Course: HSS328

Philosophy of Mathematics

Name:

Wajahat Tahir

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.