PROBLEMS EMPHA SIZED BY PEA GMA TISM 40 7 



side of the square are incommensurable. That is, he discovered that 

 certain premises imply certain conclusions. 



The process by which he discovered this we happen, though somewhat 

 indirectly, to know. The instance is a remarkable one of the fecundity 

 of the deductive process. The conclusion when the Greek reached it was 

 a novelty. It seemed to him so novel and uncanny a conclusion that the 

 Pythagorean school is said by a later tradition to have long regarded 

 the whole matter as a mystery which must not be mentioned to the 

 vulgar. It was reported that the man who revealed this mystery came 

 to a bad end and received special penalties in the underworld, so closely 

 in those days was exact science linked with tabu and with superstition. 

 Yet no one who once goes through the little process of reasoning by 

 which the ancient geometer established his result can remain without 

 interest in the psychology of such a process. 



You see throughout how my account of the whole matter is of course 

 colored by my logical interests, and yet I freely admit that the psycho- 

 logical problems at issue ought to be considered without any undue 

 reference to anybod^'^s philosophical prejudices or concerns. I admit 

 my own bias in the matter, merely because I am thereby enabled to call 

 attention to what the live process of deduction is, and to point out that 

 the recent psychology of reasoning has profoundly neglected to take 

 account of some of the most elementary facts with regard to the nature 

 of this process. 



Charles Peirce long ago called attention to the general nature of 

 the psychological processes which are in question in deduction. They 

 are processes of the nature of ideal experiments. The instance of the 

 one-sided strip of paper readily shows how many intermediate steps 

 there may be between such ideal experiments and physical experiments 

 with a strip of paper. On the other hand, as soon as you begin to 

 reason, and to predict what will be true about a given strip of paper if 

 certain hypotheses are met, with regard to its structure, you get an 

 insight into the whole situation which you can not possibly get by cut- 

 ting the strip of paper without adding the deductive process. And in 

 general, wherever deduction is worth while, the testing of hypotheses 

 after deductions have carefully been made whereby we predict deter- 

 minate results of such hypotheses, has a wholly different value and 

 interest from that which results from the testing of hypotheses without 

 previous deduction. 



The psychology of deduction may then well be characterized as the 

 psychology of the GedanJcen-experiment. The peculiarity of the experi- 

 mental processes in question is that whether we use symbols, or dia- 

 grams, or mental images, our reasoning depends upon the fact that the 



