Mathematics and War.

[sjastro]

Given the recent emotive threads regarding ANZAC day here is the cold and clinical application of mathematics to war and peacetime strategies.

Game theory was originally developed by mathematicians as an application to economics but was adopted by both camps of the Cold War to analyse the potential outcomes of various military strategies.

The use of Game Theory to the Cuban missile crisis led to the adoption of a peaceful solution rather than the onset of World War III.

http://science.howstuffworks.com/game-theory.htm

Regards

Steven

[madbadgalaxyman]

Good maths, and the cruise missile flies through the window of the terrorist's bedroom, hitting his bed with a small explosive charge, thereby despatching him to the hereafter.

Bad maths; a large explosive charge delivered to the target inaccurately, with massive casualties to non-combatants.

"Money forms the sinews of war" (and also good mathematics!!)

[GeoffW1]

Well,

The application of maths to war goes back further. Many know of Bletchley Park, Alan Turing and the code cracking that went on there. I've always found that morbidly fascinating after having to study Operations Research (similar sort of thing) in my misspent youth.

Some cruder things this sort of analysis was used for:

- how deep to set the depth charges, the better to puree people
- was it better to have many machine guns or fewer but powerful cannon on a Spitfire, the better to ventilate people
- how far apart to put the land mines (ghastly) the better to blow people up (or yourself, sometimes)
- what was the optimum number of ships in a convoy, the better to avoid getting sunk
- and more, sadly

On this Anzac weekend, I wish they had analysed whether it was better to shoot or talk.

To all those who did not come home...:thanx:

[Weltevreden SA]

A certain kind of mind will turn anything into something to hurt with. The first stone tool scraped leather one day and slashed somebody the next. The first sharp stick dug a furrow for seeds in the morning and wounded somebody in the afternoon. I can't imagine what goes on in a planning meeting to design a cluster bomb or phosphorous grenade, but I have a feeling mathematicians weren't there. As astronomers, we represent the noblest aspiration humans have come up with: to understand why the microcosm and macrocosm are one. It feels great to admire the sky's glory, but feels better to look in the mirror and respect what's looking back.

[sjastro]

This is an example where Game Theory is applied.
The ideal "payoff" (a term used in Game theory) is the missile strikes the target, kills all the terrorists with no collateral damage to the non combatants.

The least ideal payoff is the missile completely misses the target, none of the terrorists are killed, non combatants are killed, and the incident creates a breeding ground for more terrorists.

Then there is in-between case, some terrorists and some non combatants are killed in the strike.

Each case has a particular payoff, game theory analyses whether the ends justifies the means from a purely clinical viewpoint, without moral or ethical considerations.



One of the earliest applications of mathematics to war was calculation of artillery trajectories in the 16th century, in particular when the target was at a higher elevation than the artillery pieces.
This was particularly impressive given it was done a 150 years before Newton and the birth of modern physics.

Regards

Steven

[gary]

Hi Steven,

This is not pure coincidence.

Game Theory first hits the big time when John Von Neumann and
Oskar Morgenstern published the seminal 616-page book of the
"Theory of Games and Economic Behaviour" in 1944.

This is the book that originates the term "zero-sum game".

When it was first published it was greeted with reviews that have
been described as extraordinary in their volume and quantity.

Von Neumann had published an earlier paper on the theory of games
in 1928.

In the year "Theory of Games and Economic Behaviour" was
published, Von Neumann was working at Los Alamos on the atomic
bomb.

Later, he was a consultant for the RAND Corporation in Santa Monica,
which was the US think tank for the Cold War.

So it is no coincidence with this connection that Game Theory was
further heavily developed at RAND.

John Neumann was not just a genius, but by numerous accounts of any of the
scientists and mathematicians that ever encountered him during his life,
he is usually described as the smartest person they ever met.

[madbadgalaxyman]

As you said in your post, he was keenly interested in applying mathematics to really tough problems in economics and politics and military strategy. Yes, his politics were 'somewhat right of Genghis Khan', but one has to understand that he came from a country that was several times the plaything of totalitarian ideologies such as Nazism and Communism, therefore the feeling was at the time that a muscular and aggressive military posture was the only way to deter further aggression from these sorts of extremist regimes (as distinct from a pacifist posture in which you let your country be overrun by a large gang of organized and fanatical thugs)

Even very smart people like the great physicist Hans Bethe sometimes wondered if von Neumann was actually a new species, as the great speed of his thought and his remarkable memorization abilities were constantly remarked on, even by other super-smart people.

We learn in IT class that the stored program computer in common use has the "von Neumann" architecture.
Of course this could eventually end up as being one of those "oft repeated attributions" that ends up being proved wrong after further historical analysis.....however, I think that all would agree that John von Neumann was a very important figure in many and diverse fields such as pure mathematics, information theory and computing, quantum mechanics, game theory, cellular automata (self-replicating machines and robotics), economics, military theory and policy, and even in the life sciences and meteorology!!

Apparently he did not have the classic nerdy and introverted "mathematician's personality" when in company; he was extroverted, 'quick' with a quip, and sociable.

[gary]

Hi Robert,

Thanks for the post.

It proved to be an accidental and unfortunate attribution.

The successor to ENIAC was to be EDVAC and Von Neumann came to the
project by chance.

ENIAC had used an awkward plug-board to program it. When Mauchly
later developed the mercury delay line, it became practical in an
engineering sense to afford the stored-program paradigm and
it became incorporated into the plans for EDVAC.

Von Neumann volunteered to draft an internal memorandum documenting
EDVAC's design and Eckert, Mauchly and the team accepted, grateful
to have him on-board.

Unfortunately Goldstine, who was administrator of the project, more
widely circulated this incomplete draft report which only had Von Neumann's
name on the cover and attributions were to be added later.

When Von Neumann left the ENIAC group along with Goldstine
to work on the Institute of Advanced Studies (IAS) machine in the
town of Princeton, they further elaborated the stored-program
concept. This is why Von-Neumann architecture machines are also
often referred to as Princeton architecture machines.

The stored-program paradigm lends itself to the idea of self-modifying
code and this was elaborated in several reports by the IAS group.
However, they mainly restricted themselves to the idea of using it
for a computed address modification similar to what an index register
does on some machines.

The concept of a linear address space and a Program Counter would
all flow out of this work.

I've often wondered if Von Neumann had lived longer whether he would
have formulated a computer that exploited the self-modifying code
concept much further in some weird and wonderful way, much akin
to his work on self-reproducing automata.

Toward the end of his life he wrote a beautiful little book called "The
Computer and the Brain". There use to be a copy in the UNSW library
which I read decades ago.

Getting back to the mathematics and war theme, Von Neumann's
interest in ENIAC was not only academic but for him of pressing
importance. He was a major part of the brain power behind the
implosion device at Los Alamos and designing and computing
the spherical charges to uniformly implode the plutonium sphere
was a incredibly challenging problem. Von Neumann brought to
the project the concept of solving the nonlinear partial-differential
equations numerically and they were employing rooms full of
woman with Marchant calculators crunching the numbers.

When he had a chance meeting with Goldstine at a railway station
and Goldstine told him he was working on ENIAC, he instantly
recognized the utility it had to the problem at hand. A problem that
could bring about a quick end to the war.

[madbadgalaxyman]

Thanks Gary, that is very very informative and very interesting.

In this case, I think that calling computers "von Neumann machines "is obviously a bad thing to do. I wonder if they still teach this at uni?

Credit is always a complicated thing in science and technology, as we found out when we looked into Hubble's discoveries in an IIS science thread. In Hubble's case, he probably discovered noticeably less than he is usually credited with, but the von Neumann case seems like a straightforward case of attributing credit in a way that is grossly unfair to the people who did the work.

[avandonk]

If you look at game theory it is better to cooperate than be very greedy. Then you maximise a good outcome for all.

With all out war ie non cooperation everybody loses apart from a few!

CSIRO had the first computer in the early fifties with a magnetic storage drive at Melbourne Uni.

It's clock speed was 1.1 kHz?

Bert