1 as according to Galileo, mathematics is the language

1 – An introduction: Is mathematics a construct of
the human imagination that we tailor to describe our reality?You cannot deny the
fact that everything has a numerical counterpart; everything, especially that
which is material, have properties, patterns and structures. Is the way we
write these just notations? If, as according to Galileo, mathematics is the
language of science does that mean there could be other languages? Or is the
universe inherently mathematical in the way it is constructed and doesn’t have any
other way to behave other than according to mathematical rules?Mathematics is a
natural phenomenon. Even without our understanding of numerical relationships, nature
would still conform to our laws of mathematics. Although the way we perceive
and describe these relationships may be a human construct, is it that mathematics
is just an effective way of describing the physical reality; it’s not a
complete description, nor the only one.Axioms based on the notion of simple counting are
not innate to our universe, but what exactly are numbers? Different examples of
counting systems have occurred across history, all having different strengths
and weaknesses. For example the Sumerians used the sexagesimal system, a system
with 60 as its base; a modified version of this system is
still used today for measuring time, with 60 seconds to a minute, and 60
minutes to an hour, and angles, with 360 degrees to a circumference (Navarro, 2017) The Romans used the Roman numeral system
which in itself does not have a value for zero, yet they used the word “nulla”
meaning nothing.  Modern day human
society cannot function without numbers, they are vital for things we take for
granted – the binary language of computer programming, radio, television and even
electricity. The presence of numbers is overpowering. If we look back to the
most basic civilisations, they all developed a numerical system as a result of
a need to comprehend basic, everyday tasks, yet they were represented in a
different way and they all had the same functions: counting, ordering,
measuring and codifying (Corbalán, 2016RB1 ). This suggests that the concept
of numeracy is transcendent, although the way we transcribe them is ultimately trivial. 2 – Irrationality and the Golden RatioIn mathematics, the
word “irrational” has a different meaning than in literature. The concept stems
from the word “ratio”. A rational number can be written as a ratio of two
integers, a fraction. Therefore, it follows that an irrational number is defined
as a number that cannot be written in the form of a ratio. Irrational numbers
are never-ending, never-repeating decimals such as , a number at the very heart of the
Pythagoreans obsession with matheRB2 matics (Navarro, 2017). One of the most famous
irrational numbers of all time, one that has fascinated mathematicians and
artists alike, is the golden ratio, often denoted by the Greek letter F.
The symbol, phi, was given to the golden ratio when Mark Barr wanted to link
the ratio to Phidias, builder of the Parthenon in Athens, by borrowing his
initial, for he believed the Parthenon was built to conform to the mathematical
beauty of the ratioSB3  (Corbalán, 2016). One of the most
reprinted books of all time: “Euclid’s Elements of Geometry”, written around
300BC, is a collection of definitions, axioms, propositions (theorems), and
mathematical proofs for said propositions. RB4 Within it Euclid defined what we know today as the
golden ratio: “A straight line is said to be cut in the extreme and mean ratio
when, as the whole line is to the greater segment, so is the greater to the
lesser.”  The most irrational number “the
extreme and mean ratio” is the golden ratio (Euclid,
300BC).                                      The positive solution
to the equation  is    This is the Golden Ratio.The golden ratio is found almost
everywhere we look and is believed to represent what is perceived as perfectly
proportionate by the human eye. For example, the Mona Lisa’s face has been
framed in a succession of Golden Rectangles, to create arguably the most
elegant and visually pleasing portrait of the Renaissance (Corbalán, 2016).Leonardo’s Vitruvian Man assumed that the golden
ration was reflected in the animal world and signifies a way of thinking that
the ratios of proportion on the human body and produced a set of ideal
measurements, the ratios of which falls closely to the golden ratio. Studies
have shown that mouths and noses are positioned at golden sections of the
distance between the eyes and the bottom of the chin. The same proportions can
be seen from the side, and even the eye and ear itself, which follows along a
logarithmic spiral.  Everybody’s different, but the average
proportions across populations lean towards the golden ratio, the closer our
proportions get, the more attractive those traits are perceived. “The senses
delight in things duly proportioned”  –
Saint Thomas Aquinas (1225-1274). For example, the most scientifically
beautiful smiles are those in which central incisors are 1.618 wider than the
lateral incisors, which are 1.618 wider than canines, and so on. (Corbalán,
2016)One sequence of
numbers which is found a lot in nature is the Fibonacci sequence (Adam,2003).
The sequence starts with two ones and you proceed by adding the previous two
numbers together to obtain the next. For Example: 1, 1, 2, 3, 5, 8, 13, 21, 34,
55 … etc. Divide any Fibonacci number by the preceding number in the sequence,
and you will obtain an angle very close to the golden ratio. The most common
number pairs seen in nature are 21 and 34 or 59 and 89, sequential Fibonacci
numbers. The ratios between them to 3 decimal places are 1.619 and 1.618
respectively, very close to the golden ratio. You can never achieve the golden
ratio exactly with Fibonacci numbers, no matter how large the numbers used
within the sequence, because the golden ratio is irrational, so by definition,
it cannot be displayed as a ratio of two whole numbers (a fraction).The Golden Ratios
beauty is not just a human perception; Many plants produce leaves, petals and
seed formations in sequential Fibonacci numbers. For example, if you count the
seed spirals in a sunflower you will find that the amount of spirals in each
direction adds up to sequential Fibonacci numbers (Rehmeyer, 2007). This is not
the only place we find the link between nature and mathematics in the structure
of plants. We also find this connection in the arrangement of branches on a
tree, the number of petals and or leaves on a stem, or even their shapes. Why does nature
behave so predictably and resonate with the laws of mathematics? Scientists
theorise that it’s a matter of efficiency. When, for example, a sunflower has
each seed separated by an irrational numbered angle it can pack in the maximum
number of seeds in the space availableRB5 , particularly if the space is circular (Bassa,
2017). In geometry, a golden spiral is a logarithmic
spiral whose growth factor is F, the golden ratio, meaning that a golden spiral
gets wider (or further from its origin) by a factor of F
for every quarter turn it makes. A golden spiral with initial radius 1 has the
following polar equation:   (Corbalán, 2016). The
Nautilus shell is one of the best natural examples of the golden spiral. The
spiral occurs as the shell grows outwards and tries to maintain its
proportional shape. Unlike humans and other animals, whose bodies change
proportion as they age, the nautilus’s growth pattern allows it to maintain its
shape throughout its entire life.  On the
other side of the spectrum, you have the spiral of the arms of the galaxy.  A new section on the outskirts of the Milky Way
Galaxy RB6 was recently discovered which suggests that the
galaxy is a near-perfect mirror image of itself (Smithsonian Astrophysical Observatory,
2011). The basic structure of the Milky Way is comprised of two main spiral
arms protruding from the bar centre: the Scutum-Centaurus, and Perseus. These
arms, as well as being mirror images, follow the pattern of a logarithmic
spiral (Wethington, 2009).Why is it that nature follows this pattern on the
micro and macro scale, as well as everything in-between? Well if the Fibonacci
sequence is nature’s numbering system, and for an organism to grow and maintain
its shape it must conform to the growth of a logarithmic spiral.  With the Fibonacci numbers you can obtain the
golden ratio, and therefore can easily create a golden spiral. So the golden
spiral is really a result of efficiency and ease, nature has got to be
conservative with its approach to problems, to minimise energy. After all, it’s
a matter of life or death. 3
– Mathematical models for Nature and FractalsWhat is a mathematical model? One basic answer is
that it is the formulation in mathematical terms of the assumptions and their
consequences believed to underlie a particular “real world” problem. (Adam,
2003). In the
caseRBM7  of fractals, they
display particular characteristics which we see reflected in nature, making
them the main area of interest for modelling nature. “These shapes fractals
are extremely involved, however, and are strikingly unlike anything in the
familiar discipline of classical geometry, or “Euclid.”” (Mandelbrot
1982).A fractal is a
mathematical set that exhibits a repeating pattern displayed at every scale. Mandelbrot used his fractal geometry to model patterns
found in nature, like
the texture of
barkRBM8 . In mathematics, a self-similar
object is exactly or approximately similar to a part of itself at any scale.
Self-similarity is a typical property of fractals. Scale invariance is an exact
form of self-similarity where at any magnification there is a smaller piece of
the object that is similar to the whole (Basa,
2017).
The best examples of organic forms displaying fractal properties are Romanesco
broccoli or the theoretical Barnsley Fern, seen in the image on the right. “A
particular set of complex numbers that has a highly convoluted fractal boundary
when plotted.” The Mandelbrot set is the set of complex numbers c for which the
function does not diverge to infinity when iterated from z = 0,
where z is a complex number in the form   The world of Fractals is a harsh
and complex place, which is a prominently unknown territory with which we can
describe the patterns seen in nature. The problem we face is whether we can
decode this landscape and make sense of it.Crystalline formations are another example of nature adhering to its
economical approach to structures (Greuel, 2014). In 1611, Kepler attempted to describe the
hexagonal structure of a snowflake as a structure of miniscule particles with
minimal distance between them, causing him to study the maximum density of
circles and spheres, ultimately forming the Kepler conjecture. 4 – Control or a matter of InfluenceRBM9 An interesting philosophical
debate is “does nature influence maths, or does maths influence nature?” Maths
can be used to predict how nature will behave under certain conditions, but is
this just a mathematical model that describes this process? Or is it the
working mechanism behind it? This theoretical
conundrum is similar to that of Schrodinger’s cat, a thought experiment that
challenges the definition of existence and whether it is possible to
prove/disprove existence. Maybe it is both and how can you distinguish between
them?  As mathematics represents
the discovered properties of nature and is not an invented discipline, it is
not unreasonable to assume that these mathematical “rules”, or axioms, are just
part of how our universe works; therefore we have just observed them at play.
Despite this, it’s possible that there is a complex formula behind these
natural wonders, and the theories we have come up with so far, such as
Newtonian Mechanics and Special Relativity, don’t have the depth to fully
comprehend the inner workings of nature, they are merely insufficient models to
describe it. Maybe there is a higher level of mathematics we are yet to
discover, just like the leap in understanding that occurred when trigonometry
was discovered; we might just be waiting for a new level of maths that will allow
us to explain these phenomena in greater detail where our current theories and
formulae fall through. After all, this is true of all things in life, it is
inevitable that things will advance and adapt to our ever-changing needs. The
very nature of science and mathematics is to change as our beliefs are
challenged in order to overcome them and advance our understanding. Galileo
concluded that to model the fall of bodies toward the Earth, one needs a
different curve—a parabola. And he proclaimed that “the great book of nature
. . . is written in mathematical language and the characters are triangles,
circles and other geometric figures . . . without which one wanders in vain
through a dark labyrinth.” (Mandelbrot 1982).
However, making this argument we may be guilty of a fallacy first
mentioned in the work of Aristotle, known as “the fallacy of composition”. The
fallacy is the assumption that because something is true of the parts, exactly
the same must be true of the whole. Aristotle first discovered this
argumentative flaw when he argued that because every part of the body has a
function, man as a whole must also have a function (reference). We cannot go from our desire for something to be the
case to the assumption something is the case. We cannot assert this assumption
until we are presented with evidence, and we must not get confused by the
notion of what we can imagine and what we can conceive. We cannot assume that
because the inner working of the universe, mathematics and science, may conform
to the so-called “Laws” of nature then so must the universe as a whole.It is hard to
pinpoint the reason for the existence of mathematics, but using platonic logic
we can deduce there must be an underlying cause. In a famous radio debate with
Bertrand Russell in 1948, F. C. Copleston said: “Therefore, I should say, since
objects or events exist, and since no object of experience contains the reason
of its existence, this reason, the totality of objects, must have a reason
external to itself.”  (Seckel, 1994-2017) The reason for the
existence of mathematics and its roots must be external to itself.RBM10 Alternatively, our
notation for describing mathematics could be fatally flawed. The maths itself
is constant across the universe, but the way we perceive it, describe it and
notate it is not universal. Within different cultures mathematics is notated
differently; we can write it out fully, in different languages even, but it all
means the same. Unlike maths which is constant, our way of describing it is
invented. 5 – The Mathematical Universe HypothesisThe
idea that the universe is a mathematical structure of some degree has been
discussed extensively in literature and dates back to at least before the time
of the Pythagoreans. As we heard earlier, Galileo wrote about the great book of
nature written in the language of mathematics. The Mathematical Universe
Hypothesis (MUH), defined as “our external physical reality is a mathematical
structure” (Tegmark, 2007) and based upon the “unreasonable effectiveness of
mathematics in the natural sciences” as reflected by Wigner in 1960. In his
Hypothesis, Tegmark argues that, with a sufficiently board definition of
mathematics, the physical world is an abstract mathematical structure. In his
1967 essay, Wigner said the following; “the enormous usefulness of mathematics
in the natural sciences is something bordering on the mysterious… there is no
rational explanation for it”. Tegmark proposed the MUH as this mysterious link
between our physical world and mathematics. As he put is, “It MUH explains
the utility of mathematics for describing the physical world as a natural
consequence of the fact that the latter is a mathematical structure”. In simple
terms, mathematics describes our universe and physical reality so well because
our universe and physical reality is mathematical by nature. The
alternative view to the MUH is the non-Platonic position, the view that the only
reason mathematics is so suited to describing the physical world is that we
invented it to do just that. It is a product of the human intuition and we make
mathematics up as we go along to suit our purposes. If the universe
disappeared, there would be no mathematics in the same way that there would be
no football, tennis, chess or any other set of rules with relational structures
that we contrived. Mathematics is not discovered, it is invented. In response to Wigner’s article, Derek Abbott
published “The reasonable ineffectiveness of mathematics” and argues that
mathematics is actually very ineffective at describing our reality and the only
reason it appears to be so effective is that we only focus on the successful
examples and that there are many more cases of when it is ineffective than
effective, we have cherry picked the problems for which we have found a way to
apply mathematics and ignored the probably thousands of failed attempts at
mathematical models. He concludes that mathematics is a human invention that is
useful, limited, and works about as well as expected. 6 – ConclusionIn response to Abbott’s point about the numerous
failed attempts at mathematical models, I think that this seems to be an
unreasonable criticism. That math maps onto the universe does not mean that all
the universe has to map onto math (i.e. that for every possible expression of a
mathematical/physical theorem there has to be a corresponding reality…that
would be Platonism)