“Given a triangle. Is it isosceles?” a teacher
asks strictly. “It looks like it,” the student replies.
This is a rather absurd
dialogue. Anyone who knows a little about mathematics knows that “looks” don’t
prove anything. Two procedures are allowed: a suitable proof and a suitable
measurement, or more generally: a suitable observation. What constitutes a “suitable
proof” is far from a simple question, but we’ll put that in parentheses for now
(trusting we can return to it later).
But what constitutes a
suitable observation and measurement? In this area – until now – there has been
no real order or law, but we are now beginning to address this gap. There are
two major theoretical fields of science where observation plays a serious role:
geometry and physics. The role of the observer has become particularly
spectacular with the revolution of modern physics. It is as if processes have
lost their absolute or objective character, and the way they are observed has
become a decisive element. (It’s true that Galileo also dealt with the
paradoxes of observing relative motion.) It is a well-known fact that quantum
physics has accepted as a law of nature that the foundations of the world
behave according to whether someone observes them. Not to mention the true
failure of human observation: for thousands of years, people were unable to
correctly interpret the structure of the cosmos they observed day and night.
Returning to the important and
gentle world of geometry, let’s divide our research area into two parts:
simple, “bare-eyed” observations, and constructions (“already known to the
ancient Greeks”) with a ruler, a compass, and chalk. It is worth making a mythical
figure, a certain (fictional) Theoristos, the common denominator of these two
areas, whose task is to observe and construct. The big question is: what
(human, superhuman, or even “artificial”) abilities should we give him? What
should be allowed and what should be forbidden? This is none other than the
science of observation and perception, Teorology.
The rules of geometric
construction are well known, but perhaps not sufficiently systematized in a way
that an axiomatic theory would require. For example, it must be stated that
Theoristos can place his compass on a given point as many times as he wants,
that he can always decide whether a point lies on a line or a curve, etc. But a
number of more difficult questions arise, e.g., can the compass be placed on a
line or curve if there is no marked point on it? An interesting
(“multiple-chance”) question is whether placing the compass on a point that was
not “given” before marks the point.
An even more difficult
question seems to be: can a tangent line be drawn to two circles? It would be
particularly important to list the observations we “allow” Theoristos to make,
such as drawing a line and marking two points on either side of the line. This
quickly reveals the different categories and competencies of observation. This
theoretical experiment is an innocent spiritual adventure of the human mind.
Today it is difficult to decide whether it promises little or much, but
especially now, in the turbulent period of the rapid development of artificial
intelligence and machine learning, it can bring about surprises.
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