It is impossible to construct mathematical truth machines. All interesting mathematics displays independent creativity and leads to unpredictable consequences which cannot be deduced from the given premises but must be studied empirically in a wider frame of reference. Willful juggling with these concepts leads to paradoxes – a sure sign of the inadequacy of the language game. Faith in science as an infallible truth machine is thus a fallacy.
The impressive achievements of the natural sciences indubitably suggest close
contact with the essence of reality. Consequently, it has been widely assumed
that these insights must be based on a set of axiomatic truths which would
constitute the rational grounds for our faith in science. Following Pierre Simon
de Laplace (1749–1827) and Ernst Mach (1838–1916), the logical positivists, with
Rudolf Carnap (1891–1970) in the van, made frantic efforts to realize this
programme by applying rigorous scientific reasoning – all of which sounds very
much like Baron Munchausen’s method of pulling himself up by his own bootstraps.
Like infinity or the microcosm of quantum physics, the inmost substance, ‘the reality’ of mathematics cannot be strictly defined. At the basic level we can articulate reasonable rules of the game within a formalized language. But the introspective capacity of a self-referential idiom is necessarily limited: no interesting language can explain or justify itself. A higher-order language may put the basic idiom into perspective but is even more opaque and impenetrable due to the broadened axiomatic base.
These inherent limitations of mathematics are fundamental to all deductive thinking. No interesting language, not even mathematics, can explain itself, its own foundations or consequences. In one way or another they all come up against the classic liar paradox – “this proposition is not true” – which was refined by Gödel with the help of the most sophisticated tools of his trade. In the same vein the mature Wittgenstein suggested that ordinary language can be perceived as an open-ended, self-organizing game. The relationships between the different word-moves create a pattern which to the trained player conveys real but mostly rather inaccurate information.
Game theory appears to preclude fully rational decision-making, except in the
simplest of circumstances. Any meta-rules that might be agreed upon, as a matter
of expediency, will be torn by self-contradictions and burdened with profound
inconsistencies. All socially relevant norm systems suffer from game-theoretical
contradictions; moral principles tend to run into conflict with themselves.
The search for objective norms for human behaviour is in vain; we must always absorb a great deal of personal uncertainty. Rational clarity on questions of values is a sure sign of self-deception; the creativity of the culture game is not compatible with a completely lucid account of the meta-rules. The definitive verification of any specific value system is impossible but systematic falsification may be feasible.
However rational the means, the ultimate ends will elude our probing intellect. The enigma of existence can never be captured in a single formula, and we are bound to face a whole set of mutually conflicting frames of reference. Painful disorientation and irresolution is thus a natural predicament for those in the vanguard. Accordingly, creative achievement is tied to competitive pluralism and pestered by schismatic discontent. Only in the fair play of conflicting views can we establish a dynamic equilibrium, which compels mutual consideration while steering clear of stagnation.
Reason must be consulted as a talented and cunning counsellor, eager to make himself master in the house. If we succumb to intellectual intoxication, the self-contradictory outcome is dogmatic fanaticism – a caricature of genuine faith, The fanatic anaesthetizes his conscience with huge doses of doctrinaire self-deceit: the infallibility of the pope, the party or the prophet – opiates for every season.
Our hypertrophic analytical intellect has triumphantly torn our norm systems
apart, corroded our values and relativized the rules of the game. Recently the
intellect in its other incarnation has been forced to recognize both the
inevitability of transcendent premises and its own subordinate instrumental
role. Voltaire’s ironic proposition: “If God does not exist, he must be promptly
invented”, was watered down with an expedient disclaimer: “but the whole of
nature cries out that he exists.” Anyhow, Voltaire´s quip should be turned
inside out – we are the invention desperately looking for a purpose.
We cannot become better ‘by ourselves’, the infinite regress of self-simulation soon exhausts our powers of introspection. Chaitin’s proof (GvH 1993, 9.1.5) corroborates St. Augustine’s dictum that the self-perfection of man is a hopeless undertaking. No programme can improve itself beyond certain rather narrow limits; no one and no thing can know the truth about itself.
Solely mathematics, and above all formal logic, can maintain the absolute mental purity that is essential to avoid logical fallacies. In Tractatus Logico-Philosophicus (1921) Wittgenstein methodically demolishes the hope of deducing anything but tautological truths from a priori arguments. The final conclusion has gained fame: “Wovon man nicht sprechen kann, darüber muss man schweigen.”*