Kripke tries to demonstrate the problem with ‘plus’ and ‘quus’ functions by exploiting the fact that each of us has done only finitely many addition problems in our lifetimes. He says that unless each of us had an infinitely large set of numbers and their sums in our heads any time we added two numbers, there would be no fact that justifies our stating that we’re using ‘+’ the same way we did in the past. However, one could successfully add two numbers even without having an infinitely large table of values memorized. Instead, one only need memorize a single finite algorithm for addition.

The first time Kripke addresses the algorithm objection, he gives only the most basic method for adding two numbers. Namely, the method whereby you add a and b by counting out one pile of a items and another of b items, and then counting the number of items in a pile made by combining these two piles. He easily refutes this objection by attacking the definition of ‘counting,’ pointing out that each of us has counted only finitely many piles of things, so there is no justification for our saying that the rule we previously employed while counting didn’t make an exception for the numbers we are now adding.

Later, he brings up the more efficient and more useful algorithm whereby single digits are added and 1’s are carried over to the next digit’s place when necessary. The only attempt Kripke makes to refute this objection is as part of his general refutation of the dispositional objection to Wittgenstein’s extreme skepticism. He says that many people are disposed to make certain mistakes while adding. According to his argument, then, no justification can be made for the claim that a present ‘correct’ response was following the same rule as was used before. This is the case because the previous dispositional rule may have involved a ‘mistake’ in the procedure for the particular pair of numbers currently being added.

It seems to me that he would have had a much harder time dealing with this algorithm had he brought it up in place of the more simple counting algorithm. Kripke fails to address the fact that, with such a concise algorithm, many people are perfectly capable of visualizing the whole process completely and without any of the dispositional ‘mistakes’ he brings up later. Assuming that all a person’s mental states are recorded somewhere, it would then be very easy to see whether or not an addition operation done in the present follows the same rules as addition problems done previously.

What makes the carrying algorithm objection even stronger is the fact that it can be represented almost entirely visually, without the need for words. The version of this algorithm depicted below is far from perfect because it requires words to explain what could be easily visualized entirely in terms of numbers. For example, the steps in the process involving concepts of “preceding” or “on the left” could be pictured mentally without the need to resort to either of these concepts. Furthermore, if a person’s mental picture also includes some aspect of movement, the process could be clearer still. It is much harder to attack the visualization of a process in the same way as one can attack a verbal definition, because mental pictures don’t need to be nearly as conceptual as mental collections of words.

One would most assuredly need to resort to concepts of “number” and “digit” in order to fully understand what the algorithm is accomplishing, but that’s not the issue here. Regardless of whether or not the person understands what’s happening, the visualized process consistently produces a set of digits corresponding to what we consider the “correct” result. Kripke’s skeptic argues that there is no fact about a person that can be used to justify the present procedure used to add two numbers, but any sufficiently aware observer could easily ascertain if a person is using the same procedure as he or she used before. Furthermore, because this algorithm addresses all ten digits and what to do with any arbitrary number of digits in each of two values to be added, the skeptic would have a much harder time finding some part of the procedure that would be affected by the fact that it had only been done finitely many times.

Addition algorithm

Write the two summands one above the other, with the rightmost digits aligned. Define all digits to the left of those defined as 0:

abcde

00fgh

Then starting at the rightmost digits in each number, and moving left until all remaining digits to the left are 0’s in both numbers, refer to the following table, which has the top number’s digit along the top and the bottom number’s digit along the left side. Write what is at the intersection of the row and column referred to by these digits directly below the digits themselves:

	0	1	2	3	4	5	6	7	8	9
0	0	1	2	3	4	5	6	7	8	9
1	1	2	3	4	5	6	7	8	9	‘0
2	2	3	4	5	6	7	8	9	‘0	‘1
3	3	4	5	6	7	8	9	‘0	‘1	‘2
4	4	5	6	7	8	9	‘0	‘1	‘2	‘3
5	5	6	7	8	9	‘0	‘1	‘2	‘3	‘4
6	6	7	8	9	‘0	‘1	‘2	‘3	‘4	‘5
7	7	8	9	‘0	‘1	‘2	‘3	‘4	‘5	‘6
8	8	9	‘0	‘1	‘2	‘3	‘4	‘5	‘6	‘7
9	9	‘0	‘1	‘2	‘3	‘4	‘5	‘6	‘7	‘8

When this has been done for every relevant digit in the summands, you end up with a string of digits, some which are preceded by ‘. For these digits, change the digit to the left by the following map (ignoring ‘ if it happens to already precede the digit to the left):

0 -> 1

1 -> 2

2 -> 3

3 -> 4

4 -> 5

5 -> 6

6 -> 7

7 -> 8

8 -> 9

9 ->‘0

Continue doing this to each digit, moving from right to left, until you reach the point at which all digits to the left are understood to be zero.

For example, to compute 57+68:

…0 0 0 5 7 7 and 8 yield ‘5

…0 0 0 6 8 5 and 6 yield ‘1

‘1‘5

‘5 makes the 1 become 2, and ‘1 makes the implied 0 to the left become 1: ‘1‘5 -> ‘25 -> 125