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I think that there is a germ of a reasonable question here, but I'm not sure the present question is appropriate for MO. I have not yet cast a vote to close. Perhaps we could discuss it here on meta.
I'm very sorry, but I couldn't resist. The coincidence of timing made it irresistible.
I had already casted my vote to close. This question is at once subjective and argumentative, nonsense, and not of interest to research mathematicians.
But I like Andrew's answer!! :-D
This question is at once subjective and argumentative, nonsense, and not of interest to research mathematicians.
I tend to agree with most of it, except possibly for the "nonsense" part. I started this thread because I could not immediately dismiss the question. Speaking from the perspective of a mathematical physicist, I am not aware of any place where the irrationality of π is used explicitly. In fact, although π appears everywhere, it is usually swept under the carpet by choosing the "right" units. On the other hand, it appears implicitly in fundamental properties of, say, trigonometric functions,...
Please do not close this. There are a few interesting answers already and I'm sure more would come up.
@JME: I thought Andrew's comment was frivolity and not meant to poke fun at the questioner. Granted, it showed that he (Andrew) did not take the question seriously, but I would argue that it his prerogative to do so and everyone else's to vote him down or leave critical comments. Neither has happened yet; perhaps this will change in due course. Until then, I don't see that his "answer" was meant to belittle you personally.
@JME: Because the book Andrew linked you to is fiction.
Here is one possible answer to the question:
Suppose there were some universe in which pi were rational. If this universe admits a notion of say, second order logic, then it also admits a proof of Lindemann-Weierstrass, and we obtain a contradiction. Every statement is then easily proved both true and false. In other words, mathematics as we practice it cannot possibly be a useful tool for describing the physical nature of this universe.
I hope this drives home the point that mathematical counterfactuals can be dangerous ground for thought experiments (and fertile ground for logicians...). I suspect that many mathematicians have internalized the above argument to such a degree, that these thought experiments are immediately dismissed as nonsense. I think this explains (but does not entirely excuse) some of the flippant responses to the question.
I suspect that the questioner (who has contributed some very insightful questions and answers about physics and algebraic geometry) may wish to express a question with different emphasis, e.g., is the irrationality of pi ever explicitly used in physics? Some of the more enlightening answers seem to have tacitly interpreted the question this way. If the question is suitably edited, I may vote to reopen.
JME: Firstly, I am not a moderator. I have no official connection to MathOverflow and anything I do is not indicative of general policy. Nor is Andrea a moderator. We both have acquired sufficient reputation on this website to be granted some moderator-like privileges, but we are allowed to use our own judgement in these matters.
Of course I meant no offence by my answer. The phrase "honourable gentleman" comes from the British parliament and was meant as an indication that the answer was completely frivolous.
But the question, as originally stated, made no sense. Pi is a mathematical construct and has the properties that it has. To say "imagine a universe in which pi were an integer" means that mathematics itself would have to be different and who knows what changes would have to be made in order to ensure that pi was an integer? As has been said, logic itself would have to be different. If you had said, "where the ratio of the diameter of a circle to its circumference were an integer" then that would have at least made sense but would have been a question about detecting which geometry we live in.
The question as it stands now does at least make sense (though the question is now slightly divorced from its motivation). But I would still have voted for closing it. It is not a "research level" question, it is what has been termed a "fishing expedition". Whilst such questions are of interest, indeed to many, they belong on a blog and not on MO.
I've been described as being a bit "hard-line" on this, so this should definitely be taken as my own opinion and not the opinion of anyone official from MathOverflow. My view is that MathOverflow is for questions relating to a person's actual research that can be definitely answered without significant effort on the part of the answerer.
This question is not related to someone's research and cannot be given a definite answer. So I consider it not suitable for MO.
However, as I have already cast a vote to close on this question, I cannot vote to close it again even if it is reopened.
I will not delete my answer as I see nothing wrong with it but if you truly do consider it offensive then I suggest that you contact a real moderator and ask them to delete it.
Since I cast the first vote to close, let me say a little bit about my thought process.
As Scott Carnahan suggests, my reaction to the question was: well, since pi is transcendental, if it were also an integer or an algebraic number then mathematics would be contradictory. So what would go wrong? Everything.
When I clicked through to look at the profile of the OP, I noted that s/he has asked and answered sophisticated, research-level questions in (e.g.) algebraic geometry, so I realized that surely s/he was familiar with the "answer" I had in mind above. So probably there was more to the question than was apparent to me.
In this case, my vote to close should be interpreted to mean "Please clarify" not "Please remove this question from my sight". It is a well-known problem that it is hard for a user to hear the closure of their question as "Please clarify", but at least in this case some helpful clarification after closure does seem to have taken place.
Dear JME, I should make clear that I am not a moderator. I have some powers, granted by the reputation system, but I'm in every respect a common user of MathOverflow. That said, I called the question nonsense exactly because of what Scott said. If pi was rational, since we have a proof that it is not, we could draw contradiction. So the only possible answer I see is: if pi was rational, then we can prove anything you want. For instance if pi was rational, 64 would be odd. I guess this is not was you expect from an answer.
The problem is that every sentence like "what if..." only makes sense if the hypothesis is conceivable, that is, not logically contradictory. I wrote "nonsense" with precisely with this meaning: the question as asked has no meaning.
What bothers me is the way you and Andrea feel ok to bully the next person while being supported by some members of the community
"bully"? I fail to see how my answer can be construed as bullying! It was frivolous, completely so, and I tried by my tone to ensure that no-one reading it could take it otherwise.
I think that it is wrong and I made my point clear
Yes and no. I agree completely that bullying is wrong but no, you did not make your point clear. Which part of my answer do you construe as bullying?
I also think that you know it yourself in view of your first comment ...
Wrong. My first comment referred to the fact that I left a frivolous answer and that I know that MO is a place for serious research so not the place to indulge in frivolity. If you look at the most recent discussion that I started here you will see two things: I do like being frivolous from time to time (though I usually confine myself to being frivolous here rather than on MO) and that I have just read the book Going Postal so the episode referred to was uppermost in my mind. That was the "coincidence of timing" that I referred to in my first comment.
NOW WE SEE THE VIOLENCE INHERENT IN THE SYSTEM (reprise)
P.S. In case it needs to be said, I agree wholeheartedly that the doling out of magical swords by aquatic females is not a sound basis for government. [Can't be bothered to look up Idle's exact line.]
I'm happy to oblige...
Listen, strange women lying in ponds distributing swords is no basis for a system of government. Supreme executive power derives from a mandate from the masses, not from some farcical aquatic ceremony.
Dear JME, there must be an enormous mistunderstanding. I have no intention whatsoever to bully, and I'm sorry that you perceived as such. I tried to explain better in my next answer here, but apparently I didn't get it right.
By nonsense I mean that the question as stated has no possible interpretation. This is not to dismiss the question, it is a statement of a fact. I am really not able to attach any meaning to a question which starts "What if (insert any logical contradiction here)...?". It is a feature of the English syntax that it allows to build hypothetical clauses that have no semantic meaning, since the hypothesis cannot be satisfied, not even hypothetically.
I hope this clarifies my issue. I do not have any intention to draw a personal attack or be dismissive.
By the way, for what concerns Andrew's answer, I think you should allow some humour. :-) I'm certain he wrote that in good faith.
As far as I am concerned the controversy is over
I'm afraid that it isn't. You've accused me (and Andrea) of bullying and of others as being complicit in it: "while being supported by some members of the community". In addition, you do not use your name nor do you provide an easy way to contact you to deal with the matter privately, so this is the only avenue that I have available to resolve the matter.
I have just deleted my flippant answer because flippancy really does have no place on MO and this matter has underlined that for me. Yet again I am reminded of why I like it that discussions are not good on MO.
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