Monster-barring, exception-barring and the method of lemma incorporation

This essay is about to discuss some of the methods of proof analysis/adaption which occur in ”Proofs & Refutations” by Lakatos, a discussion on how to properly adjust proofs when they appear to be incomplete. Amongst these methods are monster-barring, exception-barring and the method of lemma incorporation. After discussing these three in detail, I will give an outline of which one of these is promoted by Lakatos himself and give his reasons therefore.

Monster-barring
The method of monster-barring deals with occurring incompleteness of a proof found through a counter-example by excluding the counter-example explicitly in the definition of an object. A good example would be to define prime numbers not only as ”numbers which are only divisible by 1 and itself”, but also explicitly excluding 1 to have the theorem of unique prime factorisation up to ordering come true. (Note that, if 1 is not excluded we immediately find a counter-example to the unique prime factorisation theorem: by multiplying a given prime factorisation of a number n with 1 an arbitrary amount of times (> 0) we would receive another prime factorisation of n.) However, it should be noted at this point that after studying a bit of Algebra it becomes natural to define prime numbers this way, since ±1 are the only units of the ring Z. A related method is the one of monster-adjustment which, after a monster appeared”, reinterprets/rephrases the former definition in a way that the monster in fact isn’t a real monster any more and claims that the definition has always been thought of in that way, but (mistakenly) hadn’t been made more explicit before because there has been no need to do so. An example for this phenomenon would be to count the zeros of a polynomial with multiplicity to keep the fundamental theorem of algebra a true statement.

Exception-barring

In a somewhat similar way, but with some small differences, the method of exception-barring deals with occurring monsters: It leaves the conjecture as it is and claims it is still true, but for some (explicitly mentioned) exceptions. This way of dealing with counter-examples appeared to be common among earlier mathematicians, so even great mathematicians as Lagrange stated theorems in it their biggest generality at first, only to mention some exception right after.

The method of lemma incorporation
When using the method of lemma incorporation to deal with counter-examples, mathematicians analyse at which sublemma the theorem really went wrong and aims to really only adjust this lemma. As a result, the monster is excluded aswell. However, as Lakatos argues, we also gain some more information which is to be mention in the next paragraph.

Lakatos’ prefered method of dealing with counter-examples
As one might have noticed by my formulation of the described methods of proof-adjustment when ”threatened” with counter-examples, Lakatos prefers the latter one and gives convincing reasons for his opinion: He more than once underlines that we do not learn anything when applying exception- or monster-barring, since we don’t take the counter-examples too serious, while in the method of lemma incorporation we need to consider the counter-example carefully to and find out quite some properties the counter-example has by searching for the lemmas which need to be adapted. Also, Lakatos believes that we bring mathematics forward by adjusting only the lemmas which need adaption, since the other lemmas might still be true in more general cases and may also be used for these. As an example we saw Euler’s theorem about polyhedra turning into a theorem in topology. Last, but not least we face the problem that there might always be further counter-examples yet to be found. To give a further argument for the method of lemma incorporation and against exception- and monster-barring, I want to mention some of my own thoughts on the stated problem: Because of the mentioned uncertainty of whether or not there are more counter-examples out there, mathematics in a certain sense comes down to an approach similar to the one physicians take when trying to describe the world around us: Build up a (consistent) theory and work with it as long as you haven’t found a counter-example (experiment that disproves the theory). For mathematics theorems would take over the role of theories in physics. However, using this metaphor the method of lemma incorporation can be seen as not ignoring upcoming counter-examples, but as adjusting the theory (as far as this is possible) to the newly gained information about the world. As we refer to the latter as being ”good”practice in physics, taking a similar norm to decide between ”goodänd ”bad”mathematical practice seems natural.