Let’s not allow ourselves to become beguiled by words.

Ahh… I am afraid you are too late for me. I’ve been word-beguiled for a long time now!

Thanks for your comment, Stephen. It provoked an interesting debate at my work at lunch-time today. I was clearly in the minority by effectively claiming that, while logic may well be “the very essence of mathematics”, that mathematics wasn’t the very essence of logic.

While I did foresee that this could devolve into a debate about definitions, and tried to prevent it:

any definition of the term “mathematics” that might apply in a daily newspaper

What I didn’t foresee was the level of acceptance (amongst around three people that I sampled randomly) for the very broad definition of mathematics to include all of logic and reason.

On the other hand, that same group of three people included two others who had (completely independently of this blog and of each other) chosen to write their own Sudoku-solving software in their spare time, so perhaps my sample wasn’t exactly unbiased.

In conclusion, while I still feel that The Australian was acting beyond rebuke in suggesting that Sudoku doesn’t require “mathematics”, I have to accept that, to many people, the term “mathematics” can be stretched out to include all the logic and reasoning used in solving Sudokus.

The whole thing is simply the meaning of the term “mathematics”. It does not refer to a single entity. Rather different people use it to refer to different things. Mathematicians, so called, know perfectly well their art is one of reason and logic. That is what they do. The rest of us learnt arithmetic at school when we were 9 and called it mathematics. That is what we mean by the term. For us (if I may include myself for now) Sudoku require no arithmetic, so they require no mathematics. But it another sense of the term, sudoku are quintessential mathematics of the highest order requiring nothing more or less that logic, the very essence of mathematics. Between the two there is really no argument. Rather there appears to be because the same word is used in two different ways. Let’s not allow ourselves to become beguiled by words.

Alastair,

Hmmmm…

I have simplified his argument. You are effectively suggesting I have over-simplified it. That seems a fair complaint. So let me try again, with less simplification to show his argument is still invalid, even if my initial argument didn’t adequately demonstrate it.

Gagen’s argument follows this structure:

The Australian’s Premise: The skill required is logic and reason alone, but not mathematics.
Gagen’s Premise: Mathematics is logic and reason alone.
Conclusion (by simple substitution) : The skill required is mathematics, but not mathematics.

This is clearly a contradiction – so either one of the premises must be false, or the substitution must be invalid.

Gagen argues the Australian’s premise is false. He goes further and corrects it by saying mathematics IS the required skill. [Alastair pointed out he doesn't explicitly make that final claim. I think he implies this by thanking The Australian for the mathematics.]

I argue that the problem is either with the soundness of the second premise or with validity of the substitution – it depends on how you interpret the ambiguous wording of the second premise.

If Gagen’s premise meant “Mathematics is a subset of logic and reason,” I would accept his premise but the act of substitution would be invalid. The conclusion should instead read something like “The skill required is included in the “logic and reason” superset that also includes mathematics, but the skill required is not mathematics.” No contradiction.

If Gagen’s premise meant “Mathematics is equivalent to logic and reason.” I wouldn’t accept his definition. This very discussion, as a perfect counter-example, uses logic and reason, but not mathematics (in any definition of the term “mathematics” that might apply in a daily newspaper.)

From your extract, the only claim that Gagen is making is number 2 on your list. He’s not claiming 1 or 3, at least from my reading. He’s repeating claim 1, but disagreeing with it.