Math in the Modern World (C)
Math in the Modern World (C) MATH 1319
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What is quotHow Manyquot 19972008 Math Academy Online Platonic Realms Number is the within of all things Pythagoras FOUNDATIONS begin with arithmetic or plane geometry rather than numbers after all everyone knows what a number is light Not so fast it can be remarkably dif cult to say exactly what a number is Go ahead try it If you are tempted to say it39s what tells you how much or how many of something then that39s a good eff01t Only trouble is that39s not a de nition but a characterization It39s like saying a chair is something you sit on Useful information no doubt but we39d be a bit disappointed if we looked up chair in a dictionary and found no more than something you sit on After all one can sit on lots of things that aren39t chairs 1 t would be natural to suppose that mathematical foundations might Number proceeds from unity Aristotle The problem of what is a number is an old one To tackle it mathematicians in the late 19th and early 20th centuries palticularly Cantor Dedekind Frege Peano Russell and Whitehead turned to a new at the time branch of mathematics called set theory They didn39t solve the problem but they developed a beautiful theory which can be used to model and extend our primitive ie given or intuitively obvious sense of number A good deal of modern mathematics is now founded on this work using at root nothing more sophisticated than the set operations of union and intersection with which you may already be familiar We39ll be wanting to use these operations shortly so let39s review them now We begin with defining the notion of a set A set is any welldefined collection of objects In other words any collection considered as a single thing By well defined we mean that we can always tell when something is an element of the set in question or when it isn39t no ambiguity By object we mean absolutely anything physical objects ideas colors abstractions and anything else you care to think of that can form part of a well de ned collection We want to be able to write our sets down and there is an established way of doing this We rst designate a symbol to stand for the set itself usually a capital letter like A or S or some such Then we use curly braces to enclose some representation of the elements of the set as follows A elements ofA How do we represent the elements inside the braces There are two ways The rst and best is simply to list them For example if A is a set of colors we could write it down this way A red green blue Sometimes however listing the elements is not convenient or even possible In that case we would use a rule method using a statement like all shades of green or even all colors to represent our set When something is an element of a set we denote this with a special symbol that looks kind of like a curvaceous E blue 6 A This means blue is an element of the set A Much of the power of set theory arises from the fact that we can form sets whose elements are other sets For example if A B and C are sets of colors we could form a set of sets of colors A red green blue maria B purple blue orange sets C red yellow green I S A B C 777T j a set of sets red green blue purple blue orange red yellow green Notice that the sets A B and C have some elements in common For instance A and B both contain the element blue If it happens that every element of a set is also contained in some other set then we say that the rst set is a subset of the other set and we denote this with a big U shape lying on its side For instance if D is the set containing only blue then we could write DCA or equivalently blue C red green blue We consider every set to be a subset of itself Funny thing to do really but it makes sense sort of At least it matches the defmition of subset Also there is one set that is a subset of every set namely the empty set the set with no elements This is often denoted by a circle with a line through it or a pair of braces with nothing between them 53 525 C X for every setX Finally we are ready for the set operations of union and intersection The union of two sets A and B is the set containing all the elements that are in eitherA or B Thus if A and B are the two sets of colors above then we have A U B red green blue U purple blue orange red green blue purple orange The intersection of two sets is the set containing only elements that are in both For example the intersection of A and C would be denoted as follows A quot1 C red green blue 391 red orange green red green Armed with these ideas we may now turn to our real purpose nailing down numbers NATURAL NUMBERS mathematical thing we learn to do as children is to say this many and hold up our ngers When we do this we mean that there is the same many as the many of fmgers we are holding up Then we learn that the many39s have names and are ordered and slowly we memorize the names and the order they come in in base ten starting with one two three These numbers are called the natural numbers and are denoted as follows W e will begin where every child begins with counting The rst The set of natural numbers is always symbolized by a boldface or chalkboard capital N as above Notice the set notation This is critical and provides us with an important characterization A number is an element of a set Thus the counting numbers one two seventy three a million and so on are elements of the set of natural numbers The set of natural numbers has some properties that should be noted First of course is that this set is ordered This means that given two different natural numbers one always comes after the other and the other comes before This isn39t true for example of the set of colors One color doesn39t come after another in any necessary sense This may seem so obvious as to be beneath our notice but we will find as we start to really learn mathematics as opposed to just memorizing procedures that such niceties can sometimes take on surprising significance In fact we can do better with the natural numbers than saying merely that they are ordered They have a property that we call being wellordered A set is said to be wellordered if every subset of it has a smallest element So in other words given any collection of natural numbers say for instance 67 4 9 345 22 then there is always a smallest one in the set 4 in this instance Notice that there is not always a largest element the set of all even numbers the set of all multiples of 5 and the set of all the natural numbers greater than 37 are examples of sets that have no greatest element This business of not having a largest element is something every child notices at some point and then experiences her or his rst brush with the idea of infinity We know that there isn39t a largest natural number because intuitively at least we know the following principle which is sometimes called the Archimedian principle If n is a natural number then n 1 is a natural number Another way of saying this is that the natural numbers are closed under addition That is take any two natural numbers and add them and you get another natural number GETTING FORMAL Addition can39t take you outside of the set Notice that the natural numbers are also closed under multiplication which makes sense since multiplication is just repeated addition The natural numbers are the only numbers we need for one of the most imporant results in classical mathematics which comes down to us from antiquity it is found in Euclid39s Elements This result is called the Fundamental Theorem of Arithmetic which every numerate person should know Students often ask why zero isn39t included in the set of natural numbers Many texts describe a set called the whole numbers which is just like the set of natural numbers except that it also includes zero This set however is not used much and it is as well to separate zero conceptually from the natural numbers because it is really a very different kind of thing When you count a collection of objects you don39t begin by saying zero one two after all Its historical development is quite different We can establish in a concrete way that the natural numbers have no largest element by reasoning as follows suppose that there was a largest natural number and let39s designate it by B Then by the principle that you can always add one B 1 is also a natural number Observe that B 1 is larger than B But we said that B was the largest This is a contradiction and whenever a chain of reasoning leads to a contradiction we conclude that one or more of our premises was false The only premise in this case was that there does exist a largest natural number so this must be false ie there does not exist a largest natural number too People were counting for millenia before zero was ever thought of In fact its rst use is thought to have been in India in the 6th or 7th century and came to us like so much of the mathematics that we use in the western world by way of Arabic culture in about the 11th century It39s notable that native Americans speci cally the Mayan civilization also developed a concept of zero independently of the old world So anyway we don39t include zero in the natural numbers We will nd it however in our next set INTEGERS multiplication but of course there are other operations with numbers Subtraction for instance If we take away two from three then there is no problem because the remainder is one and one is in the set of natural numbers What happens however if we want to go the other way and take three from two We get a negative number and negative numbers aren39t included in the set of natural numbers In order to talk about negative numbers we will need to introduce a new set W e said that the set of natural numbers is closed under addition and z 3 2 1 o 1 2 3 This is the set of integers and is always denoted by a bold faced Z The Z is from the German word zahlen which means to count Notice that it includes all the natural numbers zero and all the negatives of the natural numbers This means the natural numbers are a subset of the integers ie NCZ Historically negative numbers didn39t come into wide use until the late middle ages circa the 14th century Before that time negative quantities weren39t considered real and so it was thought that one shouldn39t try to calculate with them After all when did you last see a negative quantity No one is minus ve feet tall for instance However negative numbers can be very handy for calculations involving debt and the Italians who invented banks were the first to recognize their imp01tance in finance and to use them for that purpose RATIONAL NUMBERS but what about division If we divide two into four we39re all light but what about dividing two into three Then we get catapulted light out of the integers and into the world of ratios or rational numbers 6 he integers are closed under addition subtraction and multiplication Q pquql0 1 This looks much more complicated than anything we39ve done before but don39t be alarmed we39ll take it apaIt piece by piece What it says is the set of rational numbers is the set consisting of all numbers of the form p divided by q where p and q are elements of the set of integers and q is not zero symbol for the set p and q are integers Q p 162 1113 I Tl J74 quotk I ff such that numbers of this Tenn 1 is not zero This is an example of using the rule method to designate the elements of a set rather than the list method we used before We need to use the rule method because there is no way to list rational numbers that suggests the complete list in an unambiguous way Even using an elipsis doesn39t help Historically the rational numbers are nearly as old as the natural numbers They go all the way back to the ancient Babylonians and Egyptians and the Greeks were particularly fond of them though none of these cultures used our notation which is Arabic The Pythagoreans of ancient Greece even believed that everything in creation could be understood and analyzed in terms of natural numbers and their ratios As we39ll see below this idea wasn39t to last it39s days so to speak were numbered Notice that the natural numbers and the integers are both subsets of the rational numbers since any integer can be expressed as a ratio 94 ll HIM WHY NOT What makes giving a quotlistquot of the rational numbers difficult is their denseness Suppose a and b are rational numbers Then since the rational numbers are closed under addition and division the number a bl2 the average of a and b is also rational and it lies between a and b Thus between any two rational numbers you can find another one This is what quotdensequot means in this context Although as we show in the In nig MiniText it really is possible to make a one aftertheother list of the rational numbers no such list is really suggestive of the entire set in an unambiguous way The rational numbers are closed under all the arithmetic operations and if all we ever needed to do was arithmetic we39d never need any other numbers at all However sometimes we need more than arithmetic to construct adequate models of the world around us For this reason much of the work we do in mathematics will require us to add to the rational numbers a new set a set of numbers which is the topic of our next section IRRATIONAL NUMBERS were very keen on geometry Among the most important results in all 1 n addition to their fascination with numbers and ratios the Greeks of mathematics is the famous Pythagorean Theorem The Pythagorean Theorem In a righl1rinngll39lhc are nl39 n square consumed on me Ilnmlenusc is equal In In sum ufllm mus unhe squirts 1 quot2 hz m summl rm um um mm the nhvinlls questinn haw lung is the dizgnnal39quot We see Lhztthelength an dizgnnalisthe Squarermn unwn since 11112 m twn so inhe Square rnnt nftwn zrztinnal number That is can ithe 39 enumhernr I r w m m 1 lo a irrz nnalquot 39 Pythagoreans who were at sea at the time rst heard about it they were so overcome by their feelings that they took the poor man who discovered this fact and threw him overboard drowning him The upshot is that we need more than just rational numbers if we wish to work with many kinds of abstract quantities such as length and proportion in idealized space for instance We need irrational numbers too We don39t tend to bother much about the set of irrational numbers usually denoted by a bold faced I in and of itself but focus instead on what we get when we mix the rationals and the irrationals together that most beautiful strange and wondrous of sets the real numbers REAL NUMBERS e said above that the set of real numbers is obtained by collecting together the rational numbers and the irrational numbers ie RQUI Thus every rational number is a real number and every irrational number is a real number The real numbers taken all together form what is called a continuum They are closed under all the arithmetic operations and they are also closed geometrically This last point is important because it means that we can use geometrical pictures to represent the real numbers and vice versa The principal and most useful picture we use is called the real number line and will no doubt be familiar Here every point on the real number line corresponds to a unique real number and every real number corresponds to a unique point on the line We see that the real numbers are totally ordered and include every kind of number we might need A beautiful set Notationally we often represent real numbers using decimal notation in which we write a real number as its integer and non integer parts separated by a dot Notice that the non integer part is actually a sum of fractions so many tenths plus so many hundredths plus so many thousandths and so on integer part 3125 tenths thnusandths hundredth T trueifthe decimal repeats the same pattern endlessly In this latter casewe n part 123232323 LE H u u m u a few digits tn represent that the sequence cnntinlles 12 1415 Nntice new nninnzginzhly dznxe the set nfreal numbers is We already knew annther hnt nuw Insider the inzdnnals e are they dense nr what Just as we r d 39l a 145623 83975 14561383975113 145623 839751234564356743 145623 83 751246 145623 3975200000187456 145623 83976 dam nmhm ratinnal and irrztinnal numbers between In nities within in nities The Stable Marriage Problem by Harry Mairson originally appeared in quotThe Brandeis Reviewquot Vol 12 No 1 Summer 1992 Here is the problem of stable marriage Imagine you are a matchmaker with one hundred female clients and one hundred male clients Each of the women has given you a complete list of the hundred men ordered by her preference her first choice second choice and so on Each of the men has given you a list of the women ranked similarly It is your job to arrange one hundred happy marriages It should be immediately apparent that everyone is not guaranteed to get their first choice if a particular man is the first choice of more than one woman only one can be matched with him and the other women will have to make do with less Rather than guarantee the purest of happiness to everyone a promise that almost surely would subject you to eventual litigation your challenge is to make the marriages stable By this we mean that once the matchmaker has arranged the marriages there should be no man who says to another woman You know I love you more than the woman I was matched with let s run away togetherquot where the woman agrees because she loves the man more than her husband In the spirit of equality no woman should make such a successful proposal to a man should she so propose we want the man to respond Madam I am attered by your attention but I am married to someone I love more than you so I am not interestedquot Is it always possible for a matchmaker to arrange such a group of marriages regardless of the preference lists of the men and women If so how Were it not for computers no one might have thought of the solution we will describe While finding and keeping a mate is a good deal more complicated then the mathematically simple problem stated methods for achieving stable marriage have real uses Perhaps the most well known is The Matchquot spoken of with fear and reverence by medical students everywhere in the United States When a student finishes medical school and wants to specialize in say cardiology she interviews for cardiology residency programs at hospitals across the country After all the interviews she makes a list of the programs she visited in order of preference Each of the medical programs after having interviewed many candidates for the job makes a similar preference list of students Everyone sends their list to be processed by a big computer which matches students and jobs Once again no medical program or student is guaranteed their first choice the matching is done to achieve stability so that no student and hospital can successfully conspire to outwit the national medical establishment Once we understand how to compute a stable marriage we will return to the politics of residency selection because there is a very interesting story to be told an unusual controversy about resident assignments that actually spilled over into the pages of the New England Journal of Medicine A method for computing a particular value for example a stable marriage is called an algorithm The word comes from the name of a Persian textbook author Abu Ja far Mohammed ibn Musa alKhowarizmi ca 825 who wrote Kitab aljabr w39al muqabala Rules of Restoration and Reductionquot Another familiar word algebra derives from the title of his book The stable marriage algorithm we describe due to D Gale and H S Shapely originally appeared in the American Mathematical Monthly in 1962 under the title College Admissions and the Stability of Marriagequot Rather than explain the algorithm in Arabic or even worse a computer language let s do so in English The matchmaker arranges marriages in rounds where in each round he instructs certain men to propose marriage In the initial round he tells all the men to quite sensibly go out and propose marriage to their f1rstchoice women Each man then proposes to the woman he loves most Each of the women then receives either no proposal if she was not the first choice of any man one proposal if she was the first choice of exactly one man or more than one proposal if many men nd her to be their first choice The matchmaker instructs the women to respond to the proposals according to the following rules If no one proposed to you don t worry says the matchmaker I promise someone will eventually If exactly one man proposed to you accept his proposal of marriage the man and woman are then considered to be engaged If more than one man proposed respond affirmatively to the one you love most and become engaged to him and reject the proposals of the rest Surely nothing could be more reasonable This concludes what we ll call the rst round After one round certain contented men are engaged and the other discontented men are unengaged In round two the matchmaker says to the unengaged men Do not despair Go out and propose again to your second choice While the engaged men do nothing the unengaged men send out another round of proposals This time the matchmaker says to the women use the same rules as before with one important change if you are currently engaged and receive proposals of marriage from men that you love more than your fiance you may reject your current intended and reengage yourself to the new suitor that you love most Thus a man who is happily engaged at the end of the first round may find himself suddenly unengaged at the end of the second round After two rounds once again the men are divided into the engaged and unengaged In the next round the matchmaker tells each unengaged man to propose to the woman he loves most among those women to whom he has not yet proposed Again the matchmaker tells each woman that she can change her mate if she instead prefers one of the new proposers Each time a man proposes it is with greater desperation since he begins by proposing to his true love then his second choice third choice and so on Each time a woman changes her fiance she becomes happier because her new intended is someone she loves more This continues in round after round until finally there is no one left to propose or be proposed to But is this indeed the case Does this succession of rounds ever come to an end And is everyone engaged at the end of this romantic variation on musical chairsquot And are the arranged marriages indeed stable It is not hard to prove mathematically that the story does indeed have the happy ending we suggest First does the process ever end Of course If there are one hundred men and one hundred women each man can only make a hundred proposals During each round some man proposes reducing the nite supply of proposals by at least one If the rounds continue long enough then the supply of proposals will descend to zero and the game has to come to an end because there is no one left to propose At the end is everyone engaged Notice that at the end of each round the number of engaged men is equal to the number of engaged women Computer scientists like doctors have a name for everything and call this kind of assertion an invariant Notice also that once a woman becomes engaged she is always engaged though not necessarily to the same man So suppose that all the rounds take place and yet there is some man let39s call him Bob and some woman named Carol who are both unengaged Is this possible No If Carol is unengaged no one ever proposed marriage to her All the other men may not have proposed to Carol if each of them found a woman they loved more than Carol but the same cannot be said of Bob who went through his whole list which has to include Carol somewhere and supposedly came up emptyhanded Clearly he had to propose to Carol at some time and Carol thus had to accept Now we know that everyone gets engaged by the matchmaker There is only one thing left to verify stability Again suppose that Bob and Carol were engaged by the matchmaker as were Ted and Alice Is it possible that Bob loves Alice more than Carol and Alice loves Bob more than Ted This would be an example of what we have called an instabilityquot Were this indeed the case Bob must have proposed to Alice before he proposed to Carol because the matchmaker made Bob send out proposals according to Bob39s preference list What then did Alice do with Bob39s proposal One of two things she accepted it or rejected it Let s consider the first case when Bob proposed to Alice she accepted Then why isn39t she now engaged to Bob There is only one possible reason why she dumped him to get engaged to someone she loved more Since every time Alice changes frances it is to increase her love in life she is certainly now engaged to someone she loves more than Bob As a consequence even though Bob loves Alice more than his intended Carol Alice could have no interest in dumping her mate Ted to run off with Bob Now let39s consider the second case Alice rejected Bob s proposal The only possible reason she rejected Bob39s proposal was her engagement to someone she loved more than Bob Once again Alice must still be engaged to someone she loves more than Bob namely Ted so Bob has no hope of convincing Alice to run off with him While this excursion into the mathematics of love may seem to have a perfect symmetry about it the above algorithm has a nasty characteristic that women should object to it favors men over women It is merely a social custom that men propose marriage to women there is certainly no reason why women can propose instead to men and the matchmaker could have arranged his directions so that the women indeed did so rather than the men The following example will show that whoever does the proposing gets a better deal Suppose that the men and the women hopelessly disagree about who their first choice is For instance imagine that Bob39s first choice is Carol and Ted s first choice is Alice while Carol s rst choice is Ted and Alice39s rst choice is Bob It should then be clear for each person who their second choice is When the matchmaker instructs the men to propose as described above in the rst round Bob proposes to Carol and Ted to Alice Since each woman received exactly one proposal they accept Game over Bob and Ted get their rst choice while Carol and Alice get their second choice If the matchmaker exchanged the directions he gave to the men and women and let the women propose instead Carol would propose to Ted and Alice to Bob Since Ted and Bob each get one proposal they have to accept Game over Carol and Alice get their rst choice while Bob and Ted get their second choice It now takes no imagination to imagine why there appeared two articles about The Matchquot in the New England Journal of Medicine some time ago addressing inequities in the matching procedure used to assign graduating medical school students to internships See Sounding Boards The Matching Programquot and An Analysis ofthe Resident Matchquot NEJM 30419 1981 pp 11631166 and further correspondence in NEJM 3059 1981 pp 525526 The principal anomaly criticized in these articles was precisely the rst choicesecond choice asymmetry just outlined that the stable marriage algorithm is male optimalquot As described earlier in The Matchquot medical students list their preferred jobs in order of desirability while hospital programs do the same and everyone feeds their list to a computer programmed with the stable marriage algorithm Now when the stable algorithm is run is it the hospitals or the students who get to play the role of the men The hospitals of course The authors of the NEJM articles asked that either the mathematicians and computer scientists work to nd a more equitable matching algorithm or that the national medical establishment let the students at least occasionally do the proposing In the words of the authors of the second article all parties are entitled to be informed of the bias of the present algorithm toward hospital programs and of the availability of workable although differently biased alternativesquot While there has been continued research in this genre of matching problems no suitably unbiased replacement for the stable marriage algorithm has been found To the best of my knowledge there has been no wavering on the issue of alternating students and hospitals as the proposers Perhaps the new President of Brandeis as a spokesman for national health and education issues would care to begin by addressing this clear and unconscionable inequity in the training of our nation39s doctors The mathematics and politics of love should now be clear but there are more lessons to be learned about computer science by studying this algorithm The rst and most obvious point to make is that the matchmaker doesn t need to know anything about men women or love to do his job In spite of this what was once said of philosophy is even more true of computer science to paraphrase even when all the computational and algorithmic dif culties of marriage have been solved the real and most profound questions about this most compleX form of human relationship remain and most likely will remain unaddressed and unresolved Of these larger and more important questions which in part give life its mystery and its interest computer scientists remain decidedly silent Moreover this algorithm does not account for the fairness of What is Mathematics By Paul Cox 2001 Part 1 An in depth review of Pi in the Sky Counting Thinking and Being by john D Barrow quotA mathematician is a blind man in a dark room looking for a black cat which isn 39t there quot Charles Darwin The May 7 2001 issue of Newsweek had as its cover story an article about how religious experiences may be explained as a function of the brain At the same time scientists are looking into the possibility that math is also a basic function of the brain as I mentioned in the introduction Coming full circle is John D Barrow39s 1994 book Pi in the Sky Counting Thinking and Being which has as one of its primary themes the idea that mathematics has much in common with religion Here is the intriguing first paragraph A mystery lurks beneath the magic carpet of science something that scientists have not been telling something too shocking to mention except in rather esoterically refined circles that at the root of the success of twentiethcentury science there lies a deeply 39religious39 belief a belief in an unseen and perfect transcendental world that controls us in an unexplained way yet upon which we seem to exert no influence whatsoever What this world is where it is and what it is to us is what this book is about Op cit pg 1 As Carl Sagan said quotincredible claims require incredible evidencequot and I will cut to the chase and say that Barrow fails to prove his initial thesis Math is not really a religion nor is it built on a religious foundation What Barrow does provide are numerous ideas that will inspire deep thoughts on the paradoxical nature of mathematics and its place in human knowledge For the next 100 pages Barrow gives a brief account of the early history of mathematics He begins at a time when math really was a religion the time of Pythagoras Pythagoras believed that all is number and that the world behaved mathematically While many of Pythagoras39s ideas have proven faulty his cult of WGHW ELKOQ matters learned contributed the first three books of Euclid s Elements including the first known formal proof of the Pythagorean theorem the theorem actually was known to the Egyptians and Babylonians way before Pythagoras Their idea that everything has as its basis in number in uenced Greek philosophy especially Socrates and his student Plato At least three dialogues of Plato discuss the mystical nature of mathematics M eno The Republic and The Laws It should be noted that today very few mathematician takes Plato39s views literally but it has become a powerful metaphor It is why we label new theories in mathematics as quotdiscoveriesquot Math is metaphorically described as existing in an independent world we can explore with our minds This is mathematical Platonism and it has existed longer than Christianity So why Platonism has few true believers today it is so ingrained in how we think about math that we cannot help thinking of new theories as discoveries instead of what they probably are quotcreationsquot That mathematics is often described in metaphorical terms is not the problem The real problem is that mathematics does not have a good description of what it really is except in metaphorical terms Barrow describes it this way Just as our picture of the most elementary particles of matter as little billiard balls or atoms as mini solar systems breaks down if pushed far enough so our most sophisticated scientific description in terms of particles fields or strings may well break down as well if pushed too far Mathematics is also seen by many as an analogy But it is implicitly assumed to be the analogy that never breaks down Our experience ofthe world has failed to reveal any physical phenomenon that cannot be described mathematically That is not to say there are not things for which a description is wholly inappropriate or pointless Rather there has yet to be found any system in Nature so unusual that it cannot be fitted into one ofthe straitjackets that mathematics provides This state of affairs leads us to the overwhelming question Is mathematicsjust an analogy or is it the real stuff of which the physical realities are but particular reflections This leads us to our first glimpse of the mysterious foundation of modern science It uses and trusts the language of mathematics as an infallible guide to the way the world works without a satisfactory understanding of what mathematics actually is and why the world dances to a mathematical tune pp 21 22 The Central Problem of the History of Numbers Approximately a third of the book is dedicated to the study of the history of numbers Why Because the best way to find out what mathematics is is to look at where mathematics comes from This is the primary source of contention in the debate over the philosophy of math The history of mathematics seems our best chance at answering the following questions from the book Is there an innate human propensity for mathematical concepts Does the human brain contain a natural structure 39hardwired39 into its makeup some way Does the human mind therefore possess a natural intuition for simple mathematical concepts If so does this mean that our mathematical picture of the physical world is primarily a mind imposed projection of our own mental structure upon the outside world Did we discover counting or did we invent it Could we have just missed it and developed in a literate but innumerate fashion Did an intuition for counting arise all over the world wherever there was language and society of any sort or was it come upon rather infrequently and then only by those in the midst of sophisticated cultural developments And what of counting itself it appears to be necessary for mathematics but is it sufficient to guarantee the sophisticated abstract notion of number and structure that forms the basis ofthe modern scientific picture of the world pp 101102 The new brainmath theories make a big deal of the fact that so many diverse and independent civilizations came up with symbolic numbers and arithmetic principles on their own It is hinted that this is not possible unless math is something that comes natural to the human brain Their answer to at least the first four questions would be quotyesquot Barrow39s own conclusions are not so simple quotWhilst we have seen that rudimentary counting systems were almost universal in the ancient world they were not completely soquot Furthermore he points out quotWhilst not every society could count they could all speak Language predates the origin of counting and numeracyquot pg 102 This implies a notion of distinction for counting and number which developed solely as a feature of language If you develop words for hot and cold boy or girl you are bound to develop words for one and two Barrow contradicts the brainmath theories all together by pointing out The existence of a natural human propensity for counting would lead one to expect that counting would spring up independently all over the globe But we have seen that the rival picture in which distinctive counting practices develop once in an advanced cultural centre and then spread to lesser societies has much evidence to support it pg 104 Another contradictory point has to do with the development of higher math skills which unlike counting and adding is extremely rare Barrow points out Having a notion of quantity is a long way from the intricate abstract reasoning that today goes by the name of mathematics Thousands of years passed in the ancient world with comparatively little progress in mathematics It is not good enough to possess the notion of quantity One must develop an efficient method of recording numbers more crucially still the adaptation of a place value system with a symbol for zero was a watershed A good notation permits an efficient extension to the ideas of fractions and the operations of multiplication and division Again we find these discoveries are deep and difficult almost no one made them pp 103104 Even more damaging evidence can be found in the fact that most ancient cultures lacked the ability to abstract numbers Most ancient counting systems varied in vocabulary depending on what was counted We do a little of that today with ordinals and other counting methods Besides one two three four we sometimes use first second third fourth or mono or uni bi tri quad or whole half third quarter or primary secondary tertiary quaternary or singular dual trio quad or solo duet trio quartet etc it is interesting to note that each of these English counting systems have different languages of origins with French German Latin and Greek all represented in one form or another Many ancient languages are worse having separate counting words for people and cattle and time etc This made it hard to abstract oneness twoness etc Without the ability to see numbers as abstract entities regardless of what is being counted mathematical insight of any kind is very difficult at best As a result most ancient counting systems are simply incapable of being used to describe higher level math Algebra was not possible until the advent of zero as a place holder in a numbering system Numbering systems have been around thousands of years before India came up with a place value system of counting needed for higher levels of mathematics In other words it was a miracle that high level math was ever developed Looked at this way there is a sense that mathematical truth eXists independently of the brain and that there are mathematical truths that the brain cannot comprehend I previously reviewed a book called Uncommon Sense by Alan Cromer that says similar things about the development of science It would be negligent of me as a reviewer not to point out that Barrow39s history of counting is far from comprehensive nor are his conclusions solidly proven to the point of being free from alternate interpretation The books that I will be reviewing later also contain their own telling of the history of numbers again not comprehensive with understandably different conclusions This is the central problem of the history of numbers there simply is no singular way to interpret the evidence Having made these conclusions about the history of math and counting Barrow proceeds to tackle the philosophy of math which covers the last two thirds of the book This is divided into four chapters each covering a major philosophy of mathematics Formalism Inventionism Intuitionism and Platonism After a de nition and brief history Barrow proceeds to punch holes in each philosophy None of the schools of thought come out unscathed including his obvious favorite Platonism Here is a brief summary of each Formalism Formalism is the view that mathematical statements are not about anything but are rather to be regarded as meaningless marks The formalists are interested in the rules that govern how these marks are manipulated Mathematics in other words is the manipulation of symbols The fact that a b c a b c is simply a rule of the system The principle protagonist of this philosophy is David Hilbert Unfortunately this philosophy was proven unfit by Godel39s incompleteness theorem I described this con ict between Hilbert and Godel in my Greatest Math Mistake of the Centurv essay Godel39s theorem suggests that the truth of a mathematical statement cannot merely consist in its proof from a set of axioms Hence Formalism has been defeated A break off of Formalists school is the Logicists school championed by Bertrand Russell and Gottlob Frege who sought to show that are knowledge of mathematical truth was as certain as our knowledge of logical truth They attempted to define mathematics in the language of logic Their efforts resulted in some important ideas such as the relationship between number theory and set theory but ultimately this enterprise was found faulty as well due to paradoxes such as Russell s Paradox an important principle of set theory which could not be based on logic Inventionism Inventionism also sometimes called Constructivism holds that true mathematical statements are true because we say they are Mathematicians do not discover mathematics as the Platonists claim they invent new mathematics This is actually pretty close to the brainmath philosophy so I will not go into detail However here are Barrow39s arguments against this philosophy We have explored the case for regarding mathematics as a human invention shaped primarily by the structure of the human mind and its particular ways of processing and organizing information and responsive to the ways of human society and culture The products of human thinking must necessarily be fallible at some level On this picture we do not discover mathematics 39out there39 it need not exist in the absence of mathematicians and the form it takes is strongly associated with our own genetic makeup To pursue the inventionist philosophy is to make mathematical truth dependent upon time and history We are forced to an antiCopernican stance which sees mathematical truth changing with the evolution ofthe human mind lnventionism is a wonderful philosophy for the arts and humanities where we see the fruits of imaginative subjectivity their nature and practice contrast so drastically with that of mathematics that the objective element seems to have failed to be adequately incorporated into this view of mathematics lnventionism fails to provide insight into the fact that Nature is best described by our mental inventions in those areas furthest divorced from every day life and from those events that directly influence our evolutionary history In the end one cannot help but feel that humanity is not really clever enough to have 39invented39 mathematics lbid pp 176177 In other words if mathematics was invented how is it possible that mathematics explains the real world so well The history of science tends to support this view For example Non Euclidean geometry was discovered in the 19th Century separately by Gauss and Riemann No one saw any use for these geometries until Einstein based his General Theory of Relativity on them Real space is NonEuclidean in shape and operation If the inventionist view is held then NonEuclidean Geometry should have been invented as a result of the General Theory of Relativity not the other way around This is a central problem in the philosophy of Mathematics Mathematics explains too well the nature of reality as new scienti c discoveries are found In truth I nd this argument to be weak It does not take into consideration the nature of science and scienti c progress I will explain my views in detail in part 2 but let me just say that Inventionism has more credibility than at rst glance When you take into account cultural considerations and scienti c discovery and evolution in general it is easy to see Barrows point quotone cannot help but feel that humanity is not really clever enough to have 39invented mathematicsquot But when you look deeper into how the process of discovery works in the scienti c community you can see direct parallels with the math community and it turns out humanity is clever enough after all Intuitionism Barrow39s chapter on Intuitionism is probably the most interesting in the entire book despite the fact that intuitionism is probably the weakest of all the mathematical philosophies The easiest way to de ne intuitionism is that it is the corollary of logicism The Logicists want to de ne mathematics in the language of logic The intuitionists want to de ne logic in the language of mathematics Barrow credits the school of intuitionism on 19th Century Dutch mathematician Luitzen Brouwer though other books take the philosophy back to 18th Century German philosopher Immanuel Kant Barrow de nes intuitionism as follows Because of Brouwer39s concern about the uncertain and subjective influence ofthe mind upon our mental constructions he sought to found mathematics in a conservative manner as possible upon the smallest and surest island of those intuitions which he believed we all share For Brouwer the island consisted of the natural numbers 123 and simple counting processes From this basis he defined mathematics to be the edifice that can be constructed from them by stepbystep deductions using a finite number of steps pg 185 If you do not understand this last sentence neither did I at first The intuitionists basically are trying to separate mathematics from the fallacies of the human thought process Sometimes we humans have a tendency to jump to conclusions ahead of step by step logic The intuitionists are basically against these jumps to conclusions The problem with ruling out jumping to conclusions is that you are forced to rule out acceptable means of mathematical proof like reductio ad absurdum also known as proof by contradiction Many important mathematical proofs like the irrationality of the square root of 2 are based on such proofs so are proofs about the nature of infinity Brouwer got around this by introducing a three valued logic system true false and 39not proven39 The consequences of following Brouwer39s demand that mathematics consist only of those statements that can be constructed in a finite number of steps from the properties of the natural numbers are very great It produces a mathematics that is far smaller in extent far more limited in power and far more predictable than the conventional mathematics which employed a twovalued logic in which every statement was either true or false The mathematics according to the intuitionists was just a part of the ocean of mathematical truths that were accepted by other mathematicians Any truth of intuitionism would be a truth oftraditional mathematics but not necessarily vice versa Threevalued logic produces a host of changes to traditional assumptions Tautologies of twovalued logic need not be tautologies in three valued logic An obvious example is the principle ofthe excluded middle itself that is 39a statement is either true or false39 pg 186 Barrow goes on in this chapter to write about the nature of proof and as I said before this is one of the most interesting sections in the book Especially entertaining are the stories of Ramanujan the Indian math protege who came up with brilliant number theories after reading a study guide to the Cambridge entrance exams as well as Cantor and his wild theories about in nity and the computer that proved the four color conjecture The ideas of what constitutes proof are too wide to be restricted as the intuitionists would have it Platonism Like Sherlock Holmes saying that once you eliminate the impossible whatever is left no matter how improbable must be the truth The shortcomings of Formalism Inventionism and Intuitionism lead us to take a new look at Platonism Barrow explains Platonism this way Plato39s philosophy of mathematics grew out of his attempts to understand the relationship between particular things and universal concepts What we see around the world are particular things this chair that chair big chairs little chairs and so on But the quality they share let39s call it 39chairness39 presents a dilemma It is not itself a chair and unlike all chairs we know it cannot be located in some place or at some time But that lack of a place in space and time does not mean that 39chairness39 is an imaginary concept When you replace the concept of 39chairness39 with the concepts of number like 39threeness39 you start to see Plato39s point Three is not a physical object it is a universal concept like 39chairness39 Plato39s approach to these universals was to regard them as real In some sense they really exist 39out there39 The totality of his reality consisted of all the particular instances ofthings together with the universals of which they were examples Thus the particulars that we witness in the world are each imperfect reflections of a perfect exemplar or 39form39 pg25 The view as pointed out earlier is this Mathematics exists It transcends the human creative process and is out there to be discovered Pi as the ratio of the circumference of a circle to its diameter is just as true and real here on Earth as it is on the other side of the galaxy Hence the book s title Pi in the Sky This is why it is thought that mathematics is the universal language of intelligent creatures everywhere Yet Platonism is far from a perfect philosophy of mathematics and Barrow agrees The Platonic picture is one of those ideas that at first seems eminently simple and unencumbered by abstraction but which is infested with all manner of problems It is a philosophy of mathematics that is popular amongst physicists but less so among mathematicians and is almost universally regarded as an irrelevance by consumers of mathematics like computer scientists psychologists or economists One immediate consequence of the abstract world that it introduces is the downgrading of the meaning of traditional mathematical objects like numbers Whereas the Pythagoreans saw them as symbols imbued with irreducible meaning Plato sees them as empty vessels whose significance lies entirely in the relationships they have with other symbols pg 255 Let us take a long hard look at what the Platonist is asking us to believe We must have faith in another 39world39 stocked with mathematical objects Where is this world and how do we make contact with it How is it possible for our mind to have an interaction with the Platonic realm so that our brain state is altered by that experience pg272 Barrow goes on to discuss Platonic views in detail The most interesting idea is what Platonist mathematics has to say about Arti cial Intelligence it does not think it is really possible The nal conclusion of Platonism is one of near mysticism Barrow writes We began with a scientific image ofthe world that was held by many in opposition to a religious view built upon unverifiable beliefs and intuitions about the ultimate nature ofthings But we have found that at the roots ofthe scientific image ofthe world lies a mathematical foundation that is itself ultimately religious All our surest statements about the nature ofthe world are mathematical statements yet we do not know what mathematics quotisquot and so we find that we have adapted a religion strikingly similar to many traditional faiths Change quotmathematicsquot to quotGodquot and little else might seem to change The problem of human contact with some spiritual realm oftimelessness of our inability to capture all with language and symbol all have their counterparts in the quest for the nature of Platonic mathematics pg 296297 Ultimately Platonism also is just as problematic as Formalism Inventionism and Intuitionism because of its reliance on the existence of an immaterial world That math should have a mystical nature is a curiosity we are naturally attracted to but ultimately does not really matter Platonism can think of a mathematical world as an actual reality or as a product of our collective imaginations If it is a reality then our ability to negotiate Platonic realms is limited to what we can know if it is a product of our collective imaginations then mathematics is back to an invention of sorts True or not our knowledge of mathematics is still limited by our brains Do there exist mathematical theorems that our brains could never comprehend If so then Platonic mathematical realms may exist if not then math is a human invention We may as well ask quotIs there a Godquot The answer for or against does not change our relationship to mathematics Mathematics is something that we as humans can understand as far as we need In Part II I will look at a recent experiment in understanding how a group of individuals can work together to achieve a level of success that far exceeds what any single person is capable of doing This experiment demonstrates how the math community could quotinventquot mathematics capable of complimenting science so well What is Mathematics Part 2 Multiperson social problemsolving arrays considered as a form of quotartificial intelligencequot ln part1 ofthis essay we looked at the four dominant philosophies of mathematics with the conclusion that they are all flawed in some way In the book Pi in the Sky author John D Barrow concluded that Platonism is the least flawed ofthe mathematical philosophies despite being the most mystical in nature Click here to read Part1 In part 2 we will reexamine lnventionism or Constructivism The principle arguments against the idea of math being a human invention is that most human inventions art and literature for example come from human brains and therefore contain cultural biases and are fallible at some level Mathematics is too useful and too universal to be something that is simply invented What I will demonstrate in this essay is that under the right circumstances humans can create something universal and bias free In short mathematics was created by the scientific community using the rules of the scientific method It is not a cultural construct forced into universal acceptance by imperial decree It is rather something that has evolved over time and over many cultures and has gained universal acceptance because of its usefulness and universal application A recent bizarre experiment emphasizes how this is possible M ulti person social problem solving arrays considered as a form of quotartificial intelligence quot is the title of a report by Dr Jeanine Salla of Bangalore World University It is a report on an amazing sociological experiment which will have many philosophical implications Among the philosophical questions to be answered with this experiment are questions about the nature and limits of the scienti c method the effectiveness of democracy and rational consensus The experiment s conclusions will have major implications on economic theory and the possibility of sentient arti cial intelligence Most importantly of interest to the philosophy of mathematics is whether or not the quotunreasonable effectiveness of mathematicsquot problem can be explained in purely human terms This important report will be published in the year 2142 Did I mention that Dr Jeanine Salla is a ctional character Maybe I should also mention that the experiment is being conducted as part of an advertising campaign for the new Stephen Spielberg movie A Despite its less than prestigious origins the experiment disguised as a clever online game is real and the ndings while lacking scienti c rigor will be discussed for years to come long after the movie it is advertising has retired to the bargain shelf at Blockbuster Cloudmakersorg The origins of this experiment is still a mystery but it is believed to be the brain child of legendary lm maker Stanley Kubrick who starting with a short eight page story by science fiction author Brian Aldiss developed a rather complex futuristic setting for a movie which he never lived to make The Arti cial Intelligence project was instead handed over to his friend Stephen Spielberg The next contributing factor to the experiment was the low budget movie The Blair Witch Project which became a 150 million dollar success on a shoestring marketing campaign based entirely on the intemet The third suspected contributing factor was Microsoft39s interest in developing video games with movie tie ins for Microsoft39s soon to be released XBox Some interesting detective work has revealed that the intemet game experiment is basically developed and being run by programmers within Microsoft who go by the name quotpuppet mastersquot within the game community Unbeknownst to anybody sometime in early march the quotpuppet mastersquot put up a series of elaborate web sites 30 at last count that pretend to be websites of people and organizations set in the year 2142 These web sites were never advertised anywhere Then a recent advertisement for the movie A listed among its credits of producers directors etc quotJeanine Salla Sentient Machine Therapistquot An intemet search of the name revealed her home page no longer available which begins a hyperlinked tree that eventually links to all 30 other web sites Getting to these other sites requires sending emails calling phone numbers and solving complex puzzles These sites have two things in common they are all set in the year 2142 and they all have something to do with the mysterious death of an engineer named Evan Chan That is the hook that gets you started The discovery of this mystery and the difficulty of the puzzles led to the spontaneous creation of Cloudmakersorg Starting with a discussion group on Yahoo where puzzle solvers could compare notes it eventually led to the creation of a web site that explains to anyone who wants to join what is going on so far These 30 web sites are constantly updated by the puppet masters and as they are updated they reveal new plot twists into the murder mystery the nature of artificial existence and some far more interesting stories about a technologically dependent society on the brink of collapse from the rebellion of sentient AI s yearning to be treated as equals The Hive Mentality The philosophical interest in this game experiment has nothing to do with the story revealed on these web sites but rather the behavior of the group of individuals trying to solve the puzzles This is of great interest because it is one of the best controlled examples I know of a human based hive mentality A hive mentality sometimes called a quothive mindquot is similar to an insect colony ie ants or bees which individually behave seemingly independent and almost unpredictably random but when thought of as a whole they manage success far exceeding what any one of them could accomplish Examples of human based hive minds include the scienti c community including the mathematics community governments and charitable organizations Most of these quothive mindquot societies are too large or too complicated to study up close and nd out what makes them successful or failures Cloudmakers is a fairly controlled environment it shows all the signs of success and it numbers between 10006000 participants world wide over a span of just a few months By studying the behavior of this group a lot can be learned in understanding the behavior of much larger groups over longer periods of time I should point out that there is a difference between quothive mentalityquot and quotmob mentalityquot that difference is an informed hierarchy One of the things the Cloudmakers did early on was establish two groups a free for all discussion group and a moderated group that featured the most important and informative messages of the free for all group If you want your point to get attention you need to convince a moderator and it will be forwarded to the quotimportan quot group The moderators are not there to dictate they are there to keep things productive and civil If they wanted to take control I doubt it would be possible All quothive mindquot structures have similar structures In the sciences you have publishers of prestigious journals If you want your opinion read or recognized you need to convince an editor rst Representative democratic governments have a hive mind structure which may explain their superiority over other forms of government A true democracy is a free for all In a representative government proposed laws and policies have to go through the legislators for consideration quotIt is not whether you win or lose it39s how you play the gamequot Initially the game was set up to be played by individuals Some of the early puzzles were solvable by anyone with enough brainpower A couple of weeks into the game however it seemed that the puzzles were being solved too quickly A puzzle or riddle would be posted then 1000 people each had their own ideas on how to solve it would tackle the problem the problem was solved in no time A thousand people working independently on a common goal is a erce intelligence Depending on how you look at it it is either highly efficient or highly inef cient On the ef cient side it gets answers quickly Far quicker than any single human could On the inef cient side there is a lot of quotwastedquot effort on bad solutions and redundant puzzle solutions If you had an idea on how to solve a particular puzzle you can bet there were 20 others thinking along the same lines doing the same work you are doing My own participation is pretty typical I have got credit for one or two puzzle solutions but to get that credit I have put in a lot of time on other puzzles ultimately solved by others Even the puzzles I got credit for solving were solved by seeing what other people were doing and just nding their mistakes No puzzle was considered solved until the quotsolutionquot could be explained and replicated by others The way the game was structured once a puzzle was solved new information was revealed which eventually led to more puzzles The game was updated weekly with new pages and information This parallels perfectly the scienti c method What we puzzle solvers were doing was behaving the way scientists have behaved over history Scientists observe natural phenomenon and ask questions other scientists speculate on how to explain the phenomenon requiring data to be collected and experiments to be conducted Someone comes up with a solution and explains their results Other scientists collect more data and perform identical experiments to verify the results Eventually there is enough evidence to make the explanation widely acceptable This new quottheoryquot ultimately leads to new questions and mysteries to be solved With the puzzles being solved so quickly there was a noticeable increase in difficulty with later puzzles to the point where it was no longer possible to solve these puzzles alone Later puzzles involved expert knowledge of Shakespeare T S Elliott HTML coding CGI scripting foreign languages including French German Japanese and Kannada a fairly rare Hindi dialect art history religious history architecture British cuisine psychology Morse code and the WWII German Enigma code Someone who knows all of these topics is rare but in a group of nearly 5000 with Internet access you can find someone with the expertise that is needed Again this parallels the history of Science and Math There is simply no way for any one person to know everything about a major scientific specialty let alone be an expert in all disciplines 19th century mathematician David Hilbert is often considered to be the last person who knew everything about mathematics Today it is impossible to know everything about mathematics the topic is too broad But it is possible to know everything about one or two specialty topics and have enough general knowledge to communicate with experts of other specialty topics The unanswered questions of Math and Science are solved today by groups of experts working together Einstein consulted dozens of famous scientists and thinkers to formulate his General Theory of Relativity Today many new discoveries and inventions are rarely even credited to an individual but are credited to Universities Corporations think tanks and independent laboratories The Reasonable Effectiveness of Mathematics In a way this answers the quotunreasonable effectiveness of mathematicsquot problem Why is it that so much of science can be explained mathematically Because so much of mathematics is speculative brainstorming The unreasonable effectiveness is an illusion For every mathematical theory which has gone on to be a good fit for reality there are hundreds that are just theories and do not fit any reality Thus we see that of all of the mathematical philosophies Inventionism also known as Constructivism has much more merit when we consider the quothive mentalityquot of mathematicians The biggest argument against Inventionism was that the quotcreationquot of mathematics would be culturally biased and would not fit with reality as well as math actually does When a diverse group with different biases and expertise get together bias tends to get weeded out The most elegant solutions eventually get universal acceptance The invention of the zero is a perfect example Indian mathematicians discovered that using a place holder made multiplication and division much easier This innovation spread to neighboring Persia and Babylon who quickly adapted it Our quotArabicquot numerals came from them Once a good idea gets invented it spreads quickly Another argument in favor of Inventionism is the fact that there is some cultural bias in mathematics Despite its universal appeal we made a mistake adapting a base 10 numbering system A base 8 or base 12 system would be much more ef cient We are stuck with base 10 because of cultural bias it is universally accepted because most of us have 10 ngers to count with Mathematics and the Brain I started this inquiry into what mathematics is by comparing two popular theories the old school quotPlatonistquot view and the new school where math as a product of the brain The quothive mindquot model as a creator of mathematics kind of puts a major dent in both of these theories The Platonist view that mathematics is too useful to be invented gets hit with the realization that when you have thousands of inventors the good bubbles up to the top and the weak falls off the scope Under a quothive mindquot model the Platonist argument that Inventionist math would be culturally biased and limited to what we can understand both prove faulty The good news is that the underlying mysticism of a Platonic realm that we can communicate with and explore no longer needs to be a physical reality it can remain just a metaphor On the other hand those that say that mathematics is a product of the brain believe that mathematics is ultimately limited by what the brain is capable of understanding If mathematics is the total of what all mathematicians understand then the quothive mindquot is capable of far more than any individual brain This leads to a bizarre possibility that there are mathematical objects or concepts which are only understood in the collective unconscious of the mathematical community but which are too complex for any individual mathematician to comprehend This possibility is something the Platonists can conceive of and understand Could there be a mathematical idea so compleX no single mathematician could understand it but as a group the world community of mathematicians might be able to understand and work with this idea None eXist yet but there is at least one possible candidate Arti cial Intelligence The Dif th Problem of making Arti cial Intelligence a Reality Since I have managed to mention quotArti cial Intelligencequot twice already in two completely unrelated contexts I thought I would nish up with a brief introduction to the topic A more detailed look will be forthcoming in a future essay While the idea of creating an arti cial intelligence can be traced back through the centuries the rst person to de ne the idea concretely was Alan Turing the theoretical inventor of the computer It seems the idea of making a computer that could quotthinkquot is as old as making computers that could only do simple math Turing proposed an experiment popularly called the quotTuring Testquot in which you type messages on a terminal and either a person or a computer is on the other end If a computer can imitate a person so awlessly that it is impossible to tell whether they are real or not then it is said that the computer program is capable of thought I will not go into all of the details and philosophical problems associated with the quotTuring Testquot there are plenty of other great sources Suf ce it to say that in the fty plus years since the Turing Test was rst proposed we have not even come close There are too many problems that have to be overcome rst A major one is natural language processing Have you ever used a computerized translation program to read web sites in other languages such as babel shcom or worldlingocom Have you found such programs severely lacking This is the natural language processing problem This problem reared its ugly head back in the 1950 s when the Defense Department tried using computers to automatically translate intercepted Russian communications Translation was thought to be an easy problem enter a Russian word look up its English equivalent and print the results The translations were nowhere near as comprehensible as that produced by a human translator It turns out that translation is far more difficult than just looking up words For example read the following paragraph lfthe balloons popped the sound would not be able to carry since everything would be too far away from the correct floor A closed window would also prevent the sound from carrying since most buildings tend to be well insulated Since the whole operation depends on a steady flow of electricity a break in the middle ofthe wire would also cause problems of course the fellow could shout but the human voice is not loud enough to carry that far An additional problem is that a string could break on the instrument Then there could be no accompaniment to the message it is clear that the best situation would involve less distance Then there would be fewer potential problems With face to face contact the least number of things could go wrong Bransford and Johnson 1972 Do you understand it If not is there a word that you do not know There is not a single difficult vocabulary word in the paragraph So why are you having trouble with the paragraph The problem is that you do not recognize the context that the author is talking in Need some help Click here to nd what this paragraph is talking about In order to process natural language computers need to understand the world around them rst There is the the central problem how do you make computers understand Considering the difficulty of programming something so simple we do it without even noticing gives you just a glimpse of the enormity of the arti cial intelligence problem as a whole I believe that creating the rst arti cially intelligent program will require an organized quothive mindquot style coordinated effort with many computer programmers linguists neurobiologists and behavioral psychologists working many decades on the project The rst one to create a satisfactory intelligence will be billionaires in no time at all with such a motivation it might actually happen within a century How Arti cial Intelligence can work is something too complex to be solved by a single individual and if and when it becomes a reality no single individual may be able to understand how it works It will be a mathematical object too big for an individual brain to comprehend only understandable by the quothive mindquot that created it Conclusions There is an aspect of the Arti cial Intelligence problem that has some bearing on our pursuit of the nature of mathematics Computers have a dif cult time understanding simple language something that comes easy to us Computers have an easy time solving math problems something that often times seems dif cult for us There are two ways to analyze this observation First we could conclude that Computers and Humans have such completely differing natures computers will never be like humans and humans will never be like computers If so then mathematics is counter to human nature which explains why it took so long to develop The other possible conclusion is that computers are way below us on the intelligence chain This is the conclusion we must accept if we believe that computers will one day have human like intelligence If this is the case then mathematics is too easy for our complicated brains There is evidence to support both conclusions Either way we can see that math is not dif cult it is merely contrary to our normal way of thinking So far in the quest to answer the question quotWhat is Mathematicsquot we have managed to at least to say what mathematics is not Mathematics is not mystical nor can it de ne itself It is not a product of individual human minds but rather a product of consensus among all mathematical thinkers Ultimately mathematics is something that has been created over time as a means of conceptualizing the natural world We should not be surprised by its effectiveness at doing what it is designed to do What then is mathematics It is the study of patterns or more accurately it is a language to describe patterns Identifying patterns in the world around us comes almost as easy as identifying objects in the world around us Realizing this is the key to breaking our dif culties with math What is Mathematics Part 3 The Science of Patterns An in depth review of The Math Gene How Mathematical Thinking Evolved and Why Numbers are Like Gossip by Keith Devlin This is the final for now chapter discussing the philosophy of mathematics In part one I reviewed Pi in the Sky which is an overview of Math Philosophy which favors Platonism In part two I looked at new ideas about group think vs the power ofthe individual mind We eliminated the quotmysteryquot of how math came to be but we still did not answer the fundamental question quotWhat is Mathematicsquot In this chapter we look in depth at the quotbrain mathquot theories specifically how does the brain do mathematics Basically the brain does mathematics the same way the brain handles language Which leads to the inevitable question How come we can39t do math as easily as we speak The linguist in me knows one answer spoken language uses simpler grammars than mathematical grammars Cognitive Psychologists will tell you another mathematics generally has a higher level of abstraction than everyday language If you did not understand those last two sentences don39t worry I will explain later There is however a much more surprising answer We can do math as easily as we speak Once we recognize what math really is it becomes clear that many kinds of math come very naturally Unfortunately what we normally think of math arithmetic and algebra are not one of those topics that come naturally The point is that studying how our brains handle mathematics not only will help us understand what mathematics is it also will help us find better ways of teaching mathematics Let s start this essay with a couple of logical problems Problem 1 There are four cards laying on a table Each card has a number on one side and a letter on the other Face up you see these symbols E K 4 7 The cards are printed according to the rule lfa card has a vowel on one side it has an even number on the other side Which cards do you have to turn over to be sure that all the cards satisfy the rule Got an answer Are you sure it is right Before I reveal the answer let me give you another problem Problem 2 You are in charge of a party where there are young people Some are drinking alcohol some are drinking soft drinks Some are old enough to drink some are under age You are responsible for ensuring that the drinking laws are not broken so you ask that they have their lD39s on the table At one table there are four young people who may or may not be legal drinking age One has a beer and another has a coke but their lD39s are face down so you cannot see their ages The other two have their lD39s up one is a legal drinking age the other is under age but both are drinking a clear fizzy liquid that may be either 7up or vodka tonic Which lD39s and drinks do you need to checkto make sure everyone is legal The answer to both questions are the same you check the rst and the last The two problems are in fact are identical with these substitutions vowel alcohol consonant soft drink even number legal drinking age odd number too young to drink But if you are like me and most people you probably found problem 2 much easier to answer than problem 1 Why Because the rst problem is a higher level of abstraction than the second problem It is similar to another study I saw in which two word problems were given to kindergarten and rst graders There are nine birds and five worms How many more birds are there than worms There are nine birds and five worms How many birds do not get worms This was a study into how the difficulty of word problems has to do with how well they are written It should come as no surprise that the students got the second one right more often then the first The biggest problem people have with word problems is that they are not written very well which makes me question whether or not learning to solve word problems does anything to help students to apply math to the real world That is a question for another time In this essay we will be exploring the connection between language and mathematics A good book on this topic came out just last year called The Math Gene by Keith Devlin The paperback version came out in June What follows is a brief summary of the main ideas found in this book I will not go into detailed arguments that support these ideas you will just have to read the book First a quick review This is not a book written only for math people People who are interested in the current problems of math education would benefit as well This is not a scholarly work it is easy to read without math psychology or educational jargon Keith Devlin explains his ideas easily and occasionally with humor without getting too technical On with the overview How the brain does math After an introductory chapter Devlin has two chapters devoted to what psychologists know about how our brain does math It follows that old genetics vs environment argument some of our math ability comes naturally though the majority is learned Chapter 2 is devoted to fairly new ideas about our natural ability to do basic math such as counting It seems from birth we have the ability to understand the difference between one two three and many Psychologists have confirmed this through rather clever testing of infants as young as four weeks old For example one test involved showing babies a series of puppets going behind a curtain The testers show the babies a couple of puppets going one at a time behind a curtain The curtain is then lifted If there are one or three puppets revealed behind the curtain the babies will stare longer at the puppets than if there are only two puppets which is what the baby expects Babies stare longer at things that do not make sense to them so when they see two puppets go behind a curtain they expect two puppets when the curtain is dropped A basic inherent quotnumber sensequot is not just a human trait it has been confirmed in mammals and birds as well There is an old story about a bird that was nesting in a tower and the owners of the tower wanted to get rid of it When they sent someone into the tower to capture the bird the bird would leave to a nearby tree and would not return to the tower until the man left So they came up with a clever plan of sending two men into the tower then soon after one man left but the bird stayed in the tree and would not return to the tower until it saw both men leave it saw two men enter and it waited until both men were gone They tried again with three same thing They tried with four same thing Finally it got confused with five men entering the tower and four men leaving the fifth man was able to capture the bird I can attest to the fact that this number sense also exists in cats I have a cat and sometimes I give her some kitty treats just before I leave this distracts her so I can go out without her darting outside If I toss three treats on the oor with my cat watching she will systematically go after all three treats If I toss ve she will get three easily then she has to sniff around and search for the others she knows there are more than three If she nds one she may stop looking because she lost track of how many I threw So the number sense is a naturally occurring ability of knowing one two three and many It is an instinct developed for survival Many primitive human languages actually only have words for one two three and many Remnants of this can be found in English like the ordinal numbers rst second and third Later ordinals fourth through twentieth all have identical endings which makes the rst three ordinals unique Language ability and math ability As language re ects the number sense it is necessary to anything higher in mathematics That is to say anything beyond one two three and many requires language and counting ability to understand This is the focus of chapter three of the book the close ties between the brains ability to handle language and the brains ability to do higher math Devlin gives some interesting examples of this connection He relates the story of meeting a math wizard one of those people who can do large calculations in his head Before the demonstration he asked that the air conditioning be turned off because the noise it makes interfered with his ability He does the lightning calculations by hearing them in his head and certain kinds of noises hampered this process Devlin relates We all use the human ability to remember a spoken linguistic pattern when we learn our multiplication table We learn by reciting the table over and over Even today fortyfive years after I quotlearned my tablesquot I still recall the product of any two single digit numbers by reciting that part ofthe table in my head I remember the sound ofthe number words spoken not the numbers themselves Indeed I believe the pattern I hear in my head is precisely the one I learned when l was seven years old pg 59 Devlin is right on the money I must confess that whenever I multiply three by any number greater than ve in my head I sing the chorus to quotThree is a magic numberquot from Multiplication Rock More evidence can be found studying bilingual learners Mathematical ability seems to be tied to what language you learn it in Devlin mentions a study done with bilingual RussianEnglish speakers They were taught various mathematical concepts some in English some in Russian They were then given a timed test of these concepts some of the questions in English some in Russian The results were that if they were given a concept in one language and tested on it in another it took them longer to do the question because they had to translate the concept in their head Having once been uent in Spanish I can attest to this fact When you are uent in a second language you develop the ability to think in that language If some asks you a simple question like quotComo esta ustedquot you can answer quotMuy bien graciasquot without doing any translating into English Math concepts don t work that way If someone asks me quotCual es seis por sietequot I cannot come up with the answer quotCuarenta y dosquot without thinking in my mind quotsix times seven is forty twoquot in English While Devlin does not get into this controversy I could not help thinking that Bilingual Education at least as it applies to math may be a major disservice to students For those who do not know Bilingual Education is the practice of teaching native foreign students basic topics math reading social studies etc in their native language while they learn English as a second language If we do better in math when we do it in the language we are taught in then Bilingual Education students will struggle with math in the English classroom Just a thought One last interesting point about math and language Devlin points out that some languages are more suitable to mathematical thinking than others Chinese is maybe the best language for mastering math Doing arithmetic and in particular learning multiplication tables is simply easier for Chinese and Japanese children because their number words are much shorter and simpler generally a single short syllable such as the Chinese 5139 for 4 and qi for 7 The grammatical rules for building up number words in Chinese and Japanese are also much easier than in English or other European languages For instance the Chinese rule for making words for numbers past ten is simple 11 is ten one 12 is ten two 13 is ten three and so on up to two ten for 20 two ten one for 21 two ten two for 22 etc Think how much more complicated is the English system It s even worse in French and German with their quatre vingt diXseptfor 97 and vierundfunfzig for 54 A recent study by Kevin Miller showed that language differences cause English speaking children to lag a whole year behind their Chinese counterparts in learning to count By the age of four Chinese children can generally count up to 40 American children ofthe same age can barely get to 15 and it takes them another yearto get to 40 How do we know the difference is due to language Simple The children in the two countries show no age difference in their ability to count from 1 to 12 Differences appear only when the American children start to encounter the various special rules for forming number words The Chinese children meanwhile simply keep applying the same ones that worked for 1 to 12 American children often apply the same rules but they find they have made a mistake when they try to use words like twentyten and twentyeleven In addition to being easier to learn the Chinese number word system also makes elementary arithmetic easier because the language rules closely follow the base10 structure of the Arabic system A Chinese pupil can see from the linguistic structure that the number quottwo ten fivequot ie 25 consists of two 10s and one 5 An American pupil has to remember that quottwentyquot represents two 10s and hence that quottwenty fivequot represents two 10s and one 5 Pg 65 This is a pretty good explanation why Asian students are better at math than American and European students it is easier for them What is Mathematics So if math is closely tied to language but at the same time simple math is instinctual then what came first math ability or language ability It seems obvious that we need to speak in order to do math but the existence of a quotnumber sensequot seems to contradict that Devlin s own answer is that math came first but before he explains how this is possible we have to answer that nagging question that we have been struggling with for three months namely What is Mathematics Devlin s short one sentence answer is that mathematics is the science of patterns He attributes the de nition to WW Sawyer s 1955 book Prelude to Mathematics Forthe purposes of this book we may say quotMathematics is the classification and study of all possible patternsquot Pattern is here used in a way that not everyone may agree with It is to be understood in a very wide sense to cover almost any kind of regularity that can be recognized by the mind Life and certainly intellectual life is only possible because there are certain regularities in the world A bird recognizes the black and yellow bands in a wasp man recognizes the growth of a plant follows the sowing of a seed In each case a mind is aware ofthe pattern Sawyer Prelude to Mathematics pg 12 When we think of what a quotpatternquot is we usually thinking wallpaper or oor tiles but as it refers to mathematics we are referring to a property a category a type or a kind of something Pattern here refers to relationships between things not the things themselves This is a very broad definition of mathematics but a fairly accurate one Every topic of mathematics has to do with patterns or relationships though the reverse is probably not true there are patterns and relationships which are not mathematical though they certainly fall into some branch of science This is also a good explanation of why mathematics is quotthe Queen of Sciencequot because no scientific discipline is complete without a study of patterns of behavior and interactive relationships of whatever is being studied and most of these patterns can be expressed in a mathematical language Devlin gives more specific examples that are interesting but which I have no time to go into here One of the great things about a book like this is I now have a few essay topics to follow up with in the future On the Origin of Mathematics As I pointed out in part 1 all books dealing with the philosophy of mathematics inevitably give a history lesson to support this theory This book is no exception but where Pi in the Sky started with primitive counting methods that existed 10000 years ago The Math Gene goes back to Homo Erectus some 35 million years ago Devlin does this to support his controversial theory that mathematical ability may have developed prior to linguistic ability While this theory is intriguing I am not convinced it is true Devlin himself acknowledges the fact that there is no way to prove or disprove the theory s accuracy thus it is purely an academic argument a philosophical idea with no hard facts to back it up Lack of evidence aside he makes a convincing case Here is a terribly simplified summary of the theory which in fact covers the last half of the book 1 Language follows a rather complex pattern which no primate can duplicate Some may argue that some apes have demonstrated sign language or picture languages in truth apes have only demonstrated an ability for quotprotolanguagesquot which while an impressive feat by itself it is not as complicated as all human languages Strangely enough and this is a prominent theory of linguistics all human languages have virtually identical grammars which hints at some common origin at some unknown part of our development Devlin discusses this point at length because it is necessary to understand his theory 2 Early humans did actually have some basic skills necessary for mathematical development From the book We have already seen that many creatures have a number sense and that this sense yields obvious survival benefits ranging from recognizing which tree has the most fruit to knowing whether your group is outnumbered in a potential confrontation In addition to number sense living in trees with all that swinging from branch to branch demands a good threedimensional spatial sense and survival on the open plains requires a twodimensional spatial sense including the ability to judge distance I doubt that anyone would call such abilities geometric but they are necessary prerequisites the first steps if you will for geometric ability the beginnings of a quotgeometry sense analogous to number sense Another necessary component of mathematical thinking is an awareness of cause and effect Since contemporary primates all appear to have such a capacity it is reasonable to suppose that their and our ancestors in the forests and on the savanna likewise realized if only in a restricted sense that one thing can cause another Thus as much as seven million years ago there were brains having some ofthe capacities necessary for mathematical thought This is not to imagine that those early primates possessed anything like mathematical ability We have no reason to suppose that they had any capacity for reflective or abstract thought at all Rather my point is that mathematical thinking as practiced today makes use of mental capacities that were developed hundreds ofthousands and in some cases millions of years ago Doing mathematics does not require new mental abilities but rather a novel use of some existing capacities Of course strictly speaking being able to use those existing capacities in a new way does in fact constitute a quotnew mental ability pp 179180 3 The ability to use language seems to have come out of nowhere about 200000 years ago yet our brains were developing steadily for 3500000 years prior How could we evolve an ability to speak before we were capable of speaking Far more likely is that one development set the scene for the other that language arose as a byproduct of some more fundamental ability the one that drove the initial brain growth over a 3500000 year period So what is that ability If what I have been saying is correct the answer should be staring us in the face so obvious that we fail to see it so commonplace and accepted that we do not pause to regard it as anything out of the ordinary Something we simply take for granted An ability we have had from the moment we were born An ability that lies behind language Any ideas Here s another clue It was when I realized what this key ability must be that I knew everybody has the math gene the ability to do mathematics If you are still puzzling over the matter then almost certainly you are a victim of a view of the human mind propagated by the famous French philosopher Rene Descartes during the seventeenth century The modern incarnation of Descartes view is that the mind is a computing machine which thinks by following a progression of discrete logical steps According to this view the key to understanding all ofthe other things people do with their minds such as recognizing faces or understanding stories is to express those mental activities in terms of logical rules As I argued in my book Goodbye Descartes the assumption that all mental processes can be captured as logical rules is false Moreover it is precisely the falsity ofthat assumption that explains the failure of the many attempts to program digital computers to recognize scenes handle natural language and exhibit artificial intelligence At the end of Goodbye Descartes I proposed an alternative view ofthe human mind as a device for recognizing patterns visual patterns aural patterns linguistic patterns patterns of activities patterns of behavior logical patterns and many others Those patterns may be present in the world or they may be imposed by the human mind as an integral part of its view ofthe world pp 186187 In a way this is a throw back to the Arti cial Intelligence problem I mentioned in part 2 Computers have the reputation to do math very well and for most kinds of math this is true But the ability to recognize patterns in general is something computers have difficulty doing The important connection you need to make here is that if the human mind is a quotdevice for recognizing patternsquot and mathematics is quotthe science of patternsquot then our brain s primary functions are mathematical in nature 4 If what Devlin implies is true then the skills needed to learn language are greater than skills needed to learn math Math skills at least in the case of pattern recognition developed necessarily ahead of linguistic skills 5 If math skills are more basic than language skills then why are we not all gifted at doing math Because not all math skills are basic Devlin makes a list of skills necessary for math ability A number sense A sense of cause and effect The ability to construct and follow a causal chain of facts or events Relational reasoning ability Spatial reasoning ability Numerical ability The ability to handle abstraction Algorithmic ability Logical reasoning ability PWFQWPP N The abilities necessary to develop language are l to 5 6 to 9 came later in fact according to Pi in the Sky they developed much later only 3000 years ago or so 6 The principle ability that stands in our way from becoming good at mathematics is 7 the ability to handle abstraction 6 8 and 9 all require 7 As demonstrated at the very beginning of this essay problems involving lower levels of abstraction are simpler than higher levels of abstraction Devlin lists four levels of abstraction Level 1 abstraction is where there is really no abstraction at all The objects thought about are all real objects that are perceptually accessible in the immediate environment Level 2 abstraction involves real objects that the thinker is familiar with but which are not perceptually accessible in the immediate environment Level 3 abstraction are the objects ofthought which may be real objects that the individual somehow has learned of but has never actually encountered or imaginary versions of real objects or imaginary versions of real objects or imaginary variants of real objects or imaginary combinations of real objects Level 4 abstraction is where mathematical thought takes place Mathematical objects are entirely abstract they have no simple or direct linkto the real world other than being abstracted from the world pg 121 These levels of abstraction seem to parallel Plato39s worlds pointed out in part 1 of this essay The real world the objective world the world of the imagination the world of mathematical thought We can all handle levels 1 to 3 it is only at level 4 that many get confused Level 3 abstraction is necessary to speak thus we can all do it Level 4 abstraction does not come naturally it has to be learned and practiced This is what is standing in our way from being good at math The key to being able to think mathematically is to push this ability to quotfake realityquot one step further into a realm that is purely symbolic level 4 abstraction Mathematicians learn how to live in and reason about a purely symbolic world By quotsymbolic worldquot I don39t mean the algebraic symbols that mathematicians use to write down mathematical ideas and results Rather I mean that the objects and circumstances that are the focus of mathematical thought are purely symbolic objects created in the mind Although it does not require a different kind of brain to deal with this world it does involve considerable mental effort All mathematicians can solve the four cards problem ifthey put their minds to it But like everybody else they find it harder than the party problem pg 123 emphasis mine The point being which I have been trying to stress since April39s essay is that everyone can do math It just takes a certain amount of effort on the learners part to practice How Numbers are like Gossip The ability to do level 4 abstraction or what Devlin calls quotthe math genequot is something our brain developed when it developed language skills about 200000 years ago but it lied dormant for about 195000 of those years because we did not need it Then the Chinese Egyptians and Babylonians all started using and developing math almost simultaneously quotThe same cannot be said of language Human beings found a particularly important use for language the moment it arrived and they have been using it for that purpose ever sincequot says Devlin quotWhat is that purpose Answering that question will lead us to the explanation of why the language gene and math gene are one and the samequot The answer is that we gossip Sociologists have done studies that say at least half of all our conversations are gossip about other people Devlin explains that this is just human nature and in fact it plays a vital role in our survival So gossip is not necessarily a bad thing As abstraction goes gossip is actually pretty high It is speculation and imagination about people who are not around us Sometimes about people we never even met So gossip exists between level 3 and even level 4 abstraction Not only do we acquire and maintain this vast amount of information about others we can all reason about their lives We can have opinions on the actions of others we can understand explain and pass judgment on things they do we can guess or predict what they will do next Again not only can we do all of these things we do do them and without effort