\section{Pythagoras and the Presocratics}
\label{pythagorasandthepresocratics}

_There_ are *multiple* branches of Presocratic Greek philosophy which all start with the Milesians as the main trunk and work within the productive Milesian philosophical Research Programme. Remember the three important Thales commitments:
\begin{enumerate}
\item the universe is consistent, 
\item it's knowable, and 
\item it is derived from a single, common substance.
\end{enumerate}
The multiple strings differ in the details and include for our purposes of physics the
\begin{itemize}
\item mathematical branch of Pythagoras,
\item the branch which singles out change as the important concept of Heraclitus,
\item the branch which singles out stasis and logic of Parmenides, and 
\item the branch which identifies atomism and discreteness as essential of Democritus.
\end{itemize}


We'll take them in turn and point out the enduring features which form important underpinnings of physics today.


\subsection{The Pied Piper of Samos}
\label{thepiedpiperofsamos}

\begin{flushleft}
\begin{verse}
\textit{Once more he stept into the street; \\
And to his lips again \\
Laid his long pipe of smooth straight cane; \\
And ere he blew three notes (such sweet \\
Soft notes as yet musician's cunning \\
Never gave the enraptured air) \\
There was a rustling, that seemed like a bustling \\
Of merry crowds justling at pitching and hustling, \\
Small feet were pattering, wooden shoes clattering, \\
Little hands clapping, and little tongues chattering, \\
And, like fowls in a farm-yard when barley is scattering, \\
Out came the children running. \\
All the little boys and girls, \\
With rosy cheeks and flaxen curls, \\
And sparkling eyes and teeth like pearls, \\
Tripping and skipping, ran merrily after \\
The wonderful music with shouting and laughter }\\
\vspace{0.15in} from: \textit{The Pied Piper of Hamelin} by Robert Browning \\
\end{verse}
\end{flushleft}
\vspace{0.1in}

\noindent In 1284, as the story goes, a flute-wielding ratcatcher extracted revenge on the German town of Hamelin because of their refusal to pay for his rodent-eradication services. His revenge? The additional eradication of all of the town's children--some 130--who followed him and his music down a road still called Drumless Lane. According to some versions, he led them into a mountain, according to others, into a tunnel leading to Transylvania.  Following one person as a part of a herd is the hallmark behavior of a cult and almost always leads to a bad result. Some cults are short-lived and dramatic, but some last for centuries. Perhaps surprisingly, one of the most famous---and best-subscribed---cults in history is attached to the person most responsible for the deep importance of Mathematics. While math can be habit-forming, the equating of it with a nearly religious society, seems farfetched, right?


But, this was precisely the condition enjoyed (!) by those of the 300 year old School of Pythagoras. That old fellow attached to that famous theorem was leader who mesmerized a group of followers who were absolutely loyal to his strange ideas and to the absolute power of mathematics. Pythagoras of Samos (ca. $-582$ -- $-497$) was the first to begin to attach meaning to numbers beyond merely counting. Prior to him, ``3'' meant nothing more than the count of things between two and four. After him, the \textbf{concept} of Three, became an idea in and of itself. He said, ``All things have form, all things are form; and all forms can be denoted by numbers,'' and to be considered ``Pythagorean'' is to subscribe to this declaration that ``all things have form{\ldots}and can [therefore] be defined by numbers.''


Again, we have a cosmopolitan Greek  responsible for new ideas. Pythagoras' life is documented in four or five ancient accounts and they agree that he was born on the Island of Samos to a father who traveled widely and perhaps took his son along. He was said to have traveled and studied in Melitus and influenced by Anaximander and maybe an elderly Thales himself. There is also reason to believe that he spent a couple of decades in Egypt and perhaps was there when it was subjugated by the Persians in $-525$. He also spent time in the sophisticated city of Babylon, perhaps as a Persian prisoner, but returned to his home eventually, only to find that he could not abide its politics. This part of his life is known for sure: he relocated to the Greek colony of Croton in southern Italy, another land that he might have visited with his father as a young man. But, by now he was the head of a society of scholars and they came along with him, lock stock and barrel, like a flock. Like he was playing a flute.


Pythagoras' influence was due to his originality, but also due to the fact that he too continued the Milesian tradition. All three of Thales' important ideas were woven into Pythagoras' commitments, only the details were different. Rather than water, or the Aperion{\ldots}the pure form of mathematics was the substrate responsible for the Universe. And now, the abstraction of mathematics and its importance in physics is directly descended from his influence. But, his was more than a commitment to an application of mathematics to reality. It was mystical. How he came to this comes from a story told about him. An  apparently accomplished musician, he found that that when the four strings of a Greek lyre were tuned to the most pleasant combination, the lengths of the strings were in precisely the ratios of 1:2 (the octave), 2:3 (the fourth), and 3:4 (the fifth). This response to pleasing cords is borne out in numerous physiological tests and we tend to ascribe this to our anatomy and the psychology of hearing. Not Pythagoras. He concluded that the pleasant harmony that we experience was a feature of the {\itshape tones themselves}, and that the numerical ratios responsible for the harmony were related to the sacred nature of integers. Once you've bought into that, it's hardly a stretch to extrapolate to the entire harmony (orderliness?) of Nature itself is due to the orderliness inherent in Number. Surely, if not explicitly by leading a parade with a flute (or lyre!), Pythagoras was the piper to  a 2000 year long parade guided by the harmonies of mathematics.


Further, this orderliness could be represented as a geometrical form{\ldots}always with integers as the key players. Take a handful of four kinds of colored poker chips. Place one blue chip down and lay three red ones around it, one right, one below, and one in the diagonal to make a square of chips. Notice that you've created the following little equation:
%
\begin{equation}
\label{eulersidentity}
1+3=4=2^{2}
\end{equation}
%
One blue plus three red. The left side of the equation simply counts the ``inner'' (blue) and ``outer'' (red) chips while the right side relates to the geometrical {\itshape form} of the resulting rectangle{\ldots}in this case, a square. Adding another row of green chips around the outside of the red, and one can make the square bigger through the addition of five more. Again, continuing the same equation,
%
\begin{equation}
\label{eulersidentity}
1+3+5=9=3^{3},
\end{equation}
%
and so on, each time the number of numbers added on the left corresponding to the edge-count of chips, which when squared, equals the sum. This geometrical picture of addition is the origin of the word ``square'' for taking a number and multiplying it by itself. Notice one more thing, if you haven't already. The sums of our colored chips are the sums of the incrementally increasing odd numbers. These were  called the ``square numbers'' and were destined to be critical to Galileo when he applied them to the mechanics of free-fall before any of us had use of algebra.


In addition to the square numbers, one can construct a similar pattern of ``rectangular numbers'' like this: take two blue chips side by side and add four red ones, one to the right and three directly below. The result is now a rectangle and the sums and products as before show:
%
\begin{equation}
\label{eulersidentity}
2+4=6=2\times 3.
\end{equation}
%
Do it again, and find:
%
\begin{equation}
\label{eulersidentity}
2+4+6=12=3\times 4.
\end{equation}
%
Notice now that it's the even numbers which form the sums. The two kinds of integers--even and odd--seem to be related to rectangles and squares. The first melding of arithmetic and geometry and a sacred union it was to the Pythagoreans--and to their natural descendants: Plato, Kepler, Descartes, Einstein, and do on. 

Do this construction  one more time and a magic number occurs: if you take the sequence to 4 and then divide it all by 2:
%
\begin{equation}
\label{eulersidentity}
1+2+3+4=10=\frac{4\times 5}{2}\times 4.
\end{equation}
%
This{\ldots}is the magic tetraktys, a different geometrical construction from integers--the number 10, arranged in the magic bowling-pin triangle, a sacred pattern and sacred number. If one builds the pyramid of numbers in this pattern, putting in the numbers, in the places where they match the count of chips, you find a 1-4 ordered, top to bottom as a representation of the increasingly complex construction of the Universe, or so they thought. The tetraktys is a sacred pattern for more than bowlers, it formed a pattern revered by Pythagoreans and early kabbalists. In fact, the Pythagorans {\itshape prayed} to this pattern. We'll see the importance of the number 10 in play when we consider Pythagoras' cosmological theory, which I'll consider below.


The essential feature of all of these geometrical constructions is the need for integers and a countable number of chips. Integers were deemed sacred and since there was this relationship between the harmonious and ordered universe and number (the lyre), the orderliness was only preserved in a countable number of pieces. So, {\itshape all numbers} had to be expressible as at least ratios of whole integers{\ldots}because Pythagoras said so. 


The Society of Pythagoras was not just a club, enjoyed at special meetings like the Rotary Club. It was a way of life with strict rules of behavior, dress, diet, belief, and proscribed loyalty. Men and women were equal, but in this case not equally free, but equally bound to the code of Pythagorean life. Meat was forbidden in the diet because of a belief in reincarnation. The Harmonies of the Spheres were apparent for all to see through the regular motion of the stars and planets. In fact, the planetary harmonies were supposedly real sounds which normal humans could not hear as simply a familiar and ignored background experienced from birth. But, Pythagoras? He, alone among men, could hear the sounds! The strict code of the Society of Pythagoras was:
%
\begin{enumerate}
\item that at its deepest level, reality is mathematical in nature,
\item that philosophy can be used for spiritual purification,
\item that the soul can rise to union with the divine,
\item that certain symbols have a mystical significance, and
\item that all brothers of the order should observe strict loyalty and secrecy.
\end{enumerate}
%
The Pythagorean Theorem, familiar to all from elementary geometry was not a product of Pythagoras, nor his following. (In fact, it's been impossible in later millennia to unravel what mathematical pronouncements were Pythagoras' or from one of his hundreds of disciples.) In any case, the notion that in a right triangle, the square of the hypothenuse was equal to the sum of the squares of the sides was known for a millennium before Pythagoras got the credit. Perhaps he provided one of the many proofs, but it had been a part of Babylonian and Egyptian commerce for hundreds of years. 

But, there is a famous story about the right triangle which casts an even more sinister cloud over the Leader and his followers. The statement of the Theorem can be realized as a geometrical construction, easily seen for the famous 3-4-5 triangle. 


![The Triangle][]


Notice how neatly the three squares line up on the triangle{\ldots}the squares representing the square numbers and the theorem. A simple geometric argument shows that the sums of the areas of the two squares against the perpendicular triangle legs is the same as the area of the square against the hypothenuse's leg. That's the Pythagorean Theorem. 

Well, woe to the poor fellow who took this to the logical limit of one, which is, after all the loneliest number. If you construct a triangle with perpendicular sides of one unit, then the hypothenuse is unpleasantly{\ldots}irrational. Enough was understood about mathematics to know that this hypothenuse--realized in nature because you could simply construct such a triangle--could not be represented as a ratio of whole integers. We would see such a number as a never-ending string of decimal places, like $\pi$. The poor fellow who first came on this discovery was drowned for his discovery and all who were privy to its existence were sworn to secrecy due to the violence that such knowledge does to the divinity of the integer. Not your normal reaction  to a mathematical discovery, but such is the provence of Cult Leaders: they seldom enjoy being wrong.

Pythagoras' life was long, but ended badly in a fight following the torching of his home by Cylon who was a powerful civic leader, but a tyrannical man who had been spurned  in his attempts to join the Society of Pythagoras. According to legend the elderly Pythagoras himself interviewed Cylon and turned his application down, only to suffer a very bad reaction to the negative news. Some people never leave high school it appears. 

\subsection{Talking to Your House}
\label{talkingtoyourhouse}

\noindent We know almost nothing of his life: of a noble family, perhaps destined for rule, but passed it up in favor of a younger brother. This in a time which is not altogether specific. Most place him between Pythagoras (whom he refers to in the past tense) and Parmenides (who seems to some to indirectly refer to his ideas), but others are not sure about the latter ordering. Why is he important?
% 
\begin{verse}
\textsf{Nature loves to hide.}
\end{verse}
%
Heraclitus (or Herakleitos) of Ephesos (ca. $-540 $-$ -480$) must have written persuasively, as his influence extended at least as far as Plato and Aristotle. While it's not quote so simple, he's usually credited with being the expositor of Change, as opposed the the equally persuasive and influential-to-Plato, Parmenides, the hero of Permanence. Change versus Permanence. A classic confrontation and again one of those sets of concepts which seem to weave their way throughout the history of physics, sometimes one dominating, sometimes the other{\ldots}sometimes coexisting in an uneasy alliance simultaneously.
%
\begin{verse}
\textsf{The sun is new every day. \\
It is the opposite which is good for you.}
\end{verse}
%
He is said to have written at least a book, and what we have of it is mere fragments{\ldots}obscure aphorisms in the form of sayings or proclamations. The kind of thing that makes you go, ``hmm.'' In fact, The Obscure, or The Riddler are the sorts of names he was known by later.
%
\begin{verse}
\textsf{In the circumference of a circle the beginning and end are common. \\
It rests by changing.}
\end{verse}
%
In the story told so far, the underlying emphasis was on what Is. Certainly, Pythagoras valued a world and set of rules which are unchanging, and to violate the rules could be dangerous. He came on this sensibly by following the lead of the Ionians, for whom the underlying fabric of the world was at once universal and unchanging. Through that steadfastness, was a kind of importance that would seem unnatural if it were variable and ephemeral. This seems a natural outgrowth of the new way of thinking that they were busily inventing. A way of thinking which was counter to the vagaries and randomness that was a part of the Greek life before, one in which Man was constantly being buffeted and bounced around by the gods and their whims. To counter that, they proclaimed a oneness and sameness and argued for it in spite of their surroundings which are never the same and always in variation.
%
\begin{verse}
\textsf{We step and do not step in the same river twice.}
\end{verse}
%
Yet along comes this self-renounced Priest who claims that the only thing that Is Constant is Change. That what holds the world together is a balance in tension between opposites. Here is a philosophy in which Process holds importance over Substance. There are things that Are, but what's most important is how they change and fight with one another. The world, according to this newly formed view, is what it is because of its constantly evolving nature and to understand it is to appreciate and understand Process. Becoming.
%
\begin{verse}
\textsf{The eyes are more exact witnesses than the ears.}
\end{verse}
%
This was a threat to those for whom accounting for unity was the primary project, but it's awfully hard to disregard your eyes which are bombarded constantly with reminders of the unceasing Differences and Motions of all in the world. 
%
\begin{verse}
\textsf{Praying to a statue is like chatting with your house.}
\end{verse}
%
You have to like someone like this, right? Obviously, in-your-face philosophy, challenging in the best sense. Appealing to your common sense (your eyes tell you change is everywhere), but fighting with your commitments (the world is underpinned by the universal and permanence). And doing it with style. 

In the much later Greek times, after the time of Alexander and after the time in which Greek dominated the known world, the library at Alexandria in Europe was burned to the ground (in one version) by frightened Christians unhappy at challenges of the sort that Heraclitus would have hurled with bumper-sticker like clarity. Much was said to have been lost in that fire. The entirety of all writing known to that time and maybe the actual texts belonging to Heraclitus from which we could piece together a more accurate context. 

But, the challenge persists through the millennia and was clear enough in its pithiness: You want permanence? Then you had better explain what you see!


\subsection{The First Lawyer}
\label{thefirstlawyer}

\noindent The antidote to Heraclitus, at least in the nutshell in which it's sometimes portrayed today, came in the form of the first legal argument. The stakes were high and in order to preserve the important argument that the world is orderly and reliably constant, drastic measures were required.

Traditionally, the hero of Permanence is Parmenides of Elea (b ca. $-510$), the first of a string of Eleatic Philosophers in a tradition of considerable influence in what was to come later. He took the straightforward route to preserving Permanence and accepted the consequences of the common sense argument that what you observe is always changing. The  only route that was available to him, he took: Change is Illusion, or{\ldots}you can't believe your eyes.

Parmenides left us one poem, one document in which he presents his case to the jury. And, in doing so the way he did, he also opened the door to a tradition of argument by indisputable logic. A technique to learning (really?) which itself ebbed and flowed through the first one and half millennia after he first developed it. But, it was important as he took a step beyond the techniques of the Ionians. Remember, they Proclaimed. They Pronounced the Way It Is. It sounded nice, it sounded even plausible. But, you had to take Thales and Co. at their word that somehow they had stumbled (how?) across ideas which they were unobligated to defend. They simple Stated.

What Parmenides did was different. With only a few reasonable sounding presumptions, he proceeded to develop a step-by-step argument. Accepting each step, required only reason. Adding together the sum total of accepted steps, then required accepting of the whole argument itself. Indisputable. That's the essence of the logical argument. At risk of being accused of irrationality or madness, all must accept the conclusions.

The essence of the argument is the commitment: Reality is One Permanent Thing. It's knowable through rational thought and it's true by virtue of its Permanence and the infallible route by which it's ascertained. It's through thought, rational thought, that Reality is apprehended. All else that comes to our attention through our senses is Mere Opinion, and so it's the senses which are suspect. Blame the messanger! One has examples of this all the time: the senses betray and in order to find Truth, you have to learn to distrust your senses and the world of change that they constantly present us with.

How, he asked, can anything come from nothing? Why, no way. So, the universe Is. It could not have come from the opposite of what Is, namely what Is Not and so the universe is unchanging and has always to have been real. So, since the world is what is, and could not have come from what is not, namely nothing, then nothing Is Not. ``What is, is. What is not, is not.'' Hard to argue with that sort of logic, right? But the import is huge: Since what Is cannot have come from what is not, Creation is in itself impossible.

Furthermore, to Become, requires a motion of sorts{\ldots}to have somewhere to come from, and somewhere to go to which is not currently there. So, since nothing can come from nothing, nothing can go to nothing and Nothing does not exist and hence, Becoming is impossible. Only Being is possible and the state of nothing, namely the Void, cannot exist.

But wait, there's more. ``Thou canst no know what is not--that is impossible--nor utter it: for it is the same thing that can be thought and that can be.'' He goes a step further and proscribes in essence the limits to what rational thought can grasp. You can know, or think of, what Is. But, you cannot know or think of what Is Not. Because, you cannot think of what does not exist.

\subsubsection{Smackdown}
\noindent Now, the line is drawn. In this corner is the pragmatist, common sense proponent: Change is everywhere, understanding the world requires the understanding of Process. In the other corner, the rationalist: Change is an illusion, what's real is what's permanent  and what's real is true and what's true is only discernible through rational thought.

Now, this may sound like an artificial distinction at this point. But, think about it. We can't have a science if there is not something that's believably constant. I can't do science if I cannot believe that an experimental result of today, cannot be reliably re-determined tomorrow. Yet, I can't deny that the dynamics of the world, and the constituents in it, are an important part of the story as well. The rules by which science is understood are almost never rules about just what Is. Except for the (rare) early stages of a new branch of science, the rules  are not merely an inventory or organizing of the objects in the universe---the mere facts. They are more interesting and rich when they are rules of engagement{\ldots}how the objects of the world interact with one another and how they change in time. Can't have one, without the other in having a mature science. 

Before you have sleepless nights over Parmenides' argument, here's at least a part of where he went wrong. He forbad thinking of that which was not, but he didn't allow for thinking of that which Might Be. I can't do science without imagining the consequences of a circumstance which Might Be. 

But, here's where he went right. Among the most sought-after features of the physical world are the things that remain the same. We've come to call these regularities Conservation Laws, and, while allowing for the anachronistic use of the word ``Law,'' the preservation of some measurable quantity is always an indication of insight into something powerful and important about how the world works. Mix up constituents, heat them, speed them, pressurize them, that they will preserve quantities like energy and momentum, respect quantities like the homogeneity of space and time is always an indication of deep understanding. And, in recent decades, the discovery of a Conservation Law is also connected with a deep symmetry buried within the mathematics used to describe those circumstances. A connection about which I'll have much more to say later, but a connection which is as important as any idea in the past 300 years.


\subsection{The Atomists}
\label{theatomists}
%
Ask any educated person on any western street what the world is made of and you'll almost certainly be told: atoms. Without knowing chemistry, most people have an idea of the elements, compounds of elements, crystalline combinations and structures of elements, and a building of matter from these little objects. It is no small victory for the scientific view that such a concept would be largely accepted from the beginnings of the 20C, for such a view has traditionally been highly suspect and irreligiously materialistic.

The Universe can be only of a few kinds of substance. It can be continuously  joined together in an ooze of plastic parts, leaving no space anywhere, just more ooze. Or, it can be disjoint, discontinuous, and particulate. The problem with that view was immediately evident when it was first proposed (you guessed it: by a Greek), was that where there were then not-atoms, there had to be nothing.

The Atomistic view is a strangely collaborative one, almost conspiratorial--a political event of sorts in which proponents sing or write of the doctrines of the forefathers' views, restating, remolding, but always reaffirming. The originator of the Greek Atomistic view is often attributed to Democritus, but that's not accurate...a case of the student receiving credit for the teacher's idea. However, in the best tradition of that relationship, here the student took no credit personally, and extended his mentor's idea and brought it out-front in a way that probably did credit to its inventor. The real father of Atomism, then, is Leucippus (fl. $-440$).

The collaboration referred to above, comes from the retelling of the story by successive proponents and reporters. In this case, Epicurus, that strange larger-than-life personage for whom ``Epicurean'' is coined, was an Atomist and follower of Democritus. But, we know most of these fellows from the Roman digest writers. While Rome was responsible for much impressive mechanical and civil engineering, they were not exactly original thinkers of depth when it came to scientific matters. But, with a growing upper-middle class and the need to educate citizens in the finer and nobler things, they relied on summaries, encyclopedias, and digests of Greek thought prepared by learned, and distilled and lightened up for the non-majors.

Of these, were two who stood up for Atomism: From Diogenes Laertius (fl. 3C) we learn: ``Leucippus was born at Elea, but some say at Abdera and others at Miletus. He was a pupil of Zeno. His views were these: The sum of things is unlimited, and they all change into one another. The All includes the empty as well as the full. The worlds are formed when atoms fall into the void  and are entangled with one another; and from their motion as they increase in built arises the substance of the stars...Leucippus was the first to set up atoms as first principles.'' 

Now, this is interesting, for Zeno was a follower of Parmenides and is of course famously responsible for nearly 50 paradoxes, often intended to show the impossibility of motion or movement logically. Of course, the other reason that the Eleatics distrusted the idea of movement comes from the need to postulate a void--an Is Not--in order for bodies to have somewhere to move to. The notion of the void was too terrible to contemplate, so they took the next best step, that there was no movement possible and if you think that there is, you're fooled by believing in your senses. Zeno was Parmenides' posse and added absurdity to the argument by demonstrating that if you believed in motion, then among other inexplicable circumstances, Achilles could never catch a tortisse in a foot-race. Follow the argument, reach an absurdity...must reject the argument. Of course, this is interesting from a biographical point of view in that if Leucippus was really a student of Zeno, he took a different direction from his teacher, which is a bold thing to do. Since motion happens (he believed his eyes, apparently), then there must be a void. So, the two substances for Leucippus---and Democritus, who seemed to be indistinguishably in agreement with his teacher---were ``stuff'' and not-stuff: atoms and a void.

The other Roman commentator was Lucretius ($-99$ - $-55$) who wrote in style, that is he wrote poetry. As such he was a popular expositor of those he summarized and brought much of the Roman world, and the later Latin-reading Enlightenment world (he was translated into English in 1682 by Thomas Creech) into the Atomistic-know with De Rerum Natura (On the Nature of Things).
\begin{verse}
\textit{Yet bodies do not fill up every place: \\
For besides those there is an empty space, \\
A void. This known, this notion framed aright \\
Will bring to my discourse new strength and light, \\
And teach you plainest methods to descry \\
The greatest secrets of philosophy. \\
A void is space intangible, thus proved: \\
For were there none, no body could be moved; \\
Because where're the brisker motion goes, \\
It still must meet with stops, still meet with foes, \\
'Tis natural to bodies to oppose. }
\end{verse}


%% An example of a body-quotation
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%\begin{quote}
%\textsf{\footnotesize {}
%``A simple example  would be the proposition
%that there are mountains on the other side of the moon. No rocket
%has yet enabled me to check this, but I know it to be decidable by
%observation. Therefore this proposition is verifiable in principle
%and is accordingly significant. On the other hand with such metaphysics
%as \char`\"{}the Absolute enters into, but is itself incapable of,
%evolution and progress\char`\"{} {[}F.H. Bradley] one cannot conceive
%of an observation which would determine whether the Absolute did or
%did not enter into evolution; the utterance has no literal significance.''
%}{\footnotesize \par}
%\end{quote}
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%\sidenote{
%There is a particularly poignant story of a slightly older and more established mathematician named Gottlob Frege who was similarly pursuing a logical derivation of mathematics. Russell's discovery of the paradox ruined Frege's life work. In reply to Russell's respectful letter informing him of the difficulty, Frege wrote back, ``Your discovery of the contradiction has surprised me beyond words, and I should like to say, left me thunderstruck because it has rocked the ground on which I meant to build arithmetic...I must give some further thought to the matter.'' Later, Russell took pains to highlight Frege's many contributions to mathematics, but the older man never was able to rebuild his system in light of the Russell Paradox.
%}
%%========= example of sidenote ==================================

%
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%\begin{boxer}{Bertrand Russell (1872-1970)}

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%\noindent More words. 

%\end{boxer}
%\end{figure*}
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%\label{box:mathematics}\vspace{0.5cm}

%\begin{boxer}{Geometry, late 1800's}

%The late 19th century was 

%\indent In 1853 in a career-making move,  
%\end{boxer}
%\end{table*}
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%\fig{LadiesInBlue.jpg}{
%Bertrand Russell shortly after being released from his five month prison sentence for his vocal oppossion to Britain's participation in WWI.%
%\label{cap:russell_40}} 
%}
%%========= example of marg ==================================