Very short mathematics history. Also wrong, but approximately right.

Babylonians - the Early Mathematics

Babylonians used natural numbers and fractions, albeit usually in a base 60 numeral system. It is possible to count to 12 on one hand, using the thumb to point to one of the three finger bones in the remaining 4 fingers, and to 60 using two hands.

The advantage of base 60 and of 12 is that they have many factors, making common fractions (e.g. half, third, quarter), therefore division, easy. That’s important because division is an order of complexity compared with multiplication, which itself is an order more complex than addition and subtraction.

Left overs to this date are the fact we still use 60 seconds to the minute, 60 minutes to the hour, eggs are sold to the dozen, full rotation is 360 degrees, etc.

Pythagoras - the Greek Breakthrough

3 4 5

Triplets such as \((3, 4, 5)\), \((5, 12, 13)\), \((6, 8, 10)\), \((7, 24, 25)\) have been known since about 2000 BC by the Egyptians and Babylonians, first on the basis of ‘here are some values that have a certain property’, later as a rule, introduced to the Greeks by Pythagoras (around 500 BC) as:

In a right-angled triangle the square of the side opposite the right angle equals the sum of the squares of the other two sides

It is believed that one early proofs of the rule involved sliding around same sized right angled triangles inside a square:

a b c b 2 a 2 c 2

For a while it looked like there is a beautiful link between numbers and geometry.

By numbers they meant strictly positive integers and rational numbers (i.e. ratios). However it soon become evident that they are not enough to reflect sizes in geometry. The diagonal of a square, does not have a common unit with the side of the square. In modern language we would say that they discovered that \(\sqrt{2}\) is irrational, i.e. that it has an infinite number of decimals.

1 1 sqrt(2)

The links between geometry, numbers and infinity, is a theme that runs deep throughout the mathematics history.

Euclid - the Greek Apex

Mathematics flourished in the ancient Greek culture, and in Euclid’s Elements (13 books) it reached it’s peak. The Pythagorean theorem is captured in Euclid’s Elements in Book I Proposition 47.

A B C D E F G H K L

Proposition 47 uses Proposition 46 and not the other way around, because of the underlying belief that proofs must not be circular, and so on until axioms and definitions, reflecting the underlying belief that proofs will not go on forever, instead will reach statements taken for granted.

Some axioms in Book I, are geometry specific, but some, the Greeks believed, are of wider generality, such as:

If two things are equal to another, they are equal between themselves.

For example, the attention given to definitions is amazing.

They made the observation that to the learner, the natural way of perceiving is solid objects, surfaces are boundaries of solid objects, lines are intersections of surfaces, points are intersections of lines. However points, lines, surfaces are more primitive than solid objects, and the logical presentations should start with them.

Hence the first definition is that:

A point is that which has no part.

They questioned if it’s appropriate to give definitions in the negative. In general that would be a poor choice, but it’s appropriate here because the point is the only one, in geometry, meeting the criteria of having no part.

Another example is the definition of a square:

Definitions beside being a label, could wrongly imply that the thing defined exists. The square for example, defined at the beginning to Book I, is not used until it is proved that it can be constructed, in Proposition 46.

And if a proposition asks for a construction, the solution ends in QEF, “quod erat faciendum”, while if the proposition requires a proof, then then the solution ends in QED, “quod erat demonstrandum”.

Euclid marked an apex of logic that was not matched until around the 19th century. Until most of the contents of the Elements became part of the standard maths eduction in the 20th century, reading Euclid’s Elements was The Thing that every truly educated man was supposed to know.

The Greek legacy

From the ancient Greeks we got a lot of inconsistent, invalid, confused text. But we got Euclid. And other than Euclid there are still plenty of gems.

Zeno is famous for his paradoxes such as Achilles and the tortoise:

Achilles will never reach the tortoise as by the time Achiles reaches the place where the tortoise has been, the tortoise has already moved away.

On the surface, it seems nonsense for which the calculus seems to offer an explanation with regards to adding infinite sums of ever smaller quantities. But I think Zeno’s paradoxes are deeper than that. For example the paradox above suggests that maybe modelling the word using Euclidean geometry does not work at tiny scales. We really can’t just divide a line in smaller and smaller parts forever.

Diophantus (around 300 AD) left Arithmetica, a collection of problems.

One such problem states that a person had a was a boy for a sixth of his life, a twelve more till his beard started to grow, and so on, asking for the age of a person.

This kind of problems look like equations for which the solution must be an integer, the age of the person in years for the problem above, or rational numbers.

Each problem had a unique way of being solved. Solving 100 of such problems give little insight on how to solve the 101th. However this book will inspire the development of algebra.

The fifth postulate also known as the parallel axiom haunted mathematicians for the simple reason that it was more complex that any of Euclid’s axioms. For about a couple of thousands years there were attempts to prove it from the other axioms, and Euclidian geometry was regarded as the Truth (the only possible geometry). Eventually different axiomatic choices were shown to be possible in the 19th Century, resulting in non-Euclidian geometries where the name non-Euclidian however refers to the choice of axioms.

al-Khwarizmi - the Arabs

Unlike the Romans, which, when encountering the Euclid’s Elements did not preserve or care much about the proofs, the Arabs, when encountering the Greek culture, preserved and studied as much as they could get their hands on.

Through the Arabs, around the 12th century AD, the Europeans rediscovered the Greeks, got the Arabic numerals and the idea of zero, originating from India, and also additions such as algebra.

Peano - Formalization

As mathematics started to flourish again in Europe following the rediscovery of Greeks, it generally still suffered by poor notation and lack of logical rigour.

This changed in the 19th century, when mathematicians started to work on formal systems.

Peano’s axioms are an example of such formalization efforts, introducing notation that formalizes integer arithmetic axioms in a familiar form on the lines of:

\[\begin{align} &\forall x\in\mathbb{N}~(0 \neq S(x))\\ &\forall x, y\in\mathbb{N}~((S(x) = S(y) \Rightarrow x = y)\\ &\forall x\in\mathbb{N}~(x + 0 = x)\\ &\forall x, y\in\mathbb{N}~(x + S(y) = S(x + y))\\ &\forall x\in\mathbb{N}~(x \times 0 = 0)\\ &\forall x, y\in\mathbb{N}~(x \times S(y) = x \times y + x)\\ &\forall \bar{y}(\varphi(0, \bar{y}) \land \forall x(\varphi(x,\bar{y}) \Rightarrow \varphi(S(x),\bar{y})) \Rightarrow \forall x~\varphi(x,\bar{y}) \end{align}\]

The axioms above define the natural numbers in terms of rules with regards to a successor function \(S\). For example the first one states that zero is not a successor for any other natural number. The last one is the induction rule, \(\bar{y}\) stands for \(y_1,\dots,y_k\) :if some formula is true for 0, and if true for x is true for the successor of x, then it’s true for all natural numbers.

In general, a formal system would use symbols like \(\forall\), \((\), \(x\) etc. and build sentences. There is a grammar as to what a valid sentence is. e.g. \((\) is not valid because for example it does not have a closing parenthesis. Some sentences are given as axioms. And then there are rules on how to build further sentences.

Around the end of the 19th century, a number of such formal systems were developed.

An example would be using sets. The empty sets corresponds to zero, the set containing the empty set to one, and so on.

Cantor - Counting

Cantor had amazing insights to the simple question of: given two sets, how do we compare them for size.

If the elements can be counted then the counts can be compared. For sets of infinite number of elements he proposed using cardinality:

  • Two sets have the same cardinality if there is a function that maps 1-to-1 between elements of the two sets.
  • The cardinality of the set of natural numbers is \(\aleph_0\). Sets of this cardinality are set to be countable.

Based on these definitions, he derived (in increasing order of complexity):

  • The set of even numbers and the set of odd numbers have the same cardinality
  • The set of even numbers is countable. It has the same cardinality as the set of natural numbers, despite the former being a proper subset of the latter. This was known as the Galileo’s paradox.
  • The set of rational numbers is also countable.
  • The set of rational numbers between 0 and 1 is also countable.
  • The set of points in the square of side 1 with rational coordinates is also countable.
  • The set of polynomials with integer coefficients is also countable.
  • The set algebraic numbers (solutions that cause the polynomials with integer coefficients to be zero) is also countable. This set includes all rational numbers, but also numbers like \(\sqrt{2}\). However this does not include transcendental numbers like \(\pi\) and \(e\).
  • However real numbers are not countable, they have a higher cardinality, \(\aleph_1\).
  • But there are as many real numbers between 0 and 1 as there are real numbers.
  • And there are as many real in the square of side 1 as there are real numbers.

At that point in time, it was not clear if there are additional infinities between \(\aleph_0\) and \(\aleph_1\).

David Hilbert - We will know

However despite formalisation efforts, paradoxes coming all the way from the ancient Greeks continued to haunt mathematicians.

One such was “This sentence is false”. If true, then it states that it’s false. If false, then the negation is true, stating the contrary.

Another is the barber’s paradox: “On an island, a barber shaves all those, but only those, who do not shave themselves. Does the barber shave himself?”. If yes, then he shaves a man that shaves himself. If no, then there is a man not shaved by the barber.

So what does a barber have to do with maths? Well, a similar question arises in set theory about the set of all the sets that do not contain themselves. Does this set contain itself?

Inconsistencies are also important, based on the assumption that if a formal system can prove both a statement and it’s negation, then it’s inconsistent, and an inconsistent system can then prove everything, making it useless.

Enter David Hilbert a highly influential German mathematician that led a program with the aim of clarifying the foundations of mathematics that were found to suffer of paradoxes and inconsistencies.

Bertrand Russell - Principia Mathematica

One such attempt was Bertrand Russell’s Principia Mathematica published around 1910, that famously takes hundreds of pages to prove that \(1 + 1 = 2\). It builds on Peano’s formalism with great care on avoiding self referencing, the idea being that the issue with the paradoxes above is that they allow statements about the statement itself.

Kurt Gödel - the Limits

Gödel took on Bertrand Russell’s Principia Mathematica and started with two major insights.

First is that statements can be associated to numbers and on top of statements about numbers one can make statements about statements. Today we know that in computing we associate text, statements to a binary representation that can be interpreted as a number, but at the time Gödel used prime numbers.

The second insight is that truth and provability are different things, therefore one can ponder about the truths of statements like “This statement is not provable”.

This led to his two inconsistency theorems:

In a formal system that includes a certain amount of arithmetic, either the system is inconsistent (i.e. useless) or there are statements that cannot be proved (i.e. potential additional axioms)

and

In a formal system that includes a certain amount of arithmetic, either the system is inconsistent or it’s consistency cannot be proved within the system itself

On one side the theorems seem to have a depressing meaning, setting limits to what can be achieved ultimately by the formalisations based on elementary natural numbers arithmetic and “what we can know”.

With the first theorem one faced with a statement that cannot be proved (or disproved), the statement can be added as a new theorem creating a new formal system, but then either the new system becomes inconsistent, or another (unprovable) statement can be added creating yet another formal system and so on.

With the second theorem we can’t prove the consistency of a formal system based on natural numbers. We could prove it using a superset, but then the superset might be inconsistent.

But then there is a optimistic interpretation. Maybe for example we can prove the inconsistency for a formal system using another formal system that does not have the kind or arithmetic that implies the inconsistency theorem. Or maybe we need to go back to foundational questions like: “what are zero and one?” or “what does «for any» means?”.

Turing, Alonzo Church - the Computers Link

Alonzo Church’s lambdas and Alan Turing’s machine, that turned out to be influential in computer theory, are actually mathematical work exploring the limits of logic and maths on the trail set by David Hilbert and Kurt Gödel. In particular their objective was the clarity and simplicity of the proofs, not the runtime efficiency (and it’s highly likely they were aware of that).

Fun fact: The story goes that Alonzo Church’s usage of lambdas was accidental. Originally he wanted to use a caret i.e. ^ over one or more letters like x and y. But the printers had a technical issue, so the closest compromise option was to use the Greek letter λ as a prefix in front of x and y.

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