The problem is that mathematical ideas
are just that – ideas. They are
not contrived as tools to better understand the world around us. Instead, they exist external to ourselves and are developed within a human mind
simply for the sake of further thought.
However, more often than one might otherwise suppose, these abstract
mathematical ideas show their face in the fabric of the physical world in ways
that were never imagined in the mind where the mathematical idea first took
root. Here, we will look at a few of
these mathematical “surprises” and see just how far their influences can reach.
First, let us
consider a perfect circle.
Mathematically defined, a perfect circle is the set of all points in a
plane that are at a given distance from a given point, the center. While we can all imagine such a perfect
circle, it is actually quite difficult to find one in reality. Even a circle drawn with a computer or a
compass, pen and paper will have slight variations in it. Indeed, at the atomic or quantum level, there
can be no continuous curve that would constitute a circle existing! Yet, the idea is simple and clear in our
minds.
From this concept of
a perfect circle, many of the ancients attempted to find the ratio of its
circumference to its diameter. The
Egyptians, around 1650 BC, thought this to be a fraction slightly above 3.[1]
The ancient Israelites also had a figure around three. Their understanding of this ratio is recorded
in their sacred texts: “Then he made the sea of cast metal. It was round,
ten cubits from brim to brim, and five cubits high, and a line of thirty cubits
measured its circumference.”[2]
Ultimately, after centuries of increased precision, we came to know this ratio
as the irrational number named after the Greek letter pi. Thus, pi was the
outworking of a completely mental exercise to determine the ratio of one
dimension to another of an abstract object that does not exist in the physical
world. Yet, this is not the end of the
story regarding pi. Not even close. Pi unexpectedly
and repeatedly shows up in a plethora of physical applications. Mario Livio relates the example of Buffon’s
Needle.[3] In this example, a sheet of paper with a
series of parallel lines each one inch apart is laid on a table. A needle that is also one inch long is then
randomly dropped on the sheet of paper.
What is the probability that the needle will land in such a way so as to
intersect one of the lines?
Surprisingly, it is exactly pi divided
by two.
Another ratio, about
which Mario Livio wrote an entire book because of its surprising applications,
is known as phi, or the Golden
Ratio. This ratio is found when one takes
a line segment and cuts it into two sections such that the ratio of the
original to the longer section is the same as the ratio of the longer section
to the shorter section. This ratio
(calculated at approximately 1.61803) was originally found by Euclid of
Alexandria around 300 B.C. for purely geometric purposes. Centuries later, and totally apart from
Euclid’s endeavors, a man named Leonardo Fibonacci generated an infinite series
of numbers (0,1,1,2,3,5,8,13,… ) in which the sum of the previous two numbers
equals the value of the next number in the series. The Fibonacci numbers and Euclid’s phi have nothing to do with one
another. However, if one takes the ratio
of a certain Fibonacci number with its direct predecessor, the series of these
numbers approaches the value of phi exactly! This is indeed a surprising result. Moreover, the Fibonacci numbers and phi are found consistently in
nature. Flower petals, seed heads,
pinecones, sea shells, spiral galaxies, hurricanes, human fingers, and even DNA
molecules all exhibit the golden ratio and the Fibonacci numbers.[4] This is indeed surprising. Two separate ideas generated solely in the
minds of men have unexpectedly shown themselves in hundreds of ways in nature.
A more complex and
abstract mathematical idea is that of symmetries. Although it is too technical for the purpose
of this paper, we can recognize that a square has a set of 4 rotations that
will reproduce the same square. An
equilateral triangle has three. A circle
has infinitely many. Thus we can say a
circle is somehow more symmetric than a square than a triangle. We can start with a circular table, but if we
generate a mathematical group of its symmetries, we have something different. When considering this, mathematician Edward
Frenkel writes, “We cannot touch or hold the set of symmetries of a table, but
we can imagine it, draw its elements, study it, talk about it.”[5] Yet objects seemingly do not generate such
symmetries. In other words, symmetric
objects merely have the objective and unchanging mathematical property of being
symmetrical. If there were no
symmetrical objects in existence, humans could still imagine such objects and
study them. However, we know that
symmetry runs rampant in nature. Almost
all life exhibits some form of symmetry.
This is another mathematical surprise that mankind takes for granted,
but there is no reason to expect this to be the case.
The
Platonic Answer
Einstein once asked,
“How is it possible that mathematics, a product of human thought that is
independent of experience, fits so excellently the objects of physical
reality?”[6] This is the question we will attempt to
answer in the remainder of this paper.
Indeed, many have attempted to answer this question. From these attempts, two main answers have
arisen. I will show that both of these
answers are lacking in lesser or greater degrees, and I will show that
intelligent design seems to hold the key.
The first proposed
answer is given by those that have been called the mathematical Platonists, or
mathematical realists. Regarding
mathematical knowledge, Edward Frankel remarks, “The fact that such objective
and enduring knowledge exists is nothing short of a miracle. It suggests that mathematical concepts exist
in a world separate from physical and mental worlds – which is sometimes
referred to as the Platonic world of mathematics.”[7]
According to such a view, if humanity went extinct overnight, two rocks plus
another two rocks would continue to be four rocks. The ideal of a perfect circle, the Fibonacci
numbers and ratios such as pi and phi would persevere in our absence. Furthermore, man has no role in the creation
of mathematics. Oxford mathematical
physicist Roger Penrose says, “Mathematical truth is not determined arbitrarily
by the rules of some ‘man-made’ formal system, but has an absolute nature, and
lies beyond any such system of specifiable rules.”[8]
Man does not generate mathematics but merely discovers it as he goes
along. This is the position of the
Platonist.
Such a position
requires that there be some “other” world apart from the physical world where
we reside where mathematical entities “exist” in some very real sense. However, the Platonic view by itself does
nothing to explain why these two words should overlap. Even if one were to adopt the view that the
Platonic mathematical world somehow underpins the physical world, there still
is not an explanation as to why that would be the case. We might see evidence for it, but it is just
as “surprising” as it was before. I
propose there to be a simple and intuitive solution to this Platonic problem,
but let us first consider the naturalists’ answer to the surprising
effectiveness of mathematics.
The
Naturalist Answer
It
should be clear by this point that the Naturalist simply cannot accept the
Platonic answer to the effectiveness question.
Naturalism asserts that the physical world is all there is. Mathematician and naturalist Yehuda Rav has
written that “there are no preordained, predetermined mathematical ‘truths’
that just lie out or up there.
Evolutionary thinking teaches us otherwise.”[9]
Thus, to assert that there is a completely different world that contains
objective and eternal mathematical truths would akin to denying
Naturalism. So what option is left to
the Naturalist?
Clearly, the first
thing a naturalist is inclined to do is reject platonic mathematical
realism. John Byl notes that while
mathematicians are practical realists, this isn’t the case with other
academics. “Nowadays,” write Byl, “most philosophers of mathematics reject
mathematical realism, primarily because of its perceived ill fit with
naturalism.”[10] It should not be overlooked that it is
inherently not the case that
mathematics leads to naturalism. Rather,
the presumption of naturalism’s truth has led some to reject the platonic
ideals foundational to objective mathematics.
But if that is the case, what is left of mathematics? How does one explain the apparent
objectiveness and universalness of mathematical truths?
The naturalist
answers by saying that mathematics is a manmade invention. Furthermore, since man is the direct result
of natural selection acting upon random variation, mathematics is likewise a
product of evolution. It is easy to see
how an individual who “puts two and two together” in an appropriate way has a
survival edge over one who does not. As
a survival tool, math became genetically ingrained into the human brain. According to such a naturalist perspective,
math has no existence outside of the human brain. Furthermore, since humanity has supposedly
evolved from one or very few common ancestors, this genetically imbedded
mathematics is common to all humanity.
Thus, mathematics merely appears universal. Citing different psychological experiments,
Mario Livio relates, “Experiments show that babies have innate mechanisms for
recognizing numbers in small sets and that children acquire simple arithmetical
capabilities spontaneously, even without much formal instruction.”[11] Also, legions on the inferior parietal cortex
in the brain adversely affect one’s mathematical ability. According to the naturalist, these physical
evidences lend support to their conclusion that math is a product of the brain
and nothing more.
Problems
with the Naturalist Answer
While the Platonic
view of mathematics has problems of its own, the naturalist has problems that
seem insurmountable. How, for example,
does Darwinian evolution account for abstract algebra and complex
analysis? Paul Davies notes that “While
a modicum of intelligence does have a good survival value, it is far from clear
how such qualities as the ability to do advanced mathematics… ever evolved by
natural selection.” Indeed, evolution does not even guarantee that our
mathematics is correct. It merely
asserts that the mathematics we developed were the best as helping us survive.
Moreover, Davies adds, “Most biologists believe… human brain has changed little
over tens of thousands of years, which suggests that higher mental functions
have lain largely dormant until recently.”[12] If the human brain has remained unchanged for
millennia, then mathematics (including the most abstract varieties) must have
been genetically imbedded long before the time of the Greeks. This seems at best to be wishful thinking
spawned by cognitive dissonance. At
worst, it’s simply absurd.
A second problem
with the naturalistic perspective is its implicit denial that mathematics in
objective. When one investigates a
crime, they do so with the understanding that there is an objectively true
series of events that happened. Without an objective truth “out there,” there
is substantially less motivation to do mathematical research and
investigating. Even if a mathematician
were to adopt the anti-realist philosophy of the naturalist, they would have to
pretend that math was objective in order to carry out their research with any
real purpose. Furthermore, if math is
not objective, then how can we have any confidence that intelligence elsewhere
in the universe would have formulated anything remotely similar to the
mathematics humanity has developed?
Indeed, SETI has been sending out signals for years based on
mathematical communications. If our
concepts of prime number, for example, are merely evolutionary, then we can
have absolutely no confidence that a “prime number” is even a concept some
extraterrestrial civilization would have developed. The final, and perhaps most fatal, problem
with naturalism’s lack of mathematical objectivity is that almost all of our
science is founded upon mathematics.
Physics has been so successful primarily because it is basically
mathematical. Quantum mechanics is
almost entirely explained using equations.
If, however, mathematics is the result of random variation and natural
selection, then so is science. This
makes science less than objective. But
science is the bedrock on which naturalism is founded. In a sense, then, the rejection of the
objectivity of mathematics by the naturalist is self-refuting.
This rejection of
objective mathematics becomes even more problematic when we see that
mathematics is intimately tied with logic, which is itself critical to
reasonable human thought. When we say
that some a statement “A” and the statement “not A” cannot be both true at the
same time and in the same meaning (this is the logical law of
non-contradiction), this seems to be an objective truth rooted in an external
platonic world of logic. If, however,
there exists no such platonic world and if evolution explains the existence of
mathematics and the logical rules it plays by, then the law of
non-contradiction is the product of random variation and natural
selection. In other words, we can have
no confidence that the law of non-contradiction is a law at all outside of the
biological human brain. This is clearly
an enormous problem for the naturalist who wishes to do away with platonic
ideals.
A third problem for
the naturalist is that is simply does nothing to address the problem at the
outset of this paper. Namely, why is
mathematics so powerful in explaining the universe? Why do abstract mathematical ideas formed
centuries ago show up in completely separate physical studies and the sciences
of today? Outside of evolutionary
hand-waving, which is less than objective as we have seen, there does not seem
to be a satisfactory answer.
With all of these
shortcomings in mind, I believe it is time we consider the solution that
Intelligent Design offers.
Intelligent
Design as the Solution to the Applicability Problem of Mathematics
Thus far I have not
spoken much of intelligent design and its role in abstract mathematics. I have reserved its role for the present
simply because intelligent design, once entertained, is a simple and obvious
solution to the problem of the applicability of mathematics. Indeed, there was once a time when an
intelligent designer was simply assumed, and during that time there were not
mathematical “surprises.” Johannes
Kepler believed that geometry “supplied God with patterns for the creation of
the world, and passed over to Man along with the image of God; and was not in
fact taken through the eyes.”[13] The great physicist and mathematician
Pierre-Simon de Laplace wrote in 1812:
Given for one instant and
intelligence which comprehends all the forces by which nature is animated and
the respective positions of the beings which compose it, if moreover this
intelligence were vast enough to submit these data to analysis, it would
embrace in the same formula both the movements of the largest bodies in the
universe and those of the lightest atom.[14]
The thinkers from
centuries past may not have been evangelical Christians, but then again many of
the members of the current intelligent design movement are not Christian
either. What they all have in common is
the belief that behind the physical universe, there exists some sort of
intelligent and purposeful designer. In
such a scheme, the fact that platonic mathematics influences and describes
physical phenomena is hardly surprising.
However, with the
rise of philosophical naturalism, there ceased to be any valid account for a
platonic view of mathematics and why it should be applicable to the
universe. Indeed, even a realist
perspective without such a designer still lacks the necessary step to go from
the platonic world to the physical world.
Thus we have arrived at the critical point of this analysis: The problem is not that mathematics is inexplicably effective for describing the
universe. The real problem is that philosophical naturalism forces us to believe
that the applicability of mathematics is a problem. John Byl writes, “within the context of
Christianity it makes sense to talk about the objective existence of numbers
and mathematical truths. Once God is
removed, there is nowhere to place abstract mathematical concepts and nobody to
guarantee their truthfulness.”[15]
By incorporating the
possibility for an intelligent designer into our understanding of the world,
the objectivity of mathematics gains a firm foundation. Furthermore, we gain confidence that our laws
of logic are grounded and are able to lead us to objective truth. The importance of this cannot be
overstated. This entire paper, and many
others like it, has the purpose of arguing for the truth of one position over
another. If our laws of logic are not
universal, then this argumentation is pointless and its fruit will be
random.
An intelligent
designer also allows us to think objectively about an actual infinity instead of just a possible infinity. Why is this the case? Modern quantum mechanics has shown that there
exists the smallest of discrete sizes in the universe. Big Bang cosmology asserts that the universe
is finite in age and is finite in size.
Taken together, we see that physically
the universe does not contain the infinite or the infinitesimal. Why is this such a big deal for
mathematics? The entirety of the
calculus is based on there being a conceptually real idea of the
infinitesimal. Taking an integral or a
derivative of a mathematical function demands that the function not be
discrete. Such calculus is used in
almost all of our engineering today.
Even though an actual infinity does not exist, we can still
conceptualize it and apply it in everyday life.
This seems illogical if there is not some external world where an actual
infinity exists. It also seems logical
that if an intelligent designer exists, it would be infinite in nature (though
such arguments lie outside the scope of this paper).
But what should we
make of the physical evidences offered in favor of the naturalistic view of
mathematics? How is it that brain
legions affect mathematical ability? Why
are children seemingly born with an innate ability to do mathematics? Doesn’t this all seem to point to the truth of
naturalistic evolution and physicalism?
I assert that it does not. In these
cases, the evidences favor intelligent design just as much as naturalism if not
more so. Intelligent design allows for
the anti-naturalist position of mental dualism, the idea that the mind and the
body are distinct. In such a framework,
the brain and the mind are related. If
the brain is physically injured, this could create difficulties for the
mind. More profoundly, an intelligent
designer could uphold not only objective mathematics, but also the
“pre-programming” of the human mind.
Thus the innate ability of children to mathematics is far from a “proof”
that mathematics is the result of human evolution.
Conclusion
Mathematics is
pervasive and dense in the world around us.
It is everywhere in the universe and even seems to lie behind everything. Ancient mathematical ideas continue to show
themselves in the physical world in areas that were never even thought of
before. The overwhelmingly successful
application of mathematics to the physical world generates an important
question. Why is this the case? Platonism gets close to answering this question,
but cannot quite explain why an external mathematical world should have
anything to do with the physical world.
Naturalism fairs much worse that Platonism as it casts doubt on the
objectivity and universality of mathematics and logic. In the end, the best possible answer is that
there is an external world of objective mathematical truth, and there exists
and intelligent designer that actively incorporates these mathematical truths
into the very fabric of the universe.
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[1] G.
Donald Allen, “Pi, A Brief History.” (Texas A&M University: College
Station, TX). Web. http://www.math.tamu.edu/~dallen/masters/alg_numtheory/pi.pdf
Accessed, Nov. 28, 2016.
[2] 1
Kings 7:23, ESV.
[3]
Mario Livio. The Golden Ratio: The Story
of Phi, the World’s Most Astonishing Number. (New York, NY: Broadway Books,
2002). P. 2.
[4]
Mario Livio discusses these and many other natural occurrences in his book.
[5]
Edward Frankel. Love and Math. (New York, NY:Basic Books, 2012). P. 21.
[6] Max
Jammer, Einstein and Religion, Princeton University Press, 1921.
[7]
Frankel, 23.
[8]
Roger Penrose. Shadows of the Mind,
Searching for the Missing Science of Consciousness. (Oxford, England:
Oxford University Press, 1994). P. 418.
[9]
Yahuda Rav. Math Worlds: Philosophical
and Social Studies of Mathematics and Mathematics Education. (Albany, NY:
SUNY Press, 1993). P 100.
[10]John
Byl, The Divine Challenge: On Matter,
Mind, Math & Meaning. (Carlisle, PA: Banner of Truth,2004). 137.
[11]
Livio, 244.
[12]
Paul Davies. Are We Alone. (New York,
NY: Basic Books, 1995). Pp. 85-86.
[13]
Johannes Kepler, E. J. Aiton, Alistair Matheson Duncan, Judith Veronica Field. The Harmony of the World, Vol. 209. (Philadelphia,
PA: American Philosophical Society, 1997). P. 304.
[14]
Pierre-Simon de Leplace. Philosophical Essay on Probabilities (1814), 5th edition (1825), trans.
Andrew I. Dale (1995), 2.
[15]
Byl, 255.
