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"Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This was the position I was placed in by a letter of Mr. Bertrand Russell, just when the printing of this volume was nearing its completion."
So begins the appendix of the 1903 Volume 2 of The Basic Laws of Arithmetic, part of Gottlob Frege’s epic work to ground arithmetic in the laws of logic.
Russell had communicated a paradox that implied a contradiction in Frege’s axiom system. This wasn’t the first time a paradox had undermined proposed foundations for mathematics – nor would it be the last.
Russell’s paradox is about sets that can or cannot belong to themselves. Here’s a variant: Divide adjectives into two complementary categories, (A) self-referential, and (B) non-self-referential. For example, long is not a long word and thus belongs to B, while verbal is verbal and belongs to A. Where would you place unpronounceable? That doesn’t quite give rise to a paradox, but trying to classify the adjective non-self-referential does!
Here’s another variant: In a certain town, the barber shaves all (and only) those who do not shave themselves. Who shaves the barber?
It’s not quite a resolution of the paradox, but you could identify the barber as the person who is perpetually raising a razor to his chin and then setting it down!
Lest you think such paradoxes are simply linguistic games, here are some other conflicting ideas:
• The green and blue in the headline image are the same colour (the neighbouring pink and orange stripes fool the eye into believing otherwise).
Various solutions to these paradoxes have been proposed and debated. Some believe the problem lies in the use of models which, by definition, are imprecise. Others believe the problems runs deeper and our very rules of logic and inference are flawed. Still others would say that all communication is relative and hence the problem is with absolute concepts such as “all” or “blue”. Some would like to isolate and contain problem elements, believing order could be maintained elsewhere.
We could chart these competing ideas but, starting in the late 1920s, Kurt Gödel and subsequent researchers established the futility of seeking meaningful mathematical systems that are devoid of paradoxes. Indeed, one could claim that the power of mathematics is intrinsically tied in with its paradoxes.
It is ironic that for a number of us it is the certainty of mathematics, that 1+1=2, which makes it attractive. On the point of 1+1=2, despite Russell’s undermining of Frege’s axiomatisation of arithmetic, it didn’t prevent Russell from attempting the same – although it took Russell and Whitehead over 360 pages of dense symbolism in their Principia Mathematica before they were even in a position to consider 1+1=2.
[Excerpt from page 379 of Volume I, 1st edition of Principia Mathematica by Whitehead and Russell]
One is not suggesting giving up exactness or accuracy. But to err is to be human. And mathematics is the art of interpreting, quantifying, and working with error and uncertainty.
Among mathematicians there are at least two tribes: those who seem not to dwell on internal inconsistencies and who are more interested in building new structures or discovering new truths, and those who would look more closely at the connections and contradictions that lie at the foundations of our mathematical methods. We need mathematicians who can be both, aiming to reach beyond existing horizons while still being reflective and critical about the models we use. We should not be paralysed by the paradoxes at the heart of our structures, but recognise that the paradoxes may be a source of power, and clarify and leverage them.
I would like to leave the final words to an insightful collection of essays, Discrete Thoughts: Essays on Mathematics, Science, and Philosophy by Mark Kac, Gian-Carlo Rota and Jacob Schwartz.
The giants of Victorian certainty had their feet mired in hypocrisy and wishful thinking, and the hatchet job of negative mathematicians had a liberating influence that is just beginning to be felt. “Suppose a contradiction were to be found in the axioms of set theory. Do you seriously believe that that bridge would fall down?” asked Frank Ramsey of Ludwig Wittgenstein. What if we were never able to certainly found set theory on a non-contradictory set of axioms? Maybe it is our Victorian ideas about the need for certainty of axiomatization that are naïve and unrealistic. Maybe the task ahead of us is to live rigorously with uncertainty, and bridges need not fall down in the attempt.
[Image acknowledgements: The THIS IS FALSE image has been constructed by me but I would like to thank Professor Akiyoshi KITAOKA as it is a derivative of his work www.psy.ritsumei.ac.jp/~akitaoka/color12e.html]