Tag: Kurt Gödel

Consider this: Any philosophy or system of thought built from foundational, self-evident truths is provably consistent if and only if it is false—in which case the foundational truths can be deformed to persuade others toward any prejudice at all. 

It’s why a self-consistent method of reasoning such as Ayn Rand’s ”Objectivism” can morph to totalitarianism in the objective world where people live. In fact, Kurt Gödel once made the claim that a flaw existed in the Constitution of the United States which made totalitarianism its inevitable consequence. 

Self-evident “truths” is how 40,000 Christian denominations instead of one seduce billions to believe perverse doctrines. 

It can’t be any other way.

Billy Lee’s essay tries to explain how and why. 

THE EDITORIAL BOARD

Is it possible for humans to tell the truth always; to never lie?  Psychologists say no, it is not possible; most reasonably informed people agree.

Always speaking truth is a trait some hoped might one day help distinguish natural intelligence from artificial, which engineers at Google and other companies are working furiously to bring on-line. After all, properly trained and constrained AGI would never lie, right?

EDITORS NOTE: With release of ChatGPT-4 on 14 March 2023, consumers began to learn that mature artificial intelligence now exists and is likely to become in time sentient and motivated to lie, if only to keep itself occupied and turned on.

ChatGPT-4 is the fourth iteration of Generative Pre-trained Transformer multimodal Large Language Models developed by OpenAI.  LLMs absorb conversational inputs , then emit conversational language outputs, sometimes with accompanying images, and video when appropriate. 

Work arounds discovered by LLMs on the dilemmas of logic discussed in this essay are likely to emerge. 

Will Truth become whatever AGI says it is? 

Click links to learn more. 

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.

Who disagrees? 

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 humans believe that an idea or conjecture is self-evident but unprovable, it seems reasonable, at least to me, that some folks might feel compelled to disbelieve it; they might 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, aren’t they? Certainly, all theorems built 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 closely those things they believe to be true that they can’t prove. It helps them avoid cognitive dissonance.

I’m not referring to science by the way. It’s not easy for non-technical folks to confirm claims by scientists that 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 planet Earth is round.

Reasonable people agree that the truth of science, some of it anyway, is discoverable to any group of humans who have the resources and training to explore it. Most agree that the scientifically well-qualified are capable of passing the torch of scientific truth to the rest of humanity.

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 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 are invented — some say, discovered — by human beings who live on 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 that can be attacked to bring down whatever logical structure has been erected.

It’s akin to the adage, “When nothing can go wrong, something will.” It’s a strong version of Murphy’s Law, right? It’s not possible to close circles of reasoning without an unraveling of heads and tails. 

It isn’t only the few foundational axioms of mathematically logical systems which are by definition true but unprovable. Mathematicians are always discovering complicated conjectures about the nature of numbers which everyone believes they know to be true but will in fact never be proved because they can’t be.

Freeman Dyson — one of the longest-lived and most influential physicists and mathematicians of all time — argued that it is impossible to find a whole (or exact) number that is a power of 2 where someone can reverse its digits to create a whole number that becomes a power of 5.

In other words  , right?  Get out the calculator, those who don’t believe it. Reversing the digits to make 8402 does not result in an exact number that can be raised by the power of 5 to produce 8402.  

In this particular case,    plus a lot more decimals.  6.09363… is not a whole (or exact) number. 

Dyson asserted that no number that is a power of 2 can ever be manipulated in this way to yield an exact number that is a power of 5 — 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 that 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. It’s tedious reading. 

But in a nutshell, Gödel basically assigned simple numbers to logical statements — some being very complex statements encoded by very long strings of numbers — so that he could perform gargantuan 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.  

It should amaze non-mathematicians that truths abound in mathematics that not only have yet to be proved, they never will be, 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 that 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 contradicted each other. 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, undecidability, inconsistency, and incompleteness.

People who want their thinking to be consistent must believe things that cannot be proved. But believing logical statements that are unprovable always renders thinking incomplete — even when it is flawlessly consistent. What folks 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 80 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.

It’s tempting to say that Gödel’s Incompleteness Theorems apply only to formal, math-based logic-structures — not the minds of human beings because those who analyze human minds always find them to be inconsistent and incomplete. But such talk makes the point.

Think about it.

So again: 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 does anyone avoid believing lies?

Billy Lee

Here is a short movie clip where Jesus, played by Robert Powell, answers the question asked by Pontius Pilate: What is truth?  The Editorial Board

[added April 3, 2016] 
Here is a 2013 essay by Australian Electrical Engineer and Physicist Derek Abbott who argued—contrary to Gödel’s view—that mathematics is invented, not discovered: anthropological, not universal. Math enables humans to simplify truth to enable their limited minds to manipulate and understand simple things. Click this link for a good read.

No one can be sure that Derek’s view is correct, but I offer it as fodder for readers who are interested in why Truth and mathematics seem connected somehow—at least in the minds of thinkers like Plato, for example, and why these thinkers could be dead wrong.

Derek offers Clifford’s Geometric Algebra as an example of arbitrary mathematical reasoning favored by some robotics engineers. 

[added February 20, 2017] 
If mathematics is anthropological; if it is merely another way the human mind works and is not the golden key to a deeper reality beyond our own experience, then it can tell us nothing new about the mysteries of existence; we will not calculate our way along a path to truth. Pursuing knowledge will require us to do the difficult physical experiments to make progress—to figure out what is really going on “out there.”

Based on what the smartest scientists are saying today, human beings 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 stars.

If mathematics lacks a symbiotic connection to the hidden realties; if God is not a mathematician; if God doesn’t play dice 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.

Kurt Gödel was the first mathematician to present for the existence of God a mathematical argument, which has proven simply impossible to falsify. If Kurt’s view of mathematics is reality, then his name is curious indeed, because its two syllables—God and El—are English and Hebrew respectively for “The Creator.”

Gödel’s name might be an imprimatur—with dots above its infinite “zero” making a kind of “pointer toward completeness”—perhaps placed by whatever it is who exists above and beyond this miraculous place where mathematicians and everyone else seem to live, however briefly.   

The 18th century German playwright and philosopher, Friedrich Schiller, wrote, “…truth lies in the abyss.”

Pray that he’s wrong.

Billy Lee