Objectivity And Mathematics

"There is a general principle that a stupid man can ask such questions which one hundred wise men would not be able to answer." - Vladimir Arnol'd, famous Russian Mathematician.

"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!

On Privatization

Well, this is not an article on public economic policy written from a magisterial perch. This is ... well, let us get going...

Pune is reputed to have one of the worst public transport systems in the country. The service is run by the city government. Run-down buses, inadequate buses, unreasonable fares... it is a full list of woes. Of late, things have shown some improvement but...

Cochin is an urban area of comparable size (including long strings of satellite towns and urbanized villages radiating from the down-town Ernakulam area) and has, on the other hand, a fully privatized bus network (yes, allegedly commie Kerala). In rush-hours, the frequency of buses is impressive and their service is often impressive as well, with energetic 'kilis' canvassing and grabbing potential commuters with gusto. And the fares are considerably less than in Pune.

Last week. I touch down at Ernakulam railway station at 5 am and look for city buses to Tripunittura, a suburb about 9 kilometers away. No luck. The autowallahs ask for "150 bucks". I decide to walk, not all the way (the luggage is heavyish) but until the first city bus catches up with me and which I could flag down... Cochin is a fairly early rising city and there are plenty people around but no buses... Almost hour later, I reach Vyttila junction (more than halfway to my destination), give up (the weather has been rather humid, even in 'winter') and wait among a group of nearly a dozen people for the still pending first bus - which arrives just past 6 am, almost full.

Today. I reach Pune station at almost precisely 5 am. The weather is not very chilly, just energizingly cool. So, I decide to walk to my apartment, almost 7 kilometers away. The luggage begins to weigh me down after 2-3 kilometers (when was the last time I did some exercise for these ageing shoulders?) but I press on, ignoring a few autowallahs who hover around hopefully. At 6.15 I am home. And yes, I counted. No less than 6 city buses overtook me during the walk, at fairly regular intervals. They were all very sparsely filled and must have all been loss-making trips, even at the Pune bus rates.

'Lateral Thinking'

1.Here is a cricket puzzle circulating in the sms circuit:
One day match. 3 balls to go; 7 more runs to get. Two batsmen Rahul (on strike) and Sachin (runner) are 'in'. Both are on 94. The match is won and both batsmen get *unbeaten* hundreds. How is this possible?

Here is a 'lateral thinking' answer floating around:
The two were playing *different* matches. So, what is the problem??

And here is a rather straightforward *straight* answer, which anyone who knows his cricket can get:

- The first of the 3 remaining balls is bowled. Rahul has an almighty swing and connects. The ball goes for six. The scores are tied. But alas, he has pulled a muscle very badly and collapses in a heap; he can't even get up and celebrate his century, let alone face the next ball. Rahul helped out of the ground. A new batsman Sourav takes his place.

- The second ball bowled. Sourav pulls and top-edges. The keeper runs around to hold; Sourav is out; no run added, scores still tied. But while the ball was swirling in the air, the batsmen have crossed. Sachin is on strike, on 94. 1 run to win, 1 ball to go.

- The rest is simple. The last ball is overpitched on the legstump. Sachin (when was the last time he did such a thing?) flicks it high and it just about sails for six. That is that.

2. Another problem from IIT-JEE of around 1980: An anti-aircraft gun fires a volley of four shells in quick succession at an intruding enemy fighter. The first shell has a probability of 0.9 of finding the target, the second 0.8, the third 0.7 and the fourth, 0.6 to knock out the plane. What is the probability that the gun was able to hit the plane?

I will only give a *laterally thought* answer for this: "Zero!" The *gun* can never hit the plane, although the *shell* might!!"

This reminds me of another set of questions, maybe less serious(?) but requiring similarly lateral thinking to crack:

Q1: "What is a XXXX (let me leave this word blank for obvious reasons) throws a grenade at you?"
Answer: "Don't worry. (S)he won't have pulled out the pin!"

Q2: "But what if the XXXX pulls out the pin from a grenade and throws it at you?" Answer: "Again; don't worry. (S)he has thrown the pin!"

'Jigrr!'

I am old enough to be among those were fond of the Russian bear 'Misha' - the mascot/symbol of the 1980 Moscow Olympics - and our own elephant 'Appu' (of the Delhi Asiad, 1982). Indeed, one of the debates of those days was whether Misha or Appu was 'better'.

And there was big-time disappointment when the bald eagle 'Sam' (1984 LA Olympics) was unveiled. Things went downhill thereafter and the mascot of every single Olympics/Asiad that followed has been a total washout (Sam is remembered only in comparison to Misha). From the entire period: 1980-present, only two other mascots have stayed in my memory. 'Raju' of the 1985 (inaugural) National Games and 'Nandu' of the Bangalore-Mysore National Games of around 2000 - Raju was a rather stiff-looking tiger but he was still funny in a vague 'Hobbesian' way. I remember Nandu the bull for the pun in the name (with 'Nandi' of mythology; and the bull is a symbol of both Mysore and Bangalore)and also as a 'namesake!

Yesterday's Pune Marathon featured another memorable mascot - 'Jigrr' the tiger. Appearance-wise he is pretty standard (a jolly tiger dressed as a runner) but his is an absolute winner - a mix of the Hindi/Urdu 'Jigar' (literally, 'liver' but also 'courage') and 'tiger' and of course an emphatic 'grr!'. But he is a sad victim of under-exposure - a google search gave exactly 3 pages and there don't seem to be more than half a dozen hoardings in the entire city declaring "Jigrr welcomes you to the Pune Marathon!".

'Not A Minute More, Not A Minute Less'

Well, that has been the story of the Pune Marathon, for self, this year.

I had clocked approximately 65 minutes in the 10 km segment last year - approximately' because I had no watch and went by the timing of chaps who finished 'around' me. This year, I had an analog watch and it showed 65 minutes almost to the second. There is a bit of disappointment there; I was at the 7 kilometer mark at just over 40 minutes - the same pace maintained could have given a sub 60 minute timing; and *that* would have called for a few 'Planter's Punches' (as of now, my favorite drink). But then, the last few kilometers were painfully leaden footed and the final nearly 1 kilometer straight stretch, seemingly interminable - I just about ambled over the finish (I never *walked* though).

A very beefy Salman Khan and Sunil Gavaskar, all smiles, flagged us off and that was a solid shot of adrenalin (there were a few pretty faces as well, up there in the VIP box) but the rest of the event gave less of a high than it did last year. There were fewer spectators and flag-waving kids on the way and fewer bands playing 'inspirational' songs. And there was a discordant note as well - a smartly dressed man came across the road carrying a kid, halted a second on seeing me approach and then suddenly strode right across my path - and nearly knocked me over.