4 September 2012 by Francis Goode
Perhaps, like me, you sometimes find yourself uttering these words, especially during some heated debate to express just how obvious and self-evident our viewpoint is. To us, at least.
Logic occupies a special place in our lives, and not just when we argue. The core of rational thinking it provides the bricks and mortar from which the whole edifice of human understanding is built. More basic even than the natural number system, logic runs like a common thread linking together disciplines as diverse as philosophy, mathematics, engineering and IT.
The discipline of formal logic was established by Aristotle, but its history over the last hundred years provides a unique illustration of how potent ideas grow and spread from one subject area to another. And how ideas that initially appeared to be purely abstract notions can be developed into useful theories about the world, and then implemented in physical products that change the way we all live.
Around the beginning of the twentieth century many traditional notions about the world were crumbling. Darwin had blown away the old idea that we humans were given a special place to live in a clockwork universe, designed, built and wound up by a God, while Einstein blasted into (sub-)atoms our deepest certainties about time and space. Faced with such disruption to the old certainties, what more natural human reaction than to go back to basics and seek a way to reconstruct our knowledge on more solid footing? And what more solid ground than reason and logic?
And so the logicians set to. In mathematics, David Hilbert laid down a challenge to the whole community in the form of twenty-three unsolved problems whose solutions would provide a firm, logical foundation for their discipline. Meanwhile Bertrand Russell built upon Gottlob Frege's work in analytic philosophy, and his own Principles of Mathematics proposed to reconstruct the whole edifice of mathematics on top of this foundation. But these efforts relied on set theory, and even as he wrote, Russell spotted a tiny flaw in this construction that would, in time, bring the whole project crashing to the ground. For at the heart of set theory was a tiny, but fundamental, paradox: could a set be a member of itself?
But undaunted, they carried on with their project. Russell continued his work at Cambridge where, one day, in through his door walked a young Austrian aeronautical engineer who had an even more ambitious programme of logical reconstruction in mind. Perhaps being a member of the second richest family in Europe helped, but Ludwig Wittgensteins's supreme self-confidence in announcing himself to the leading mathematician and philosopher of his day was not misplaced. Russell soon recognised the young man's immense talent, and also the breathtaking ambition of his project. For Wittgenstein realised that using logical to construct a complete, coherent and consistent model of a system need not be confined to mathematics. He would apply it much more widely - to the whole of existence!
His vision was simple enough to fit into one slim volume, Tractatus Logico-Philosophicus (1921), but wide enough to take in every event and every feature of the known universe. All could be represented in symbolic logic; the relationships between them were subject to known laws (science and mathematics) which were expressed as logical functions; hence, the whole of being could be represented as, modelled and understood, using the logical manipulation of well chosen symbols. The book employed the "geometric form," starting with basic definitions and axioms and then using logic to build up the full set of propositions and their corollaries. (This is also the format that Spinoza for his own masterwork, The Ethics and the book's name echoes that of Spinoza's own Tractatus Theologico-Politicus.)
Wittgenstein's ideas inspired a philosophical movement known as the logical positivists, who grew out of that group of thinkers known as the Vienna Circle and, with the rise of Nazism, scattered around the world taking their ideas with them. There they galvanised a whole new generation of thinkers who realised their ideas need not be constrained to philosophical and mathematical abstractions; people such as Van Neumann and Turing who recognised that the symbols manipulated by the new class of machines known as computers could represent anything in the real world, not just numbers.
But even as the ideas began to seep into tangible products, the philosophers and mathematicians who spawned them were having a harder time. The flaw spotted by Russell had grown into a fissure before Gödel's incompleteness theorems brought the whole edifice crashing down. Effectively, Gödel proved that no system can be full defined within itself. Without an external reference point, all was circular logic and tautology.
Wittgenstein recanted his Tractatus and turned his attention to language. He realised that, like mathematics, language itself is a closed system, where words have no meaning outside the context they're used in. "The limits of my language means the limits of my world." Philosophical problems only arise when philosophers attempt to bestow upon them some deeper, fundamental meaning.
Now philosophers were in a spot. They had killed God in the certain hope of finding a coherent and complete explanation of the universe based on reason logic and logic. To lose one certainty may have been regarded as a misfortune; to lose two within the space of a century could look like carelessness, or worse. Nihilism, existentialism and absurdism rushed to fill the void from which certainty had been driven, and claimed there could be no real meaning in a purposeless universe.
But engineers have no time for such angst, and they worked on with a new purpose. For them, the uncertainties of quantum mechanics could be harnessed to create miniaturised circuits capable of handling information on a scale Turing and Van Neumann could only have dreamt of. Their ideas were proved in straightforward commercial applications such as accounting and payrolls, but early computer technology also helped to put man on the moon. Just as Wittgenstein had said, symbolic logic could be used to represent any feature on earth, or beyond.
I wonder what that young man who burst into Russell's room that day would make of it all? Just one hundred years ago all these ideas were just esoteric academic abstractions, so what language could we possibly use to explain supercomputers, smart phones, games stations, media players, virtual worlds, 3D games, internet Google and Wikipedia to someone from that era? I like to think that, given a little time to take that young aeroengineer through all the wonders that have followed behind his ground-breaking work, step by step, he would soon catch on.
After all, it's only logical.
Posted By: Francis Goode @ 04 September 2012 02:41 PM General
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