|Abstract: ||There are two areas of mathematics, namely, arithmetic and geometry. They
are independent, yet clearly separated. Arithmetic deals with numbers, geometry
deals with space. Whereas the notion of number is rooted in our thinking
that most creators of mathematics were inclined to accept it without discussion,
views on space have always been subject to deep splits. Whether space should
be treated as a mathematical object — that is as an object of thought — or as
a physical object is a question which we will not answer. Parmenides, one of the
first philosophers of nature whose views we will have occasion to investigate,
identified space with ideal existence, and thus with existence that is invariant,
homogeneous, infinite, and forming an entity.
The people noted more specific characteristic of space. One of them is
This characteristic of space is so much part of our notions that we lose our way
in its analysis. In ancient Greece the continuity of space meant like the possibility
of subdividing it indefinitely. This was the view of Anaxagoras who said that
“there is no least in the small.” Translated into non-archaic language, this means
that one can subdivide every part of space. Aristotle took this characteristic of
space as the starting point of his investigations. But there is another characteristic
of continuity which ensure the cohesiveness of continuous existence: two parts into
which we separate it mentally adhere to one other. A mathematical formulation of
this characteristic was discovered only a little more than a hundred years ago.
A continuous object, that is, one infinitely divisible and cohesive, has been
called already in antiquity a continuum. The root of this word is the Latin
continere, whose Greek prototype is syn-echein, which roughly means to bond.Space is not the only object to which we ascribe continuous structure. The
intensity of stream, or of color, seem to have this quality. But, above all, it is
the flow of time that is continuous.
A loose and free structure, composed of isolated elements, is the opposite
of a continuous structure. Such a structure is said to be discrete. The word
“discrete” is derived from the Latin discretus, separate, detached from other
things. “Discrete” thus means “consisting of, or pertaining to, distinct and
1, 2, 3, …
form a discrete structure.
Could space be discrete? This cannot be ruled out a priori. Nor can we rule
out of possibility that the flow of time might be discrete.
Geometry, the mathematical science of space, has also another, more mundane
origin. The two relevant Geek word are gea — land (we mean arable land)
and metrein — to measure. Proclus (ca. 410—485), a commentator of works of
his predecessors, wrote that “Many people assert that geometry was invented
by Egyptians for measurement of land. They needed it because the inhabitants
of the Nile washed out balks.”
From balks to infinitely divisible existence — a breathtaking span.
Space is a composite object made up of elements that enable us to realize
the nature of the whole. We single out points — places in space. This is not
a definition but just another term of language. Points are not parts of space: we
do not attribute them a material nature even when we are prepared to attribute
a material nature to space. They are not a raw material out of which space, or
a part of it, is composed. When we think of a point, we think of its location.
A point is a synonym of its neighbourhood. Only if the space is not uniform,
these neighbourhoods may be different.
Nevertheless, we are willing to imagine points as independent existences,
and the thought that they could be the raw material of space does not always
strike us as alien. This dilemma is one of the difficulties we encounter when
we think of the notion of a continuum.
Another difficulty is the infinitude of space, a notion which suggest itself
irresistibly when we think of straight lines, yet another element of space.
After a few attempts we give up the idea of defining a straight line. It seems
to be as primitive as the concept of space. One can also adopt the reverse view point: it is straight lines that suggest to us the notion of space. We see and
move along straight lines. Moving along the straight line, we move towards
an objective. We are not always sure of the possibility of reaching it. Hence
straight lines give us the initial sense of the possible nature of the infinite.
Planes are yet another element. We see in space at least one plane, the plane
we seem to be in. The initial stage of geometry codifies our notions related to
our staying in that plane. Space notions came later. Then we begin to notice
other planes as well.
The mutual disposition of points, straight lines and planes is subject to
definite rules (such as say, that two different straight lines can have at most
one common point, that they adhere to planes, and so on). That are truths that
must be accepted without proof (which does not mean on faith). Such truths
are called postulates. It is arguable whether postulates are facts so obvious
that nature thrusts them before our eyes and all we need do is note them, or
whether they statements are the result of slowly growing knowledge that is
finally spelled out, knowledge of which we do not know whether it is final
and beyond doubt. The evolution of geometry tells us that what is true is the
latter rather than the former.
It is also arguable whether the formation of geometric postulates belongs in
the domain of mathematics, or philosophy, the guide of learning. Aristotle was
believed that the issue belongs to philosophy. This statement should be interpreted
as saying that the issue is metamathematical, i.e. lies beyond mathematics.
We attribute the quality of continuity to plans and straight lines.
But straight lines are continua with the earnest structure. A point divides
straight line into two parts, each of which is again a continuum. This property
of a straight line enables us to order the set of its points. We say that a straight
line is an ordered continuum. We also say that it is one-dimensional. Neither
a plane nor space have this property.
What is space? Why does it exceed our imagination and why must
a child learn about it? Why do even accomplished painters lose their way
when dealing with perspective, a subject whose knowledge is only a few
century old, and produce either “flat” paintings or “space” paintings that are
frequently flawed? Why can’t we exit from space into an extra dimension
the way we exit from a plane? Is it because of a limitation of our senses
or is it because of the nature of space? While the first of these views is
very popular and opens the door to a variety of speculations, the three dimensionality
of space is a physical fact; no mathematical premise supports
the number 3. Kant linked the number of dimensions with the form of the
law of gravitation. Can it be that counting dimensions is a necessity of our
thought processes? Time is very troublesome. The 19th century provided a simple mathematical
description of time but behind it hides a physical phenomenon that is hard to
grasp. There is also a subjective sense of time. The two are connected. Explanation
of this connection is a task of natural sciences: for physics, physiology, and
psychology. In spite of its vagueness, time is subjectively the most continuous
of all continuities: if we cannot imagine a break in the space then we cannot
possibly imagine a break in time.
It seems that time is a stream of events with a direction. It isn’t clear whether
the notion of direction of time is due to our senses or is part of the nature of
things. Time seems to flow continuously. If not much is going on, then we notice
changes of the intensity of its flow, momentary atrophies and turnings. We seem
to flow with the stream. We do not know if the flow of time is everywhere
the same and whether it will always be the same. We cannot imagine its ever
coming to an end and its ever beginning. We experience the physical nature
of time most having intensely when we can turn time back. Preconditions for
this are: a small number of phenomena and not much happening. Then we can
turn the time back by restoring earlier positions of moved objects. To turn the
time back in the full sense of the word we would need all the energy in the
world, if not more. Aristotle, with Plato in mind, said that “Some claims that
time is the motion of the whole world.” St. Augustine agreed with Plato and
thought that time began at the moment of creation, and added that before that
moment eternity ruled.
We tend to think of a moment as a point separating the past of the future.
This means that we are willing to treat time as the ordered continuum, a universal
continuum for all phenomena, but, strictly speaking, we never ascertain
this universality. Each range of phenomena seems to have its own time stream.
The time notion we use is always a strand we attribute to the stream of phenomena
in which we move. In that strand a moment seems to have a definite
content. In mathematical problems we restrict phenomena so that time takes
on the structure of a straight line.
The ancients removed time from the range of mathematics. Their geometry
— as Aristotle stated succintly — was limited to consideration of motionless
existences. They had definite reasons for so doing. We will talk of these reasons.
Modern mathematics has included time in its deliberation as a schematic
existence devoid of all the varied properties suggested by its nature.
We speak of space and time as of things. We have no right to do this because
these are qualities of things rather than things, qualities we might call
spaceness and variability. But when speaking about qualities of things we sometimes
find it convenient to elevate them to the level of things. Then we forget about the origin of the new existences and treat them like things. Plato called
these existences ideas, and maintained that they are the only things worthy of
deliberation. Let’s not argue about this. An issue more worthy of argument is
probably the issue of the origin of ideas. In spite of the fact that we are their
makers (or, at least, we think we are), we make them as a result of the pressure
of phenomena, and this endows them with a quality of objectivity. If we do not
want to limit ourselves to the manipulation of objects and events, then ideas
are indispensable for our thinking. We fix their properties so as to enable to
think about these properties as if they were characteristics of external objects.
But it is an exaggeration to follow the believers in Plato for whom the world
of objects and phenomena is a mere reflection of the world of ideas. We can
go further in this opposition to Plato, like Aristotle we can say that ideas are
the only things we can investigate in a rigorous manner.
In spite of the fact that ideas evolve, the evolution of mathematical ideas is
very slow. This gives the impression that the structure of mathematical knowledge
grows like a building. The notion of number does not change, and when
we look at the three millennia of the evolution of geometry, to the period for
which we have documentary evidence, the changes of concepts are minimal.
The concepts of physics are less durable. But we hasten to add that it took two
millennia to replace the physics of Aristotle with its opposite, the physics of
Newton. Some claim that the most durable principles are the principles of logic.
Time to pose a more basic question. To what extent are the mathematical
notions we form independent of the way we observe or even of the nature of
This question was posed by Kant. Roughly speaking his answer was that
in our choice of motions bearing on time and space we are limited by our
nature. Once equipped with such notions — whether inherited, learned at an
early age, or picked up with the rest of the culture of our environment — we
use them in fixed form.
According to extreme views connected with this orientation, man is
equipped with a sense of time and space which imposes a definite pattern on
the knowledge he forms. We cannot completely reject this possibility, but in
line with what we’ve said thus far, we state a reservation. Even if it is true
that our sense of time and space depends on the limitations of our nature, this
sense was shaped under the influence of the outside world, and thus contains
a general cognitive element. To use Kant’s terminology, this is a cognitive
element a priori.
Kant’s views are a good reference to a veritable maze of presentations philosophy
which can serve and that admits to mathematics. We took a step away from Kant’s view in a direction that admits the evolution of what Kant called
reason. But one can take a step in a direction that ascribes to reason in Kant’s
sense invariability and absolute infallibility. The invariability of mathematical
truths seems to justify this view. Many thinkers, such as the Pythagoreans,
Parmenides, Plato (the key representative of this viewpoint), St. Augustine,
and among more recent figures, Bolzano and Cantor, are inclined to accept it.
For the Greeks, the notion of continuum emerged from the philosophy of
nature, that is, contemporary physics. Attempts of its mathematization failed.
The famous aporia of Zeno of Elea paralyzed these attempts. Such failed attempts
are found in the works of Aristotle, which include an accent of his
own view. Aristotle concentrated the key difficulty in the question whether the
continuum can be viewed as made up of points.
An affirmative answer leads to difficulties. Aristotle was sufficiently open
minded to admit that it also leads to a logical contradiction. But the negative
answer deprives us of methods.
Attempts were made to get around these difficulties by erecting certain
thought barriers. The construction of Euclid’s Elements rules out the possibility
of stating Zeno’s aporias in the language used there.
We know more of the continuum than the Greeks, but the area of ignorance
has not decreased. Every now and again discoveries are made. They are
undoubtedly important but are unnecessarily advertised as epochal, discoveries
that claim to have solved the problem.
We will try to show that this view is false. We will give a historical account
of the problem and show how philosophers and mathematicians, both famous
and not very famous, lost they way in the labyrinth of the continuum, what
was the outcome of their efforts, and in what sense their labors, so seemingly
Sisyphian, were actually not.|