Is it possible for humans to tell the truth always; to never lie? Psychologists say it is not possible; most reasonably informed people agree. It’s a trait that can distinguish humans from some forms of artificial intelligence, which engineers at Google and other companies are working furiously to bring on-line.
People’s ideas—their belief systems—are inconsistent, incomplete, and almost always driven by logically-unreliable, emotionally-laden content, which is grounded in their particular life experiences and even trauma. Does anyone disagree?
Cognitive dissonance is the term psychologists use to describe the painful condition of the mind that results when people are unable to achieve consistency and completeness in their thinking. Every person suffers from it, to one degree or another. An unhealthy avoidance of cognitive dissonance can drive people into rigid patterns of thought. Political and religious extremists are examples of people who probably have a low tolerance for it.
Decades ago, mathematicians like Kurt Gödel proved that any math-based logic-system that is consistent can never be complete; it always contains truthful assertions—including but not limited to foundational truths, called axioms—which are impossible to prove.
Whenever an idea or a conjecture that seems to be self-evident can’t be proved, it seems reasonable, at least to me, to assume that some people might feel compelled to disbelieve it; they might even believe they are trapped in what could turn out to be a lie, because no one should be expected to embrace a set of unprovable truths, right? Axioms that can’t be proved are nothing more than assertions, right? Certainly any theorems built-up from unprovable assertions (axioms) must carry some inherent risk of falsifiability, shouldn’t they?
Someone unable to convince themselves that an assertion or axiom they believe is true actually is true might necessarily feel uncomfortable; even incomplete. Folks often teach themselves to not examine too closely those things they believe to be true, which they can’t prove. It helps them avoid cognitive dissonance.
I’m not referring to science by the way. No easy way exists for non-technical folks to confirm claims by scientists that the earth is round, for example. The earth looks flat to most people, but scientists who have the right tools and techniques can reach beyond the grasp of non-scientists to prove to themselves that the earth is round.
Reasonable people agree that the truth of science is discoverable to any group of humans who have the resources and training to explore it. Most agree that those who are scientifically well-qualified are fully capable of passing the torch of scientific truth to the rest of humankind.
But this essay isn’t really about science; it’s about truth itself—a concept far more mysterious and elusive than any particular assertion a scientist might make that the Earth is not the center of the Universe, or that the Moon is not made of cheese.
All logically consistent ways of reasoning that we know about have been invented—some say, discovered—by human beings who live on planet Earth. Humans can and often have argued that the unprovable assertions which form the basis of any consistent way of thinking are an Achilles heel, which can be attacked to bring down whatever logical structure has been erected.
But it isn’t only the few foundational axioms of any mathematically logical system that are by definition true, but unprovable. Very complicated conjectures about the nature of numbers, for example, are always being discovered, which everyone believes they know to be true, but will never be proved, because they can’t be.
Freeman Dyson—one of the longest-lived and most influential physicists and mathematicians of all time—argues that it is impossible to find a whole (or exact) number that is a power of two where someone can reverse its digits to create a whole number that becomes a power of five.
In other words , right? Get out the calculator, those who don’t believe it. Reversing the digits to make does not result in an exact number that is a power of five. In this particular case, plus a lot more decimals. No matter how many decimal places anyone rounds-off to, the number raised to the power of five won’t return exactly. A calculator will confirm it.
Dyson asserts that no number which is a power of two can ever be manipulated in this way to yield an exact number that is a power of five, no matter how large or unlikely the number might be. Freeman Dyson and all other super-intelligent beings—perhaps aliens living in faraway galaxies—will never be able to prove this conjecture, even though they all know for certain inside their own logical brains that this particular statement must be true.
All logically consistent methods of reasoning which can be modeled by simple (or not so simple) mathematics have these Achilles heels. Gödel proved this truth beyond all doubt; he proved it using a method he invented, which allowed him to circumvent the dilemmas posed by the unprovable truths of the system of thinking he contrived to demonstrate his discoveries.
I’m not going to get into the details of Gödel’s Incompleteness Theorems; books have been written about them; most people don’t have the temperament to wade through the structures he built to make his point. He basically assigned simple numbers to logical statements—some being very complex statements encoded by very long strings of numbers—so that he could perform garantuan operations of logic using rules of simple arithmetic on ordinary whole numbers. Take my word, his method requires traveling over unfamiliar mathematical roads; it takes getting used to.
But it should amaze non-mathematicians that truths abound in mathematics, which not only haven’t been proved; they never will be proved, because no proof is possible. A logical path to the truth of these statements does not exist; indeed, it cannot exist. But it is useful and necessary to believe or at least accept these statements to make progress in mathematics.
The late mathematician Paul Cohen—at one time a friend to Gödel—said that Gödel once told him he wondered if it might be true that any and all conjectures in mathematics could be solved if only the right set of axioms could be collected to construct the proofs. Cohen is best known, perhaps, for showing that indeed—in the case of the Continuum Hypothesis, at least—he could collect two reasonable, self-evident, and distinct sets of axioms that led to logically consistent and useful proofs. One small problem, though—the proofs were completely contradictory. One proved the conjecture was true; the other proved it was false.
His result is often explained this way: the consistency of any system of mathematical reasoning cannot be proved by its foundational axioms alone. If it can, the system must necessarily be incomplete; its conjectures—many of them—undecidable.
Cohen showed that a consistent and sound axiomatization of all statements about natural numbers is unachievable. Many such statements, in his view could be true, but not provable. Cohen introduced the concept that all systems of logic built on numbers have embedded within them some combination of ambiguity, un-decidability, inconsistency, and incompleteness.
People who want their thinking to be consistent must believe things that cannot be proved. But believing logical statements that can’t be proved always renders their thinking incomplete, even when it might be flawlessly consistent. What they believe to be true depends fundamentally on what they believe to be self-evident: it depends on statements no one can prove: on axioms, and a little bit more.
For those who decide to believe and accept only statements that can be proved, their thinking will necessarily unravel to become inconsistent or incomplete; most likely both. Their assertions become undecidable. It can’t be any other way, according to Gödel, whose proof has withstood the test of eighty years of intense scrutiny by the smartest people who have ever lived.
Paul Cohen jumped onto the dilemma-pile by showing that the incompleteness made necessary by a particular choice of axioms can turn a logically-consistent proof to rubble when a mathematician tampers-with or swaps-out the foundational axioms. A sufficiently clever mathematician can prove that black is white; and vice-versa.
Some might be tempted to say that Gödel‘s Incompleteness Theorems apply only to formal, math-based logic-structures—not the minds of human beings, because minds and the way they operate when analyzed are always found to be inconsistent and incomplete. But such talk makes the point. Think about it.
So again, we ask: what is truth? How do folks determine that a particular statement is true, if it happens to be one of those assertions that lies beyond the reach of logic, which no one—no matter how smart—will ever be able to prove?
What good do collections of so-called self-evident axioms serve, if different collections can lead to contradictions in theorems?
Most important: how do we avoid believing lies?
Post Script: This post is one of those living-essays Billy Lee writes that is likely to be modified and added to as he thinks about the subject and learns more. Check back for updates from time to time.
[added April 3, 2016] I found a 2013 essay by Derek Abbott, the Australian Electrical Engineer and Physicist, who argued that our mathematics is invented, not discovered; anthropological, not universal. It enables us to simplify truth to enable our limited minds to find ways to do and understand simple things. Click this link for a good read.
I don’t know if Derek’s view is correct, but offer it as fodder for readers who are interested in why Truth and mathematics seem connected somehow in the minds of so many thinkers like Plato, for example; and why these thinkers could be dead wrong, at least in Derek Abbott’s opinion. He offers Clifford’s Geometric Algebra as a practical example of a useful system of mathematics used by robotics engineers, which is built very differently than the mathematics most people use.
[added February 20, 2017] I will offer this opinion: if mathematics is anthropological; if math is just another way our minds work and not the golden key to a deeper reality beyond ourselves, then it can tell us nothing new about the mysteries of existence; we are not going to be able to calculate our way along a path to truth; we will have to do the hard physical experiments to figure out just what the heck is going on.
Based on what the smartest humans know to be true today, we can’t build the kind of instruments required to answer the mysteries of the very large and the very small. Getting answers will take detectors the size of galaxies; it will demand the energy supply of thousands of suns.
If mathematics lacks a symbiotic connection to realty; if God is not a mathematician, if He doesn’t play dice after all, as Einstein insisted, well, we won’t get to a deeper understanding of how the universe works—or why it exists—through clever use of mathematics. It just isn’t going to happen—not now; not anytime soon; not ever.
The 18th century German playwright and philosopher, Friedrich Schiller, wrote …truth lies in the abyss.
Pray that he’s wrong.