Monday, 28 July 2008

Lawrence of Lindores on the Royal Road to Geometry

In his questions on Aristotle's Physics, written in Paris at around the turn of the 15th century, the Scottish arts master Lawrence of Lindores asked whether one could arrive at knowledge of effects from knowledge of their causes (I.4, utrum ex cognitione causarum contingat devenire in cognitionem effectuum, ed. Dewender 2002).

One of the arguments under consideration involved the claim that, if knowledge of a cause was sufficient for knowledge of its effect,

sequeretur quod cognito aliquo statim cognoscerentur simul omnia possibilia cognosci ex illo.  Consequens falsum, quia tunc sequeretur quod cognitis principiis geometriae statim cognoscerentur omnes conclusiones eius.  Probatur, quia notitia principiorum geometriae est causa sufficiens ad habendum notitiam primae conclusionis, et notitiae principiorum una cum notitia primae conclusionis esset causa sufficiens ad habendum notitiam secundae conclusionis, et sic ulterius de tertia, quarta et quinta, et sic de aliis, ergo propositum.

‘it would follow that, once something was known, all that could be known from it would immediately be known at the same time.  The consequent is false, because in that case it would follow that once the principles of geometry were known, all the conclusions of geometry would be known immediately.  Proof: knowledge of the principles of geometry is a sufficient cause for having knowledge of the first conclusion, and knowledge of the principles together with knowledge of the first conclusion would be a sufficient cause for having knowledge of the second conclusion, and so on for the third, fourth and fifth, and so for the others, QED.’

Lindores was more down to earth in his response:

Et si dicatur "Ponatur quod Socrates habeat notitiam omnium principiorum geometriae et velit agere toto conatu suo ad disponendum illa principia in debito modo et figura ad inferendum omnes conclusiones geometriae, et non habeat impedimentum extrinsecum nec ex parte famis nec sitis aut frigoris vel quocumque alio extrinseco, tunc in isto casu, ex quo velit agere, sequitur quod statim cognosceret omnes conclusiones geometriae", respondetur admisso casu negando consequentiam.  Et causa est ista, quia, quamvis Socrates non haberet impedimentum extrinsecum, tamen haberet intrinsecum, quia actualis consideratio circa illationem unius conclusionis impediret actualem considerationem circa illationem alterius conclusionis.  Quo dato poneret magnum tempus ad inferendum duas conclusiones, ut pateret experientia, igitur et cetera.

‘And if someone says "Suppose Socrates had knowledge of all the principles of geometry, and wanted to put all of his effort into setting those principles in the mode and figure necessary for inferring all the conclusions of geometry, and had no extrinsic impediment from hunger, thirst, cold, or any other extrinsic thing – then in that case, from his wanting to do this, it follows that he would at once know all the conclusions of geometry", the response is to allow the case and deny the consequence.  And the reason is as follows: although Socrates would not have an extrinsic impediment, he would still have an intrinsic one, because actual consideration concerning the deduction of one conclusion would impede actual consideration concerning the deduction of another.  Given which, it would take a long time to infer two conclusions, as should be clear from experience; therefore etc.’

If only knowledge was closed under implication!