For us to understand any idea, then, we pursue three aims:
- 1. We find out what it is called.
- 2. We collect true statements about it.
- 3. We apply those statements—we use them in situations.
We can do this about any idea. Therefore, we can understand any idea, and the degree of our understanding increases as the number of true statements mastered increases.
Perhaps this seems to be a rash claim. Of course, it may take a good deal of time to collect true statements about many ideas. In fact, a scientist may spend thirty years of his life trying to find out from experiment the truth or falsehood of one statement, though, when he has succeeded, the fact can be swiftly told to others. Also, we all vary in the speed, perseverance, skill, etc., with which we can collect true statements and apply them. Besides, some of us have not been taught well and have little faith in our ability to carry out this process: this is the greatest obstacle of all. But, there is in reality no idea in the field of existing science and knowledge which you or I cannot understand. The road to understanding lies clear before us.
Supplement 2
MATHEMATICS
In the course of our discussion of machines that think, we have had to refer without much explanation to a number of mathematical ideas. The purpose of this supplement is to explain a few of these ideas a little more carefully than seemed easy to do in the text and, at the end of the supplement, to put down briefly some additional notes for reference.
DEVICES FOR MULTIPLICATION
Suppose that we have to multiply 372 by 465. With the ordinary school method, we write 465 under the 372 and proceed about as follows: 5 times 2 is 10, put down the 0 and carry the 1; 5 times 7 is 35, 35 and 1 is 36, put down the 6 and carry the 3; 5 times 3 is 15, 15 and 3 is 18, put down the 8 and carry the 1; ... The method is based mainly on a well-learned subroutine of continually changing steps:
- 1. Select a multiplicand digit.
- 2. Select a multiplier digit.
- 3. Refer to the multiplication table with these digits.
- 4. Obtain the value of their product, called a partial product.
- 5. Add the preceding carry.
- 6. Set down the right-hand digit.
- 7. Carry the left-hand digit.
We can, however, simplify this subroutine for a machine by delaying the carrying. We collect in one place all the right-hand digits of partial products, collect in another place all the left-hand digits, and delay all addition until the end.