The late prolific writer Isaac Azimov addressed many subjects in his books and essays. Among those subjects were Euclid’s axioms. In an essay called “Euclid’s Fifth,” Azimov partly addresses the concepts of truth and proof.* A point he makes is that Euclid’s five axioms and five postulates, particularly the Fifth Postulate, ultimately rely on some unquestioning faith and assumptions about assumptions. Euclid’s axioms stood virtually unchallenged for over a thousand years though Nasir Eddin al-Tus tried to rework the Fifth, and building on his work, Girolamo Saccheri also tried to rethink the postulate during the Renaissance. Eventually, all of Euclid’s foundations of geometry developed some cracks as new geometries arose in the nineteenth and twentieth centuries.
But for over 2,000 years and even in today’s high school geometry textbooks, Euclid’s axioms have been accepted as “incapable of contradiction” and, as Azimov argues, representative of “…absolute truth. They seem something a person could seize upon as soon as he had evolved the light of reason. Without ever sensing the universe in any way, but living only in the luminous darkness of his own mind, he would see that things equal to the same thing are equal to one another (one of the axioms) and all the rest [of the axioms]” (143). To further make his point about truth, proof, and assumption, Asimov uses Socrates’ discussion in Meno with an “uneducated” slave. Under the assumption that the slave can reach a valid conclusion because he is imbued with knowledge a priori, Socrates draws truths about a geometric figure from the slave’s mind, evidence, we should think that there lies within us some a priori truths that are self-evident, those truths in question here being Euclid’s axioms.
But are there truly self-evident truths available to any rational, even uneducated, mind? Are there any absolutes, such as Euclid’s axioms? And if there aren’t, are we required to ask Pilate’s question indefinitely, “What is truth?” Does that mean that in the dark recesses of each mind there is some sort of flashlight all of us can turn on to illuminate what we alone—in our loneliness—can see?
The term that catches my eye (and maybe yours, also), is Azimov’s oxymoronic “luminous darkness of…mind.” Aside from the relevance of the phrase to Azimov’s point, “luminous darkness of…mind” encapsulates a fault that lies in me and probably you. It’s easy for us to assume our “common sense” and our experiences are sufficient enough for us to know through reason the essence of truth—our truth that we assume is THE TRUTH. In our isolation, do we bring to light Socrates’ a priori knowledge? is there a priori knowledge that we can discover on our own?
That brings us back to what I wrote at the beginning of this little discussion, that we are ultimately isolated and that in our isolation, the dark mind self-illuminates and then determines what is meaningful, at least, what is meaningful to us.
So, then what do we do when someone like Euclid comes along and offers us a set of principles that appear so self-evident that for thousands of years no one can think of a valid challenge to them? Have all those mathematicians of the past two-plus millennia simply taken Euclid’s axioms on faith, and if they have, does that mean that even the brightest of us are locked by faith into axiomatic thinking? Does it also mean that in order to accept assumptions, we have to make further assumptions?
And given the penchant we have to simplify in light of our darkness, will we forever be closed off from discovering how our own truths might fall and others’ truths might rise? You might say this is all hogwash, some drivel from a prattler. But think of what has occurred in our halls of enlightenment, universities where speakers of recent have been subjected to censorship because the self-supposed enlightened do not wish to hear any challenge to their truths.
The dark truth is that in censoring free discussion and rational debate, a feeble light of “truth” will shine only internally. No challenges to axioms will illuminate the validity or invalidity of postulates that lie in the darkness of isolated minds.** And like Euclid's parallel lines that remain isolated forever, those with opposing views never meet on any point.
*Asimov, Isaac. "Euclid's Fifth," in The Left Hand of the Electron. New York. Dell Publishing Co., Inc., 1974. Pp. 140-153. By the way, don’t confuse “Euclid’s Fifth” with Beethoven’s (Da-da-da daaaaaaaaaa; da-da-da daaaaaaaa).
**No debate is, well, no debate.