“*Either mathematics is too big for the human mind or the human mind is more than a machine*.”

“*Science without epistemology is – insofar as it is thinkable at all – primitive and muddled*.”

“*Consciousness is connected with one unity. A machine is composed of parts*. *The brain is a computing machine connected with a spiri*t.”

“*Materialism is false*.”

Kurt Gödel

At the end of the nineteenth century, scientists gathered at the Royal Academy in London and agreed that almost everything had been discovered, and that the era of scientific enquiry would soon end.

Five years later, Albert Einstein published his theory of general relativity, throwing every notion of space and time into question. One of the few remaining areas of certainty was mathematics.

In 1931, Kurt Gödel (1906-1978) demonstrated that no logical system can capture all the truths of mathematics; nor can any logical system for mathematics, from its own axioms, be shown to be free from inconsistency.

Later, in America, Gödel and Einstein became close friends. With their respective theories of relativity and incompleteness, the two had helped subvert scientific certainty. They also become estranged from the scientific mainstream, for Einstein never accepted the quantum theory of Niels Bohr and Werner Heisenberg, and Gödel believed that mathematical abstractions were every bit as real as tables and chairs. Both also commented on metaphysics – including God, consciousness and eternity – though Gödel was more private about them than Einstein. The quotes on the left were published posthumously.

In the decades that followed, notions of relativity and uncertainty – both scientific and popular – transformed modern consciousness, facilitating the widespread abandonment of organized religion and changing the way people saw the universe, space, time and life.

By the nineteen sixties, Western post-secondary education was shifting its focus from the pursuit of knowledge to specialized job training. In search of existential answers to a soulless world, thousands of students ‘dropped out’ and trekked to Asia to explore ancient ways of wisdom. They brought back a good deal.

Of all the Eastern thought that was to take root in the West, Buddhism was most prominent. Its central tenets of *sunyata* (emptiness), and Buddhism’s juxtaposition of relative and conventional truth proved both amenable and soothing to young Westerners raised on notions of relativity and undecidability. Asian teachers, who warned them of the great difficulty of these ideas, were as surprised by the facility with which they grasped them as they were troubled by their unruly minds and intense appetites.

Such was the inner predicament faced by Stephen Schettini as he set out on his quest to find meaning. Read his story in The Novice: *Why I Became a Buddhist Monk, Why I Quit and What I Learned*.

*Kurt Gödel, a leading voice in the generation that
coaxed
science from exactitude to uncertainty*

In 1931 the mathematician and logician Kurt Gödel proved that within a formal system questions exist that are neither provable nor disprovable on the basis of the axioms that define the system. This is known as Gödel's Undecidability Theorem. He also showed that in a sufficiently rich formal system in which decidability of all questions is required, there will be contradictory statements. This is his Incompleteness Theorem.

In establishing these theorems Gödel showed that there are problems that cannot be solved by any set of rules or procedures; instead for these problems one must always extend the set of axioms. This disproved a common belief at the time that the different branches of mathematics could be integrated and placed on a single logical foundation.

Alan Turing later provided a constructive interpretation of Gödel's results by placing them on an algorithmic foundation: Some numbers and functions that cannot be computed by any logical machine.

More recently, Gregory Chaitin, a mathematician working at IBM, has stressed that Gödel's and Turing's results set fundamental limits on mathematics.

These results, along with quantum uncertainty and the unpredictability of deterministic (chaotic) systems, form a core set of limitations to scientific knowledge that have only come to be appreciated in recent years.

© The Exploratorium,

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