Is There Such a Thing as a Private Discovery?
Mathematics is (or purports to be) a deductive science. Unlike the empirical sciences, whose conclusions are only ever tentative, once a mathematical theorem is proved, it is proved for all time. But what does it mean to “prove” a theorem? I suppose it means proceeding in logical steps from a set of premises to a conclusion. As long as the steps taken accord with rules that are accepted as valid, then the steps will be valid, and thus the conclusion will be valid.
In the book Fermat’s Enigma, Simon Singh tells the story of Andrew Wiles, a brilliant mathematician who proved Fermat’s Last Theorem, a seemingly intractable problem that had baffled mathematicians for over three hundred years. The problem can be stated simply enough: the equation a^n + b^n = c^n has no positive integral solutions where n > 2. But to solve it, Wiles had to use a number of highly sophisticated mathematical concepts – like modular forms and elliptic curves - and his proof was so esoteric, so impenetrable, that only a handful of the world’s brightest mathematicians had the knowledge and skill to check his work.
That got me thinking. In Proofs and Refutations, philosopher Imre Lakatos argues that one never really proves what one sets out to prove. The reason is that, when a proof is offered, other mathematicians will offer counterexamples that attempt to show that one of the assumptions the proof relies on is false. At this point, disagreement arises as to whether these are actually counterexamples, and deciding this questions depends on how broadly or narrowly certain concepts in the proof are defined. Define them broadly, and it becomes easy to find counterexamples. Define them narrowly, and it becomes harder. But don’t define them too narrowly, because then the “proof” covers so few cases that it ceases to be interesting.
The neat proof that results from this process hides the fact that many of its key concepts’ definitions have had to be renegotiated by the community along the way. “Discovery does not go up or down, but follows a zig-zag path: prodded by counterexamples, it moves from the naïve conjecture to the premises and then turns back again to delete the naïve conjecture and replace it by the theorem. Naïve conjecture and counterexample do not appear in the fully fledged deductive structure: the zig-zag of discovery cannot be discerned in the end-product.”
Put more bluntly, Lakatos argues, “one does not prove what one has set out to prove.”
Lakatos’ argument is reminiscent of Wittgenstein’s discussion of what it means to obey a rule. One example of following a rule is using a word correctly, but can one do this privately? Wittgenstein makes the point that following a rule is not the same thing as thinking one is applying a rule, since in the latter case one may be mistaken. Without other members of the community checking our usage of words, we could never say we were using them consistently, and every act of recognition would be a new act of definition. Similarly, Lakatos underscores the public nature of proofs and the dialectic essential to arriving at a consensus definition of concepts central to the proof.
In the book Fermat’s Enigma, Simon Singh tells the story of Andrew Wiles, a brilliant mathematician who proved Fermat’s Last Theorem, a seemingly intractable problem that had baffled mathematicians for over three hundred years. The problem can be stated simply enough: the equation a^n + b^n = c^n has no positive integral solutions where n > 2. But to solve it, Wiles had to use a number of highly sophisticated mathematical concepts – like modular forms and elliptic curves - and his proof was so esoteric, so impenetrable, that only a handful of the world’s brightest mathematicians had the knowledge and skill to check his work.
That got me thinking. In Proofs and Refutations, philosopher Imre Lakatos argues that one never really proves what one sets out to prove. The reason is that, when a proof is offered, other mathematicians will offer counterexamples that attempt to show that one of the assumptions the proof relies on is false. At this point, disagreement arises as to whether these are actually counterexamples, and deciding this questions depends on how broadly or narrowly certain concepts in the proof are defined. Define them broadly, and it becomes easy to find counterexamples. Define them narrowly, and it becomes harder. But don’t define them too narrowly, because then the “proof” covers so few cases that it ceases to be interesting.
The neat proof that results from this process hides the fact that many of its key concepts’ definitions have had to be renegotiated by the community along the way. “Discovery does not go up or down, but follows a zig-zag path: prodded by counterexamples, it moves from the naïve conjecture to the premises and then turns back again to delete the naïve conjecture and replace it by the theorem. Naïve conjecture and counterexample do not appear in the fully fledged deductive structure: the zig-zag of discovery cannot be discerned in the end-product.”
Put more bluntly, Lakatos argues, “one does not prove what one has set out to prove.”
Lakatos’ argument is reminiscent of Wittgenstein’s discussion of what it means to obey a rule. One example of following a rule is using a word correctly, but can one do this privately? Wittgenstein makes the point that following a rule is not the same thing as thinking one is applying a rule, since in the latter case one may be mistaken. Without other members of the community checking our usage of words, we could never say we were using them consistently, and every act of recognition would be a new act of definition. Similarly, Lakatos underscores the public nature of proofs and the dialectic essential to arriving at a consensus definition of concepts central to the proof.
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