Few people remember that as well as being a prolific author, Lewis Carroll was also an accomplished philosopher and logician. At the dawn of the 20th century, trying to elucidate the ‘foundations of mathematics’ had become something of a craze amongst mathematicians and philosophers. Simple truths such as 2+2=4 seem obvious and incontrovertible, yet the source of their truth is less concrete. Carroll's

In the article a pretentious tortoise challenges Achilles to use logic to force him to accept the conclusion of a logically valid argument:

A. Things that are equal to the same are equal to each other.

B. The two sides of this triangle are equal to the same.

Z. The two sides of this triangle are equal to each other.

The tortoise claims to accept the first two propositions, but not the third. Achilles takes up the challenge of trying to set the tortoise straight. First, he points out that if somebody accepts A and B, then they must also accept Z. This seems quite sensible to the tortoise, so he adds it to his premises.

A. Things that are equal to the same are equal to each other.

B. The two sides of this triangle are equal to the same.

C. If somebody accepts A and B, then they must also accept Z.

*What the Tortoise Said to Achilles*is one of the most insightful (and amusing) attempts to grapple with this foundational crisis.In the article a pretentious tortoise challenges Achilles to use logic to force him to accept the conclusion of a logically valid argument:

A. Things that are equal to the same are equal to each other.

B. The two sides of this triangle are equal to the same.

Z. The two sides of this triangle are equal to each other.

The tortoise claims to accept the first two propositions, but not the third. Achilles takes up the challenge of trying to set the tortoise straight. First, he points out that if somebody accepts A and B, then they must also accept Z. This seems quite sensible to the tortoise, so he adds it to his premises.

A. Things that are equal to the same are equal to each other.

B. The two sides of this triangle are equal to the same.

C. If somebody accepts A and B, then they must also accept Z.

Z. The two sides of this triangle are equal to each other.

The tortoise, unconvinced, claims to accept A, B and C, but not Z. Achilles, seeing triumph is in reach, exclaims, “Aha! If A and B and C are true, Z must be true.” Here, the beginnings of an infinite regress are apparent. The tortoise seems to have a point. Though we would normally simply accept the move from A and B to Z, the intermediate steps all seem necessary to license the inference, even though we would not normally state them; indeed, they are the sort of things we would say if we were teaching a child to reason.

Wittgenstein’s response to this paradox was to say that when we follow a rule, such as a rule of inference, it cannot be the case that we require intermediate steps, or interpretations of the rules, such as C, to lead us along. If it were, then we would never get anywhere, as Carroll’s paper shows, for "any interpretation still hangs in the air along with what it interprets, and cannot give it any support” (Wittgenstein’s

This solution to how it is we are able to go by a rule, which I think is the right one, does not leave us with much less of a sense that mathematics - indeed, any rule based practice - is a precarious affair. There is nothing more underlying the fact that 2+2=4 than the contingencies of human nature; the fact that because of the things most humans share - biology, culture (in a very broad sense), primitive desires, ect., - when it comes to mathematics we mostly find it more natural to go on in one way rather than another.

Given that Carroll foreshadows the most celebrated philosopher of the 20th century, amongst the puzzles his paper raises is - Why is Carroll not regarded as highly in philosophical circles as he is in literary circles?

The tortoise, unconvinced, claims to accept A, B and C, but not Z. Achilles, seeing triumph is in reach, exclaims, “Aha! If A and B and C are true, Z must be true.” Here, the beginnings of an infinite regress are apparent. The tortoise seems to have a point. Though we would normally simply accept the move from A and B to Z, the intermediate steps all seem necessary to license the inference, even though we would not normally state them; indeed, they are the sort of things we would say if we were teaching a child to reason.

Wittgenstein’s response to this paradox was to say that when we follow a rule, such as a rule of inference, it cannot be the case that we require intermediate steps, or interpretations of the rules, such as C, to lead us along. If it were, then we would never get anywhere, as Carroll’s paper shows, for "any interpretation still hangs in the air along with what it interprets, and cannot give it any support” (Wittgenstein’s

*Philosophical Investigations*, §198). Instead, human beings, being the sort of creatures we are, when we have been given the appropriate training, we see immediately, without interpretation or further premises, that Z follows from A and B. This is what comes naturally to most people after minimal instruction. To somebody like the tortoise, who despite plenty of instructions and illustrations, cannot be brought to see that Z follows from the premises, there is nothing to do but say, “I am sorry, but I cannot make you understand.”This solution to how it is we are able to go by a rule, which I think is the right one, does not leave us with much less of a sense that mathematics - indeed, any rule based practice - is a precarious affair. There is nothing more underlying the fact that 2+2=4 than the contingencies of human nature; the fact that because of the things most humans share - biology, culture (in a very broad sense), primitive desires, ect., - when it comes to mathematics we mostly find it more natural to go on in one way rather than another.

Given that Carroll foreshadows the most celebrated philosopher of the 20th century, amongst the puzzles his paper raises is - Why is Carroll not regarded as highly in philosophical circles as he is in literary circles?

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