Synthesis Paper

Mathematics in the Modern World

THE UNREALITY OF MATHEMATICS NATURE’S NUBERS

 A Synthesis Synthesis Paper  Presented to the Central Bicol State of Agriculture. In the Fulfilment Of the Requirements for the Mathematics in the Modern orld

Prepared by:  Ra  Ralp Ste!e" Bra"#$ela S t $ de " t

C!AP"#R $ THE NATURAL OR(ER

Passes to: Mr% Ar"$l&o L% Nebres Lee I"st$r'tor  

Mathematics in the Modern World !uman mind and culture ha%e de%eloped a formal system of thought for recogni&ing' classifying' and e(ploiting patterns. e call it mathematics. Our )orld is full of patterns thus' )e li%e in a *ni%erse of Patterns. e can see patterns in many different )ays. "igers and &e+ras are co%ered in patterns of stripes' leopards and hyenas are co%ered in patterns of  spots. "he tiger,s stripes and the hyena,s spots attest to mathematical regularities in +iological gro)th and form. Patterns possess utility as )ell as +eauty that )e can see in nature. By using mathematics to organi&e and systemati&e our ideas a+out patterns' )e ha%e disco%ered a great secret- nature,s patterns are not ust there to +e admired' they are %ital clues to the rules that go%ern natural processes. "here is much +eauty in nature,s clues' and )e can all recogni&e it )ithout any mathematical training. Rain+o)s tell us a+out the scattering of light' and indirectly confirm that raindrops are spheres. "he regular nightly motion of the stars is also a clue' this time to the fact that the #arth rotates. "he planets )ere clues to the rules +ehind gra%ity and motion. /epler )as fascinated )ith mathematical patterns in nature' and he de%oted much of his life to loo0ing for them in the +eha%ior of the planets. !e de%ised a simple and tidy theory for the e(istence of precisely si( planets 1in his time only Mercury' 2enus' #arth' Mars' 3upiter' and Saturn )ere 0no)n4. /epler found that if you ta0e the cu+e of the distance of any planet from the Sun and di%ide it +y the square of its or+ital period' you al)ays get the same num+er. "hrough numerical patterns5 one of the 0ey steps to)ard Isaac 6e)ton,s theory of gra%ity' /epler,s neat' tidy theory for the num+er of planets has +een +uried )ithout trace. "here is also +eauty in the mathematical stories that start from the clues and deduce the underlying rules and regularities. But it is a different 0ind of +eauty applying rather than things. e are still learning to recogni&e ne) 0inds of pattern. Only )ithin the last thirty years has humanity +ecome e(plicitly a)are of the t)o types of pattern no) 0no)n as fractals and 'aos. Fra'tals are geometric shapes that repeat their structure on e%er7finer scales' and I

)ill say a little a+out them to)ard the end of this chapter5 chaos is a 0ind of apparent randomness )hose origins are entirely deterministic. 6ature 80ne) a+out8 these patterns +illions of years ago' &or 'lo$ds are &ra'tal a"d )eater *s 'aot*' . It too0 humanity a )hile to catch up. "he simplest mathematical o+ects are num+ers' and the simplest of nature,s patterns are numerical. N$+erolo,y  is the easiest7and consequently the most dangerous7 method for finding patterns. "he difficulty lies in distinguishing significant numerical patterns from accidental ones. 6ature,s lo%e of stripes and spots e(tends into the animal 0ingdom' )ith tigers and leopards' &e+ras and giraffes. "he shapes and patterns of animals and plants are a

Mathematics in the Modern World happy hunting ground for the mathematically minded. 9 In addition to patterns of form' there are patterns of mo%ement. In the human )al0' the feet stri0e the ground in a regular rhythmleft7right7Ieft7right7Ieft7right. hen a four7legged creature7a horse' say7)al0s' there is a more comple( +ut equally rhythmic pattern. . "his comprises patterns that )e ha%e only ust learned to recogni&e7patterns that e(ist )here )e thought e%erything )as random and formless. "here are more elusi%e of nature,s patterns are +eginning to re%eal their secrets. Already )e are seeing a practical impact as )ell as an intellectual one. Our ne)found understanding of  nature,s secret regularities is +eing used to steer artificial satellites to ne) destinations )ith far  less fuel than any+ody had thought possi+le' to help a%oid )ear on the )heels of locomoti%es and other rolling stoc0' to impro%e the effecti%eness of heart pacema0ers' to manage forests and fisheries' e%en to ma0e more efficient dish)ashers. But most important of all' it is gi%ing us a deeper %ision of the uni%erse in )hich )e li%e' and of our o)n place in it.

C!AP"#R : -HAT MATHEMATICS IS FOR

Communing )ith nature does all of us good- it reminds us of )hat )e are. Painting pictures' sculpting sculptures' and )riting poems are %alid and important )ays to e(press our feelings a+out the )orld and a+out oursel%es. One of the strangest features of the relationship +et)een mathematics and the 8real )orld'8 +ut also one of the strongest' is that good mathematics' )hate%er its source' e%entually turns out to +e useful. Mathematics has de%eloped alongside our understanding of nature' each reinforcing the other. #ach of nature,s patterns is a pu&&le' nearly al)ays a deep one. Mathematics is +rilliant at helping us to sol%e pu&&les. It is a more or less systematic )ay of digging out the rules and structures that lie +ehind some o+ser%ed pattern or regularity. hate%er the reasons' mathematics definitely is a useful )ay to thin0 a+out nature. e )ant to understand ho) they happen5 to understand )hy they happen' )hich is different5 to organi&e the underlying patterns and regularities in the most satisfying )ay5 to predict ho) nature )ill +eha%e5 to control nature for our o)n ends5 and to ma0e practical use of )hat )e ha%e learned a+out our )orld. Mathematics helps us to do all these things' and often it is indispensa+le. ithout mathematics )e )ould ne%er ha%e con%inced oursel%es that matter really is made from atoms' or ha%e )or0ed out ho) the atoms are arranged. "he disco%ery of genes7and later of the molecular structure of ;6A' the genetic material7relied

Mathematics in the Modern World hea%ily on the e(istence of mathematical clues. Many different pieces of mathematics )ere in%ol%ed in the disco%ery that ;6A has the cele+rated dou+le7helical structure. e can use mathematics to relate the resulting geometry to all the different %aria+les7such as gro)th rate and eccentricity of gro)th7that are in%ol%ed. !umanity )as in a %ery curious position as regards the calculus. "he story of calculus +rings out t)o of the main things that mathematics is for- pro%iding tools that let scientists calculate )hat nature is doing' and pro%iding ne) questions for mathematicians to sort out to their o)n satisfaction. "hese are the e(ternal and internal aspects of mathematics' often referred to as appl*ed  and p$re +ate+at*'s. "he function of mathematics is to organi&e the underlying patterns and regularities in the most satisfying )ay. Another function of mathematics is prediction. e 0no) ust as much a+out the mathematics of )eather as )e do a+out the mathematics of tides' +ut )eather has an inherent unpredicta+ility. "he role of mathematics goes +eyond mere prediction. Once you understand ho) a system )or0s' you don,t ha%e to remain a passi%e o+ser%er.
Mathematics in the Modern World unpredicta+le. It is their %ery unpredicta+ility that ma0es them important- they change our )orld in )ays )e didn,t see coming. Moreo%er' goal7oriented research often runs up against a +ric0 )all' and not only in mathematics. "here is nothing )rong )ith goal7oriented research as a )ay of achie%ing specific feasi+le goals. But the dreamers and the ma%eric0s must +e allo)ed some free rein' too. Our )orld is not static- ne) pro+lems constantly arise' and old ans)ers often stop )or0ing. =i0e =e)is Carroll,s Red >ueen' )e must run %ery fast in order to stand still.

C!AP"#R ? -HAT MATHEMATICS IS ABOUT

Most people do not ha%e an accurate picture of mathematics. For many )ays' mathematics is torture of test' home)or0' and pro+lems and pro+lems. "he %ery )ord pro+lem suggest unpleasantness and an(iety. @"he mathematics is not the pro+lem 1Burger' :4. hen )e hear the )ord 8mathematics'8 the first thing that springs to mind is num+ers. N$+bers are the heart of mathematics. @Some people %ie) mathematics as a set of formulas to +e applied to a list of pro+lems at the ends of te(t+oo0s. "oss that idea into trash. e need to see that mathematics is a net)or0 of intriguing ideas 1Burger' :4. Its almost impossi+le to get through a day )ithout using maths in some )ay' +ecause our )orld is full of num+ers to handle and pro+lems to sol%e. Studying the maths in e%eryday life pro%ides you )ith the tools to ma0e sense of it all' ma0ing life that little +it easier. 6umerical and logical thin0ing play a part in each of these e%eryday acti%ities' and in many others. A good understanding of maths in e%eryday life is essential for ma0ing sense of all the num+ers and pro+lems life thro)s at us. Bet)een D and $: A;' the concept of &ero )as in%ented and accepted as denoting a num+er. For a long time 8one8 )as not considered a num+er +ecause it )as thought that a num+er of things ought to +e se%eral of them. Mathematics uses sym+ols' +ut it no more is those sym+ols than music is musical notation or language is strings of letters from an alpha+et. "he real num+ers are one of the most audacious ideali&ations made +y the human mind' +ut they )ere used happily for centuries +efore any+ody )orried a+out the logic +ehind them. Mathematics is not ust a+out num+ers.
Mathematics in the Modern World landscape5 it has an inherent geography that its users and creators employ to na%igate through )hat )ould other)ise +e an impenetra+le ungle. "he creator of mathematics e(plores its un0no)n mysteries' maps them' and +uilds roads through them to ma0e them more easily accessi+le to e%ery+ody else. @ Mathematics ma0es our life orderly and pre%ents chaos. Certain qualities that are nurtured +y mathematics are po)er of reasoning' creati%ity' a+stract or  spatial thin0ing' critical thin0ing' pro+lem7sol%ing a+ility and e%en effecti%e communication s0ills. Mathematics is the cradle of all creations' )ithout )hich the )orld cannot mo%e an inch. Be it a coo0 or a farmer' a carpenter or a mechanic' a shop0eeper or a doctor' an engineer or a scientist' a musician or a magician' e%eryone needs mathematics in their day7to7day life. #%en insects use mathematics in their e%eryday life for e(istence 1"ime of India :$E4.

C!AP"#R D THE CONSTANT OF CHAN.E

*ni%erse o+eys fi(ed' immuta+le la)s' and e%erything e(ists in a )ell7defined o+ecti%e reality. "he opposing %ie) is that there is no such thing as o+ecti%e reality5 that all is flu(' all is change. 6ature' the creation of higher +eings' is +y definition perfect' and ideal forms are mathematical perfection' so of course the t)o go together. And perfection )as thought to +e un+lemished +y change. e 0no) that o+ects or particles can mo%e on different types of traectories. "hose )hich are mo%ing along a straight line ha%e the mathematical property of mo%ing at 'o"sta"t rate o&  'a",e. "he 'o"sta"t rate o& 'a",e de&*"*t*o" is %ery simple and logical- 8An o+ect

mo%ing uniformly )ith respect to time' is said is mo%ing at a constant rate of change.8 hat is happening loo0s li0e a retreat- if it is too hard to find a formula' then try an appro(imation5 if  appro(imations aren,t a%aila+le' try a qualitati%e description. But it is )rong to see this de%elopment as a retreat' for )hat this change of meaning has taught us is that for questions li0e the three7+ody pro+lem' no formulas can e(ist. e can pro%e that there are qualitati%e aspects to the solution that a formula cannot capture. "he search for a formula in such questions )as a hunt for a mare,s nest. @e desire for a lot of things. hat is important today may not hold the same place tomorro). 6othing is constant in life' not e%en that heart +eating inside you. It also +eats )ith a rhythm.

Mathematics in the Modern World Belie%e in the fact that )hate%er happens in your life is al)ays for a reason7 +e it good or +ad. It )ill al)ays ma0e you a +etter indi%idual7 you )ill either ha%e a lesson to learn or a memory to share. And )hat is LIFE )ithout CHAN.E9 ithout change life )ill +e stagnant. It )ill +ecome dull and sluggish. Imagine if e%erything )as same' man )ould still ha%e +een an Ape' li%ing in ungles. "heory of #%olution is the +iggest e(ample of change for good. =et your life ta0e a natural turn. Be spontaneous. Be ali%e. Be fun. Be the magic you )ant in your life. Be the change you )ant to see in the )orld. Accept the fact that things )ill not al)ays +e the )ay you )ant them to +e. "here )ill al)ays +e things +eyond your control 1 E!eryt*", )*ll 'a",e o"e day/ A$,% 0123 4

6o)adays' it can +e o+tained from theories that deal directly and precisely )ith the main qualitati%e aspects of the motion. For the first time' )e are starting to understand nature,s patterns in their o)n terms.

C!AP"#R E FROM 4IOLINS TO 4I(EOS

Mathematics re%eal the simpliest of nature' and permits us to generali&e from simple e(ample to generali&ed from simple e(amples to the comple(ities of the real )orld.  As the de%elopment of mathematics has +een separated into t)o distinct su+ disciplines la+elled p$re +ate+at*'s a"d appl*ed +ate+at*'s. "he society gro)s through the continuous impro%ement of technology +y using mathematics. *nfortunately' it is then %ery tempting for the outside )orld to assume that the only useful part of mathematics is applied mathematics5 after all' that is )hat the name seems to imply. "his assumption is correct )hen it comes to esta+lished mathematical techniques- anything really useful ine%ita+ly ends up +eing considered 8applied'8 no matter )hat its origins may ha%e +een. But it gi%es a much distorted %ie) of the origins of ne) mathematics of practical importance. ood ideas are rare' +ut they come at least as often from imaginati%e dreams a+out the internal structure of  mathematics as they do from attempts to sol%e a specific' practical pro+lem. It too0 many people from many different areas of human acti%ity to turn a mathematical insight into a useful product. But the ne(t time you go ogging )earing a al0man' or s)itch on

Mathematics in the Modern World your "2' or )atch a %ideotape' pause for a fe) seconds to remem+er that )ithout mathematicians none of these mar%els )ould e%er ha%e +een in%ented . o ahead and challenge yourself today. Commit to e(panding your mind' continuing your  education and +ecoming a student of life. *tili&e the )orld as your classroom' and no matter  ho) +ig or small' al)ays come a)ay )ith a lesson. Remem+er to culti%ate your mind so it is prepared to e(pand' +lossom and gro). And share your fountain of 0no)ledge. 1May+erry' :$E4 Re&ere"'e:

https-GGcismasemanuel.files.)ordpress.comG:$G:Gian7ste)art7numerele7naturii.pdf  1Ian Ste)rarts 6ature 6um+ers4 Add*t*o"al Re&ere"'e:

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"he !eart of mathematics an in%itation to effecti%e thin0ing' Burger :.