Invention, Discovery and Creativity in Mathematics

Non-philosophers I meet sometimes ask: do I think mathematical facts are invented or discovered? IMO, This is a weird question - and not one that comes up much in the phil math literature- because the contrast between invention and discovery is not very well defined. For example, did Alexander Gram Bell *invent* the telephone, or did he *discover* that putting components together in a certain way would build a telephone? Intuitively, one might say both.

Maybe what people mean to be asking by this question is just this: do mathematicians bring new mathematical objects into existence, or do they discover already existing objects? For, paradigmatic cases of invention typically do involve creating a new physical object, while paradigmatic cases of discovery involves visiting an already existing physical object. So e.g. Columbus discovered America (he went to visit it) whereas Bell invented the telephone, by physically building the first telephone. However, the contrast between invention and discovery doesn't really capture the distinction between cases where a new object is made vs. not. This is because making a new thing isn't required for invention *or* discovery. Consider a thought experiment where Bell just thought up a plan for a telephone, and told someone else who physically constructed the first one years lated. Bell would still have invented at telephone, if he though up the plan and then worked out from known principles that the plan would work, but never made one.

While we are talking about invention and discovery, I think there's a third notion -artistic creation (e.g. what happens when someone composes a story or a poem)- which bears an interesting relationship to mathematical discovery. When a writer writes a story, they are putting down a sequences of sentences which already exists as an abstract object. I mean, suppose that the story teller composes a story today. If a linguist said yesterday 'no intelligible sequence of English sentences has property P', the and the sequence or sentence which the story teller writes down today has property P, then then the linguist's claim yesterday was false. The domain of potential counterexamples to linguistics claims today, already contains all sequences of English sentences which literary ingenuity could ever devise. Note also that to compose a story or poem doesn't require writing it down anywhere, (the person in the Borges story who has time stop so he can finish writing a poem before he gets shot, still counts as creating the poem). For this reason the task of literary "creation" doesn't really seem to involve creating anything, (neither a physical artifact, nor an abstract string of sentences), but rather directing your attention to an abstract object that already exists - carefully sorting out which string of sentences will combine certain varied and subtle properties in the right way.

Now, if I'm right about this- the creativity of a poet or novelist doesn't need to involve creating any new object, but rather amounts to discovering a pre-existing string of sentences which has a certain property - this suggests a potential confusion about the relationship between mathematical creativity and ontology. Arguably, mathematical creativity is much like literary creativity. But, if mathematical creativity is like literary creativity, it does not follow from this that the mathematician creates the mathematical objects he describes, or that he creates anything else. For (if the above is right) literary creativity isn't a matter of bringing new objects into being, but rather a matter of discovering, amid the combinatorial explosion of possible sequences of english sentences, one that has a certain special features.

A Proof that e is Irrational

e is one of those special numbers in mathematics, like pi, that keeps showing up in all kinds of important places. For example, in Calculus, the function f(x) = c(e^x) for any constant c is the one function (aside from the zero function) that is its own derivative. It is the base of the natural logarithm, ln, and it is equal to the limit of (1 + 1/n)ⁿ as n goes to infinity. In the proof below, we use the fact that e is the sum of the series of inverted factorials.

Like Pi, e is an irrational number. It is interesting that these two constants that have been so vital to the development of mathematics cannot be expressed easily in our number system. For, if we define an irrational number as a number that cannot be represented in the form p/q, where p and q are relatively prime integers, we can prove fairly easily that e is irrational.

The following is a well known proof, due to Joseph Fourier, that e is irrational.

The Beginnings of Probability...

Archaeologists have found evidence of games of chance on prehistoric digs, showing that gaming and gambling have been a major pastime for different peoples since the dawn of civilization. Given the Greek, Egyptian, Chinese, and Indian dynasties' other great mathematical discoveries (many of which predated the more often quoted European works) and the propensity of people to gamble, one would expect the mathematics of chance to have been one of the earliest developed. Surprisingly, it wasn't until the 17th century that a rigorous mathematics of probability was developed by French mathematicians Pierre de Fermat and Blaise Pascal. The Problem of Points The problem that inspired the development of mathematical probability in Renaissance Europe was the problem of points. It can be stated this way: Two equally skilled players are interrupted while playing a game of chance for a certain amount of money. Given the score of the game at that point, how should the stakes be divided? In this case 'equally skilled' indicates that each player started the game with an equal chance of winning, for whatever reason. For the sake of illustration, imagine the following scenario. Pascal and Fermat are sitting in a cafe in Paris and decide, after many arduous hours discussing more difficult scenarios, to play the simplest of all games, flipping a coin. If the coin comes up heads, Fermat gets a point. If it comes up tails, Pascal gets a point. The first to get 10 points wins. Knowing that they'll just end up taking each other out to dinner anyway, they each ante up a generous 50 Francs, making the total pot worth 100. They are, of course, playing 'winner takes all'. But then a strange thing happens. Fermat is winning, 8 points to 7, when he receives an urgent message that a friend is sick, and he must rush to his home town of Toulouse. The carriage man who has delivered the message offers to take him, but only if they leave immediately. Of course Pascal understands, but later, in correspondence, the problem arises: how should the 100 Francs be divided? In a letter to Pascal, Fermat proposes this solution: Dearest Blaise, As to the problem of how to divide the 100 Francs, I think I have found a solution that you will find to be fair. Seeing as I needed only two points to win the game, and you needed 3, I think we can establish that after four more tosses of the coin, the game would have been over. For, in those four tosses, if you did not get the necessary 3 points for your victory, this would imply that I had in fact gained the necessary 2 points for my victory. In a similar manner, if I had not achieved the necessary 2 points for my victory, this would imply that you had in fact achieved at least 3 points and had therefore won the game. Thus, I believe the following list of possible endings to the game is exhaustive. I have denoted 'heads' by an 'h', and tails by a 't.' I have starred the outcomes that indicate a win for myself. h h h h * h h h t * h h t h * h h t t * h t h h * h t h t * h t t h * h t t t t h h h * t h h t * t h t h * t h t t t t h h * t t h t t t t h t t t t I think you will agree that all of these outcomes are equally likely. Thus I believe that we should divide the stakes by the ration 11:5 in my favor, that is, I should receive (11/16)*100 = 68.75 Francs, while you should receive 31.25 Francs. I hope all is well in Paris, Your friend and colleague, Pierre

Prime Numbers

One of the most important and beautiful fields of mathematics is number theory - the study of numbers and their properties. Despite the fact that mathematicians have been studying numbers for as long as humans have been able to count, the field of number theory is far from being outdated; some of the most exciting and important problems in mathematics today have to do with the study of numbers. In particular, prime numbers are of great interest.

Definition: A number p is prime if it is a positive integer greater than 1 and is divisible by no other positive integers other than 1 and itself.

Positive integers greater than 1 that aren't prime are called composite integers.

Examples: 2,3, and 5 are prime. 6 is composite.

All positive integers n have at least one prime divisor: if n is prime, then it is its own prime divisor. If n is composite, and one factors it completely, one will have reduced n to prime factors.

Examples: 6=3*2, 18=3*3*2, 48=6*8=2*3*2*2*2

The following theorem was proved eloquently by Euclid.

Theorem: There are infinitely many prime numbers.

Proof:

Suppose the opposite, that is, that there are a finite number of prime numbers. Call them p₁, p₂, p₃, p₄,....,p_n. Now consider the number

(p₁*p₂*p₃*...*p_n)+1

Every prime number, when divided into this number, leaves a remainder of one. So this number has no prime factors (remember, by assumption, it's not prime itself). This is a contradiction. Thus there must, in fact, be infinitely many primes.

So, that proves that we'll never find all of the prime numbers because there's an infinite number of them. But that hasn't stopped mathematicians from looking for them, and for asking all kinds of neat questions about prime numbers.

The Beginnings of Topology...

Topology is one of the newest branches of mathematics. A simple way to describe topology is as a 'rubber sheet geometry' - topologists study those properties of shapes that remain the same when the shapes are stretched or compressed. The 'Euler number' of a 'network' like the ones presented later in this discussion is an example of a property that does not change when the network is stretched or compressed. The foundations of topology are often not part of high school math curricula, and thus for many it sounds strange and intimidating. However, there are some readily graspable ideas at the base of topology that are interesting, fun, and highly applicable to all sorts of situations. One of these areas is the topology of networks, first developed by Leonhard Euler in 1735. His work in this field was inspired by the following problem: The Seven Bridges of Konigsberg In Konigsberg, Germany, a river ran through the city such that in its center was an island, and after passing the island, the river broke into two parts. Seven bridges were built so that the people of the city could get from one part to another. A crude map of the center of Konigsberg might look like this: The people wondered whether or not one could walk around the city in a way that would involve crossing each bridge exactly once. Problem 1 Try it. Sketch the above map of the city on a sheet of paper and try to 'plan your journey' with a pencil in such a way that you trace over each bridge once and only once and you complete the 'plan' with one continuous pencil stroke. Solution Problem 2 Suppose they had decided to build one fewer bridge in Konigsberg, so that the map looked like this: Now try to solve the problem. Solution Problem 3 Does it matter which bridge you take away? What if you add bridges? Come up with some maps on your own, and try to 'plan your journey' for each one.

Finding the Value of Pi

Historians estimate that by 2000 B.C. humans had noticed that the ratio of circumference to diameter was the same for all circles. This discovery hinged on the idea of proportion - in this case humans noticed that if you double the distance "across" a circle, then you double the distance "around" it. In today's algebraic notation this implied the formula

where Pi was constant. (It wasn't until 1706 that this notation, using the Greek letter seen in the above equation - often written Pi and pronounced like the English 'pie' - was introduced by William Jones). The significance of this discovery is clear: Circles are everywhere - in the sun, the moon, the pupils of our eyes, the most basic religious rituals and the earliest man-made structures. Achieving a greater mathematical understanding of Pi would lead to scientific and technological advances that would further the development of civilization, as well as creating some very interesting problems in pure mathematics. But one problem remained - what is the numerical value of Pi?