"How can it be that mathematics, being after all a product

of human thought which is independent of experience, is so

admirably appropriate to the objects of reality (Einstein)?"

"How can it be that mathematics, being after all

a product of human thought which is independent of

experience, is so admirably appropriate to the objects of

reality (Einstein)?" This issue has troubled philosophers

as well as scientist since Einstein first posed it. To

answer what Einstein meant when he asked the question, a

person must understand what mathematics really is. The

physical world is very important to understand when looking

at the question. When deriving a meaning from Einstein's

statement it is necessary to restate the quote in a

simplistic style and understand how mathematics can work

for situations dealing with the physical world. I believe

that mathematics is very useful to the understanding of

reality because we want it to be. For example, in the

movie PI, when Max Cohen, a mathematical genius, was

looking for the number 216 and he kept finding that number

everywhere he looked. All of the problems that humans

face are there because they are blocking the answer society

or a particular human is looking for.

Mathematics is subvertly known as a group of related

subjects, including algebra, geometry, trigonometry and

calculus, concerned with the study of number, quantity,

shape, and space, and their inter-relationships, application

s, generalizations and abstractions (beckmann 102).

Mathematics is made up of concepts that are composed of

universal relationships. One example of a mathematical

concept is the number one. One does not exist but we use

this number to represent objects which exist. To understand

the quote, "How can it be that mathematics, being after all

a product of human thought which is independent of

experience, is so admirably appropriate to the objects of

reality?" it is important to realize that when Einstein

speaks of mathematics he is speaking of rationalism. A man

named Descartes, a mathematician who lived from 1596-1650,

first headed rationalism. Descartes argued that all

knowledge obtained through direct observation of nature was

suspect because it comes through the senses, and such

knowledge can notoriously be deceptive, as all kinds of

hallucinations and dreams show (Jeans 34). Descartes has a

very good point but it is very skeptical to believe that

everything is suspect to deception. There must be some

puritanical things in the world. Two reasons why Descartes

believed that knowledge from mathematical proofs may be

deceptive were 1) mathematicians have often been wrong and

2) people can never be certain that an omnipotent God may

not have decreed that people should be deceived even in the

things people think they know the best (Jeans 35). I agree

with him for the most part but what about people that do

not believe in God. How would number two be applicable for

them. Also, these statements would discredit the entirety

of scientific knowledge because it might be from unreliable

sources. Descartes also believed that were a number of

principles already imbedded into the brain i.e. the

existence of God. All rationalists believe that all things

can be derived through intuition. Kant, a philosopher,

believed so strongly that truth is obvious that he claimed

that it would be possible to construct a "pure science of

nature" (Jeans 35). That is a very bold statement but I

think that most of the philosophers are just too black and

white. If they would just find a medium they would be able

to solve more questions. Another word for rationalism

would be a priori which means previously to, and independent

of, all actual experience of the world. Mathematics is

completely a priori and is derived from intuition.

Mathematics is very important but it is also a

necessity to understand the physical world. Physics'

purpose is to identify the most elementary building blocks

of Nature and the laws that govern them (Barrow 65).

Physics are the physical processes and phenomena of a

particular system. In this case, it would be the universe

that embodies humans and other lifeforms. The physical

world deals with empirical information and is made up of

very complex rules. To understand the quote, "How can it

be that mathematics, being after all a product of human

thought which is independent of experience, is so admirably

appropriate to the objects of reality?", it is necessary to

understand Einstein is talking about empiricism when he say

s the objects of reality. Empiricism is derived from

experiences rather than from logical principles alone.

Empiricism is also known as a posteriori. A posteriori

means that there are sources in experience (Jeans 35). As

I stated earlier, this black and white formula which most

intellectuals tend to work with is not the best way to

handle the issue. There should be a word in between a

priori and a posteriori. Empirical logic is strictly,

not able to be known independently of experience. This is

an epistemological property, and hence distinct from the

logical property of being synthetic (Jeans 36).

Epistemology is the study or a theory of the nature and

grounds of knowledge especially with reference to its

limits and validity. In opposition to rationalism,

empiricism is based on experience alone. The two most

prominent empiricists were philosophers John Locke and

David Hume. A reason to why people were persuaded to be

empiricists was because many believed that a person could

not imagine a mathematical point, line, or triangle unless

we had first made the acquaintance of their imperfect

representations in the outer world. Locke believed that

the truths of pure mathematics were not the only issues

which were constructed through empirical views. He believed

that the existence of God and ourselves and the morality

ought to be admitted to the class of intuitive truths

(Jeans 36). These philosopher are journeying into a very

touchy subject which has been questioned since man could

question. This qoute be the existence of God. Locke

statement about morality and truths can only be answered

to the extent that people can define morality and truth.

The truth of empiricism is what the physical world is based

on.

When Einstein said, "How can it be that mathematics,

being after all a product of human thought which is

independent of experience, is so admirably appropriate to

the objects of reality?", he was questioning how something

as rationalistic as mathematics can explain something as

empiricalistic as the physical world. The concepts of

mathematics and the empirical physical world are so

different most people would find it difficult to be able

to comprehend that mathematics, a conceptual view, could

explain something as complex as the physical world. Some

philosopher believe the physical world is composed of

many complex equations. I believe that many of these

equations we find are simply there because we want to find

them. For example, when a person buys a new car, that

person suddenly sees his car all over the place. The

number of cars did not simply enlarge when he bought his

new car but simply the equation appeared more translucent.

In the book, Character of Physical Law, Richard Feynman

says that mathematics is related to physics by stating,

"it is perfectly natural that the mathematics will be

useful when large numbers are involved in a complex

situation i.e. the physical world (Feynman 35). In

contradictory to that statement, Feynman also says that in

biology the action of a virus on a bacterium is

unmathematical (Feynman 35). Feynman and many other

scientists and philosophers have tried to answer the how

mathematics can explain something so complex as the

physical world. This answer will come in time just like

the knowledge to these topics will eventually be complete.

The question concerning abstract and empirical

topics and how one can explain another is simply formed

by a biased opinion just like every other philosophical

question. Rationalism will always be prominent as long

as people believe in intuition. Empiricism will also be

just as prominent as long as there are mathematicians and

philosophers that believe that everything is a posteriori.

One explanation for the two separate ideas could be

because the Continentals loved their abstract ideas and

the British's love for practical investigation. The

complex physical world is and always will be explained by

the a priori mathematics as long as there are people that

want it to be explained that way.

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