Chris Fenwick's webpage

This page contains (or will contain) three parts to an essay relating to why mathematics was arithmetised. The essays are currently in draft form but are sufficiently complete to be read as is.

The question of why maths needed to be arithmetised arose when I asked myself (all those years ago) why things like exp(x) needed to be defined, particularly since, as school students, we have been using it at school for some time. We know exp(x) is a function which can be plotted, we know the number e = 2.7182818..., we know how to differentiate and integrate it, etc. So, from the perspective of an A-level student I asked myself why on earth one would need to say that exp(x) was defined as the limit (1+x/n)^(n/x) as n → ∞, or as the Taylor series exp(x) = 1 + x + x²/2! + x³/3! + x⁴/4! + ..... And, as for ln(x) we know this to be the inverse of exp(x) so there is no need to define ln(x).

The only things we accepted as being defined were the three basic trig functions of sin, cos, tan based on the geometry of a right-angled triangle, and the idea of a complex number. We would never have thought of defining ln(x) as an integral. We can integrate ln(x) and we know that ln(x) is the result of an integral, but to define ln(x) by an integral would not have crossed our minds. Even less would we have thought to define the number pi as an integral. And even less would we think that exp(x) could be defined in five different ways (as a Taylor series, as the result of a particulat first order ODE, as a functional equation, as the inverse of ln(x) (but this requires us to define ln(x)), and as the limit stated above).

The result of thinking about this lead me not to write about the nature of mathematical definition but about why maths needed to be arithmetised. In other words why did we go from sin(x) = opp/hyp as related to a right-angled triangle to, for example, sin(x) = x − x³/3! + x⁵/5! − .... This lead me down the path of going back to look at episodes in the history of maths from the time of Pythagoras through to end of the 19^th century in order to understand the development of maths, and why its foundation moved away from geometry and towards number and arithmetic. The essays below are my synthesis of this development as a way of explaining why mathematics is the way it is now, i.e. arithmetic rather than geometric.

Quick access: Part I | Part II | Part III

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Part I: On the classical attitude towards numbers, arithmetic, geometry, constructibility and infinity

Table of content 1 Introduction 1.1 A radical shift in the approach to mathematics from school maths to university maths 2 The primacy of number and arithmetic: The Pythagoreans 2.1 An initial comment 2.2 The general attitude of the Pythagoreans towards nummbers and arithmetic 2.3 Defining "1" and numbers 2.4 Defining even and odd numbers 2.5 Further studies in evenness and oddness 2.6 On numbers represented geometrically: The figurate numbers 2.7 Number patterns and arithmetic on figurate numbers 2.8 On ratios 2.9 Conclusion 3 The primacy of Geometry: Euclid's elements 3.1 Introduction 3.2 On the construction of geometric objects 3.3 Examples on how to construct geometric objects 3.4 A simple Euclidean geometric proof 4 Selected rare moments of arithmetic study from Euclid onwards 4.1 Euclid 4.2 Heron of Alexandria 4.3 Nicomachus of Gerasa 4.4 Diophantus - To come 4.5 Conclusion 5 Discovering incommensurability 5.1 Commensurability 5.2 Incommensurability 5.3 Eudoxus' theory of incommensurability - To come 6 How the issue of infinity arises naturally in geometric construction 6.1 The conception of infinity in ancient Greeks times 6.2 Achilles and the tortoise 6.3 The conception of infinity in ancient Greeks times 6.4 On points and lines 6.5 How a short line equals a long line 6.6 Galileo's hexagon wheel 6.7 Archimedes' approach to finding π 6.8 Why geometry became the foundation of mathematics? 7 On the geometric constructibility: The case of Descartes 7.1 Introduction 7.2 Constructing natural numbers geometrically 7.3 Constructing fractions geometrically 7.4 Constructing arithmetic geometrically - Addition and subtraction 7.5 On the unit line segment of arbitrary length and the principle of homogeneity 7.6 Constructing arithmetic geometrically - Multiplication and division 7.7 Constructing square roots geometrically 7.8 The geometric constructibility of curves via algebraic equations 7.9 Extending the idea of what is acceptable geometric construction 7.10 New instruments used to extend the class of geometric curves 7.11 Algebra gives absurd results 7.12 Are all numbers constructable? 7.13 A first step towards pure number? 8 On numeric constructibility (to come) 8.1 The case of Stifle 8.2 The case of Stevin 9 Appendix 1: Two examples of Archimedes using the method of exhaustion 9.1 Archimedes' approach to finding the area contained by a parabola 9.2 Why T2 = ¼T1 , etc 9.3 The error between the area under the triangle and the area under the parabola (to finish) 9.4 Archimdes' upper bound for π 10 Appendix 2: Descartes' solution to Pappus' 4-line problem 11 Appendix 3: Use of Descartes' mesolab for constructing ∛2 and a cubic equation 12 Bibliography

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Part II: The problem of infinitesimals in geometry and arithmetic (to come)

Table of content 1 Introduction: Geometric construction in finite steps and finite time 2 On impercetibles I: Geometric indivisibles 2.1 The method of indivisibles: The case of Cavalieri 2.2 Mathematical criticisms of indivisibles 3 On imperceptibles II: Geometric infinitesimals 3.1 Using infinitesimals without saying so: The case of Fermat 3.2 The idea of infinitesimals replaces the idea of indivisibles 3.3 The idea of infinitesimals considered geometrically: The case of Barrow 3.4 Using infinitesimals explicitely: The case of Newton 4 On imperceptibles III: Arithmetic infinitesimals 3.1 Treating infinitesimals arithmetically: The case of Wallis 3.2 Treating infinitesimals arithmetically: The case of Leibniz 5 Arithmetic involving infinitesimals: Infinite series 5.1 Arithmetising Newton and Leibniz 5.2 Some problems when arithmetising Newton and Leibniz 5.3 The contradictory results of arithmetic on infinite series 5.4 The problem of common rational fractions 5.5 Conclusion 6 The necessary contrivance of infinitesimals 7 Arithmetic involving infinitesimals: Infinite series 7.1 Where we have got to so far 7.2 The inconsistency of geometry 7.3 The inconsistency of algebra 7.4 The inconsistency of arithmetic and numbers 7.5 Conclusion 8 The answer to a failing mathematics 8.1 Solving the problem of number: Numeric continuity 8.2 Solving the problem of arithmetic: Convergence 8.3 The banishment of infinitesimals? 8.4 A geometric idea which can be transformed into an arithmetic statement 8.5 An arithmetic statement which can be described geometrically

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Part III: On the arithmetic construction of mathematics (to come)
Note that this part III is not a rigorous text on the foundations of mathematics nor on real analysis, but does address foundational and real analysis aspect of maths.

Table of content 1 That which is prior to number 1.1 How to think about the constructions to come 2 Constructing the set ℕ of natural numbers 2.1 Apriori logical precepts that build the structure of numbers 2.2 There is a unique starting number 2.3 Every number has a successor 2.4 The number 0 is not the successor of any number 2.5 Every element has a unique successor. Or, no element is the successor to two or more elements. 2.6 There is only one starting value, all other natural numbers can be traced back to this starting value and are themselves all natural numbers 2.7 An example of applying Peano's axioms 3 Constructing arithmetic on ℕ 3.1 Constructing addition 3.2 Constructing multiplication 3.3 On equality and ordering 3.4 On rules of arithmetic 3.5 Some basic proofs 4 Constructing the set ℤ of integers 4.1 Constructing ℤ 4.2 Constructing addition 4.3 On negation 4.4 On additive inverses and the construction of subtraction 4.5 Constructing multiplication 4.6 On equality and ordering 4.7 On the correspondence between ℕ and ℤ+ 4.8 The vexed issue of (-1)×(-1) = 1 4.9 Addendum 5 Constructing the set ℚ 5.1 Introduction 5.2 Constructing the rationals 5.3 On equality and ordering 5.4 On addition 5.5 On negation and subtraction 5.6 On multiplication, multiplicative inverse, reciprocals and division 5.7 On the correspondence between ℤ and ℚ 5.8 A final comment 6 Constructing the set ℝ 6.1 Introduction 6.2 An informal description of the construction of ℝ 6.3 Cauchy sequences 6.4 Constructing ℝ via Cauchy sequences 6.5 Addition 6.6 Multiplication 6.7 Ordering 6.8 A variant and two alternative constructions for ℝ 6.9 Loosing the ability to do arithmetic 7 Confirming certain numbers and arithmetic 7.1 On 00 7.2 The number √2 is irrational 7.3 The decimal representation of a real number 7.4 There is an irrational number between any two rational numbers 7.5 Is a√2, for a∈ℝ , a number? 7.6 The vexed issue of 0.999999... = 1 7.7 Addition of rational fractions is well defined 7.8 On exponentiation 7.9 On the correspondence between the geometric representation of numbers and the axiomatic representation of numbers 7.10 Some peoples' opinions about the nature of numbers 8 The case of coordinate geometry 9 Understanding basic mathematical things arithmetically 9.1 The definition of a limit 9.2 The definition of continuity 9.3 The definition of the derivative 9.4 The definition of the definite (Reimann) integral 10 Finally, problems involving infinities and infinitesimals can be solved 10.1 Returning to Achilles and the tortoise 10.2 Returning to Archimedes and the area under a parabola 10.3 Infinite nested triangles 10.4 The Koch snow flake: A finite area enclosed by a perimeter of infinite length 10.5 A finite volume enclosed within a perimeter of infinite size 11 Appendices 11.1 Constructing the natural number via set theory 11.2 On induction 11.3 Equivalence relations 12 References, and bibliography of real analysis books

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