Mathematics, Truth and Uncertainty
Christopher
Altman
From its birth
in ancient Greece, and for over two thousand years, mathematics
has been viewed a body of collective truth, being the basis of innumerable
scientific theories which describe the world around us. To achieve
such powerful results, early mathematicians employed deductive reasoning
in their examination of new hypotheses. But more recent realizations
in the world of mathematics have revealed that it is not the body
of truths once assumed to be, and further, that the very deductive
reasoning used to create and develop these truths contain flaws.
In his book, Mathematics: The Loss
of Certainty, Morris Kline examines these contradictions and
claims that the history and development of mathematics is rife with
error and oversight.
A
first blow to mathematics absolute nature came with Karl Freidrich
Gauss, who in the 19th century first conceived of and
developed non-Euclidian geometry. In non-Euclidian space, the angle
sum of a triangle is not 180 degreesit can be less than this sum.
Strangely enough though, Gausss geometry is fully self-consistent.
A student of Gauss, Georg Bernhard Riemann,
developed double elliptic geometry. While it was originally thought
only to apply to certain topologies, such as the surface of a sphere,
it was later shown to be equally applicable in describing three-dimensional
space in straight line as follows the ¥rulers edge.
A
next major development which struck at the underpinnings of mathematics
was the development of quaternions,
conceived of by William R. Hamilton. These new numbers, applicable
in algebra, do not follow the same algebraic rules as most numbers.
This forced a re-evaluation among theoreticians who believed the
foundations of algebra invulnerable to attack. Both
non-Euclidian geometries and quaternions
paved the way for an even greater assault on the integrity of mathematics.
Mathematicians of the 19th century had to reluctantly
accept that the geometries and algebras which they had been using
were no longer the only working models: Apparently, mankinds mathematics
was not necessarily the one true account of an inherent design in
nature.
A
further setback to the authority of mathematics was the realization
that mathematics itself was developed without strict logical rigor.
This was not an isolated occurrence Ë throughout the history of
mathematics a number of mistakes were accepted as truths. Inadequate
understanding of concepts, a failure to recognize all of the required
principles of logic, inadequate attention to proofs, and slips in
reasoning were found throughout mathematics. Intuition, physical
arguments, and appeal to geometric diagrams had frequently taken
the place of logical arguments. These developments began as early
as Euclids
Elements, ignoring certain assumptions that were implicitly
assumed in the proofs.
Axioms critical to the development of mathematics were being reviewed,
with regard to calculus in particular. Missing logical structures
were completed and the defective portions were repaired as best
could be done. The movement often described as the rigorization
of mathematics became a primary focus in the latter half of the
19th century.
But as the mathematicians rushed to mend the tears in their discipline,
another calamity struck. Just as they were toasting to their success
in reconstructing the edifice of mathematics, paradoxes were found
within the newly reconstructed systems. Many took differing views
towards these new problems, leading to a radical division in mathematics,
and breaking it into four schools of thought.
Kurt Gðdel, a German mathematician,
proved that the logical principles embraced by the schools of mathematics
could not be proven consistent. Gðdels
Incompleteness Theorem stated that no formal system
containing arithmetic and standard logic is or ever could be
complete. Even the axiomatic-deductive method so highly regarded
in the past as the approach to exact knowledge was seen to be flawed.
This revolutionary insight weakened the already fragmented state
of mathematics into a jumbled collection of differing factions,
which remain in discord with one another to this day. The present
state of mathematics, in comparison to its previous untouchable
status, is now but ¥grand illusion. Kline states,
øthe
present state of mathematics is a mockery of the hitherto deep-rooted
and widely reputed truth and logical perfection of mathematics.Ó
Conclusions
However disorganized the world of mathematics may be today, contradictions
have always existed in bodies of knowledge. Just after periods of
major revisionin which inevitable periods of uncertainty follow,
new ideas are allowed to reach fruition. A group of French mathematicians,
under the pseudonym Nicolas Bourbaki,
defend the discipline:
There
are now twenty-five centuries during which the mathematicians
have had the practice of correcting their errors and seeing their
science enriched, not impoverished; this gives them the right
to view the future with serenity.
Perhaps we place too narrow a definition upon truth Ë for despite
the seeming contradictions of mathematics and the disagreements
which characterize its past, one evident theme remains. Mathematics
has always been and remains to be a remarkably effective method
of describing the mechanics of the world around us. Even if one
is to disregard absolute certainty in mathematics, or in any body
of knowledge, we must not give up the search for Truth, or allow
our limitations to overcome us. As Kline recounts:
A
physicist had a horseshoe hanging on the door of his laboratory.
A visitor taken aback asked him whether this brought luck to him
and his work. øNo,Ó the physicist answered, øI dont believe in
superstition. But it seems to work nonetheless.Ó
In mathematics, the mind has the power to devise structures that
can øembrace the raw data of experience and provide a mode of arranging
them.Ó The source of mathematics is the continuing evolution of
the mind itself. When looking into nature for why mathematics works
so effectively, Sir Arthur Eddington,
a physicist whom Kline turns to for explanation, stated
øWe
have found that where science has progressed the farthest, the
mind has regained form nature that which the mind has put into
nature. We have found a strange footprint on the shores of the
unknown. We have devised profound theories, one after another,
to account for its origin. And at last we have succeeded in reconstructing
the creature that has made the footprint. And Lo! It is our own.Ó
