“Beware of bugs in the above code; I have only proved it correct, not tried it.” Donald Knuth

This quotation by the computer science master Donald Knuth was originally a footnote in a personal communication with another scientist, but has now reached near legendary status. The author is unrivaled in his attention to detail, perfection and thoroughness; and is widely regarded for his expositions on algorithm analysis and proofs. However, even he it seems, is bluntly aware that the transition from perfectly correct theory to actual computer code typed in by a person may easily undo all the fancy proofs that one could ever invent. Bugs will forever remain an intrinsic possiblity as long as we humans are involved in the process of software development.

“God exists since mathematics is consistent, and the devil exists since we cannot prove the consistency.” Morris Kline

This statement is in reference to the proof by mathematician Kurt Gödel that basic propositional calculi, including what we all know as general arithmetic, is not provably consistent. A logic system is inconsistent when all of the formulas in that system are true; or with a more intuitive defintion, it is inconsistent when some formula can be shown to be both true and false at the same time. Gödel's proof is remarkable because it is a formal proof about the impossibility of proving something else, in this case whether arithmetic is consistent or not. This is important because almost all fields of mathematics shares as its common core the propositional calculus. If we can never prove that such an incredibly simple system is consistent, then we will forever be cursed to never fully understand mathematics. This theorem has an equivalent unsettling effect on mathematics as the Heisenberg Uncertainty Principle has on physics, although the later is still just empirical conjecture, whereas Gödel's proof is absolute and without doubt.

“The infinite! No other question has ever moved so profoundly the spirit of man.” David Hilbert

The root problem that has confounded many mathematicians, philosophers, physicists, clergy, and even poets since the days of the early Greek philosophers has been the idea of infinity. Even the very existence of infinity is an ultimate brain twister; after all, if there is such a thing as infinity how could you ever prove it? We intuitively believe that straight lines go on to infinity, but this very question nearly toppled thousands of years of mathematical thought. Notwithstanding the question of existence, the very nature of infinity has an incredibly complex structure. Even today after the groundbreaking theories of Cantor and Russel dealing with transfinite calculus and the continuum, the concept of infinity is still one of the most puzzling concepts that the human brain can contemplate.

“Black holes are where God divided by zero.” Steven Wright

While not technically correct in an arithmetical sense, this short simple statement nonetheless is quite effective at demonstrating that the universe is more complicated than our human ability to understand. The hint is that only the Supreme Creator of the universe is allowed to break the rules by which the rest of us physical beings have to play; the rules being referenced here are the famous theories of General Relativity laid down by Einstein. Of course, what black holes really demonstrate is just the inadequacy of our own mathematical models of the universe that we have invented so far, not a fundamental paradox.

“Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.” Bertrand Russel

Pure mathematics is the study of thought at its most fundamental level. When studied to the extreme you must leave the physical world behind; even something as seemingly modern as quantum physics is horrifically bound to this physical universe and subject to emperical conjecture. Pure mathematics on the other hand does not limit itself to this universe, but rather expresses the most fundamental and invariant structures that must exist in any possible universe. So at its most refined essense pure mathematics must be devoid of all meaning and reality.

“Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin.” John von Neumann

Computer science pioneer and mathematician von Neumann wrote about the essence of programming and algorithms before the modern electrical digital computer was invented. He was perhaps the first to provide real insight into the true nature of randomness beyond its almost cavalier treatment by statisticians of his day. What is astounding about this quote is that although he wrote it before the era of the modern computer, the production of suitable random data still remains one of the most fundamental challenges of today's most sophisticated cryptographic and simulation software.

“Email is a wonderful thing for people whose role in life is to be on top of things. But not for me; my role is to be on the bottom of things.” Donald Knuth

Here is another interesting quote from computer scientist extrodinaire Donald Knuth. It concerns his retreat from intrusive electronic communications; disposing of all his email accounts in an effort to expend more concentration and time on his research. It's not exactly an earth shattering epiphany, but it does provide a somewhat unexpected glimpse into one the great minds of our time reassesing priorities in our technological age.

“The most exciting phrase to hear in science, the one that heralds the most discoveries, is not ‘Eureka!’, but ‘That's funny...’” Isacc Asimov

This lighthearted gem is a simple observation about the nature of human invention and discovery. Contrary to the popular view of scientists working hard toward a specific goal, many of the revolutionary discoveries have in fact been accidental or unforeseen side effects.

“For every problem, there is one solution which is simple, neat and wrong.” Henry Louis Mencken

Although I generally disagree with most of this author's philosophies, I enjoy this one statement because I have seen it played out far too often. I do agree that solutions should be as simple as possible, but those who thoughtlessly extol the ‘keep it simple stupid’ (KISS) princple often do so at the expense of correctness. I have little patience with such mindless thinking.

“I regret that it has been necessary for me in this lecture to administer a large dose of four-dimensional geometry. I do not apologise, because I am really not responsible for the fact that nature in its most fundamental aspect is four-dimensional. Things are what they are...” Alfred Whitehead

One of the great fathers of modern mathematics, Dr. Whitehead made this address to one of his student classes. It can be said that his goal in life was to simplify thought and philosophy through the rigor of mathematical logic; but is ironic that even he realized the limitations our human ability to simplify nature.