Why does mathematics work so well?
"As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." Albert Einstein
Mathematics has been around for a long time, and simple counting even longer. Archaeologists have discovered ancient bones dating from 30,000BC scored with notches that indicate an early form of counting. Among the ancient civilisations the Sumerians are known to have used a counting system as far back as 8000BC, using clay tokens as a method of bookkeeping. It is believed that the clay tokens were used for counting sheep, with a token dropped into a bag for every sheep that passed through a gate. The tokens were then sealed in a clay envelope and stored in a safe place. If at some later time a check was required to determine if any sheep had been stolen it was a simple matter to walk the sheep through the gate again and remove a token for every sheep that passes. If any tokens were left over in the bag, that was the number of sheep that was missing. We have a simple counting method in place here, but that was all, no arithmetic being involved at this early stage. It was only a matter of time however, before it was discovered that symbols could be scratched onto the outside of the envelope, one for every token inside, thus saving the trouble of breaking open the envelopes and sealing them up afterwards. After that it was not long before someone realised that the tokens were redundant, it was only the symbols that were needed.
Further progress was made around 3100BC when it was discovered that it was not necessary to represent 25 sheep with the symbol of a sheep repeated 25 times, it was easier and quicker to employ special signs expressing numbers in front of the symbol for a sheep. In present-day Iran a clay tablet has been found that has three wedges and three circles followed by the sign for a jar of oil. Each wedge stood for one and each circle for ten, thus these 7 symbols represented 33 jars of oil. Such was the success of this revolutionary advance that it remained in place in the Middle East for the next 3000 years. The Babylonians further improved the Sumerian system by introducing a form of place notation in which symbols took on values depending on their relative positions in the representation of a number. This place-value system was a major improvement because it enabled large numbers to be expressed in a very compact way and made arithmetical operations such as multiplication and division much easier.
Over the years the system was improved with heavy influences from the Mayans and Chinese, but it was the Indians that devised the system that laid the foundations for the system we use today. The Indus civilisation began around 2500BC in Pakistan and developed independently but in parallel with the Egyptian and Sumerian civilisations. The Indus people were invaded around 1500BC by Indo-Europeans, who adopted much of the Indus system. Their system of counting turned out to have four features that together made it superior to every other system: there were unique symbols for numbers 1 to 9; it was a fully base-10 system; it employed a consistent place-value notation and it used a zero.
John Barrow, in his book 'Pi in the Sky' describes the Indian system of counting as 'the most successful intellectual innovation ever made on our planet'. Quite a claim!
We now have a fully operational arithmetic system in place, we can use it for counting, multiplying and dividing. The chief advantage with the system as used so far is that it enables a record of large numbers of objects to be expressed using only a few symbols, such as using the number '265' instead of repeating a symbol 265 times.
With our successful counting system now in place it begins to take on a life of its own, it is no longer used merely as a system of counting and recording quantities, but develops an existence independent of the objects it was initially designed to represent. We now enter the world of mathematics.
The world of mathematics is a strange place, it contains such oddities as natural numbers, imaginary numbers, complex numbers, irrational numbers and transcendental numbers.
Natural numbers are whole numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 etc. We can use natural numbers to quantify any type of discrete entity, such as 12 sheep, 6 dogs and 1 cow.
Imaginary numbers were introduced in the 16th century to solve certain kinds of algebraic equations which required the square root of negative numbers, such as the square root of minus 1, and is represented by the symbol i. Ordinary numbers, such as 1, 2, 3, 4, 5 etc are unable to provide the answer - as there is no answer - so imaginary numbers are used to represent that value. The numbers are called imaginary in order to distinguish them from real numbers, but in fact have as much claim to 'existence' as real numbers.
Complex numbers are numbers that consist of both imaginary and real components. They take the general form of a + ib, where 'a' and 'b' are both real numbers.
Irrational numbers are those which cannot be expressed exactly by any fraction, such as 22/7 or the square root of 7. The answer would be a string of numbers without limit, they would just keep on going forever. Pi is an example of an irrational number, it too has no end, its first 50 digits being 3.14159265358979323846264338327950288419716939937510. It is not possible to calculate pi to an exact value, it is only possible to produce a figure that becomes ever more accurate as it increases in length.
Transcendental numbers are similar to irrational numbers but are a little stranger. As with irrational numbers they cannot be expressed exactly by any fraction, furthermore they cannot be expressed as the solution of an algebraic equation, such as x3 + 5x2 - 3x + 7 = 0. When in 1882 F. Lindemann succeeded in proving that pi was not only irrational but also transcendental, he proved that the circle could not be squared. Proving a number to be transcendental is difficult because you need to show that there is no algebraic equation (with a finite number of terms) that will provide the answer.
As mathematicians continue to explore the relationships between numbers, such as methods of finding prime numbers (numbers that are divisible only by themselves and 1) proving that numbers are transcendental, discovering series and patterns in number theory, it would appear that they are moving ever further away from reality and entering ever deeper into pure theory that has no application in the real world. Nothing could be further from the truth, and why this is so is a great mystery.
Mathematics has proved to be invaluable in helping us to understand quantum mechanics for example. Sub-atomic particles and atoms do not behave in the same predictable way as larger objects such as billiard balls. For example, it is possible for an object like an electron to be in two places at the same time. Quantum theory, seeWhat is Quantum Mechanics? describes electrons and other particles as behaving both like a wave and a particle. In order to understand what happens when an electron smashes into an atom, complex numbers provide the simplest way to do this.
Another strange example of the practical use of mathematics is the use of imaginary numbers to distinguish between the dimensions of space and time. According to Einstein's special theory of relativity time is regarded as a fourth dimension. When you measure distances in space-time, If you are given the xyz co-ordinates of two points in space, the square of the distance is given by the sum of the squares of the differences in these co-ordinates. This is the 3D version of the Pythagorean distance when calculating the hypotenuse of a right-angle triangle. However, If you want to measure the distance in space-time, you subtract the square of the distance instead of adding it. As a result if two events occur at the same spatial point but at different times then the distance between them is equal to the square-root of a negative number which means it must be imaginary! One interpretation of this strange finding is to regard time as an imaginary spatial dimension.
Stephen Hawking, author of the best-selling book 'A Brief History of Time', and his college James Hartle, developed a theory that could help to explain the origin of the universe by suggesting that time started off as a true spatial dimension which evolved into an imaginary dimension that gave us time. This enables them to avoid explaining what happened at the very starting point of the universe, because if the universe started out as a purely spatial object with no time there is no initial point to worry about. Although the theory is not universally accepted, such concepts illustrate the potential of mathematics and its number system in helping us to understand fundamental aspects of the world around us.
It seems odd that a system that was originally devised to help us keep track of how many sheep we had, can be used to help us understand how the universe was created, how sub-atomic particles interact, and be used to calculate the existence and properties of particles not yet discovered. Why? It has prompted John Barrow in his book 'Pi in the sky' to say. "There is an ocean of mathematical truth lying undiscovered around us: we explore it, discovering new parts of its limitless territory. This expanse of mathematical truth exists independently of mathematicians. It would exist even if there were no mathematicians at all... "
This view, known as the Platonic view, is of the opinion that mathematics has its own independent existence 'out there' and has nothing to do with mathematicians, it exists anyway, and mathematicians merely discover it, not invent it.
Others take the view that mathematics is purely man-made. For example, early in the twentieth century the Dutch mathematician Luitzen Brouwer led the school of 'intuitionists' who argued that mathematics was merely an abstraction from the physical world that is invented by human mind.
Is the difference important? Yes it is. If we assume that mathematics is not man-made but has its own independent existence, a given truth, then it will be universal in its application. It would mean that mathematics would be the same throughout the Universe, it would be the same for an alien race as it is for us. We need to ask if mathematics is a universal language. In 1974 researchers sent an elaborate message using the radio telescope at Arecibo in Puerto Rico towards a star cluster in our galaxy. This message would require knowledge of the prime numbers in order to be understood by any alien civilisation that may receive it. But what right do we have to assume aliens would use the same maths?
Why is it possible to describe reality using abstract mathematics? Paul Davis in 'The Mind of God' says:
"Scientists themselves normally take it for granted that we live in a rational, ordered cosmos subject to precise laws that can be uncovered by human reasoning. Yet why this is remains a tantalising mystery. Why should humans have the ability to discover and understand the principles on which the universe runs?"
You may wonder what it is about mathematics that so amazes mathematicians. Numbers are merely representations of objects using a particular form of man made notation. The fact that we can have either 1, 2, 3 or 4 objects is not of course man made, it is only the symbols that we use to represent the number of objects that are devised by us. We have selected the symbols that we will use to represent the number of objects and we have also settled on a system that has a base of 10, we could of course just as easily have chosen a system of base 9 or 11, or any other number, but in all probability chose 10 because we have 10 fingers. So that is all we have, ten symbols that we call numbers that we use to represent real quantities. The amazing thing is what mathematicians are discovering we can do with these numbers. It really is a world of constant discovery.
Sometimes a series of numbers can turn up in surprising places in nature. For example take the series 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 and so on. The series is simply the result of adding the previous two numbers and is called the Fibonacci sequence. It is observed that many flowers tend to have a Fibonacci number of petals, and the spiral growth of leaves on trees often exhibit the Fibonacci series. The reason why plants often develop this series is believed to arise from the way cells divide in the early stages of plant development. Fibonacci produced a model of population growth along the following lines. Suppose a pair of rabbits produced two young each breeding season but only after they have mated for one season. If you start off with a single pair of immature rabbits these mature after one season. Every season after they produce one immature pair. One season later that new pair will start reproducing and thus the original pair and its first two young will produce a pair each. The number of pairs of rabbits follows the Fibonacci sequence.
Isn't it strange that we can use mathematics to help us understand the basic principles that underpin the workings of the universe. Years ago it used to be that a scientist would discover a phenomenon while working on an experiment and would call upon the theorists to explain the event. The theorists would perform many mathematical computations and from that evolve a theory. Today, more often then not, it is the theorists who will make a 'discovery' while doing their sums and call upon the experimental scientist to carry out the experiment to prove it, such is the power of mathematics.
Paul Davis, in his book 'The Mind of God' says:
"(I) once asked Richard Feynman whether he thought of mathematics and, by extension, the laws of physics as having an independent existence. He replied: The problem of existence is a very interesting and difficult one. if you do mathematics, which is simply working out the consequences of assumptions, you'll discover for instance a curious thing if you add the cubes of integers. One cubed is one, two cubed is two times two times two, that's eight, and three cubed is three times three times three, that's twenty-seven. If you add the cubes of these, one plus eight plus twenty-seven- let's stop there - that would be thirty-six. And that's the square of of another number, six, and that number is the sum of those same integers. one plus two plus three...Now, that fact which I've just told you about might not have been known to you before. You might say "Where is it, what is it, where is it located, what kind of reality does it have?' And yet you came upon it. When you discover these things, you get the feeling that they were true before you found them. So you get the idea that somehow they existed somewhere, but there's nowhere for such things. It's just a feeling...Well, in the case of physics we have double trouble. We come upon these mathematical interrelationships but they apply to the universe, so the problem of where they are is doubly confusing...Those are philosophical questions that I don't know how to answer."
It is strange, or so it seems to me, that when investigating mathematics we end up philosophising over the nature of its existence!
Regarding the relationship between mathematics and philosophy, we have two opposing schools of mathematical philosophy, Platonism and Intuitionism. It seems at first glance to be a contradiction in terms putting mathematics and philosophy together! Platonism, as already mentioned, argues that mathematical truth exists independently of mathematicians and would exist even if there were no mathematicians at all. It seems a fairly common sense point of view.
Intuitionists, on the other hand, take a different view on the concept of existence, demanding that a definite (mental) construction be presented before it is accepted that a mathematical object actually exists. Thus, to an intuitionists, 'existence' means 'constructive existence'. Roger Penrose 'The Emperor's New Mind'.
Taking the Intuitionists view we can find a problem, for example, with the decimal expansion of pi, 3.1415926...... , we can ask if there exists a succession of twenty consecutive 7's somewhere in this infinitely long sequence of numbers. Again, quoting from 'The Emperor's New Mind'
"In ordinary mathematical terms, all that we can say, as of now, is that either there does or there does not - and we do not know which! This would seem to be a harmless enough statement. However, the intuitionists would actually deny that one can validly say 'either there exists a succession of twenty consecutive sevens somewhere in the decimal expansion of pi, or else there does not' - unless and until one has ...either established that there is indeed such a succession, or else established that there is none!"
As a matter of passing interest, it should be noted that such a sequence probably does exist. So we have two very different views on the concept of mathematics. We have the Platonistic view that mathematics has an independent existence, and the Intuitionistic view that that it is only the rules of mathematics that exists. Despite such philosophical differences regarding the nature of mathematics, it is an incredibly effective tool in helping us to understand the world around us. The only real problems within mathematic is in the practical application of what it reveals. It has to be remembered that mathematics is only symbolic of the real world, such as using symbols to record the number of sheep. When we deviate away from the real world into what may be described as 'pure' mathematics, it should come as no surprise when mathematics sometimes runs into problems when trying to describe reality. For example, take measurements.
If we are given a right-angle triangle of sides measuring 2 inches x 2 inches, what is the length of the Hypotenuse? Using Pythagorus' famous theorem we know that the square described on the hypotenuse of a right-angle triangle is equal to the sum of the squares on the other two sides. The sum of the squares 'on the other two sides' in this case will be 2x2=4 added to 2x2=4, giving a total of 8 (square inches). Therefore the area of the square described on the hypotenuse is 8 (square inches) so the length of the side of that square (which is what we are trying to discover) is the square root of 8. (Isn't it amazing how some of the stuff you learn at school just sticks in your head forever?) All we need do now is calculate the square root of 8 to find the length of the line. This is where the problem occurs. The square root of eight is 2.828427...which is an irrational number, just like pi. The mathematician will announce with some horror that the line cannot be given a precise length because it cannot be calculated! This view however, is purely academic, as the line very clearly does have a precise length, and the length of it simply depends on how accurate the measurement needs to be. If we measure the line with a ruler we get a tad over 2.8 inches. If, for some special reason, we needed a more accurate measurement we can then turn to the mathematical answer and take the measurement to any number of decimal places we like.
We need to be careful in separating mathematical problems from problems in the 'real' world. Another example in a similar vein concerns distance. It runs along the lines that we set out on a journey of known distance. We travel half way to our destination, then move forward half the remaining distance. We then again move forward half the remaining distance, and so on. As we only move forward half the remaining distance each time we will never reach our destination! Theoretically sound, but clearly nonsense in the real world, as we would soon reach a point where we would be unable to distinguish that there is any distance remaining. Again, this is simply a matter of the degree of accuracy that is either required or possible, and does not represent a problem between reality and theory. Theory, using mathematics, is capable of extraordinary precision, more precision then is capable of bring measured in the real world.
The introduction of computers has revolutionised the work of mathematicians, what previously could have taken years to calculate by hand can now be done by super-powerful computers in a matter of hours. It has also, to some extent, changed the way in which mathematicians think.
Computers, as everyone is aware, do not use a base ten number system, they use a binary system, just two digits. This is because a computer operates with only two states, it records only 'on' and 'off' represented by '1' and '0'. Any number is therefore represented by the computer as a string of 0's and 1's. The number 36, for example, is represented as 100100 in binary code. In this manner the computer is able to store information and perform calculations by using strings of 0's and 1's. When a computer displays an image on the screen, of say your dog, it is merely translating a grid of 0's and 1's into dots (pixels) of colour as determined by the programmers. We can connect digital still cameras and videos to computers and record images. We can also, by the employment of various sensors and detectors, record taste, smell, sound and touch. In other words, all the input received by our five human senses can be represented by nothing more than 0's and 1's. Isn't that strange? Because of this we can say that the information received by our senses is computational. The wonderful scent of a rose, for example, can be analysed and broken down into its constituent parts, and recorded as, say, 100111001010000111000011100.
Scientists have also discovered that they are able to express the forces of nature as a binary computer code. This enable computers to simulate all the forces that we have in nature, such as gravitation, magnetism, radiation etc. This has led scientists to believe that ultimately they will be able to simulate anything in the universe on a computer, it simply being a matter of putting in the right information in the first place. Once this has been achieved we will then be in the position of having a virtual universe running on a computer. That such a possibility exists has prompted some people to question if our universe is nothing more than a virtual universe running on some super-being's computer. This is however, nothing more than comparing the universe to the latest technology that we happen to have, and has no bearing on reality.
Pythagorus lived in the sixth century B.C. His school, of philosophers, the Pythagorans, were convinced that the cosmos was based on numerical relationships. Pythagorus discovered the mathematical relationship between the lengths of strings that produced harmonic tones, the octave, for example, corresponded to the ratio 2:1. This Pythagorean connection between musical notes and the harmony of the cosmos was expressed by the assertion that the astronomical spheres gave forth music as they turned - the music of the spheres.
Kepler, in the scientific era, described God as a geometer, and in his analysis of the solar system was influenced by the numbers involved, and gave them a mystical significance.
Newton's view of the universe was one of a Designer working through strict mathematical laws. For Newton the universe was a vast and magnificent machine constructed by God. He envisioned it as a kind of perfect clockwork mechanism that had been wound up by God and was now relentlessly ticking away following a precisely determined path.
Today we compare it to a computer programme, tomorrow it will be something else. It doesn't matter, it's just our way of trying to comprehend the incomprehensible.
What is so amazing about mathematics is that it appears to be possible to represent the entire universe mathematically.
Einstein's theory of relativity and the theory of quantum mechanics, the two great theories of modern day physics, the most powerful, accurate and descriptive theories of the universe, were both developed mathematically. Why is the universe so mathematical? Is it that mathematics - at some deep level - is an essential ingredient of the very fabric of the universe? Or is the universe nothing more than an expression of mathematics, the reality of mathematics, the ultimate truth? Some say that God must be a mathematician, but instead it may be the other way around, it may be that mathematics -being the ultimate and eternal expression of truth and perfection - is God.
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