第10章 Mathematics

I have mentioned mathematics as a way to settle in the mind a habit of reasoning closely and in train;not that I think it necessary that all men should be deep mathematicians,but that having got the way of reasoning,which that study necessarily brings the mind to,they might be able to transfer it to other parts of knowledge as they shall have occasion.For in all sorts of reasoning every single argument should be managed as a mathematical demonstration;the connection and dependence of ideas should be followed till the mind is brought to the source on which it bottoms and observes the coherence all along,though in proofs of probability one such train is not enough to settle the judgment as in demonstrative knowledge.

Where a truth is made out by one demonstration,there needs no further enquiry;but in probabilities,where there wants demonstration to establish the truth beyond doubt,there it is not enough to trace one argument to its source and observe its strength and weakness,but all the arguments,after having been so examined on both sides,must be laid in balance one against another,and upon the whole the understanding determine its assent.

This is a way of reasoning the understanding should be accustomed to,which is so different from what the illiterate are used to that even learned men oftentimes seem to have very little or no notion of it.Nor is it to be wondered,since the way of disputing in the schools leads them quite away from it by insisting on one topical argument,by the success of which the truth or falsehood of the question is to be determined and victory adjudged to the opponent or defendant;which is all one as if one should balance an account by one sum charged and discharged,when there are a hundred others to be taken into consideration.

This therefore it would be well if men's minds were accustomed to,and that early,that they might not erect their opinions upon one single view,when so many other are requisite to make up the account and must come into the reckoning before a man can form a right judgment.This would enlarge their minds and give a due freedom to their understandings,that they might not be led into error by presumption,laziness or precipitancy;for I think nobody can approve such a conduct of the understanding as should mislead it from truth,though it be never so much in fashion to make use of it.

To this perhaps it will be objected that to manage the understanding as Ipropose would require every man to be a scholar and to be furnished with all the materials of knowledge and exercised in all the ways of reasoning.

To which I answer that it is a shame for those that have time and the means to attain knowledge to want any helps or assistance for the improvement of their understandings that are to be got,and to such I would be thought here chiefly to speak.Those,methinks,who by the industry and parts of their ancestors have been set free from a constant drudgery to their backs and their bellies,should bestow some of their spare time on their heads and open their minds by some trials and essays in all the sorts and matters of reasoning.

I have before mentioned mathematics,wherein algebra gives new helps and views to the understanding.If I propose these,it is not,as I said,to make every man a thorough mathematician or a deep algebraist;but yet I think the study of them is of infinite use even to grown men.First,by experimentally convincing them that to make anyone reason well it is not enough to have parts wherewith he is satisfied and that serve him well enough in his ordinary course.

A man in those studies will see that,however good he may think his understanding,yet in many things,and those very visible,it may fail him.This would take off that presumption that most men have of themselves in this part;and they would not be so apt to think their minds wanted no helps to enlarge them,that there could be nothing added to the acuteness and penetration of their understandings.

Secondly,the study of mathematics would show them the necessity there is in reasoning to separate all the distinct ideas and see the habitudes that all those concerned in the present enquiry have to one another,and to lay by those which relate not to the proposition in hand and wholly to leave them out of the reckoning.This is that which in other subjects besides quantity is what is absolutely requisite to just reasoning,though in them it is not so easily observed nor so carefully practiced.In those parts of knowledge where it is thought demonstration has nothing to do,men reason as it were in the lump;and if,upon a summary and confused view or upon a partial consideration,they can raise the appearance of a probability,they usually rest content,especially if it be in a dispute where every little straw is laid hold on and everything that can but be drawn in any Bay to give colour to the argument is advanced with ostentation.But that mind is not in a posture to find the truth that does not distinctly tally all the parts asunder and,omitting what is not at all to the point,draw a conclusion from the result of all the particulars which any way influence it.There is another no less useful habit to be got by an application to mathematical demonstrations,and that is of using the mind to a long train of consequences;but having mentioned that already I shall not again here repeat it.

As to men whose fortunes and time is narrower,what may suffice them is not of that vast extent as may be imagined,and so comes not within the objection.

Nobody is under an obligation to know everything.Knowledge and science in general is the business only of those who are at ease and leisure.

Those who have particular callings ought to understand them;and it is no unreasonable proposal,nor impossible to be compassed,that they should think and reason right about what is their daily employment.This one cannot think them incapable of without leveling them with the brutes and charging them with a stupidity below the rank of rational creatures.