In the 18th century, Hume formulated what is now known as the problem of induction. He noted that both in science and everyday experience we use a type of reasoning that philosophers call induction, which consists in generalising from examples. Hume also pointed out that we do not seem to have a logical justification for the inductive process itself. Why then do we believe that inductive reasoning is reliable? The answer is that it has worked so far. Ah, but to say so is to deploy inductive reasoning to justify inductive reasoning, which seems circular. Plenty of philosophers have tried to solve the problem of induction without success: we do not have an independent, rational justification for the most common type of reasoning employed by laypeople and professional scientists. Hume didn’t say that we should therefore all quit and go home in desperation. Indeed, we don’t have an alternative but to keep using induction. But it ought to be a sobering thought that our empirical knowledge is based on no solid foundation other than that ‘it works’.
What about maths and logic? At the beginning of the 20th century, a number of logicians, mathematicians and philosophers of mathematics were trying to establish firm logical foundations for mathematics and similar formal systems. The most famous such attempt was made by Bertrand Russell and Alfred North Whitehead, and it resulted in their Principia Mathematica (1910-13), one of the most impenetrable reads of all time. It failed.
A few years later the logician Kurt Gödel explained why. His two ‘incompleteness theorems’ proved — logically — that any sufficiently complex mathematical or logical system will contain truths that cannot be proven from within that system. Russell conceded this fatal blow to his enterprise, as well as the larger moral that we have to be content with unprovable truths even in mathematics. If we add to Gödel’s results the well-known fact that logical proofs and mathematical theorems have to start from assumptions (or axioms) that are themselves unprovable (or, in the case of some deductive reasoning like syllogisms, are derived from empirical observations and generalisation — ie, from induction), it seems that the quest for true and objective knowledge is revealed as a mirage.