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I don't understand his idea of comparing the independence of the Continuum Hypothesis from the other axioms of set theory, which is a very deep and important re

I don't understand his idea of comparing the independence of the Continuum Hypothesis from the other axioms of set theory, which is a very deep and important result about (in essence) one of the limits of mathematical reasoning, with this little notational difficulty, which frankly is of very little interest, especially to mathematicians (you do run into this when writing parsers, sometimes...) He actually manages to say that this is "as stupid as" asking for a proof of the Continuum Hypothesis. Reminds me of people who get hung up on whether 1 is a prime number. Yes, no, maybe - people have defined it this way and that, it's a convention you can make without changing anything fundamental about mathematical reasoning. Different systems of axioms lead to very different mathematical objects (compare Euclidean and non-Euclidean geometries). Different conventions lead to the exact same thing written slightly differently - if we say that clockwise rotation corresponds to positive change in angle (the opposite of current convention), we'll exchange plus and minus signs in a bunch of formulas, and nothing else.

StevenXC 14y ago
My (intended) implication was that answering the question > What is 6÷2(1+2)? without establishing the axioms of arithmetic is as impossible as answering > Does CH hold in ZF? without establishing some axioms of set theory. Put another way, asking 6÷2(1+2) on the SAT is as ill-advised as asking an Introduction to Proofs course to tackle the Continuum Hypothesis with just some naive set theory.
- yaakov34 14y ago
  Again, I think that's a very bad comparison - resolving this difficulty is not "impossible". You can choose whatever convention you like, and all that changes is the notation. Of course it's a bad test question, but it is not "impossible" or even remotely difficult for mathematics - it's just a matter of choosing a notation. You seem to think that this question has something to do with the "axioms of arithmetic" - no, it doesn't, it's a matter of how you write things by convention. Operator precedence is not and has never been an "axiom". We can define plus to have the highest precedence, and get the exact same arithmetic we have now, written differently. On the other hand, deciding the Continuum Hypothesis is "impossible" in a very fundamental way - Kurt Godel and Paul Cohen, two of the greatest mathematical logicians in history, proved that the Continuum Hypothesis is undecidable by mathematical reasoning as we can best formulate it, i.e. it is independent of the ZFC axioms. That's not a matter of choosing a notation for it.
  - StevenXC 14y ago
    Whether you consider the properties of arithmetic to be axioms or some other word which describes a fact assumed without proof is a matter of semantics. The point was that without a precise foundation for mathematics we cannot proceed - anything further is overanalysis for an article I wrote mainly for folks without our mathematical background. :-)
    - yaakov34 14y ago
      There is a fundamental difference between conventions of notation, and properties of the thing being notated, in this case arithmetic. I'm frankly surprised that someone with a mathematical background keeps mixing up the two. There are a dozen people in this thread of comments posting different notations - prefix, postfix, what have you. Do you think these define a different arithmetic? Now of course it's stupid to ask people a question without defining what you mean by your notation - you might as well mumble something and expect them to decipher it. But that is in no way equivalent to asking a question which is undecidable in mathematical logic (which is not necessarily a stupid thing to do). And features of some particular notation do not in any way, shape, or form, or by any stretch of semantics, constitute axioms, or properties, or anything else which belongs to arithmetic itself. Do you really disagree with that?

StevenXC 14y ago

My (intended) implication was that answering the question > What is 6÷2(1+2)? without establishing the axioms of arithmetic is as impossible as answering > Does CH hold in ZF? without establishing some axioms of set theory. Put another way, asking 6÷2(1+2) on the SAT is as ill-advised as asking an Introduction to Proofs course to tackle the Continuum Hypothesis with just some naive set theory.

yaakov34 14y ago
Again, I think that's a very bad comparison - resolving this difficulty is not "impossible". You can choose whatever convention you like, and all that changes is the notation. Of course it's a bad test question, but it is not "impossible" or even remotely difficult for mathematics - it's just a matter of choosing a notation. You seem to think that this question has something to do with the "axioms of arithmetic" - no, it doesn't, it's a matter of how you write things by convention. Operator precedence is not and has never been an "axiom". We can define plus to have the highest precedence, and get the exact same arithmetic we have now, written differently. On the other hand, deciding the Continuum Hypothesis is "impossible" in a very fundamental way - Kurt Godel and Paul Cohen, two of the greatest mathematical logicians in history, proved that the Continuum Hypothesis is undecidable by mathematical reasoning as we can best formulate it, i.e. it is independent of the ZFC axioms. That's not a matter of choosing a notation for it.
- StevenXC 14y ago
  Whether you consider the properties of arithmetic to be axioms or some other word which describes a fact assumed without proof is a matter of semantics. The point was that without a precise foundation for mathematics we cannot proceed - anything further is overanalysis for an article I wrote mainly for folks without our mathematical background. :-)
  - yaakov34 14y ago
    There is a fundamental difference between conventions of notation, and properties of the thing being notated, in this case arithmetic. I'm frankly surprised that someone with a mathematical background keeps mixing up the two. There are a dozen people in this thread of comments posting different notations - prefix, postfix, what have you. Do you think these define a different arithmetic? Now of course it's stupid to ask people a question without defining what you mean by your notation - you might as well mumble something and expect them to decipher it. But that is in no way equivalent to asking a question which is undecidable in mathematical logic (which is not necessarily a stupid thing to do). And features of some particular notation do not in any way, shape, or form, or by any stretch of semantics, constitute axioms, or properties, or anything else which belongs to arithmetic itself. Do you really disagree with that?