## 1 that mathematics is just an effective way

1 – An introduction: Is mathematics a construct ofthe human imagination that we tailor to describe our reality?You cannot deny thefact that everything has a numerical counterpart; everything, especially thatwhich is material, have properties, patterns and structures. Is the way wewrite these just notations? If, as according to Galileo, mathematics is thelanguage of science does that mean there could be other languages? Or is theuniverse inherently mathematical in the way it is constructed and doesn’t have anyother way to behave other than according to mathematical rules?Mathematics is anatural phenomenon.

Even without our understanding of numerical relationships, naturewould still conform to our laws of mathematics. Although the way we perceiveand describe these relationships may be a human construct, is it that mathematicsis just an effective way of describing the physical reality; it’s not acomplete description, nor the only one.Axioms based on the notion of simple counting arenot innate to our universe, but what exactly are numbers? Different examples ofcounting systems have occurred across history, all having different strengthsand weaknesses. For example the Sumerians used the sexagesimal system, a systemwith 60 as its base; a modified version of this system isstill used today for measuring time, with 60 seconds to a minute, and 60minutes to an hour, and angles, with 360 degrees to a circumference (Navarro, 2017) The Romans used the Roman numeral systemwhich in itself does not have a value for zero, yet they used the word “nulla”meaning nothing.  Modern day humansociety cannot function without numbers, they are vital for things we take forgranted – the binary language of computer programming, radio, television and evenelectricity. The presence of numbers is overpowering. If we look back to themost basic civilisations, they all developed a numerical system as a result ofa need to comprehend basic, everyday tasks, yet they were represented in adifferent way and they all had the same functions: counting, ordering,measuring and codifying (Corbalán, 2016RB1 ). This suggests that the conceptof numeracy is transcendent, although the way we transcribe them is ultimately trivial.

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2 – Irrationality and the Golden RatioIn mathematics, theword “irrational” has a different meaning than in literature. The concept stemsfrom the word “ratio”. A rational number can be written as a ratio of twointegers, a fraction.

Therefore, it follows that an irrational number is definedas a number that cannot be written in the form of a ratio. Irrational numbersare never-ending, never-repeating decimals such as , a number at the very heart of thePythagoreans obsession with matheRB2 matics (Navarro, 2017). One of the most famousirrational numbers of all time, one that has fascinated mathematicians andartists 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 linkthe ratio to Phidias, builder of the Parthenon in Athens, by borrowing hisinitial, for he believed the Parthenon was built to conform to the mathematicalbeauty of the ratioSB3  (Corbalán, 2016). One of the mostreprinted books of all time: “Euclid’s Elements of Geometry”, written around300BC, is a collection of definitions, axioms, propositions (theorems), andmathematical proofs for said propositions. RB4 Within it Euclid defined what we know today as thegolden ratio: “A straight line is said to be cut in the extreme and mean ratiowhen, as the whole line is to the greater segment, so is the greater to thelesser.”  The most irrational number “theextreme and mean ratio” is the golden ratio (Euclid,300BC).

The positive solutionto the equation  is    This is the Golden Ratio.The golden ratio is found almosteverywhere we look and is believed to represent what is perceived as perfectlyproportionate by the human eye. For example, the Mona Lisa’s face has beenframed in a succession of Golden Rectangles, to create arguably the mostelegant and visually pleasing portrait of the Renaissance (Corbalán, 2016).Leonardo’s Vitruvian Man assumed that the goldenration was reflected in the animal world and signifies a way of thinking thatjoins artistic and scientific sensibilities. Leonardo made observations aboutthe ratios of proportion on the human body and produced a set of idealmeasurements, the ratios of which falls closely to the golden ratio.

Studieshave shown that mouths and noses are positioned at golden sections of thedistance between the eyes and the bottom of the chin. The same proportions canbe seen from the side, and even the eye and ear itself, which follows along alogarithmic spiral.  Everybody’s different, but the averageproportions across populations lean towards the golden ratio, the closer ourproportions get, the more attractive those traits are perceived. “The sensesdelight in things duly proportioned”  -Saint Thomas Aquinas (1225-1274). For example, the most scientificallybeautiful smiles are those in which central incisors are 1.618 wider than thelateral incisors, which are 1.618 wider than canines, and so on.

(Corbalán,2016)One sequence ofnumbers 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 twonumbers 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 commonnumber pairs seen in nature are 21 and 34 or 59 and 89, sequential Fibonaccinumbers. The ratios between them to 3 decimal places are 1.619 and 1.

618respectively, very close to the golden ratio. You can never achieve the goldenratio exactly with Fibonacci numbers, no matter how large the numbers usedwithin 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 Ratiosbeauty is not just a human perception; Many plants produce leaves, petals andseed formations in sequential Fibonacci numbers. For example, if you count theseed spirals in a sunflower you will find that the amount of spirals in eachdirection adds up to sequential Fibonacci numbers (Rehmeyer, 2007).

This is notthe only place we find the link between nature and mathematics in the structureof plants. We also find this connection in the arrangement of branches on atree, the number of petals and or leaves on a stem, or even their shapes. Why does naturebehave so predictably and resonate with the laws of mathematics? Scientiststheorise that it’s a matter of efficiency. When, for example, a sunflower haseach seed separated by an irrational numbered angle it can pack in the maximumnumber of seeds in the space availableRB5 , particularly if the space is circular (Bassa,2017). In geometry, a golden spiral is a logarithmicspiral whose growth factor is F, the golden ratio, meaning that a golden spiralgets wider (or further from its origin) by a factor of Ffor every quarter turn it makes.

A golden spiral with initial radius 1 has thefollowing polar equation:   (Corbalán, 2016). TheNautilus shell is one of the best natural examples of the golden spiral. Thespiral occurs as the shell grows outwards and tries to maintain itsproportional shape. Unlike humans and other animals, whose bodies changeproportion as they age, the nautilus’s growth pattern allows it to maintain itsshape throughout its entire life.  On theother side of the spectrum, you have the spiral of the arms of the galaxy.  A new section on the outskirts of the Milky WayGalaxy RB6 was recently discovered which suggests that thegalaxy is a near-perfect mirror image of itself (Smithsonian Astrophysical Observatory,2011). The basic structure of the Milky Way is comprised of two main spiralarms protruding from the bar centre: the Scutum-Centaurus, and Perseus. Thesearms, as well as being mirror images, follow the pattern of a logarithmicspiral (Wethington, 2009).

Why is it that nature follows this pattern on themicro and macro scale, as well as everything in-between? Well if the Fibonaccisequence is nature’s numbering system, and for an organism to grow and maintainits shape it must conform to the growth of a logarithmic spiral.  With the Fibonacci numbers you can obtain thegolden ratio, and therefore can easily create a golden spiral. So the goldenspiral is really a result of efficiency and ease, nature has got to beconservative with its approach to problems, to minimise energy. After all, it’sa matter of life or death. 3– Mathematical models for Nature and FractalsWhat is a mathematical model? One basic answer isthat it is the formulation in mathematical terms of the assumptions and theirconsequences believed to underlie a particular “real world” problem. (Adam,2003).

In thecaseRBM7  of fractals, theydisplay particular characteristics which we see reflected in nature, makingthem the main area of interest for modelling nature. “These shapes fractalsare extremely involved, however, and are strikingly unlike anything in thefamiliar discipline of classical geometry, or “Euclid.”” (Mandelbrot1982).A fractal is amathematical set that exhibits a repeating pattern displayed at every scale.

Mandelbrot used his fractal geometry to model patternsfound in nature, likethe texture ofbarkRBM8 . In mathematics, a self-similarobject 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 exactform of self-similarity where at any magnification there is a smaller piece ofthe object that is similar to the whole (Basa,2017).The best examples of organic forms displaying fractal properties are Romanescobroccoli or the theoretical Barnsley Fern, seen in the image on the right. “Aparticular set of complex numbers that has a highly convoluted fractal boundarywhen plotted.” The Mandelbrot set is the set of complex numbers c for which thefunction 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 harshand complex place, which is a prominently unknown territory with which we candescribe the patterns seen in nature. The problem we face is whether we candecode this landscape and make sense of it.Crystalline formations are another example of nature adhering to itseconomical approach to structures (Greuel, 2014). In 1611, Kepler attempted to describe thehexagonal structure of a snowflake as a structure of miniscule particles withminimal distance between them, causing him to study the maximum density ofcircles and spheres, ultimately forming the Kepler conjecture. 4 – Control or a matter of InfluenceRBM9 An interesting philosophicaldebate is “does nature influence maths, or does maths influence nature?” Mathscan be used to predict how nature will behave under certain conditions, but isthis just a mathematical model that describes this process? Or is it theworking mechanism behind it? This theoreticalconundrum is similar to that of Schrodinger’s cat, a thought experiment thatchallenges the definition of existence and whether it is possible toprove/disprove existence.

Maybe it is both and how can you distinguish betweenthem?  As mathematics representsthe discovered properties of nature and is not an invented discipline, it isnot unreasonable to assume that these mathematical “rules”, or axioms, are justpart 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 thesenatural wonders, and the theories we have come up with so far, such asNewtonian Mechanics and Special Relativity, don’t have the depth to fullycomprehend the inner workings of nature, they are merely insufficient models todescribe it. Maybe there is a higher level of mathematics we are yet todiscover, just like the leap in understanding that occurred when trigonometrywas discovered; we might just be waiting for a new level of maths that will allowus to explain these phenomena in greater detail where our current theories andformulae fall through. After all, this is true of all things in life, it isinevitable that things will advance and adapt to our ever-changing needs. Thevery nature of science and mathematics is to change as our beliefs arechallenged in order to overcome them and advance our understanding. Galileoconcluded that to model the fall of bodies toward the Earth, one needs adifferent 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 vainthrough a dark labyrinth.” (Mandelbrot 1982). However, making this argument we may be guilty of a fallacy firstmentioned in the work of Aristotle, known as “the fallacy of composition”. Thefallacy is the assumption that because something is true of the parts, exactlythe same must be true of the whole. Aristotle first discovered thisargumentative flaw when he argued that because every part of the body has afunction, man as a whole must also have a function (reference).

We cannot go from our desire for something to be thecase to the assumption something is the case. We cannot assert this assumptionuntil we are presented with evidence, and we must not get confused by thenotion of what we can imagine and what we can conceive. We cannot assume thatbecause the inner working of the universe, mathematics and science, may conformto the so-called “Laws” of nature then so must the universe as a whole.It is hard topinpoint the reason for the existence of mathematics, but using platonic logicwe can deduce there must be an underlying cause.

In a famous radio debate withBertrand Russell in 1948, F. C. Copleston said: “Therefore, I should say, sinceobjects or events exist, and since no object of experience contains the reasonof its existence, this reason, the totality of objects, must have a reasonexternal to itself.

”  (Seckel, 1994-2017) The reason for theexistence of mathematics and its roots must be external to itself.RBM10 Alternatively, ournotation for describing mathematics could be fatally flawed. The maths itselfis constant across the universe, but the way we perceive it, describe it andnotate it is not universal.

Within different cultures mathematics is notateddifferently; we can write it out fully, in different languages even, but it allmeans the same. Unlike maths which is constant, our way of describing it isinvented. 5 – The Mathematical Universe HypothesisTheidea that the universe is a mathematical structure of some degree has beendiscussed extensively in literature and dates back to at least before the timeof the Pythagoreans. As we heard earlier, Galileo wrote about the great book ofnature written in the language of mathematics. The Mathematical UniverseHypothesis (MUH), defined as “our external physical reality is a mathematicalstructure” (Tegmark, 2007) and based upon the “unreasonable effectiveness ofmathematics in the natural sciences” as reflected by Wigner in 1960.

In hisHypothesis, Tegmark argues that, with a sufficiently board definition ofmathematics, the physical world is an abstract mathematical structure. In his1967 essay, Wigner said the following; “the enormous usefulness of mathematicsin the natural sciences is something bordering on the mysterious… there is norational explanation for it”. Tegmark proposed the MUH as this mysterious linkbetween our physical world and mathematics. As he put is, “It MUH explainsthe utility of mathematics for describing the physical world as a naturalconsequence of the fact that the latter is a mathematical structure”.

In simpleterms, mathematics describes our universe and physical reality so well becauseour universe and physical reality is mathematical by nature. Thealternative view to the MUH is the non-Platonic position, the view that the onlyreason mathematics is so suited to describing the physical world is that weinvented it to do just that. It is a product of the human intuition and we makemathematics up as we go along to suit our purposes. If the universedisappeared, there would be no mathematics in the same way that there would beno football, tennis, chess or any other set of rules with relational structuresthat we contrived. Mathematics is not discovered, it is invented.

In response to Wigner’s article, Derek Abbottpublished “The reasonable ineffectiveness of mathematics” and argues thatmathematics is actually very ineffective at describing our reality and the onlyreason it appears to be so effective is that we only focus on the successfulexamples and that there are many more cases of when it is ineffective thaneffective, we have cherry picked the problems for which we have found a way toapply mathematics and ignored the probably thousands of failed attempts atmathematical models. He concludes that mathematics is a human invention that isuseful, limited, and works about as well as expected. 6 – ConclusionIn response to Abbott’s point about the numerousfailed attempts at mathematical models, I think that this seems to be anunreasonable criticism. That math maps onto the universe does not mean that allthe universe has to map onto math (i.

e. that for every possible expression of amathematical/physical theorem there has to be a corresponding reality…thatwould be Platonism)

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