. The Russian mathematician Grisha Perelman, who solved this problem, refused to accept the money on the grounds that monetary rewards are a vulgar acknowledgment of mathematical achievement.
Einstein was more physicist than mathematician, and although he was very competent in mathematics, others were more mathematically proficient. When formulating his theories of relativity, he sought the help of Minkowski, Grossmann and Schwarzchild. Einstein’s insights were remarkable, yet he ultimately struggled with interpretations of quantum physics that challenged the concept of determinacy and entanglement.
Problems like the Riemann Hypothesis, P = NP equivalence and the Goldbach Conjecture (every even number greater than 2 is the sum of two primes) continue to haunt mathematicians, but undermining the quest for a solution to any of these problems is the fear that the problem might be an undecidable proposition, i.e., neither provable nor disprovable (in accordance with Gödel’s Theorems).
After Isaac Newton’s great mathematical conquests, the austere and dispassionate allure of mathematical reasoning was so seductive that the intellectual community of that era embraced the vision of mathematical conquest with the bold confidence and naïveté that Alexander Pope described in his long poem An Essay on Criticism:
Fired at first sight with what the Muse imparts,
In fearless youth we tempt the heights of Arts;
While from the bounded level of our mind
Short views we take, nor see the lengths behind…
We humans have made great inroads in unraveling some of Nature’s great secrets, but as we probe more deeply into Nature’s complexities, we are not only awestricken by what we do not yet understand, but are now wondering what aspects of Nature are within our intellectual grasp.
