Friday, January 20, 2012

Studying Tactics

My friend asked me for help on military tactics. "Where'd you get the idea I know anything about that?" I asked.

"I thought game theory was known to all mathematicians," my friend sheepishly replied.

Well, he's wrong about that. But I later found Clausewitz's suggestion on examples:

EXAMPLES from history make everything clear, and furnish the best description of proof in the empirical sciences. This applies with more force to the Art of War than to any other. General Scharnhorst, whose handbook is the best ever written on actual War, pronounces historical examples to be of the first importance, and makes an admirable use of them himself. Had he survived the War in which he fell, the fourth part of his revised treatise on artillery would have given a still greater proof of the observing and enlightened spirit in which he sifted matters of experience.

But such use of historical examples is rarely made by theoretical writers; the way in which they more commonly make use of them is rather calculated to leave the mind unsatisfied, as well as to offend the understanding. We therefore think it important to bring specially into view the use and abuse of historical examples.


Now, if we consider closely the use of historical proofs, four points of view readily present themselves for the purpose.

First, they may be used merely as an EXPLANATION of an idea. In every abstract consideration it is very easy to be misunderstood, or not to be intelligible at all: when an author is afraid of this, an exemplification from history serves to throw the light which is wanted on his idea, and to ensure his being intelligible to his reader.

Secondly, it may serve as an APPLICATION of an idea, because by means of an example there is an opportunity of showing the action of those minor circumstances which cannot all be comprehended and explained in any general expression of an idea; for in that consists, indeed, the difference between theory and experience. Both these cases belong to examples properly speaking, the two following belong to historical proofs.

Thirdly, a historical fact may be referred to particularly, in order to support what one has advanced. This is in all cases sufficient, if we have ONLY to prove the POSSIBILITY of a fact or effect.

Lastly, in the fourth place, from the circumstantial detail of a historical event, and by collecting together several of them, we may deduce some theory, which therefore has its true PROOF in this testimony itself.

For the first of these purposes all that is generally required is a cursory notice of the case, as it is only used partially. Historical correctness is a secondary consideration; a case invented might also serve the purpose as well, only historical ones are always to be preferred, because they bring the idea which they illustrate nearer to practical life.

The second use supposes a more circumstantial relation of events, but historical authenticity is again of secondary importance, and in respect to this point the same is to be said as in the first case.

For the third purpose the mere quotation of an undoubted fact is generally sufficient. If it is asserted that fortified positions may fulfil their object under certain conditions, it is only necessary to mention the position of Bunzelwitz [Frederick the Great's celebrated entrenched camp in 1761] in support of the assertion.

But if, through the narrative of a case in history, an abstract truth is to be demonstrated, then everything in the case bearing on the demonstration must be analysed in the most searching and complete manner; it must, to a certain extent, develop itself carefully before the eyes of the reader. The less effectually this is done the weaker will be the proof, and the more necessary it will be to supply the demonstrative proof which is wanting in the single case by a number of cases, because we have a right to suppose that the more minute details which we are unable to give neutralise each other in their effects in a certain number of cases.

If we want to show by example derived from experience that cavalry are better placed behind than in a line with infantry; that it is very hazardous without a decided preponderance of numbers to attempt an enveloping movement, with widely separated columns, either on a field of battle or in the theatre of war—that is, either tactically or strategically—then in the first of these cases it would not be sufficient to specify some lost battles in which the cavalry was on the flanks and some gained in which the cavalry was in rear of the infantry; and in the tatter of these cases it is not sufficient to refer to the battles of Rivoli and Wagram, to the attack of the Austrians on the theatre of war in Italy, in 1796, or of the French upon the German theatre of war in the same year. The way in which these orders of battle or plans of attack essentially contributed to disastrous issues in those particular cases must be shown by closely tracing out circumstances and occurrences. Then it will appear how far such forms or measures are to be condemned, a point which it is very necessary to show, for a total condemnation would be inconsistent with truth.

It has been already said that when a circumstantial detail of facts is impossible, the demonstrative power which is deficient may to a certain extent be supplied by the number of cases quoted; but this is a very dangerous method of getting out of the difficulty, and one which has been much abused. Instead of one well-explained example, three or four are just touched upon, and thus a show is made of strong evidence. But there are matters where a whole dozen of cases brought forward would prove nothing, if, for instance, they are facts of frequent occurrence, and therefore a dozen other cases with an opposite result might just as easily be brought forward. If any one will instance a dozen lost battles in which the side beaten attacked in separate converging columns, we can instance a dozen that have been gained in which the same order was adopted. It is evident that in this way no result is to be obtained.

Upon carefully considering these different points, it will be seen how easily examples may be misapplied.

An occurrence which, instead of being carefully analysed in all its parts, is superficially noticed, is like an object seen at a great distance, presenting the same appearance on each side, and in which the details of its parts cannot be distinguished. Such examples have, in reality, served to support the most contradictory opinions. To some Daun's campaigns are models of prudence and skill. To others, they are nothing but examples of timidity and want of resolution. Buonaparte's passage across the Noric Alps in 1797 may be made to appear the noblest resolution, but also as an act of sheer temerity. His strategic defeat in 1812 may be represented as the consequence either of an excess, or of a deficiency, of energy. All these opinions have been broached, and it is easy to see that they might very well arise, because each person takes a different view of the connection of events. At the same time these antagonistic opinions cannot be reconciled with each other, and therefore one of the two must be wrong.

Much as we are obliged to the worthy Feuquieres for the numerous examples introduced in his memoirs—partly because a number of historical incidents have thus been preserved which might otherwise have been lost, and partly because he was one of the first to bring theoretical, that is, abstract, ideas into connection with the practical in war, in so far that the cases brought forward may be regarded as intended to exemplify and confirm what is theoretically asserted—yet, in the opinion of an impartial reader, he will hardly be allowed to have attained the object he proposed to himself, that of proving theoretical principles by historical examples. For although he sometimes relates occurrences with great minuteness, still he falls short very often of showing that the deductions drawn necessarily proceed from the inner relations of these events.

Another evil which comes from the superficial notice of historical events, is that some readers are either wholly ignorant of the events, or cannot call them to remembrance sufficiently to be able to grasp the author's meaning, so that there is no alternative between either accepting blindly what is said, or remaining unconvinced.

It is extremely difficult to put together or unfold historical events before the eyes of a reader in such a way as is necessary, in order to be able to use them as proofs; for the writer very often wants the means, and can neither afford the time nor the requisite space; but we maintain that, when the object is to establish a new or doubtful opinion, one single example, thoroughly analysed, is far more instructive than ten which are superficially treated. The great mischief of these superficial representations is not that the writer puts his story forward as a proof when it has only a false title, but that he has not made himself properly acquainted with the subject, and that from this sort of slovenly, shallow treatment of history, a hundred false views and attempts at the construction of theories arise, which would never have made their appearance if the writer had looked upon it as his duty to deduce from the strict connection of events everything new which he brought to market, and sought to prove from history.

When we are convinced of these difficulties in the use of historical examples, and at the same time of the necessity (of making use of such examples), then we shall also come to the conclusion that the latest military history is naturally the best field from which to draw them, inasmuch as it alone is sufficiently authentic and detailed.

In ancient times, circumstances connected with War, as well as the method of carrying it on, were different; therefore its events are of less use to us either theoretically or practically; in addition to which, military history, like every other, naturally loses in the course of time a number of small traits and lineaments originally to be seen, loses in colour and life, like a worn-out or darkened picture; so that perhaps at last only the large masses and leading features remain, which thus acquire undue proportions.

If we look at the present state of warfare, we should say that the Wars since that of the Austrian succession are almost the only ones which, at least as far as armament, have still a considerable similarity to the present, and which, notwithstanding the many important changes which have taken place both great and small, are still capable of affording much instruction. It is quite otherwise with the War of the Spanish succession, as the use of fire-arms had not then so far advanced towards perfection, and cavalry still continued the most important arm. The farther we go back, the less useful becomes military history, as it gets so much the more meagre and barren of detail. The most useless of all is that of the old world.

But this uselessness is not altogether absolute, it relates only to those subjects which depend on a knowledge of minute details, or on those things in which the method of conducting war has changed. Although we know very little about the tactics in the battles between the Swiss and the Austrians, the Burgundians and French, still we find in them unmistakable evidence that they were the first in which the superiority of a good infantry over the best cavalry was, displayed. A general glance at the time of the Condottieri teaches us how the whole method of conducting War is dependent on the instrument used; for at no period have the forces used in War had so much the characteristics of a special instrument, and been a class so totally distinct from the rest of the national community. The memorable way in which the Romans in the second Punic War attacked the Carthaginan possessions in Spain and Africa, while Hannibal still maintained himself in Italy, is a most instructive subject to study, as the general relations of the States and Armies concerned in this indirect act of defence are sufficiently well known.

But the more things descend into particulars and deviate in character from the most general relations, the less we can look for examples and lessons of experience from very remote periods, for we have neither the means of judging properly of corresponding events, nor can we apply them to our completely different method of War.

Unfortunately, however, it has always been the fashion with historical writers to talk about ancient times. We shall not say how far vanity and charlatanism may have had a share in this, but in general we fail to discover any honest intention and earnest endeavour to instruct and convince, and we can therefore only look upon such quotations and references as embellishments to fill up gaps and hide defects.

It would be an immense service to teach the Art of War entirely by historical examples, as Feuquieres proposed to do; but it would be full work for the whole life of a man, if we reflect that he who undertakes it must first qualify himself for the task by a long personal experience in actual War.

Whoever, stirred by ambition, undertakes such a task, let him prepare himself for his pious undertaking as for a long pilgrimage; let him give up his time, spare no sacrifice, fear no temporal rank or power, and rise above all feelings of personal vanity, of false shame, in order, according to the French code, to speak THE TRUTH, THE WHOLE TRUTH, AND NOTHING BUT THE TRUTH.

Addendum: one may be interested in Napoleon's Maxims, Patton's Reading List, and Roosevelt's Pigskin Library.

Wednesday, January 11, 2012


Recently I've been more interested in puzzles.

Project Euler is the classic example of puzzles which require either higher math or computational skill (or both!).

Facebook has a collection of puzzles too, motivated from the engineering perspective.

But note: Facebook uses these puzzles for hiring people. Plus, the puzzles are not always mathematically oriented.

I suppose a good mathematician should always set up puzzles for themselves. As Socrates remarked:

SOCRATES: Indeed, Lysimachus, I should be very wrong in refusing to aid in the improvement of anybody. And if I had shown in this conversation that I had a knowledge which Nicias and Laches have not, then I admit that you would be right in inviting me to perform this duty; but as we are all in the same perplexity, why should one of us be preferred to another? I certainly think that no one should; and under these circumstances, let me offer you a piece of advice (and this need not go further than ourselves). I maintain, my friends, that every one of us should seek out the best teacher whom he can find, first for ourselves, who are greatly in need of one, and then for the youth, regardless of expense or anything. But I cannot advise that we remain as we are. And if any one laughs at us for going to school at our age, I would quote to them the authority of Homer, who says, that

'Modesty is not good for a needy man.'

Let us then, regardless of what may be said of us, make the education of the youths our own education. (Emphasis added, from Plato's Laches)

For example, I know a little bit about representations of Lie groups and Lie algebras (one can always learn more!)...but what about the representation of the quaternion group induced from the irreducible representations of SU(2)? How does it decompose into irreps? Etc.

Knuth remarked somewhere what helped him understand the representation theory for the symmetric group was writing a program which generated the permutation matrix representations.

I suspect writing a program which does these sorts of computations is a great puzzle for any mathematician that's savvy with programming.

Reading Material

And now, for something completely different.

A few papers I want to read:
When physics helps mathematics: calculation of the sophisticated multiple integral, 13 pages;
Some algebraic properties of differential operators, 15 pages;
Introduction to supergravity, 152 pages;
Fermionic impurities in Chern-Simons-matter theories, 31 pages;
Spinors and Twistors in Loop Gravity and Spin Foams, 16 pages;
Quaternionic Analysis, Representation Theory and Physics, 60 pages.

Friday, January 6, 2012

Problem Notebooks

I've been looking for the infamous Kourovka notebook, but there are apparently others.

Recall the Kourovka notebook is a collection of open problems in group theory. There are other notebooks with open problems in other fields.

For example, the Dniester Notebook [] (pdf, 56 pages) has problems in ring theory.

The notebook stopped being published years ago. Now it's been open sourced.

The Sverdlovsk notebook stopped publishing back in 1989; it was a collection of open problems for semigroups, but it cannot be found.

Other problem notebooks might exist too, but I am unaware of them...

People don't read anymore :(

I have been wandering around book stores recently, and stumbled upon the Landmark Herodotus. If you recall, I wrote a few posts on Herodotus.

People ought to be reading the text, and produce their own notes, producing a similar result (as Landmark) which is personalized.

Don't mistake me: I think Landmark is a wonderful resource! It has helped with maps, and so on...but people should be doing this on their own.

The problem Landmark posed (to me) parallels Cliff Notes. The books are produced as an example of how to take notes while reading books...not a replacement for reading!

Extemporaneously producing maps is an invaluable skill. The map doesn't have to be precise, e.g. drawing Asia minor as a rectangle, or Greece as three rectangles.

There are some elements to, e.g., the Landmark Herodotus that are pleasant, and makes it a great resource to borrow from. For example: what did Herodotus get wrong?

It appears that he got a lot of Egyptian history wrong, for example.

But my point is: reading is studying! You have to make a book your own through notes, maps, references to other books, etc.

How you study is entirely up to you; how you take notes, well, that is entirely up to you. But reading without taking notes is like dancing: nobody does it unless drunk or insane.

People just confuse the mechanics of reading (parsing letters into words) with the procedure called "reading" (obtaining information from books, and evaluating it). *sigh*

P.S. Happy New Year!