Are the truths of mathematics invented or discovered?
By Vincent Wen
Imagine you are a prehistoric caveman living a nomadic life. One bright sunny
morning, a fellow caveman comes to you, borrows a few apples, and promises to give
them back. To make sure your fellow caveman will return them in the same quantity
so the days of gathering will not turn out to be fruitless, you raise your hands, bend
your fingers, and see if you can use them to represent the number of apples you lent.
Does this sound familiar to you? Yes, it is finger counting, the predecessor of our
numerical system.
Ever since the dawn of humanity, people have been using mathematics to make
our lives easier. Whether you think of math as a tool used to resolve our daily trivia; a
study aimed at explaining natural phenomena, or a system aided in the advancement
of our galloping technology, it has been and will always be an integral part of our life.
After twelve years of systematic study in this discipline, the importance of math in my
life has surpassed any other subject. Nonetheless, I still find it challenging to answer
this question: are the truths of mathematics invented or discovered? After much
thought, in this essay, I will argue that the truths of math are invented rather than
discovered.
Mathematics, a field of knowledge which has been developed by mankind for
thousands of years, is generally defined as an axiomatic study of quantity, space and
relations. It involves the formulation of new theorems by a rigorous deductive process
from several chosen axioms. These axioms are the primitive assumptions of
mathematics in which all other theorems and formulas must depend upon. These
axioms cannot be deducted nor proven using any logical means. They are considered
to be self-evident and are taken to be true (Penrose, 2004). For example, in our
arithmetic system, for every natural number x, x+0=x. This is one of Peano axioms
and it is an axiom because it cannot be proven using any other axioms; it is
self-evident; theorems such as for all natural numbers a and b, a+b=b+a can be
proven using this axiom (Hawking, 2005).
Having the definition of the truths of math established, it will not take a great
deal of mental exercise to arrive at the conclusion that these truths are invented.
Invention is the creation of new things such as computers, TV, automobiles, which
have never existed before. On the contrary, discovery is not creation. The object
discovered must exist before it is discovered. It could be something you have never
seen before or it could be a different way of thinking about something. A quote by
Albert Szent-Gyorgyi, a Hungarian Nobel Prize laureate, elaborates on the meaning of
discovery in a clever and concise manner, “Discovery consists of seeing what
everybody has seen and thinking what nobody has thought” (Good, 1963, p. 222).
Unlike discoveries, axioms did not exist before they were artificially defined (notice
the word artificially is used instead of explicitly, because the axioms can also be
implicitly defined when our ancestors did mathematical operations before they wrote
the axioms down). Therefore, using disjunctive syllogism, we can conclude the
axioms (the truths of mathematics) are invented.
At this point, many questions may arise which ponder the validity of the
arguments presented in the previous paragraph. What if someone proves an axiom to
be false? Why are electricity and gravity discovered but not axioms? Why can we not
say that axioms are taken to be true due to observation? For example, a Neanderthal
might have the experience that when one apple is put next to another apple, there are
two apples altogether. When the same principle is applied to other objects, it would
still work. Thus, the Neanderthal would come to the conclusion that 1+1=2 and use it
as an axiom for further reasoning.
To answer these questions, we must revisit our definition of the truths of
mathematics and re-examine its subtlety closely. The axioms, being the absolute
starting point of any deduction, cannot even be proven, let alone proven false. This is
somewhat similar to the uncaused cause of the universe1 except the axioms are
defined by humans instead of created by a supreme being. This also implies that if
1+1=3 was presupposed to be true, it could be used as an axiom, though all
deductions thereof must be consistent with this axiom. Although it may sound absurd
at first, there are actual mathematical systems in which 1+1=2 does not hold true. For
example, in the binary number system, 1+1=10; in the Boolean logic system, 1 AND
1 will still yield 1.
Albert Einstein (1923, p. 28) once said: “as far as the laws of mathematics refer
to reality, they are not certain; and as far as they are certain, they do not refer to
reality.” In the course of this essay, the latter part of that statement holds true. Unlike
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science, math is neither observation based nor experiment based. Unlike chemistry,
math is not concerned with matter. It could be based on axioms which are entirely
abstract. Unlike biology, math does not set hypotheses and accumulate knowledge
inductively. Unlike physics, math does not attempt to explain natural phenomena and
understand the world. Although all three sciences (biology, chemistry, physics) use
math as a tool to model relationships and draw conclusions, the math used by each
science actually depends on different axioms. Although some axioms seem to be
observed in nature and then taken to be true, nature actually “inspired” human to
define these axioms in the same way as the boiling kettle “inspired” James Watt to
invent steam engines. Although physics has always been trying to perfect our
understanding of the world, as impeccable as it may seem, it depends on the inductive
notion of causality2 which makes it unable to stand the scrutiny of mathematics and
its skepticism.
After all the struggles between these arguments and counter-arguments, I could
not help but think that this is not mathematics. It is supposed to be a complete, elegant
and dazzling system where the most rational thoughts in the world are encapsulated
within. However, each time I tried to defend my thesis, clarify the definitions, an
invisible net placed upon the realm of mathematics seemed to contract. Fortunately,
all we are looking at here is only a snippet of mathematics, namely the axioms. The
applications of mathematics are far beyond its confined domain of its axioms. As
indicated by Allegory of the Cave3, the search for truth always requires courage,
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discomfort, and sometimes even sacrifice. What is sacrificed is the flexibility of the
axioms, what is bestowed is the certainty of mathematics. To sum it up, the truths of
mathematics, or the axioms, are invented because they are defined by humans and
served as the foundation of mathematics due to our desire for certainty.
Work Cited:
Ade, H. (2009). Course Lecture. AY Jackson S. S., Toronto, ON, Canada.
Einstein, A. (1923). Sidelights on Relativity (Geometry and Experience). P. Dutton.,
Co.
Good, I. J. (1963). The Scientist Speculates. Basic Books (New York).
Hawking, S. (2005). God Created the Integers. Running Press.
Paquette, P. G., Gini-Newman, L., Flaherty, P., Horton, M., Jopling, D., Miller, H., et
al. (2003). Philosophy: Questions and Theories. McGraw-Hill Ryerson.
Penrose, R. (2004). The Road to Reality. Jonathan Cape.
Plato, The Republic. (B. Jowett, Trans.). The Internet Classics Archive. Available from
<http://classics.mit.edu/Plato/republic.html> (Original work done in 360 B.C.E).