Objectivity And Mathematics
"It is comparatively easy to make clever guesses; indeed there are theorems .... which have never been proved and which any fool could have guessed." - G.H.Hardy
I heard the statement: "Objectivity is the sum total of all subjectivities" a few months back. Although it sounds smart, it actually just a bit of sentiment. Indeed, in politics, for example, this maxim readily yields the corollary "the majority is always right" which is nonsense.
And as I seem to be discovering in recent times, even in the realm of Mathematics, where one is usually told supreme certainty ought to reign, things can be very interestingly murky. Let me get on with the story.
A Geometric guess (a 'Conjecture' to use an academic word) was floated by Ramana Rao and self well over a year ago - it was something we could neither prove nor disprove. Here it is: "Given any positive integer N, any convex polygonal region allows partitioning into N convex pieces so that every piece has the same area and the same perimeter"
I took over the 'marketing aspect' of the conjecture and wrote to several academicians for comments/insights - several of them big international names; I also discussed it with several smart folks I know.
(Note: as given above, our claim was that a certain geometric property holds for all positive integers, starting from 2, 3,... and so on. Most details are in the blog 'Tech Musings', linked on the right panel. See the 'Fair Partitions' posts)
And here are some of the responses (many of them un-edited and completely reproduced) I received over the year 2007; the following list reminds me of an earlier post here on eating beef :)
1. Interesting conjecture. Guess it is one of those claims which will be hard to prove or disprove. If I have to give a vote, I would say "True"
2. Very interesting. My intuition tells me the claim is *not true* in general. I hope to work on it when I have time; it should give a very nice paper.
3. Is the claim valid for N=2?
4. N=2 is simple. N=3 onwards is not at all obvious. It looks very difficult...
5. I am sorry I do not know of any earlier work which can help you with the problem.
6. Sorry. No idea!
7. I think the problem you pose is very pretty. If it is okay with you, I shall discuss it with some colleagues.
8. Not my field. Why don't you approach ---- or -----?
9. N=2 is easy to prove. N=3 and above are potentially very interesting. Actually, I doubt if the claim holds for N=3.
10. N=2 is obvious. N=3 is believable to me. Perhaps.... Beyond N=3, I am not sure I believe it!
11. I think the problem you pose is somewhat artificial. Not that everybody works only on 'natural' problems!
12. I think your claim is true; let me try to construct a proof.
13. I will try to work out at least a partial solution; in fact this problem looks very simple. And btw, will you be upset if I crack it in a day or two :)?!
14. I will be very surprised if nobody has ever thought of this thing. Put any reasonably smart highschooler in a library and ask him to think geometry and he would ask this question in a week!
15. Who knows, but this could be Geometry's answer to the Goldbach conjecture!
16. Yes, as you claimed, you made me understand the problem in 2 minutes flat. I certainly can't believe nobody ever thought about it. If that is indeed the case, it would be a miracle!
17. Looks like your claim can be proved by induction... but then, I like problems which I can relate to in real life, something I can construct, visualize... This is the kind of thing I really do not care about!
And yes, Response Numbers 18 to around 50: Silence.
Turing Award winner Richard Hamming has remarked that the average research paper is read by 3 people - the author, the referee and perhaps one more person. If that is indeed the case, this conjecture already is a hugely above average achievement!