No certainty exists
in search for truth

In a classic "Star Trek" episode, the crew of the Enterprise is able to escape imprisonment by a group of androids by confusing them with the seemingly simple statement, "I am lying." The android overheats while trying to ponder the oscillating contradictions of the statement.

The episode refers to an ancient paradox dating back to Epimenides in the sixth century B.C. He, being a Cretan himself, stated that "all Cretans are liars."

Three hundred years later, Socrates added another variation when he stated, "One thing I know is that I know nothing."

These and other similar paradoxes led Aristotle to consider the question, What is truth? to which he proposed a simple non-answer, "A declarative sentence is true if and only if what it says is so."

It turns out that truth is difficult to define and impossible to prove, even in the simplest and most logical cases.

Although it seems obvious that 1+1=2, to Bertrand Russell and Alfred North Whitehead, "seemingly obvious" was not enough. The system of mathematics that we rely upon for everything from simple accounting to esoteric scientific theories has been assumed to be a "true" system, and they wanted to prove it using formal logic beginning with fundamental definitions.

In 1913, Russell and Whitehead published "Principia Mathematica," a three-volume set considered one of the intellectual landmarks of the century that began from first principles and developed the laws of arithmetic (proving on Page 362 of Volume 1 that 1+1=2), but failing in the end to prove the internal consistency of mathematical logic and its ability to determine the truth or falsity of a given statement. The project drove Russell to the outer bounds of sanity.

The problem that plagued Russell began with the conceptualization of set theory by Georg Cantor in 1879, the idea that objects with similar characteristics could be grouped in sets on which logical operations could be performed, which included sets of numbers.

In 1901, Russell discovered a paradox in set theory that is related to the liars paradox. Known today as the "Russell paradox," it concerns the contradiction when considering the set containing all sets that are not members of themselves.

Consider such a set and call it "R." Then Russell's paradox would state, "If R is a member of itself, then by definition it must not be a member of itself. Similarly, if R is not a member of itself, then by definition it must be a member of itself."

The set "R" is then not a member of itself only if it also is a member of itself, a logical contradiction of the same type as the liars paradox.

That a contradiction of this severity would raise its head in a field such as mathematics, which is based on precise logic, troubled Russell because set theory underlies all branches of math.

Similarly, other inconsistencies that had been discovered in set theory caused philosophers and mathematicians alike to worry: If set theory was inconsistent, then no mathematical proof at all could be trusted, thereby creating doubt about the consistency of mathematics as a whole.

Russell and Whitehead's "Principia" is an important work in the field of epistemology (the philosophy of knowledge) and the most influential book on logic since Aristotle's "Organon," but it failed to accomplish the reduction of all mathematics to logic simply because, as it turned out, it was an impossible task.

Russell recognized that self-reference lies at the heart of the paradox, so he proposed to avoid the paradox by arranging all statements into a hierarchy of types. The lowest level consists of statements about objects, the next statements about sets of objects, then about sets of sets of objects, and so on. Then, statements apply only if they all refer to the same level of types.

This reduced the paradox to a problem of language rather than one of mathematics and admits that logic can neither prove the statement to be true nor show it to be false.

The same problem arises when trying to logically prove the truth of any statement, scientific, mathematical or otherwise. This was proved for scientific statements by Alfred Tarski in 1931 when he showed quite simply that it amounts to saying that there is no complete language of science.

Tarski's theorem basically says that as soon as you make a statement and add that the statement "is true," you are bound to generate contradictions such as the ones that arise in Epimenides' or Russell's paradoxes.

In the same year, Kurt Godel demonstrated in his famous "Incompleteness Theorem" that there will always be some propositions that cannot be proved to be either true or false using the rules and axioms within any given system of logic.

Although you could go outside a system and come up with new rules and axioms to prove every conceivable statement within the system, by doing so you only create a larger system with its own un-provable statements.

The implication is that all logical systems, regardless of complexity, are incomplete; each system contains more statements than can possibly be proved to be either true or false by using its own defining set of rules.

Godel's theorem appears to forever doom hope of mathematical certainty through use of the methods of axioms and logic. Perhaps doomed also, as a result, is the often stated ideal of science to devise a set of physical laws from which all phenomena of the physical world can be deduced.

It might also mean that we are forever incapable of understanding ourselves.

As Douglas Hofstadter said in his 1979 Pulitzer Prize-winning book "Godel, Escher, Bach," "Godel showed that provability is a weaker notion than truth, no matter what axiomatic system is involved."

This is why we say today that there are no "proofs" in science. The best we can do is to verify statements that we believe because of the observed truth of their consequences. Thus, the strengths of the system of science are in verifiability and disproof rather than in proof.