Since the earliest days of history there has been evidence of people counting. You can imagine a shepherd needing to check that none of his flock is missing. Shepherds might not have learnt to read and write but they had alternatives, for example the Yan Tan Tethera counting rhyme.
Another alternative was to use pebbles or beads to represent the things that you wanted to count. The Babylonians were doing this as early as 2700 - 2300 BC using their Sumerian abacus or counting frame. The abacus consisted of a table of successive columns. Each column indicated the next order of magnitude e.g. units, tens, one hundreds etc. The Babylonians used a sexagesimal number systems based on 60 and subdivision using the base 10. We still use part of this today in our division of time and angles (i.e 60 minutes = 1 hour). The Sumerian abacus may have been used for addition and subtraction. It could function as a number line.
The Inca's developed an alternative to the abacus called the quipu using a series of coloured, knotted strings for recording the movement of people and goods. The type of knot indicated a number, and the knot’s placement signified units of 1, 10, 100, or more. All the cords hung from a main string, and their positions and colors likely signaled what was being counted—gold, corn, or other goods.
In the abacus and the quipu the pebbles or knots are used to count and represent the number of another physical world object. One step further is to replace the pebbles or knots with symbols and you have written numbers to record your counting.
Physical World Actions
The physical world has actions that are equivalent to the key mathematical processes of addition, subtraction, multiplications and division:
Say you have 3 apples in a bowl on the dinning room table. You then go out to the apple orchard and pick 2 apples and place them in the bowl. You have added 2 apples and now have 5.
Now if you take one of the apples from the bowl and eat it. Eating the apple have subtracted 1 apple form the bowl.
Biological reproduction is a physical world form of multiplication. For example, a pair of rabbits reproducing creates many more rabbits.
Cutting a cake at a birthday party into equal portions is a type of physical world division process.
These physical world actions map to mathematical processes.
As people used numbers they started to notice that numbers had properties.For example some numbers could be made by multiplying two other numbers together 3 * 2 = 6. And yet othernumbers could only be made multiplying 1 and itself: no other number combinations worked.
These numbers were named prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43...
When counting started nobody expected prime numbers to be discovered and prime numbers are only one example of the unforeseen consequences of numbers.
New discoveries from basic definitions
As numbers were used and explored more properties were discovered. Verybasic mathematical definitions (axioms) were created which were the starting point for Euclidean geometry:
A straight line
From these there were many unforeseen consequences. For example pi.
In Euclidean geometry, π is defined as the ratio of a circle's circumference to its diameter
Pi is not just a mathematical curiosity it can be used to calculate the area of a circle.
This process is not limited to geometry and number theory and has not finished. In modern mathematics new discoveries are still being made.
Axioms Define An Abstract World
Euclid saw that newly discovered mathematical properties could be deduced from mathematical basic self-evident assumptions or axioms so he set about defining them in a set of books called theElements.
Euclid's Book 1 begins with 23 definitions — such as point, line, and surface — followed by five postulates and five "common notions" (both of which are today called axioms).
A straight line segment can be drawn by joining any two points.
A straight line segment can be extended indefinitely in a straight line.
Given a straight line segment, a circle can be drawn using the segment as radius and one endpoint as center.
All right angles are equal to one another (congruent)
If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less that two right angles, then the two lines must inevitably intersect each other on that side if extended far enough
One of the oldest and most complete diagrams from Euclid'sElements of Geometryis a fragment of papyrus found among the remarkable rubbish piles of Oxyrhynchus in 1896-97 by the renowned expedition of B. P. Grenfell and A. S. Hunt.
The diagram accompanies Proposition 5 of Book II of theElements
Plane figures: Euclid measured angles and distances and looked for relationships.e.g. If a triangle has two equal angles, then the sides subtended by the angles are equal.
Number theory: Euclide developed ideas about prime numbers and asserted that there are infinitely many prime numbers. Rational numbers: new numbers created from the numbers in a counting line (3 over 4, 3/4)
Irrational numbers: discovered - numbers that cannot be created from the counting line (0.5 = 1/2, 0.333... = 1/3, pi = ?)
These discoveries were not just interesting mathematical properties. They were useful and had practical applications in the physical world.
Solid geometry: Angles and distances give surveying, spheres in cubes gives packing and parabolas give focusing mirrors
Many of the mathematical discoveries and ideas that Euclid wrote about had been made by other people and countries however, one of his greatest achievements was to construct a framework to prove or disprove new mathematical ideas. In both early Greek and Chinese societies proofs had been used. Euclid saw that this could be greatly expanded and that the process of proving or disproving an idea was central to mathematical progress.
The ability to prove something was very attractive. Not only did the early mathematicians gain absolute confidence in a proven mathematical idea but many of these ideas also had practical applications in the physical world. This marked out mathematics when so much else appeared chaotic and unprovable.
Example of a Proofs
The following is a example of a proof of the statement:
Truth (Abstract World)
People were attracted to mathematics because they believed it gave the absolute truth. For some it was so powerful they believed it was the language of God. They thought the principles of mathematics were the principle of everything. Lamblichus of Chalcis said in a collection of Pythagorean doctrines "Number is the ruler of forms and ideas, and the cause of gods and demons"
Much of history has been characterised by uncertainty and so it was stunning to find something that appeared to provide absolute certainty and was very useful in the physical world. The importance of mathematical proofs in the Toolk Kit For Thinking is that they show us a second type of truth. An abstract world truth. We can test an abstract world statement for logical consistency. Two mathematicians arguing over a mathematical idea can appeal to a mathematical proof to decide truth.
More Unforeseen Consequences
Pythagoreans preached that all numbers could be expressed as the ratio of integers.
There are many examples rational numbers
0.5 = 1/2
1.4 = 7/5
0.333333... = 1/3
They either terminate or have a repeating decimaland Rational numbers can be ordered precisely on a number line.
This was fine except that more unforeseen consequences developed as the square root of 2 was explored. People knew that a number could be multiplied by itself to give the square and that the opposite process was the square root.
2 * 2 = 4
3 * 3 = 9
4* 4 = 16
Square root of 16 = 4
Square root of 9 = 3
square root 4 = 2
But what is the square root of 2?
This question was important to the Pythagoreans because the Phythagorean theorem (The square of the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides)stated that for a right triangle with both legs equal to one unit has a hypothenuse length equal to square root of 2.
When they calculated the square root of 2 it was found to approximate to 1.4142...
One way of calculating a square root is to use the following process:
To findr, the square root of a real numberx:
Start with an arbitrary positive start valuer(the closer to the square root ofx, the better).
Replacerby the average betweenrandx/r, that is:(It is sufficient to take an approximate value of the average in order to ensure convergence.)
Repeat step 2 untilrandx/rare as close as desired.
The problem with calculating the square root of 2 was that it did not give a rational number. In fact it gave another type of number an irrational number. This did not match Pythagorean beliefs. According to legend Hippasus was the first man to prove that the square root of two was in fact an irrational number and he was drowned at sea as a consequence.
There were now two more types of numbers:
0.5 = 1/2
Square root of 2 =1.4142 ...
1.4 = 7/5
Pi =3.1415 ...
0.3333... = 1/3
Golden ratio = 1.6180 ...
An irrational number cannot be expressed as a fraction.
Irrational numbers cannot be represented as terminating or repeating decimals.
Irrational numbers are non-terminating, non-repeating decimals.
Or put another way they cannot be exactly determined.
The Phythagoreans claimed that there must be a indivisible unit that could fit evenly into one of the lengths [of an isosceles right triangle] as well as the other. Hippasus showed that there was in no common unit of measure because if you assert there was you ended up with a contradiction. His proof was as follows:
The ratio of the hypotenuse to an arm of an isosceles right triangle isa:bexpressed in the smallest units possible.
By the Pythagorean theorem:a2= 2b2.
Sincea2is even,amust be even as the square of an odd number is odd.
Sincea:bis in its lowest terms,bmust be odd.
Sinceais even, leta= 2y.
Thena2= 4y2= 2b2
b2= 2y2sob2must be even, thereforebis even.
However we assertedbmust be odd.Therefore there is a contradition with 'a:b is in the lowest terms, b must be odd'. Hense square root of a cannot be expressed as a rational number.
Although irrational numbers were a challenge to the beliefs of the Pythagorean school they did not challenge the idea that mathematics could prove things and this has been a core part of the progress of mathematics.
Over hundreds of years many new ideas and theorems were developed and proved. New branches of mathematics were formed and new axioms defined. Euclid's axioms for plane geometry were very successful for nearly 2,000 years in answering an expanding range of geometrical problems. One question that was at the foundation of Euclid's geometry worried mathematicians: was Euclid's 5 postulate (axiom) derivable from the first 4 and therefore not an axiom? Mathematicians worked on this problem. Euclid's fifth postulate was reworded by John Playfair as "for any given line and point not on the the line, there is one parallel line through the point not intersecting the line". A Russian mathematician, Lobachevsky, also worked on this problem but rather than trying to show the 5 postulate was derivable from the first 4 he proposed a new 5th postulate stating that there is more than one line that can be extended through any given point parallel to another line of which that point is not part. This change created a new geometry (hyperbolic geometry) and in this geometry Euclid's 5th postulate was not true. There are 3 variations of the 5th postulate: no parallels through a point outside a line exist, exacctly one exists, or infinitely many exist. From these three you get elliptic, Euclidean and hyperbolic geometries.
The idea that you could have multiple geometries amongst others challenged the core belief that mathematical proofs gave absolute truth. How could you have more that one absolute truth?
Removing the Inconsistencies
Mathematicians reasoned that perhaps it was still possible to achieve one absolute truth and that the multiple geometries etc were caused by inconsistent axioms. In 1920 David Hilbert believed that the inconsistencies could be removed. His goal was to formalize all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent. Hilbert proposed that the consistency of more complex systems could be proved in terms of simpler systems. So that all of mathematics could be reduced to basic arithmetic.The finite complete set of axioms would unite all mathematics and provide one mathematical truth.
Many mathematicians concluded that Hilbert's project failed when Gödeldeveloped and proved his incompleteness theorems. His second theorem stated that basic arithmetic cannot be used to prove its own consistency. Therefore it cannot be used to prove the consistency of a more complex system.Gödel's incompleteness theoremdoes not exclude other axioms being used to prove consistency. Further work showed that the consistency of the Peano arithmetic can be proved in Zermelo-Frankel set theory (ZFC) etc.However, this still ment that no consistent finite set of axioms could be used to prove all mathematics.
Different starting definitions or axiom systems result in different deductions.
Mathematics does not reveal absolute truth.Just because the mathematical theorem is logically consistent does not mean it applies in the physical world.
Mathematical truth relates to the starting axioms. Different axioms give different truths.
Mathematics axioms exist in the abstract world. They can be tested for logical consistency "abstract world truth"
A mathematical proof does not prove something is true in the physical world
Axioms don't map to physical world objects or properties and cannot be tested in the physical world. Therefore they don't have physical world truth.
It is only when mathematics is used to describe the physical world and make predictions that the mathematical statement can be tested for physical world truth and usefulness
Maths and the Physical World
If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. Albert Einstein has stated that "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."