Limited Understanding

Understanding what we can never understand. Published by Penguin Books 1980. Cover illustration by Peter Brookes.

There is an argument that we will never be able to understand certain things.  Not because we aren’t clever enough, but because it’s built into the laws of the universe that we won’t ever be able to.

It’s supposedly a consequence of the incompleteness theorems developed by the mathematician Kurt Gödel in 1931.  He demonstrated the existence of formally undecidable elements within any formal system of arithmetic.  The basic point is, in essence, that mathematics contains statements that cannot be proved by mathematics.  And this put paid to the dreams of other mathematicians such as David Hilbert that a rigorous foundation on finite terms could be found for all mathematics.

Douglas Hofstadter discussed this complex and surprising subject in an entertaining and immensely helpful way in his Pulitzer Prize winning 1979 book Gödel, Escher, Bach: An Eternal Golden Braid.  Pulling together ideas arising from the music of JS Bach, the drawings of MC Escher and Gödel’s theorems, he develops a sort of self-referential loop which tells us much about human thought processes, which is indeed an aspect of the cognitive science that is Professor Hofstadter’s own field of specialism.  Amongst many other fascinating insights, he shows how assuming that certain systems are consistent actually forces one to conclude that they are incomplete.  That’s puzzling.

Professor Hofstadter’s book is described in the publisher’s tagline as being “in the spirit of Lewis Carroll”.  He applies one of Carroll’s logic puzzles to demonstrate how a deductive argument can get seemingly trapped in a process of infinite regression – one proposition depends on the truth of another, which depends on yet another, and so on forever.  So, perhaps, it can never be proved.

Charles Dodgson – widely known by his pen name Lewis Carroll as the author of the 1865 novel Alice’s Adventures in Wonderland – wrote extensively on mathematics during the late nineteenth century and was Mathematical Lecturer at Christ Church College Oxford for many years.   He loved a good paradox and created many puzzles which are just as entertaining today as they were then.  He, in his own way, was quite possibly putting forward thoughts on the nature of incompleteness.

In his 2007 book New Theories of Everything, John Barrow, Professor of Mathematical Sciences at Cambridge University, expresses the problem arising from Gödel’s theorems: “Superficially, it appears that all human investigations of the Universe must be limited.  Science is based on mathematics; mathematics cannot discover all truths; therefore science cannot discover all truths.”  Such a “fundamental barrier to human understanding” could, as he points out, be rather depressing.

But there’s another way of looking at this.  As Professor Barrow mentions, some scientists such as Freeman Dyson equally suggest that accepting the implications of the incompleteness theorems ensures “that science will go on forever”.  And that’s a noble aim in itself.  The endless quest for truth has been a constant feature of both fiction and research in real life throughout history.

In that seminal work for much modern science fiction, Frankenstein, published in 1818, Mary Shelley explored the use of knowledge for good and evil, and the limits of human understanding.  When considering all that had so far been achieved, “So much has been done, exclaimed the soul of Frankenstein – more, far more, will I achieve: treading in the steps already marked, I will pioneer a new way, explore unknown powers, and unfold to the world the deepest mysteries of creation.”

The human race might one day understand all there is to know, or maybe it will never achieve that position of total comprehension.  But we can try.  And the journey itself will be worth it.

Richard Hayes, Assistant Editor (Odyssey)

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