In today's lecture we leave myth, we leave story-telling, and deal for the first time with philosophy, with abstract thought. Actually, we'll be dealing with the oldest attempts at logical analysis of some of our most basic concepts. Such analysis is always difficult, so much so that at times we may get the impression we're either floating in thin air or trying to walk through molasses. So this lecture can't avoid being difficult.

We call "Presocratics" a group of thinkers who expressed themselves in various dialects of Greek during the 500's and 400's BC, that is, before the time of Socrates and his disciple Plato. Most lived in the periphery of the Greek world, in what's today the western coast of Turkey (Ionia) and the southern parts of Italy. All we have is references to them and fragments of their texts included in books by later authors: in other words, all we have is ruins, and the work of interpreting those ruins is similar to the archeologist's. Part of the problem is that the concepts behind the words they used are often not the same as the concepts used by later thinkers who quoted from them, men like Plato and Aristotle, and so translation becomes a very hard task.

First, a question: Why bother with the
Presocratics? Because in their texts we witness the origins of
Western thought—by which today we mean *philosophy* and
*science* (the distinction between those two arose much later).
Understanding Western science and philosophy is much helped by
an understanding of their origins, as understanding a person is
much helped by finding out what his childhood was like. There
is a danger, though, in what I just said. To think of the Presocratics
as children and of ourselves as sophisticated adults would be
a gross mistake. To borrow a comparison from painting: the art
of the caves is not more naive or rudimentary than that of Picasso.

One reason we can be misled into viewing the Presocratics as children is that their ideas are often presented in a naive and confusing way. The typical account goes as follows: Religion and myth taught that the gods created all and were the beginning of the cosmos; then came the philosophers, who were smarter (more modern), and said, "Forget the gods;" so Thales of Miletus, the earliest Greek philosopher, proposed that water is the beginning of all things, then came another guy who said no, it was air, then another said it was fire, another said the mind, another said love and hatred, still another guy said it was all of those things together, and so on. This simplistic account lends some justification to the erroneous belief that philosophers never agree, and that therefore one should forget about their conflicting theories. I will restrict myself to a few of those thinkers, about whom we have more than mere hearsay, in order to show you that actually they had much in common, that their theories were not in conflict, and that they were all addressing the same problems. These problems are still of central importance in contemporary thought.

There is a historic fact that may have
to do with the beginning of Western thought: in the period between
Homer (ca. 700 BC) and the Presocratics, the many Greek city-states
were giving themselves written laws, constitutions. We can view
the thought of the Presocratics as an attempt to write down the
laws governing the cosmos, the ordered universe which they likened
to an enormous city-state, and we can view these laws as an attempt
at a solution to the difficulties arising from trying to think
the concept of not-being. As I already said in a previous lecture,
philosophy starts with the opening of the possibility that *nothing*
happens after death. This introduces the horribly difficult thought
that there may be something like not-being.

To clarify what I mean by the concept of not-being, I would have to start by saying what is meant by "being," continue by what is meant by "not," and finally, somehow, put these two concepts together. This superhuman task is made even harder by our common assumption that the meanings of both "being" and "not" are so obvious that they require no analysis. But to say something is obvious, period, is to put a stop to all thought, especially philosophical thought, and even more especially, the thought of the Presocratics. As for explaining what "being" meant for the Greeks, the German philosopher Martin Heidegger built a brilliant career doing exactly that. The concepts "no" and "not" aren't any easier than the concept "being". Perhaps developmental psychology could be of help here, since those are among the first words a child learns. Exasperated parents often say to a disobedient child: "Don't you know the meaning of the word NO?" But the child could well answer: "Actually not, I haven't seen any satisfactory philosophical analysis of it." I must say I haven't either. When it comes to putting those two words, "is" and "not" together, to get "not-being", we are in a very hard position indeed.

Remarkably, when we confront the two
concepts, "being" and "not-being," we come
to somewhat more familiar territory: here logic teaches some rules,
which we may call hygienic rules, because they are designed to
keep *being* from being infected by *not-being*. The
most fundamental of the rules of logic is *the Principle of
Non-Contradiction*, which Aristotle (384-322 BC) stated as
follows:

"One cannot say of something that it is and that it is not in the same respect and at the same time."

Aristotle thought that no person in his right mind could deny this principle, for, he said, if you try to prove it false, you would be using logic and so you would be using it in your proof. In other words, if you use this rule (the Principle of Non-Contradiction) in a proof, you are implicitly accepting it. This seems irrefutable. But WAIT! For almost 2,000 years, until the birth of modern science in the 1600's, European and Arabic scholars were taking Aristotle's word as absolute truth, and doubting it was tantamount to heresy or professional suicide: are we going to behave like those scholars? No, we can reply to Aristotle: you are certainly right that we cannot logically prove that the Principle of Non-Contradiction is false, but that doesn't mean that we have to accept it, or that we're crazy if we don't. Logically proving something false and not accepting it are not at all the same. For example, we know, from our lecture on the axiomatic method, that non-Euclidean geometry does not prove that Euclid's 5th axiom is false, since that would be logically impossible to do; non-Euclidean geometry simply does not accept it, and accepts a different axiom instead.

Having said that, I must add that in a historical sense Aristotle was right: the Principle of Non-Contradiction is absolutely essential in philosophy, in all of science and mathematics. However, the Principle of Non-Contradiction itself is not without its difficulties.

To see what these difficulties are, consider
the problem of *becoming*, which means that something changes
from one state to another. This problem preoccupied all Greek
philosophers. Indeed, we may say that the problem of becoming
is the unifying obsession of all Greek philosophy. Let's consider
some examples. Let's say that an arrow is moving in space. Can
we say that the arrow *is* in here, in a certain place, without
saying, too, that the arrow *is not* in this place, since
it has already moved beyond it? Doesn't this violate the Principle
of Non-Contradiction? Well, you can reply, of course the position
of the arrow is always changing, but *at a given time* the
arrow *is* at a given place. Fine; but what is meant by that
phrase, "at a given time"? In any interval of time,
no matter how short, the position of the arrow is changing, so
when we say, "at a given time," what must be meant is
an instant, an atom of time, "a point in time," a portion
which cannot be divided into shorter portions. Is there such a
thing? Does it make any sense? A mathematician or a physicist
today will say: "Yes, an instant of time is like a real number."
I cannot tell you here the definition of "real number":
it's quite involved, and you would have to take Advanced Calculus
to understand it. Observe, though, that whatever they are, real
numbers are a mathematical construction, in other words, something
built upon logic and therefore upon the Principle of Non-Contradiction.
So when a scientist invokes the real numbers to save the Principle
of Non-Contradiction we're really going in circles, or, in other
words, *begging the question*: the principle is safe simply
because we take it for granted.

The example of the arrow as a general
analogy for motion was used by Zeno of Elea, who was a contemporary
of Socrates. He concluded that, speaking *logically*, the
flying arrow doesn't move, since you cannot say it *is* at
any position without simultaneously saying that it *is not*
there. But you shouldn't get the impression that those are old
problems, now uninteresting and out of fashion. Much of modern
math, logic and philosophy have evolved out of the effort to find
ways around questions such as Zeno's paradoxes (there are three
of them, but we don't have time to deal with the others).

After this brief introduction to the
problems of Greek philosophy, let us look at some individual philosophers.
The earliest author of whom we have some fragments of text is
Anaximander of Miletus, a city in Ionia (611-547 BC). Anaximander
is reputed to have written the first book in Greek prose; before
him Greek thinkers used Homeric verse to express their thought.
He seems to have been the first one to have explicitly used *the
Principle of Sufficient Reason*, another basic principle of
logic which states: *no event can happen without a reason for
it to happen thus and not otherwise, a reason which we call its
ground*. Even as little children we take this principle
for granted, when we ask "Why?" to almost anything we
are told. What's original in Anaximander is the use he made of
this principle, how he applied it. He wrote that the earth is
not, as some people believed, supported by something like an elephant
or Mr. Atlas, but that it stays without any support at the middle
of a spherical universe: the earth doesn't budge from that middle
because, by symmetry, there is no sufficient reason for it to
fall in one direction rather than another. Of course, we now know
that the universe is not a sphere and the earth is not at the
center—nonetheless, this is an astonishingly modern reasoning,
no different from many arguments in contemporary science.

Anaximander's most famous fragment, though,
has been preserved encrusted as a quote in a work written 1,000
years later, in the 500's AD, by Simplicius, a minor late Greek
philosopher: "Anaximander ... declared the Boundless (*apeiron*)
to be principle and element of existing things, having been the
first to introduce this very term "principle"; he says
that it is neither water nor any other of the so-called elements,
but some different, boundless nature, from which all the heavens
arise and the *kosmoi* within them; *"out of those
things whence is the generation for existing things, into these
again does their destruction take place, according to necessity;
for they make amends and pay the penalty to one another for their
offense, according to the ordinance of time,"* speaking
of them in rather poetical terms. It is clear that, having observed
the change of the four elements into one another, he did not think
fit to make any one of these the material substratum, but something
else besides these."

Out of this text we must retain two things:
(1) the word "Boundless," in Greek *ápeiron*,
and (2) what purports to be a direct quote from Anaximander (italics above):
"Out of those things..." up to "according to the
ordinance of time." First, about the word *ápeiron*:
it means "no-bounds." Not, however, in the sense of
merely outer bounds, as when we say that something is so huge
it's boundless, or infinite; rather, it means having no boundaries
at all, either outside or inside. This means that there are no
individual things in the *ápeiron*, since individuals
need boundaries to delimit one individual from another. Where
there are no bounds there can be no individuals, and there can
be no saying "this is" and "this is not."
Anaximander tells us, however, that his *ápeiron*
is infinitely fuller than anything else: you and I, these words
I'm uttering, appear, come up, shine for a little while, then
vanish again in the *ápeiron*, which is eternal. There
is no time in the *ápeiron*, since time presumes change
and becoming, but becoming requires, besides being, not-being,
and there's no such thing in the *ápeiron*. Summing
up, we can say that Anaximander's *ápeiron* is Being
where negation—the "no" and "not"—is banished,
excluded. It is the IS without the NOT.

But now we run into an insoluble problem:
how come things—you and I, these words—come out, emerge from,
the *ápeiron*? Here we arrive at point (2), the direct
quote from Anaximander. The first formulator of the Principle
of Sufficient Reason must have been keenly aware that there can
be no sufficient reason to make anything, any individual, emerge
from where there are no individuals, no distinction between the
"is" and the "is not." Thus, Anaximander calls
the groundless emergence of things, their generation, an offense,
a crime. However, unlike crimes in Greek tragedy such as we saw
in our last lecture, it is not a crime against morals, ethics
or the gods—the gods play no important role in the discourse
of the Presocratics—we should call it a crime against logic.
And, according to what I called the first axiom of tragedy in
my previous lecture, a penalty must be paid: when the time comes,
everything reverts to the *ápeiron* and vanishes.
This is, as you can see, a point where tragic thought touches
philosophical thought. And here, for the first time, we are told
that existence is a crime against logic, which means existence
is absurd—as the existentialist writers repeated two-and-a-half
millennia later.

The *ápeiron* is not the
same as chaos. Perhaps what resembles most closely the *ápeiron*
is to be found in the doctrines of Buddha, in Nirvana; it is remarkable
that he and Anaximander were roughly contemporaneous, although
living far apart, one in northern India, the other in Ionia. But
about Buddha and Nirvana Prof. Ng will talk later on.

Before we leave Anaximander, it's important
to note that in his thought we encounter, for the first time,
the question that's asked by *all* of Greek philosophy: since
change and becoming, that is, the whole world as it appears to
us, is *being* infected with *not-being*, where is Real
Being to be found? By Real Being they meant: being not infected
by not-being. In other words: never-changing being, being not
affected by becoming. Anaximander answered this question in a
particular way: for him Real Being is the Boundless, the *ápeiron*.

Let us now mention two important thinkers
whom tradition opposed to each other: Heraclitus of Ephesus (in
Ionia) (ca. 540-480 BC), and Parmenides of Elea (a town in southern
Italy) (born ca. 539 BC). Both struggled with being, not-being,
and that combination of both with which we are familiar: becoming.
To get an idea of how they struggled, you should click on either
name. Here we can only say: Heraclitus took change and becoming
to be cyclical: for him Real, Eternal Being is precisely the chain
of eternally recurrent cycles. Parmenides denied outright the
possibility of thinking the concept *not-being*.

We will next consider a different solution to the problem of not-being, that of the Pythagoreans. Pythagoras himself is a legendary figure, very much in the style of a shaman. We know that Pythagoras moved to southern Italy and gathered a group of disciples—we would call it rather a sect than a school—which became politically dominant in some cities. Their influence on Western thought has been enormous: Plato would be inconceivable without the previous Pythagorean doctrines, and Western science would be inconceivable without Plato.

What were the Pythagorean beliefs? First and foremost, that "all is number." Under the flux of becoming, these thinkers, who were the first mathematicians as we understand that word, thought they had discovered something unchanging, eternal, and therefore endowed with Real Being, as opposed to mere appearance: this something was numbers and their relations. Since all discoveries were attributed to the Master of the sect, we don't know who was the one who actually discovered, among other things, that musical intervals correspond to numerical ratios. Take a taut string with a certain thickness and density, and with a certain length. Say that when plucked this string gives the note C. If you make that string half as long, you'll get the C an octave higher. Those two strings are in the ratio 2:1. If now you make the ratio of their lengths 4:3 you'll get an interval of a fourth, like C-F; if the ratio is 3:2 you'll get a fifth, like C-G, and so on. Similar remarks apply to percussion and wind instruments. This important discovery in music suggested, too, that other phenomena, especially the motion of the celestial bodies, obeyed strict numerical rules. This is why mathematics, music and astronomy went together in the school curriculum for more than two thousand years. Of course, modern science proceeds similarly: to explain a phenomenon scientifically means to find numerical laws behind it.

The Pythagoreans also believed in reincarnation: after we die, our soul will be incarnated in another body. It is rarely noticed that this belief is intimately connected with the belief that numerical relations are the only true reality. To see why, let us quote from a modern American scientist, one of the founding fathers of computing and cybernetics, Norbert Wiener (he died in 1964). "One thing at any rate is clear," says Wiener, in strict Pythagorean spirit: "The physical identity of an individual does not consist in the matter of which it is made." What does it consist in then? In patterns. "A pattern is a message, and may be transmitted as a message. How else do we employ our radio than to transmit patterns of sound, and our television set than to transmit patterns of light? It is amusing as well as instructive to consider what would happen if we were to transmit the whole pattern of the human body, of the human brain with its memories and cross connections, so that a hypothetical receiving instrument could re-embody these messages in appropriate matter, capable of continuing the process already in the body and the mind..."

Perhaps in our computer age this is more than a dream? If we could code our identity as a long string of zeros and ones and reincarnate in a computer, we would have solved the old problem of not-being: we would have become immortal, and would not have to be bothered with the horribly difficult thought of not-being. Of course, things are not as simple as that... But we'll have much more to say about old Pythagoreans and modern ones like Wiener in future lectures.

I hope you have already understood that
the problems associated with the concepts of "being"
and "not-being" have not *only* to do with our
human concern for death. They have a lot to do with science, and
that from the earliest time. Take, as a further example, physics,
the structure of matter. The first thinkers who proposed atomism
as a physical explanation—Leucippus (ca. 420 BC) and Democritus
(ca. 400 BC)—defined the atoms as the smallest units of being.
So whatever is in between atoms, they said, is necessarily—you
guessed it: not-being. Thus, all matter, all reality, is a mixture
of being and not-being. But, through the centuries, as we will
see later when we pick up physics again, scientists and philosophers
have objected: how can whatever *is* between the atoms *be*
not-being? There seems to be a contradiction here, and this caused
a tremendous amount of controversy at the time of the birth of
modern science, in the 17th century, and even beyond.

Let me now tell you what happened to
the Pythagoreans and their solution to the problem of Real Being—that
Real Being resides in numbers. You must understand that for the
Pythagoreans the word "numbers" meant what we call the
natural numbers (1, 2, 3, etc.) and their "ratios" or
what we call fractions (1/2, 1/3, 2/3, 5/7, etc.) Now, if all
real being is number, all real being must be *these* kinds
of number: natural numbers and their ratios. We saw that musical
intervals, indeed, are at bottom just that: natural numbers and
their ratios. But it so happens that there's a very simple length,
namely the diagonal of a square, which cannot be expressed as
any of these numbers. More specifically, if you have a square
whose side measures one yard, its diagonal will not be any whole
number *nor* any fractional number of yards. But if a square
has any real being (which Pythagoreans believed), its diagonal
should be endowed with real being too. Still, it cannot be expressed
in numbers! The legend is that when this was realized, the Pythagoreans
tried to conceal the discovery, threatening with death anyone
who would divulge it. But of course, you cannot conceal logical
truths for too long.

We don't know precisely how the Pythagoreans proved that the diagonal of a unit square (a square whose side measures one unit) is not a fraction, but I will show you a proof that's probably not very different from the one they found. The Pythagoreans certainly knew what we call Pythagoras' theorem, namely that the square of the hypotenuse of a right triangle is the sum of the squares of the sides. If you have a square whose side is 1, Pythagoras' theorem then tells us that the square on the diagonal must be 2. So what we're going to prove is that there is no fraction whose square is exactly 2, and therefore the length of the diagonal is not a fraction—is not a number (as the ancients understood that word).

To do this, let's first deal with some properties of even and odd numbers, something the Pythagoreans were very fond of. An even number is divisible by 2, and therefore it can always be written as 2 times some other number = 2n. An odd number is not divisible by 2, and it's not hard to show (we'll skip it) that an odd number is always equal to 2 times some other number, plus one = 2n+1. As we see, even and odd numbers alternate, an even number is followed by an odd, and odd by an even, and so on. Next, let's see what happens when we multiply. The following are easily proved by elementary algebra:

- An even number times an even number is even—in fact, the product is a multiple of 4.
- Odd times odd is odd.
- Even times odd is even.

Now to the proof that there is no fraction
whose square is equal to 2. The proof will be by contradiction;
that is, we will assume that *there is* a fraction whose
square is 2, and we will logically deduce from this that something
*is* and simultaneously *is not*: since we accept the
logical Principle of Contradiction, we are then forced to reject
the proposition that there is a fraction whose square is 2. Let's
do it. Suppose there is a fraction (a quotient of natural numbers),
n/m, which when multiplied by itself equals 2. This logically
implies that there is a quotient of natural numbers, p/q, which
when multiplied by itself is also equal to 2, but which is now
irreducible, which means that p and q have no common factors.
This is because if the original two natural numbers n and m had
any common factors, we can cancel them all out and the resulting
quotient p/q will still be the same fraction, and its square will
still be 2. So we have: p and q have no common factors, and p/q
times p/q equals 2. This, as we will now see, entails a logical
contradiction.

Saying that p/q times p/q is 2 is the
same as saying that p2 = 2q2, the square of p is equal to 2 times
the square of q. This implies that the square of p, that is, p*p,
is an even number. But looking at the three properties of multiplication
above, we see this implies that p cannot be odd: it must be even.
Again, this means that p*p is not only even: it's a multiple of
4, so p2 = 4*some number, say 4r. Since p2 = 2q2, substituting
we have: 4r = 2q2. Now, simplifying the 4 and the 2, we get: 2r
= q*q. This means that q square is an even number, and again,
looking at the three properties of multiplication, we deduce that
q itself must be an even number. But then *both* p and q
turn out to be even, and so they share a common factor, namely
2. This contradicts the assumption that p and q have no common
factor. End of proof.

This proof shattered the ancient Pythagorean belief that real being can be expressed in numerical form. It was a good, courageous attempt at capturing Real Being, yet it failed: Not-Being crept back in. But that is not the end of the story. The difficulty of "incommensurable magnitudes," such as the diagonal of the square, was patched up--at least for the working purposes of geometers--by the great mathematician Eudoxus, a friend of Plato. Much, much later, in the 19th century, mathematicians succeeded in defining a more general concept of number: the "real numbers" which we mentioned before, and which include such things as the square-root of 2. Now all lengths, all magnitudes, can be expressed as numbers. This was achieved at a price, though. Without going into details, this is what happened: the concept of infinity had to be let in, given full citizenship in abstract thought. And, as I hope to show in future lectures, once the concept of infinity is let in, paradoxes start popping up, haunting abstract thought much as the Furies haunted Orestes.