176 



Mr DE morgan, ON THE SYLLOGISM, No. Ill, 



fact, if what either party has advanced against the other be true, the common opponent has 

 a good case against both, provided only mathematics enough for a higher kind of land-surveyor 

 be exempted from the common doom, and made a part of professional education. 



There never was in history the time at which mathematics, in any branch, wanted a pal- 

 pable separation of form and matter : and mathematicians have always seen the separation, 

 though they have not always rightly apprehended the relation of the components. They have 

 spoken much of abstraction, a word truly applied to their function : but they have not duly 

 distinguished between abstraction of colleague qualities from each other, and abstraction of the 

 instrument from the material. They have also dwelt much on generalisation, a word so truly 

 descriptive of what is always taking place within the precinct, that they have oftentimes made 

 it give name to the fence. 



The first element of mathematical process is the separation of space from matter filling it, 

 and quantity from the material quantum : whence spring geometry and arithmetic, the studies 

 of the laws of space and number. Distinctions which are of form in arithmetic become material 

 in algebra. The lower forms of algebra become material in the algebra of the functional 

 symbol. The functional form becomes material in the differential calculus, most visibly when 

 this last is merged in the calculus of operations. But, though the distinction of form and 

 matter be very certainly present to those who can see it, it is equally certain that many fol- 

 lowers of the mathematics have their ideas of the distinction as dark as those of any of the old 

 logicians. The difference is that the mathematician cannot help dealing with the thing in 

 question, though under a name of too little intension : he cannot but be sensible of abstraction; 

 but he may be unused to remember that he abstracts form from matter. The logician on the 

 other hand may, as often was the case, have his system cast in so material a mould, that he is 

 hardly sensible even of abstraction : and when the fault is not palpably committed in the 

 treatise, the individual reader may, of his own inaptitude to abstract except under symbolic 

 compulsion, convert formal logic into material. Accordingly, the separation of form is often 

 learned language to the logical student, with a bad dictionary to read it by : to the mathema- 

 tician it is as often M. Jourdain's prose, and nothing more. To the logician it is a collect 

 for certain holidays ; it is the paternoster of the mathematician, who may run it over without 

 thinking of the meaning, if he ever knew it. And these tendencies, large in amount in the 

 learner, have their sway even in the books he learns from, and in the discussions of the 

 highly informed : the great distinction of form and matter is more in the theory* of the logi- 

 cian than in his practice, more in the practice of the mathematician than in his theory. 



* 1 am fully aware of the boldness of my comparison of the 

 logician and mathematician, and of the audacious appearance 

 which it is likely to present to a class of inquirers who have 

 hitherto been allowed to distribute functions to the branches of 

 human knowledge pretty nearly in their own way. My aver- 

 ments are of that kind which nothing but success will justify : 

 and about which controversy is useless. It is not competent to 

 those who are only logicians, and to those who are only mathe- 

 maticians, to settle a question in which the alleged unfitness 

 of either to decide is a part of the matter to be decided : still 

 less is it comj-etent to the few who unite both characters to 



demand of the others that they shall see this. Time must 

 settle it ; and 1 believe this will be the way. As joint attention 

 to logic and mathematics increases, a logic will grow up among 

 the mathematicians, distinguished from the logic of the logi- 

 cians by having the mathematical element properly subordinated 

 to the rest. This mathematical logic^so called quasi lucus a 

 non nimis lucendo — will commend"itself to the educated world 



by showing an actual representation of their form of thought 



a representation the truth of which they recognise — instead of a 

 mutilated and onesided fragment, founded upon canons of 

 which they neither feel the force nor see the utility. 



