The axiomatic approach to mathematics

Modern pure mathematics is almost entirely axiomatic in approach. This is a fairly recent development: for most of its history mathematics has been ostensibly based on the natural or everyday world. Thus, whatever Euclid himself believed his system of geometry to be, it has usually been taken to be a description of physical space, while the nature of real numbers was held to be self-evident by most people. Such foundations gradually proved unsatisfactory. The assumptions behind Euclid are difficult to state clearly and since the formulation of the theory of relativity it has been found that the natural world does not obey them anyway. Analysis became rigorous only about 200 years ago, and the nature of irrational numbers was not described until 1872, while the nature of number itself is still being discussed.

The intuitive ideas behind mathematics are thus not secure, and the study of the foundations belongs more properly to philosophy: mathematics itself is concerned with the deductions obtained from the basic ideas. It is therefore more satisfactory to lay down certain initial axioms or postulates and to deduce from them according to the' accepted laws of logic (which are themselves the subject of study by philosophers). Mathematics is not concerned with the question whether the axioms are 'true' or not; all it can say is that given certain assumptions. then other results and consequences follow logically. Theoretically any set of axioms may be chosen provided it is self-consistent, but obviously the work will be unfruitful unless the choice is a careful one, and natural phenomena and the traditional fields of study can lead us to suitable sets of axioms. Thus Euclid may be put on a proper footing by assuming certain axioms, while the choice of similar sets will lead to the various non-Euclidean geometries. The fewer the axioms the greater, generality the resulting system possesses, but the fewer the results that may be deduced.

Thus modern mathematics lays down postulates and deduces from them. This can be a surprisingly fruitful process, both in practical terms (since many practical systems will obey the axioms chosen) and in aesthetic ones. It enables us to study systems that seem at first sight impossible but which often turn out to be extremely useful. For example, the study of space of four dimensions would seem useless at first sight, but it is vital to relativity theory and also in electromagnetism.

Abstract algebra lays down postulates for combining elements of sets and studies their consequences. For numbers these are the fundamental laws of addition and multiplication (the Commutative, Associative and Distributive Laws), while the choice of some only of these leads to the study of more general structures. A surprising amount of work may be done with very few axioms in this subject (group theory has only three basic laws but research is still very active in the subject).

Historical summary

The first notable algebraists were the Arabs. The Egyptians, Greeks and Hindus had all done a little work in this subject, but the Arabs were the first to concentrate on it, mainly in connection with astronomy, and progressed so far as the solution of cubic equations. The word ' algebra' is a corruption of the Xrabic 'al-jebr', meaning the transposing of negative terms in an equation to the other side.

At the time of the Renaissance, algebra became one of the main fields of mathematical study. Cubics were solved for the general case by Tartaglia (about 1499-1557) and Cardan (1501-76) and quartics by Ferrari (1522-65). Vieta (1540-1603) introduced letters to stand for unknown quantities, while the symbols + and — appear first in a book printed in 1489, and the exponential notation for powers was introduced by Descartes (1596-1650).

Newton (1642-1727) worked on the theory of equations and the binomial theorem, and about this time negatives came to be accepted as proper numbers. Complex numbers were also used but were imperfectly understood until later. Argand's famous paper on the geometrical interpretation of imaginary quantities was published in 1806, while Gauss finally put complex numbers on an equal footing with the real numbers in 1831. Gauss gave the first fully satisfactory proof of the 'Fundamental Theorem of Algebra' that a polynomial equation of the nth degree has exactly n roots, his first proof being discovered in 1797.

Determinants were studied by Wronski (1778-1853), Cauchy (1789-1857) and Jacobi (1804-51) among others, while matrices were introduced at about the same period, much of the work being by Hamilton (1805-65) and Cayley (1821-95). Hamilton also invented quaternions, the first non-commutative system to be studied intensively, which were superseded for practical purposes by matrices and tensors. The theory of invariants and linear transformations, connected with matrix theory and leading to modern linear algebra, was developed by Cayley and Sylvester (1814-97) and by Hermite (1822-1901).

A milestone in the development of modern abstract ideas was Boole's publication of The Laws of Thought in 1854. This applied mathematics to logic and marked the first real break from traditional ideas, based on practical ideas of number and space. Boole's work showed that algebra is not necessarily concerned with numbers but that the processes may be used much more generally.

The beginnings of the ideas of group theory lay in the solubility of equations. Quartics had been solved by Ferrari in the sixteenth century but nobody had been able to give a solution for the general quintic, and this was finally proved impossible by Abel in 1826. Galois simplified his solution in about 1830 and discovered a great deal about groups in connection with the solution of equations, besides investigating invariant subgroups and the theory of fields. He was the first to use the word 'group' in the modern sense. The theory was elaborated by Lagrange, Cayley and particularly Cauchy (in about 1844-46). At this early period groups were thought of in terms of permutations or substitutions, or sometimes in connection with residues (Euler) and number theory. Definitions of abstract groups were given by Kronecker in 1870 and later simplified. Other notable early workers in group theory were Jordan (composition series and conditions for groups to be soluble), Sylow (1832-1918) (subgroups), Sophus Lie (1842-99) (topological groups) and Klein (groups of the regular polyhedra).

Topology, originally known as 'analysis situs', was studied by Euler and others, but was only gradually recognised as a separate subject, distinct from geometry.

In the present century the growth of abstract ideas has been rapid. Many algebraic systems have been studied and research is still active both in the 'pure' algebra of groups, rings and fields and mоге recently, in algebra applied to topological structures.

3.1.14. The Abstract Group Concept

This article is based on a lecture given by Peter Neumann (a son of Bernhard Neumann and Hanna Neumann) at a conference at the University of Sussex on 19 March 2001 to celebrate the 90th birthday of Walter Ledermann. The talk was entitled Introduction to the theory of finite groups the title of the famous text written by Walter Ledermann. This article is based on notes taken by EFR at that lecture.

The modern definition of a group is usually given in the following way.

Definition

A group G is a set with a binary operation G×G→G which assigns to every ordered pair of elements x, y of G a unique third element of G (usually called the product of x and y) denoted by xy such that the following four properties are satisfied:

1. Closure: if x, y are in G then xy is in G.

2. Associative law: if x, y, z are in G then x(yz) = (xy)z.

3. Identity element: there is an element e in G with ex = xe = x for all x in G.

4. Inverses: for every x in G there is an element u in G with xu = ux = e.

The first point to make is that 1. is debatable as an axiom since it is a consequence of the definition of a binary operation. However it is not our purpose to debate this here and it is convention that this axiom is included.

Where did this, now standard, definition come from? Particularly we wish to examine some moves towards this definition made in the 19th century. We are not, therefore, concerned here with the bulk of the work done in group theory in the 19th century which concerned the study of permutation groups required for Galois theory. It is important to realise that the abstract definition of a group was merely an esoteric sideline of group theory through the 19th century.

Let us first note that there were two meanings of the term "abstract group" during the first half of the 20th century - say from 1905 to 1955. One meaning was that of a group defined by the four axioms as above, while the second meaning was that of a group defined by generators and relations. For example Todd used the term in this second way when he talked about "the Mathieu groups as abstract groups". Here we are only interested in the first of these meanings.

The emergence of the abstract group concept was a remarkably slow process. The story begins with the prehistory which involves Galois and Cauchy. Galois defined a group in 1832 although it did not appear in print until Liouville published Galois' papers in 1846. The first version of Galois' important paper on the algebraic solution of equations was submitted to the Paris Academie des Sciences in 1829. Rene Taton has found evidence in the archives of the Academie which suggest that Cauchy spoke with Galois and persuaded him to withdraw the paper and submit a new version of it for the Grand Prix of 1830. This is based on strong circumstantial evidence, but we do know that Galois submitted to Fourier a new version of his paper to be considered for the Grand Prix in March 1830.

Fourier died shortly after this and Galois' paper was lost. It was never considered for the prize which was awarded jointly to Abel (posthumously) and Jacobi in July 1830. Galois was invited by Poisson to submit a third version of his memoir on equations to the Academie and he did so on 17 January 1831. This version of the paper was refereed by Poisson who rejected it but wrote a very sympathetic report. Although Galois had proved the results in general, the paper only considered equations of prime degree. Poisson failed to understand the paper and suggested that the arguments were developed further. It was unclear to him how Galois' results classified which equations were soluble by radicals.

The night before he fought the duel which led to his death Galois made notes on his papers. One of these notes, made on 29 May 1832, was the following:-

... if in such a group one has the substitutions S and T then one has the substitution ST.

Although Galois had used groups extensively throughout his paper on equations, he had not given a definition. It is little wonder that Poisson found the paper hard to understand for it contains many explicit calculations in a group, yet the concept was not defined - poor Poisson!

Now in 1845, one year before Liouville published the above definition by Galois, Cauchy gave a definition. He considered substitutions in n letters x, y, z, ... and defined derived substitutions to be all those which can be deduced by multiplying these substitutions together in any order. He then called the set of substitutions together with the derived substitutions, a "conjugate system of substitutions". For some time these two identical concepts, a "group" and a "conjugate system of substitutions", were both used. However, from 1863 when Jordan wrote a commentary on Galois' work in which he used "group", it became the standard term. This was reinforced when Jordan published his major group theory text Traité des substitutions et des équations algébraique in 1870. However, Cauchy's term "conjugate system of substitutions" continued to be used by some up to about 1880.

How much was Cauchy influenced by Galois? Although he had seen Galois' papers submitted to the Academie, there was no explicit definition of a group in them. On the other hand he must have at least been subliminally influenced. Both Galois and Cauchy, of course, define groups in terms of the closure property alone. The now familiar axioms of associativity, identity and inverses do not appear. Both were dealing with permutations which means that closure is all that is necessary to define a group, the other properties all following automatically. Cauchy went on to write 25 papers on this topic between September 1845 and January 1846.

[Note by EFR. Peter Neumann did not mention in his lecture any influence that Ruffini's ideas might have had on Cauchy. In 1821 Cauchy had written to Ruffini praising his work which he had clearly read. Although there is no explicit definition of a group in Ruffini's work, again the concept clearly appears and may have had as major an influence on Cauchy's thinking as the work of Galois.]

The first person to try to give an abstract definition of a group was Cayley. He wrote a paper on groups in 1854 which he published again in two separate journals in 1878. In the 1854 paper he attempted to give an abstract definition in terms of symbols which operate on a system (x, y, ... ) so that

(x, y, ... ) = (x', y', ... ) where x', y', ... are any functions of x, y, ... .

Cayley went on to define an identity symbol 1 which leaves the members of the system unaltered. He defines the element θφ as the result of operating on the system first by θ then by φ. He notes that θφ need not equal φθ. Cayley also requires the associative law θ.φψ=θφ.ψ to be satisfied. He then says that any set of such symbols with the property that the product of any two is in the set is called a group.

This is an important attempt at an abstract definition of a group but Cayley has overreached himself. What he has here is really a mess. Why does he require the associative law to hold if his symbols are operators? As for permutations the associative law follows automatically for operators. It is also not clear that (x', y', ... ) where x', y', ... are any functions of x, y, ... will be in the system. This definition is not entirely successful.

In 1878 Cayley wrote:-

A group is defined by the law of composition of its members.

This idea was picked up by Burnside, von Dyck and others. Burnside, in his book The Theory of Groups of Finite Order published in 1897 gave the following definition:-

Let A, B, C, ... represent a set of operations which can be performed on the same object or set of objects.

He then supposes that any two of the operations are distinct in the sense that no two produce the same effect. He follows Cayley in requiring closure, the associative law, and inverses. Again his definition is subject to the same criticism at Cayley's definition. If his elements are operations then why does he need to postulate the associative law? Rather strangely Burnside does not assume the existence of an identity, although one can infer it from the fact that A and A -1 are in the set and we have closure. Burnside repeats exactly the same definition in the second edition of his book which appeared in 1911.

It is worth noting that neither Cayley nor Burnside insisted that their groups were finite. The definition deliberately allowed for the possibility of infinite groups, and Burnside in particular was interested in studying infinite groups. What we have given here is part of a sequence of development which we might call the English school. There were other bits in the sequence between these major contributions which we have omitted. We now turn to another development of group theory which was going on at the same time which we might call the European school.

In 1870 Kronecker gave a definition of a group in a completely different context, namely the context of a class group in algebraic number theory. He takes a specifically finite set, θ', θ'', θ''', ... such that from any two, a third can be derived by a specific method. He then assumes that the commutative and associative laws hold and also that θ'θ''≠θ'θ''' if θ''≠θ'''. This appears to be a separate development by Kronecker who does not tie it in with previous work on groups. However, Heinrich Weber in 1882 gave a very similar definition to that of Kronecker yet he did tie it in with previous work on groups.

Heinrich Weber defined a group of degree h, like Kronecker in the context of class groups, again to be a finite set. He required that from two elements of the system one can derive a third element of the system so that the following hold:- (θ_rθ_s)θ_t = θ_r(θ_sθ_t) = θ_rθ_sθ_t

θθ_r = θθ_s implies θ_r = θ_s

A few comments here. The associative law is put in a slightly strange way. The expression θ_rθ_sθ_t has no meaning until the associative law is defined to hold, so in a modern treatment one might write:

(θ_rθ_s)θ_t = θ_r(θ_sθ_t) so that either side may be denoted by θ_rθ_sθ_t .

One can still see that there is a little lack of clarity in the definition. What Heinrich Weber is in fact defining is a semigroup with cancellation and, given that it is finite, this is sufficient to ensure the existence of an identity and of inverses. That this fails for infinite systems was noted by Heinrich Weber in his famous text Lehrbuch der Algebra published in 1895. In this book he notes that the above definition of a group only works for finite groups and in the infinite case one needs to postulate inverses explicitly.

← Here is a diagram to illustrate the interactions between the various mathematicians working their way slowly towards the abstract definition of a group.

We should point out that there appears to be relatively little cross-fertilisation between the two lines of development. There was some, however. For instance Burnside did read Frobenius's papers, although he did so later than he should have done and had to apologise twice to Frobenius for not knowing what he had already published. Recently a letter from Burnside to Schur has been discovered and we know that Burnside corresponded with Hölder.

We do know where some of the 20th century influences came from - Emmy Noether, an important 20th century figure particularly influential through van der Waerden's Algebra book, was strongly influenced by Heinrich Weber. On the other hand Schur, another influential 20th century figure, was influenced by Frobenius.

Article by: J J O'Connor and E F Robertson

3.1.15. A History of the Calculus

The main ideas which underpin the calculus developed over a very long period of time indeed. The first steps were taken by Greek mathematicians.

To the Greeks numbers were ratios of integers so the number line had "holes" in it. They got round this difficulty by using lengths, areas and volumes in addition to numbers for, to the Greeks, not all lengths were numbers.

Zeno of Elea, about 450 BC, gave a number of problems which were based on the infinite. For example he argued that motion is impossible:-

If a body moves from A to B then before it reaches B it passes through the mid-point, say B₁ of AB. Now to move to B₁ it must first reach the mid-point B₂ of AB₁ . Continue this argument to see that A must move through an infinite number of distances and so cannot move.

Leucippus, Democritus and Antiphon all made contributions to the Greek method of exhaustion which was put on a scientific basis by Eudoxus about 370 BC. The method of exhaustion is so called because one thinks of the areas measured expanding so that they account for more and more of the required area.

However Archimedes, around 225 BC, made one of the most significant of the Greek contributions. His first important advance was to show that the area of a segment of a parabola is 4/3 the area of a triangle with the same base and vertex and 2/3 of the area of the circumscribed parallelogram. Archimedes constructed an infinite sequence of triangles starting with one of area A and continually adding further triangles between the existing ones and the parabola to get areas

A, A + A/4 , A + A/4 + A/16 , A + A/4 + A/16 + A/64 , ...

The area of the segment of the parabola is therefore

A(1 + 1/4 + 1/42 + 1/43 + ....) = (4/3)A.

This is the first known example of the summation of an infinite series.

Archimedes used the method of exhaustion to find an approximation to the area of a circle. This, of course, is an early example of integration which led to approximate values of π.

← Here is Archimedes' diagram

Among other 'integrations' by Archimedes were the volume and surface area of a sphere, the volume and area of a cone, the surface area of an ellipse, the volume of any segment of a paraboloid of revolution and a segment of an hyperboloid of revolution.

No further progress was made until the 16th Century when mechanics began to drive mathematicians to examine problems such as centres of gravity. Luca Valerio (1552-1618) published De quadratura parabolae in Rome (1606) which continued the Greek methods of attacking these type of area problems. Kepler, in his work on planetary motion, had to find the area of sectors of an ellipse. His method consisted of thinking of areas as sums of lines, another crude form of integration, but Kepler had little time for Greek rigour and was rather lucky to obtain the correct answer after making two cancelling errors in this work.

Three mathematicians, born within three years of each other, were the next to make major contributions. They were Fermat, Roberval and Cavalieri. Cavalieri was led to his 'method of indivisibles' by Kepler's attempts at integration. He was not rigorous in his approach and it is hard to see clearly how he thought about his method. It appears that Cavalieri thought of an area as being made up of components which were lines and then summed his infinite number of 'indivisibles'. He showed, using these methods, that the integral of xⁿ from 0 to a was aⁿ⁺¹/(n + 1) by showing the result for a number of values of n and inferring the general result.

Roberval considered problems of the same type but was much more rigorous than Cavalieri. Roberval looked at the area between a curve and a line as being made up of an infinite number of infinitely narrow rectangular strips. He applied this to the integral of x^m from 0 to 1 which he showed had approximate value

(0^m + 1^m + 2^m + ... + (n-1)^m)/n^m+1.

Roberval then asserted that this tended to 1/(m + 1) as n tends to infinity, so calculating the area.

Fermat was also more rigorous in his approach but gave no proofs. He generalised the parabola and hyperbola:-

Parabola: y/a = (x/b)² to (y/a)ⁿ = (x/b)^m

Hyperbola: y/a = b/x to (y/a)ⁿ = (b/x)^m.

In the course of examining y/a = (x/b)^p, Fermat computed the sum of r^p from r = 1 to r = n.

Fermat also investigated maxima and minima by considering when the tangent to the curve was parallel to the x-axis. He wrote to Descartes giving the method essentially as used today, namely finding maxima and minima by calculating when the derivative of the function was 0. In fact, because of this work, Lagrange stated clearly that he considers Fermat to be the inventor of the calculus.

Descartes produced an important method of determining normals in La Geometrie in 1637 based on double intersection. De Beaune extended his methods and applied it to tangents where double intersection translates into double roots. Hudde discovered a simpler method, known as Hudde's Rule, which basically involves the derivative. Descartes' method and Hudde's Rule were important in influencing Newton.

Huygens was critical of Cavalieri's proofs saying that what one needs is a proof which at least convinces one that a rigorous proof could be constructed. Huygens was a major influence on Leibniz and so played a considerable part in producing a more satisfactory approach to the calculus.

The next major step was provided by Torricelli and Barrow. Barrow gave a method of tangents to a curve where the tangent is given as the limit of a chord as the points approach each other known as Barrow's differential triangle.

← Here is Barrow's differential triangle .

Both Torricelli and Barrow considered the problem of motion with variable speed. The derivative of the distance is velocity and the inverse operation takes one from the velocity to the distance. Hence an awareness of the inverse of differentiation began to evolve naturally and the idea that integral and derivative were inverses to each other were familiar to Barrow. In fact, although Barrow never explicitly stated the fundamental theorem of the calculus, he was working towards the result and Newton was to continue with this direction and state the Fundamental Theorem of the Calculus explicitly.

Torricelli's work was continued in Italy by Mengoli and Angeli.

Newton wrote a tract on fluxions in October 1666. This was a work which was not published at the time but seen by many mathematicians and had a major influence on the direction the calculus was to take. Newton thought of a particle tracing out a curve with two moving lines which were the coordinates. The horizontal velocity x' and the vertical velocity y' were the fluxions of x and y associated with the flux of time. The fluents or flowing quantities were x and y themselves. With this fluxion notation y'/x' was the tangent to f(x, y) = 0.

In his 1666 tract Newton discusses the converse problem, given the relationship between x and y'/x' find y. Hence the slope of the tangent was given for each x and when y'/x' = f(x) then Newton solves the problem by antidifferentiation. He also calculated areas by antidifferentiation and this work contains the first clear statement of the Fundamental Theorem of the Calculus.

Newton had problems publishing his mathematical work. Barrow was in some way to blame for this since the publisher of Barrow's work had gone bankrupt and publishers were, after this, wary of publishing mathematical works! Newton's work on Analysis with infinite series was written in 1669 and circulated in manuscript. It was not published until 1711. Similarly his Method of fluxions and infinite series was written in 1671 and published in English translation in 1736. The Latin original was not published until much later.

In these two works Newton calculated the series expansion for sin x and cos x and the expansion for what was actually the exponential function, although this function was not established until Euler introduced the present notation e^x.

You can see the series expansions for sine and for cosine. They are now called Taylor or Maclaurin series.

Newton's next mathematical work was Tractatus de Quadratura Curvarum which he wrote in 1693 but it was not published until 1704 when he published it as an Appendix to his Optiks. This work contains another approach which involves taking limits. Newton says

In the time in which x by flowing becomes x+o, the quantity xⁿ becomes (x+o)ⁿ i.e. by the method of infinite series,

xⁿ + nox^n-1 + (nn-n)/2 oox^n-2 + . . .

At the end he lets the increment o vanish by 'taking limits'.

Leibniz learnt much on a European tour which led him to meet Huygens in Paris in 1672. He also met Hooke and Boyle in London in 1673 where he bought several mathematics books, including Barrow's works. Leibniz was to have a lengthy correspondence with Barrow. On returning to Paris Leibniz did some very fine work on the calculus, thinking of the foundations very differently from Newton.

Newton considered variables changing with time. Leibniz thought of variables x, y as ranging over sequences of infinitely close values. He introduced dx and dy as differences between successive values of these sequences. Leibniz knew that dy/dx gives the tangent but he did not use it as a defining property.

For Newton integration consisted of finding fluents for a given fluxion so the fact that integration and differentiation were inverses was implied. Leibniz used integration as a sum, in a rather similar way to Cavalieri. He was also happy to use 'infinitesimals' dx and dy where Newton used x' and y' which were finite velocities. Of course neither Leibniz nor Newton thought in terms of functions, however, but both always thought in terms of graphs. For Newton the calculus was geometrical while Leibniz took it towards analysis.

Leibniz was very conscious that finding a good notation was of fundamental importance and thought a lot about it. Newton, on the other hand, wrote more for himself and, as a consequence, tended to use whatever notation he thought of on the day. Leibniz's notation of d and ighlighted the operator aspect which proved important in later developments. By 1675 Leibniz had settled on the notation

∫ y dy = y²/2

written exactly as it would be today. His results on the integral calculus were published in 1684 and 1686 under the name 'calculus summatorius', the name integral calculus was suggested by Jacob Bernoulli in 1690.

After Newton and Leibniz the development of the calculus was continued by Jacob Bernoulli and Johann Bernoulli. However when Berkeley published his Analyst in 1734 attacking the lack of rigour in the calculus and disputing the logic on which it was based much effort was made to tighten the reasoning. Maclaurin attempted to put the calculus on a rigorous geometrical basis but the really satisfactory basis for the calculus had to wait for the work of Cauchy in the 19th Century.

Article by: J J O'Connor and E F Robertson

Notes

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3.1.14. Из книги Introduction To Mathematical Analysis. John E. Hutchinson (1994)

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