Thucydides as Geometry
Russell McNeil
October 21, 1997


In his "History of the Peloponnesian War," Thucydides 
clearly wants his readers to have us believe that the objects 
under discussion are subject to a universal law. It means 
one can generalize these observables here into a system of 
rules about human nature operating in community--politics.  
The system comes across as a "political science."  The text 
is extraordinarily seductive in this regard.   Thucydides' 
appeal to the Greek respect for geometric precision is 
purposeful but misplaced.  

This instinct however to apply the "grammar" of geometry to 
history -- and indeed to other media as well -- such as 
music, art, poetry, architecture -- is important. It might bear 
fruit. The danger in any application of a "universal grammar" 
is that the objects one trys to manipulate with geometry may 
not be subject to the grammar.  

Geometric rules might apply to history. But it is not right to 
assume that "fear, power, and self interest" are the objects 
one can manipulate geometrically -- as if equivalent to 
points, lines and surfaces.  I'll say more on this toward the 
end.

Had Thucydides been born a century later (he was born 
about 460) it is entirely possible he would have contributed 
more to Greek mathematics and science than to history. 
The strength and influence Thucydides exerts on the Greek 
mind draws in part from it's detached vantage. Euclid, and 
all the mathematical thinking that laid the groundwork for 
Euclid flourished in large part because it constituted a 
logically consistent system with explicit rules and 
assumptions in which all rational observers had to draw 
identical conclusions.  This "impulse" exerted a strong 
influence on the Greek mind and is clearly evident in the 
work of Thucydides.

Thucydides' explanation for the Peloponnesian War focuses 
on empire and power. War arises when power begins to 
shift. In fact, Thucydides provided the basis for the so-called 
"balance of power" politics which the Western tradition has 
used and still uses to underpin its thinking for over two 
millennia.  Because this amoral explanation of political 
"reality" emerges from what appears to be a geometric 
framework,  we tend to buy into the idea more readily than if 
we understood it more for what it is, the carefully contrived 
opinion of a clever thinker who applies the grammar of 
geometry to objects undefined by geometry. 

Thucydides "detached vantage" as an objective allows him 
to probe beneath the surface reasons for war to reveal those 
hidden forces (power, fear, and self interest) that are really 
responsible for events.

I think many of are impressed and persuaded that 
Thucydides really has uncovered some truths about human 
nature and war because of his "detached vantage" and 
because this account unfolds for us as "systematic and 
formulated knowledge," which in very general terms is how 
we define science.  

First, he convinces us that he has "gotten the facts straight." 

Second, he persuades us that there really are "objective" 
facts about the war that can be gotten straight.  In other 
words, if science is systematic and formulated knowledge, 
there must be a body of things "out there" that we can 
systematize and formulate! 

And third, he filters these "properly gotten objective facts" 
through a "model" of political "reality" he is persuaded is 
"right." 

Thucydides takes great pain to assure us that he has gotten 
his facts straight:

Thuc. 1.22.2-3 And as for the real action of the war, I did 
not think it right to set down either what I heard from 
people I happened to meet or what I merely believed to 
be true. Even for events at which I was present myself, I 
tracked down detailed information from other sources 
as far as I could. It was hard work to find out what 
happened, because those who were present at each 
event gave different reports, depending on what side 
they favored and how well they remembered.

This passage is often used to document the pains 
Thucydides used to ensure observational accuracy. It 
implies that Thucydides' facts are independent of his 
subjectivity: that there are objective facts separate from, and 
in theory identical for all observers. This incidentally is the 
"attitude" all "good" journalists assume when reporting on 
the world in their, "detached," "objective," "thorough," and 
"unbiased" reportage. If equally endowed observers of the 
same phenomena ever produced different results, we would 
have a problem.

I mentioned the three elements in Thucydidean process: 
getting facts straight, believing that there are objective facts, 
and the filtering of these objective facts through a "model of 
political reality."  This political "vision" is the engine that 
works on the facts; orders them; prioritizes them; classifies 
them; does geometric operations on them; and generally 
synthesizes higher order relationships. It is a complex 
intellectual device. 

The "model of political reality" is used to determine not only 
which facts are relevant but to determine how Thucydides 
reports the various speeches. It is here I think that 
Thucydides is most creative:

Thuc. 1.22.1 What particular people said in their 
speeches, either just before or during the war, was hard 
to recall exactly, whether they were speeches I heard 
myself or those that were reported to me at second 
hand. I have made each speaker say what I thought his 
situation demanded, keeping as near as possible to the 
general sense of what was actually said.

This gives the historian license to re-configure ideas to 
conform to a particular argumentative opinion, ideological 
position, or vision of human nature. There were speeches--
in most cases. What was said in those speeches may have 
included what was reported here. But, much was left out, 
and much was de-emphasized. The ordering, presentation 
and wording conform to Thucydides' vision of political reality 
and human nature.  

An important example of this is the speech of the Athenians 
during the Spartan debate, in response to the charges of 
Athenian injustice namely: Athens' siege of Potidaea, 
Athens' decision to help defend the island of Corcyra against 
Corinth, and Athens' decree restricting trade with Megara. It 
is here, in this response, that Thucydides "understanding of 
political reality" emerges. The Athenians in this speech do 
not deny "injustice," they simply notice that the concept has 
no real meaning in the world of empire. In the world of 
empire, nature and necessity take precedence. Here is the 
curious response of the Athenian delegation.

Thuc. 2.76.1 We have not done anything in this that 
should cause surprise, and we have not deviated from 
normal human behavior: we simply accepted an empire 
that was offered us and then refused to surrender it. If 
we have been overcome by three of the strongest 
motives--ambition (power), fear, and our own advantage 
(self interest)--we have not been the first to do this. It 
has always been established that the weaker are held 
down by the stronger......[ Besides we took this upon 
ourselves because we thought we were worthy of it, and 
you thought so too, until now that you are reckoning up 
your own advantage and appealing to justice--which no 
one has ever preferred to force, if he had a chance to 
achieve something by that to gain an advantage. If 
people follow their natural human inclination to rule 
over others they deserve to be praised if they use more 
justice than they have to, in view of their power.]

This then is the core of Thucydides' "model of political 
reality," his "political science."  Power, fear, and self-interest 
are primary forces on the international stage--these are the 
"geometric objects." Subject these to geometric grammar 
and you will evolve a sequence of higher order propositions 
about political nature.  This understanding presents us with 
a "framework" for politics, in effect, a "political science." 

It is however, just that, a "model," some would call it a 
"paradigm." It accounts for many of the observed facts--in 
particular those Thucydides chooses to include in his 
narrative. And to some extent the model can be applied to 
new situations. That is to say it has some predictive power. 

The label "scientific realist" has been used to characterize 
Thucydides' approach here. It is called scientific because it 
purports to report on an objective world independent of the 
observer.  Perhaps the best example of a Thucydidean 
success in observation and application of his scientific 
model and particularly its predictive power is his analysis of 
the aftermath of the civil war in Corcyra:

Thuc. 5.82.1 Civil war brought many hardships to the 
cities, such as happen and will always happen as long 
as human nature is the same, although they may be 
more or less violent or take different forms, depending 
on the circumstances in each case. In peace and 
prosperity, cities and private individuals alike are better  
minded because they are not plunged into the necessity 
of doing anything against their will; but war is a violent 
teacher...

This events in Corcyra serve as a case study for civil 
conflict. Thucydides describes the event as a general 
phenomenon. Human beings will in similar circumstances 
respond in similar ways. And they did, and they do, from the 
US. civil war right on down to the present day in Bosnia, or 
Rwanda, and hypothetically even in Quebec, if civil conflicts 
ever emerge there. But, Thucydides goes on:

Thuc. 5.82.2 Civil war ran through the cities; those it 
struck later heard what the first cities had done and far 
exceeded them in inventing artful means for attack and 
bizarre forms of revenge.  And they reversed the usual 
way of using words to evaluate activities. Ill-considered 
boldness was counted as loyal manliness; prudent 
hesitation was held to be cowardice in disguise, and 
moderation merely the cloak of an unmanly nature. A 
mind that could grasp the good of the whole was 
considered wholly lazy. Sudden fury was accepted as 
part of manly valor...

If there is a problem with Thucydidean history as 
"geometry," it is right here. Thucydides greatest predictive 
triumph reveals what might be his model's greatest flaw. If 
words are reversed, they can have no stable meaning, and 
communication breaks down. Thucydides was painfully 
aware that the same events can have very different 
meanings for different observers. 

Where words reverse their meanings, a phenomenon known 
as incommensurability takes shape. Incommensurate 
means, "no common measure." In the extreme speech loses 
its force. The ideas that  common words refer to are no 
longer held in common. In situations such as these 
communication is virtually impossible. The interlocutors in 
an argument no longer engage, they talk through each 
other.  As a detached observer, Thucydides might be in a 
position to decide or choose where, when, or who is using 
speech with twisted, distorted or reversed meanings, but the 
task seems Herculean.  The phenomenon is more than a 
matter of mere disassembly, in which speakers deliberately 
deceive--incommensurate word reversal is something much 
more--the speakers have adopted and believe in the truth of 
these new meanings and use them with as much sincerity 
and honesty as they did before the reversals occur.

So before we become seduced by Thucydidean thought, 
and generations of power politicians have, from Bismarck to 
Nixon, we need to appreciate that among the various 
problems facing the historian, incommensurability will color 
whatever claim might be made to having obtained objective 
data.

It works like this. There is an imaginary universe, a 
Thucydidean universe,  in which power, fear, and self 
interest are the forces that govern relationships between 
factions and cultures. Players in that imaginary idealized 
universe relate to one another in the ways documented in 
this book.  The real universe can often give the appearance 
of conforming to the imaginary universe especially if I am 
selective in my choice of observations and facts.  If I actually 
believe in the reality of this imaginary universe, my 
objectivity is unquestioned. I will select, report, and order 
events to conform with what I "know" to be true.

I could say the same thing about geometry. There is an 
imaginary universe populated by points, lines and figures. If I 
believe in the reality of points, lines and figures, my 
propositions flow precisely from my beliefs. Insofar as the 
real universe is not Euclidean, as we now know, any attempt 
to match the real universe to Euclid's, is at best only an 
approximation. 

At worst, and this is the scary part,  the fit between the 
imaginary Euclidean universe and the real universe is a 
complete mismatch because in the real universe points, 
lines and figures have completely different meanings. We 
use the same words to refer to points, lines and figures, but 
they contain totally different ideas.  For example, for Euclid a 
"point" is "that which has no part." In non-Euclidean 
geometries a "point" might "look" the same, but, like the 
point that situates the position of a black hole, a point 
contains "many parts," in fact, a valid non-Euclidean 
definition of a point might be, "that which has all parts." A 
black hole is a point in the real universe but a point that that 
contains all parts--another universe!!

So, is Thucydidean history a science subject to geometric 
logic or is it art? If it is a geometry does it describe an 
imaginary universe or a real one?

As a science the history falls short on many counts. 
Thucydides admits this himself. The objective facts of the 
war are difficult if not impossible to document for the very 
reason that words and ideas change their meanings most 
during the course of war. In other words the data is suspect 
because the observations are contaminated.  In other words 
there is no way we can ever know if Thucydides has gotten 
the facts straight because by his own admission objective 
facts are nearly impossible to collect. 

As a geometry the history falters too with respect to its main 
engine--the underlying assertion that power, fear and self-
interest govern the affairs of men at the international level 
when cultures or ideological factions clash in a certain way 
as for example when one party, Athens, overreaches (out of 
necessity) and the other, Sparta,  responds (in necessity) 
out of fear. 

Thucydides reveals this vision of conflicting necessities in 
the speeches he uses--the selection and construction of 
which are governed by the model of political reality he 
adheres to.  I won't deny that power, fear and self-interest 
govern the affairs of men at the international level at least 
some of the time. But that this is the way events are 
governed by necessity, because human nature works this 
way in the realm of real politics, is hard to accept. Sparta did 
not have to go to war. It could have gone to arbitration. 
Athens could have responded to the Spartan appeal to 
justice.  It did not. If it had events might have been 
otherwise. Power, fear, self interest and necessity would 
have been secondary to other influences--and Thucydides 
might have pursued geometry.
.
The danger of accepting the rather pessimistic 
consequences of Thucydidean analysis is the temptation to 
accept that what is true for human nature on the grand scale 
is true also for human nature on the smaller scale. People 
who read Thucydides take it to heart--literally. There is a 
tendency in the West to buy into this amoral paradigm as a 
formula for human success. This I think is the cruelest 
legacy of Thucydides. Outside of their very personal space 
(and in many cases even there) people actually believe that 
power, fear and self-interest govern their political lives at 
every political level, from their behavior on the job (office 
and campus politics) to their attitudes and behavior to the 
city, state and even the environment. As a consequence 
people can and often do behave in wretched ways in their 
political lives. Justice and morality have no place in the 
political life of many people. 

We will see next week in the Republic that Plato saw that 
people often behaved badly, but argued that there were real 
moral ideals which we could emulate and adhere to given 
the right education. But Thucydides was not Plato, and Plato 
did not take the Republic into the sphere of international 
events and he could never have written anything quite like 
this because he rejected empirical evidence as a basis for 
obtaining knowledge. Empirical evidence for Plato was 
nothing like the real thing.  This great shining scientifically 
determined reality Thucydides shapes, shines and sculpts 
for us here would seem to Plato as nothing more than mere 
"opinion."

[Here is an intriguing thought.  It was penned for a lecture 
series in defense of Liberal Education given in 1858 by John 
Henry Newman.]

All that exists, as contemplated by the human mind, forms 
one large system or complex fact, and this of course 
resolves itself into an indefinite number of particular facts, 
which, as being portions of a whole, have countless 
relations of every kind, one towards another.

What that statement implies to me is that at some level 
Thucydides was right. Geometry and politics are connected -
- yes -- but in a much more fundamental way.

[Knowledge is the apprehension of these facts, whether in 
themselves, or in their mutual positions and bearings. And 
as all taken together form one integral subject for 
contemplation, so there are no natural or real limits between 
part and part; one is ever running into the other; all, as 
viewed by the mind, are combined together and possess a 
correlative character one with another from the eternal 
mysteries of the Divine Essence down to our own 
sensations and consciousness...]

Liberal study -- what we do -- implicitly recognizes this. But if 
music, art, science and geometry,  are connected -- how do 
we communicate those connections? If the subject matter of 
art, music, and literature really is common, what does that 
grammar look like? Thucydides instinct -- his attempt to 
tether disparate fields together was right -- but I thing he 
grabbed the wrong objects. 

That which is common to music, politics and geometry is  
something more reductive than lines, points, power, fear, or 
musical harmony.

Beethoven offered another thought that communicates the 
same sense. "Music," he said, "is a higher revelation than 
philosophy." 

What Beethoven meant -- I think -- is that Music, when it 
operates on us, brings us to the place where truths are 
apprehended -- and that music can do this at least as well -- 
certainly in another way than conventional logic.  

But those other mediums must be able to do that too.  Music 
can bring us to truth. Art can bring us to truth. Geometry can 
bring us to truth.  Completely unrelated medium bring us to 
see different aspects of a single complex truth.

More on these things at another time.
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