Course: fundamentally mathematical. This is a strange

Course:                     HSS328
Philosophy of Mathematics

Name:
                       Wajahat Tahir

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Student
ID:               2014038

Title
of paper:         Mathematicism

 

Contents:

§  Foreword

§  Introduction
to Mathematicism

§  Origin of
mathematics and reality

§  Foundation
of mathematics

§  Mathematician
and philosopher

§  Anti-Mathematicism

§  Conclusion

Foreword

This paper is intended to investigate Mathematicism,
its origin, and verification through its analysis in light of the origin of
mathematics, comparison of mathematics with reality, nature of mathematics and
differences between mathematics and philosophy.

The origin of mathematics is itself subject to
argument. Whether the birth of mathematics was a random happening or induced by
necessity duly contingent upon other subjects, say for example physics, is
still a matter of prolific debates.

Through a brief research, one can come to a profound
puzzle that while on one hand mathematical truths seem to have a compelling
inevitability, on the other hand, the source of their “truthfulness”
remains elusive. This paper sheds some light indirectly on this vagueness as
well. Investigations into this issue are known as the foundations of
mathematics program.

This paper, in addition to the aforementioned
pursuits, also deals with the role of humankind in developing mathematics, the
sources of mathematical subject matter, the objectives of mathematical inquiry,
the source and nature of mathematical truth, the relationship between the
abstract world of mathematics and the material universe. In conclusion, the overall
aim is to deal with the topic of Mathematicism holistically from different
perspectives and verify its claim.

Introduction to Mathematicism

Mathematicism
is actually any viewpoint of philosophy that claims that everything can be
described ultimately by mathematics and that everything in the universe,
including its reality, are fundamentally mathematical. This is a strange
philosophical standpoint for its strong claim. It assumes mathematical
conformity within the universe and everything in it. Mathematics is itself a
field with many discoveries and inventions of famous people, so if the Mathematicism
is a correct idea, then mathematics itself has been a route, rather than a
tool, for discovering the truths rather than a pursuit of solutions for
problems that have been the need of time.

Mathematicism
is a form of rationalist idealist but as it started with ancient Greece’s
Pyhtagoreanism and Platonism, it has an effect from these two rationalist
idealist schools of thought. Pythagoreanism originated in the 6th
century BC, based on the teachings and beliefs held by Pythagoras and his
followers, who were influenced by mathematics and mysticism. Considering Pythagoras’
saying that ‘All things are numbers,’ and ‘Number rules all,’ one can easily
see how the idea of Mathematicism’s claim rose at that time.

Pythagoreanism
is a philosophy, which prescribed a highly structured way of life and espoused
the doctrine of metempsychosis (transmigration of the soul after death into a
new body, human or animal.) Pythagorean thought was not only mathematical but
also profoundly mystical. There are many Pyhtagorean theories but all of them
are based on the assumption that mathematics and numbers constitute the true
nature of things. There are two separate schools of thoughts in Pythagoreanism:
the akousmatikoi, or listeners, who focused on the religious and ritualistic aspects
of Pythagoras’ teachings and the mathematikoi, or learners, who extended and
developed the mathematical and scientific work he began.

The
mathematikoi group of Pathagroen later became associated with Plato and
Platonism and that is why much of the Pythagoreanism seems to overlap
Platonism. Platonism is a philosophy that affirms the existence of abstract
objects that are asserted to exist and is the opposite of nominalism. Abstract
objects cannot be seen in real world and have no physical or material existence
in reality. The primary concept of Platonism is the Theory of Forms. According
to it, the true being is founded upon unchangeable and perfect types, while all
other objects of moral and responsible sense are imperfect copies. Platonists
have fused Pythagorean speculations on number with Plato’s theory of forms and
this is how Mathematicism has propagated.

Beside
Pythagoreanism and Platonism, Mathematicism has many other forms, which emerged
with time ranging from Neopythagoreanism to Tim Maudlin’s project of
‘philosophical mathematics.’

 

 

Origin/history of mathematics and reality

Before the
modern era, all of the texts, that have been found, have the so-called
Pythagorean triples mentioned. This shows that Pythagorean theorem has been the
most ancient and widespread mathematical concept developed after basic
arithmetic and geometry. The study of mathematics as a demonstrative discipline
also started with Pythagoreans. Later on, Greek mathematics, Chinese
mathematics and Islamic mathematics further refined and contributed to
mathematics.

Prehistorically,
the origins of mathematical thought lie in the concepts of form, magnitude and
number. The idea of numbers have been evolving in many language throughout the
time. Whether these are artifacts discovered in Africa, as old as 20,000 years,
and suggesting early attempts to quantify time or the megalithic monuments in
English and Scotland, dating from 3rd millennium BC, incorporating
geometic ideas such as circles, ellipses and Pythagorean triples in their
designs, there are evidences of how humans have used mathematics for their
benefits, but the question of whether the nature of everything-including
naturally occurring things-in reality and universe is based on mathematics?

The
application of mathematics in different phases of life and work by humans
throughout the history is a proof that mathematics played a crucial role, but
how did humans learn or discovered mathematics? Has it been a response to the
need of time or has it been the obvious phenomena that humans observed in their
daily lives in nature and then used it for their own purpose from time to time?
Although the answers to these questions definitely need thorough further
investigation, this dilemma can be of utmost curiosity to every mathematical philosopher.
With powerful computers in 21st century, the developments in
mathematics are unprecedented, but the question of whether mathematics is being
discovered from nature or developed to understand the nature/universe is too
complicated to be answered in a straightforward manner.

Foundation of mathematics

Mathematics
is the study of topics such as quantity, structure, space and change. This is
just one definition of mathematics because of the variety of views among
mathematicians and philosophers about it. Mathematics work in a way that it
seeks out patterns and make use of them to make new conjectures. Mathematicians
work to resolve the truth of these conjectures through mathematical proofs.
When such work results in good models of real phenomena, then these
mathematical reasons provide us with great deal of insight and information
about nature.

Mathematics
employ abstraction and logic in order to design such structures and achieve
accuracy in them. From the essence of mathematics, it can be seen that
mathematics is nothing but process of attempting to understand nature through
various ways. Although this analysis of the nature of mathematics gives an idea
about the rise of mathematics for understanding, it leaves the answers to our
question vague in a sense that let mathematics rely on trial and error method
of modelling and understanding different natural phenomena. There is no
evidence whether all phenomena in nature can be explained and understood
mathematically.

Mathematician and philosopher

Although
there is overlap between these two categories of scholars, there are certain
differences that make each of them unique. Kant sums it up in a clear way
through his this response: “Philosophers by way of philosophizing can at most analyze
and proffer definitions, clear them of muddles via logic and attempt to explain
the world in their own ways (a priori or a posteriori or both) though many
times in vain. A mathematician on the other hand constructs definitions a
priori and then puts them onto an empirical intuition (something that can be
sensed by the outer organs of sensation), say, paper. He then apodictically
carries on and deduces other theorems or axioms from a certain set of basic
axioms. His whole enterprise deals with certainty, for in mathematics it is
absurd to hold an opinion. A mathematician cannot say that he ‘believes’ that a
square has 480 degrees instead of 360. On the other hand, philosophers can hold
opinions provided they argue for them one way or another.”

The
relation between Mathematics and Philosophy can further be seen from the way Hilary
Putnam summed up one common view of the situation in the last third of the century
by saying the following: “When philosophy discovers something wrong with
science, sometimes science has to be changed—Russell’s paradox comes to mind,
as does Berkeley’s attack on the actual infinitesimal—but more often it is
philosophy that has to be changed. I do not think that the difficulties that
philosophy finds with classical mathematics today are genuine difficulties; and
I think that the philosophical interpretations of mathematics that we are being
offered on every hand are wrong, and that “philosophical
interpretation” is just what mathematics don’t need.”

From these,
it can be seen that mathematics is based on certain definitions, proofs and explanations,
while philosophy accommodates many opinions on certain topics. From that
perspective, philosophical interpretations and stances are not fixed, while
those of mathematics are usually very specific, clear and certain.

 

CONCLUSION

            This
paper has briefly investigated Mathematicism from different perspectives, and
it can be clearly seen that it is not an easy task to verify Mathamaticism’s
claim that the nature of everything, including reality, in the universe is
mathematical and can be explained/defined mathematically. The way mathematics
originated, evolved and worked shows that its main purpose has been the understanding
of natural phenomena. It can be seen as a tool, of abstractions and
conjectures, invented by mathematicians to understand the world and reality.
However, since as of yet, humans have not been successful in modelling
everything in universe, Mathematicism can be not be considered universally
correct without full verification. However, if we consider successful mathematical
modeling so far, we are end up impressed by its accuracy and precision in defining
nature as well as reality, where Mathematicism makes sense on a small and
restricted scale.