Hadamard, in reply, argued that the central problem was that he and his colleagues had dierent conceptions of mathematics, and that their arguments resembled earlier debates around Riemann's views on what functions should be allowed into analysis. In his view, *'essential progress in mathematics has resulted from successively annexing notions'* which, for earlier generations, *'were "outside mathematics" because it was impossible to define them'.*

This correspondence (here taken from [**32**]) was published in a major French mathematical journal in 1905, and represents an early instance of formation of a French *constructivist* school of mathematics. In time, Baire, Borel, and especially Lebesgue – all of whom (like the majority of late nineteenth century analysts) had implicitly used instances of the Axiom in their own work – became strong protagonists for the conception of mathematics they had spelled out in their initial responses to Zermelo's Axiom. Lebesgue even called their approach 'Kroneckerian'.

**2.2. Earlier uses of the Axiom.** The various ways in which late nineteenth century mathematicians had used the Axiom of Choice prior to Zermelo's formal statement of the assumption are analysed comprehensively in [**32**]. The example we consider concerns the crucial concept of *continuity of a function f* : R ! R *at a point a* (cf. **Chapter 6**).

Recall that Bolzano's definition of continuity was rephrased there to read

(i) *The function f is continuous at the point a if, for given* " *>* 0 *we can find*

*>* 0 such that *|f*(*x*) *f*(*a*)*| <* " *whenever |x a| <* ;

wheras Cauchy's definition lent itself to the reformulation

(ii) *The function f is continuous at the point a if whenever a sequence* (*xn*)

*has limit a, the sequence of values* (*f*(*xn*))*n has limit f*(*a*)*.*

As noted in **Chapter 6**, (i) states the modern definition of continuity, while the formulation (ii) is now called *sequential continuity.* For real functions (though not in more general settings) (i) and (ii) are logically equivalent.

However, the proof that sequential continuity implies continuity implicitly uses Denumerable Choice. The following argument mirrors a proof given in an 1871 paper by Heine, who credited it to Cantor.

The argument goes as follows: if (i) fails for *f* at *a,* there must exist an " *>* 0 such that for any *>* 0 there is an *x* satisfying *x <a<x* + *,* for which *|f*(*x*) *f*(*a*)*|* "*.* Choose successively as the numbers *n* = 1 2*n* to produce points *xn* with 0 *< |xn a| <* 1 2*n* , but *|f*(*xn*) *f*(*a*)*|* "*.* So: lim*n*!1 *xn* = *a*, but *f*(*a*) 6= lim*n*!1 *f*(*xn*)*.* This means that *f* does not satisfy (ii) at *a.* Therefore: if (ii) is true for *f* at *a* then (i) must also be true for *f* at *a.*

But how exactly do we describe a *rule* for choosing each of the points *xn*? What has been shown is that the definition of continuity implies that if the function *f* is *not* continuous at *a*, then, for each *n* 1*,* there must be a point *xn* satisfying 0 *< |xn a| <* 1 2*n ,* but whose images under *f* are at least " apart. However, we have no explicit way of calculating the value of the number *xn.* Even a rule such as taking *xn* as the 'smallest' such point does not provide a way of *identifying* the point *xn* explicitly.

This is an example where the Axiom of Choice (at least in its Denumerable guise) cannot be avoided. This fact was only noticed in 1913 in Italy by *Michele Cipolla* (1880-1947), and picked up in Poland in 1916 by the influential *Wac≥aw SierpiÒski* (1882-1969). The latter founded a group of mathematicians in Warsaw who embarked upon an exhaustive study of the Axiom of Choice, beginning with an extensive survey by SierpiÒski of uses of the Axiom in Real Analysis. In addition to the example given by Cipolla, he described numerous key results by Borel and Lebesgue that had made implicit use of the Axiom. One striking example was that the proof of the principal characteristic of Lebesgue measure (see [**3**]), *countable additivity*, in fact depended on Denumerable Choice, whose validity Lebesgue later explicitly rejected in the correspondence summarised above!

Over the following years the Axiom of Choice became a touchstone for researchers on the foundations of mathematics. The Axiom perhaps represents the main element of what remains of Cantor's bold claim of the *'freedom'* inherent in pure mathematics as a creation of the human spirit that does not need to be kept within artifical boundaries—such as those insisted upon by Kronecker.

Such restrictions, Cantor had argued, represent a far greater danger to mathematics than did his own firm belief, that *'mathematics is completely free in its development',* and bound only by the requirement that its concepts are *'free from contradictions in themselves'* as well as *'standing in fixed relationships, ordered through definitions, to earlier concepts that are already present and have been verified'.* Even Poincaré, who argued consistently for the primacy of *intuition***,** declared himself disposed to accept the Axiom. Ever the Kantian, he declared the Axiom of Choice to be *'a synthetic a priori judgment without which the "theory of cardinals" would be impossible, for finite as well as infinite numbers'*.

Most mathematical practitioners today accept the ZF axioms as the foundation on which mathematical concepts can be built, beginning with set theory and specialising to whatever their field of interest may be. Most also use the Axiom of Choice as a valuable tool—and we will see below that, if one accepts the ZF axioms, adding the Axiom of Choice as an additional axiom creates no new logical diculties. When this is done, the resulting axiom system is simply called ZFC.

#### **3. Tribal conflict**

In the early years of the twentieth century, foundational disputes arose between various schools of mathematicians, in part over the use of Axiom of Choice and partly over more profound philosophical and methodological dierences. While these are issues of fundamental importance to the subject as a whole, they have, in large measure, tended to become the province of specialists in the foundations of the subject.

**3.1. Hilbert and Poincaré.** The contrasting philosophical perceptions of Henri Poincaré in Paris and David Hilbert in Göttingen were already apparent in their reactions to Kronecker's attacks on Cantor's work. Hilbert had admired Cantor's transfinite mathematics from the beginning, without allowing himself (at least initially) to become too disturbed by the antinomies of set theory. In 1925, more than three decades after Kronecker's death, Hilbert delivered an address entitled *On the Infinite* in Münster at an event honouring Weierstrass' work in creating more solid foundations for Real

Figure 40. David Hilbert, by an unknown photographer, 19073

Analysis, He again hailed Cantor's set theory as *'the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity'* (see [**37**]).

However, Hilbert was fully aware of the damage the antinomies would do unless ways were found to avoid them. In contrast to Kronecker's 'arithmeticisation programme', his proposal was to *'supplement the finitary statements'* of ordinary arithmetic with *'ideal statements'*, just as ideal elements had been used in parts of function theory and geometry in the past. Mathematics would then consist of two kinds of statements: those to which meaning could be attached by some external communication, and others which had no meaning in themselves, but were the ideal elements of the theory. The sole criterion for their validity would be a *'proof of consistency',* arrived at via a (formal) logical calculus. In this way, Hilbert declared triumphantly, *'No one will drive us from this paradise that Cantor has created for us!'*

Today, Hilbert's approach to foundational questions is described as *formalism,* which is summarised in *Encyclopaedia Britannica* as the belief that *'all mathematics can be reduced to rules for manipulating formulas without any reference to the meanings of the formulas. Formalists contend that it is the mathematical symbols themselves, and not any meaning that might be ascribed to them, that are the basic objects of mathematical thought'.*

3https://commons.wikimedia.org/wiki/File:David\_Hilbert,\_1907.jpg

The position of most mathematicians is somewhat dierent. Recalling Plato's insistence on the reality of his World of Ideas, one might argue that a typical mathematician today will act as an unreconstructed *Platonist* on weekdays, pausing only to recite *Formalist* liturgy when required to do so on particular feast days.

Hilbert's purpose, following on from his successful axiomatisation of Euclidean geometry, was to find a general methodology (which he called *proof theory*) for constructing a formal language in which the *consistency* of the axiom system in question could be verified. He had shown in 1899 that Euclidean geometry is free from contradictions *provided* one could assume that arithmetic and the real number system (the basis of Analysis) had this property. But this result, rather than solving the problem of consistency, had shifted it to another area of mathematics. The real goal was to obtain a proof of *absolute consistency* of certain kinds of formal systems.

This had brought the focus onto the question of the consistency of arithmetic (the system codified by the Peano axioms); in other words, formulating this system in the corresponding formal language, and using only the axioms of ZF and those of classical logic to provide a formal proof (within the formal language) that it does not contain a contradiction. The essence of the *Hilbert programme* was to construct mathematical proofs that the various branches of the subject are free from contradiction.

Poincaré, by contrast, would always emphasise the importance of what he called *intuition* in mathematical research. While praising Hilbert's *Foundations of Geometry* as a 'classic', he remarked that *'the logical point of view alone appears to interest Professor Hilbert... ...The axioms are postulated; we do not know from whence they come...'* Defending Kant's philosophical concept of *a priori* knowledge, Poincaré argued that it was not inherent notions of space or time, but rather the concept of *iteration*, or indefinite repetition (as evidenced in counting), which required our innate sense of time. This constituted the *a priori* source of extra-logical content in elementary number theory. Moreover, he argued that an intuitive grasp of 'continuity' is basic to our understanding of the continuum. It was such *a priori* precepts, rather than logical reasoning, that enabled us to understand the underlying basis of mathematical knowledge.

He rejected Cantor's views on the actual infinite, which he saw as leading to a vicious circle of self-referential statements, as illustrated by the various antinomies. His philosophy also moved him strongly to oppose the logicist programme: he did not accept that mathematics could be a mere part of logic, leaving no room for intuition as an innate characteristic of the intellect. In his view, Hilbert's relegation of truth in mathematics to mean simply the absence of contradiction, and his proposed proof theory in particular (which existed only in merest outline until after Poincaré's early death in 1912), were in danger of straying too close to the logicist perspective.

**3.2. Enter Brouwer.** A more substantive challenge to 'classical' mathematics and logic was developed between 1908 and 1918 by the Dutch mathematician *L.E.J. Brouwer* (1881-1966), whose incisive contributions to the burgeoning subject of topology had marked him out as a brilliant researcher. The publication of his ideas on the nature and foundations of mathematics began with a paper published in 1918, entitled *Founding Set Theory Independently of the Principle of the Excluded Middle. Part One, General Set Theory.*

Brouwer's stated objective was to remove the use of proof by contradiction from mathematical arguments dealing with infinite sets. He objected to the unrestricted use of the Excluded Middle as a logical tool to claim the *existence* of a mathematical object, reflecting the concerns of the French constructivists (Borel, Baire, Lebesgue, as well as Poincaré), that proof by contradiction would not identify the object in question uniquely.

Brouwer produced a number of simple, but compelling, examples to draw attention to his claim that the construction of the classical models of the continuum cannot always *determine* particular real numbers. For example, suppose that the expansion of the real number *x* starts with 0*.*33333*...* and then *either* continues in this fashion forever, *or* is terminated as soon as a string of seven consecutive sevens (*....*7777777*...*) appears in the decimal expansion of ⇡*.*

In Brouwer's time no such string had been found in the decimal expansion of ⇡, so it was not known if one existed. (If one were to be found, he could always demand a longer string, of course.) But a string of sevens must either appear at some point, or else it never appears. This means that the number *x* cannot be irrational: it equals 1 3 if the string never appears, while its decimal expansion terminates at the end of the string if it does appear. But we cannot *identify* the rational number in question, since we don't know whether or at which decimal place the expansion terminates. Similar examples, usually based on an unsolved number theory problem (see *MM* for an example) were constructed to exhibit a real number *x* for which the question whether *x* = 0 or *x* 6= 0 required the solution of the unsolved problem. Brouwer regarded the very existence of unsolved problems in mathematics as a *weak counterexample* to the Law of the Excluded Middle.

Thus, for Brouwer, the key requirement was that only sets that could be constructed were to be accorded meaning in his *intuitionist* mathematics. Everything should spring, ultimately, from our fundamental intuition of the sequential construction of the natural numbers. He was uncompromising: rather than claim that the results of classical mathematics were wrong, he argued that very many of them should be regarded as meaningless!

Moreover, it was clear that even those results that could be retained as meaningful in intuitionist terms would usually require much longer and less elegant proofs if the available mathematical tools were stripped down to those deemed acceptable by Brouwer. Hilbert memorably described the requirements of intuitionist methods as akin to *'denying the boxer the use of his fists'* when attacking a particular problem.

Throughout the 1920s, Brouwer's trenchant critique of 'classical' mathematics and logic attracted a dedicated following of talented mathematicians mainly in Holland and Germany—who were persuaded by his arguments of the need for change. Notably, he increasingly influenced Hilbert's favourite former student, *Hermann Weyl* (1885-1955), whose enthusiastic conversion to intuitionism would cause Hilbert much personal consternation. In 1918 Weyl, a highly skilled communicator, published his own version of semiintuitionist analysis, *The Continuum.* This became influential in Germany, as did his 1921 paper on the 'new foundational crisis' in mathematics.

The debates between Weyl, Brouwer and Hilbert were courteous enough at the outset, but later the dispute between Brouwer and the normally generous and fair-minded Hilbert took on a personal tone. In 1928, in an episode reminiscent of the battle between Kronecker and Cantor, Hilbert (who was seriously ill at the time and did not expect to survive) used his pre-eminence among German mathematicians to engineer the removal of Brouwer from the editorial board of the prestigious journal *Mathematische Annalen.* This led to considerable ill-feeling and Albert Einstein resigned from the board in protest. Ironically, this time the senior participant in the battle was the adherent of Cantor's 'free' mathematical style, while the outcast and principal suerer was the man who had sought to 'purify' mathematics from the eects of that style and return it to safer foundations.

Today there are various strands of intuitionism within the somewhat wider (but still quite limited) group of *constructivist* mathematicians, among whose notable achievements the work begun by the founder of modern constructive analysis, the American *Errett Bishop* (1928-1983), perhaps stands out as having attracted most interest. The great majority of mathematicians, however, continues to work within 'classical' mathematics.

#### **4. Gödel's incompleteness theorems**

The final act in the 'foundational crisis' occurred only three years later. In 1931 the Austrian logician *Kurt Gödel* (1906-1978) dealt a fatal blow to the Hilbert programme by publishing his two (now famous) *incompleteness theorems.* An understanding of what these theorems said and why they had such a devastating impact upon the formalist ideal necessitates a more detailed description of that ideal.

**4.1. Hilbert's programme.** In Hilbert's view, the provision of an axiomatic basis for a branch of mathematics—as he had done for Euclidean geometry in 1899—ensured that one could ignore the traditional 'meaning' of the concepts encountered in the theory: as he often emphasised, notions such as *point, line* or *plane* should be capable of being replaced by *tables,* *chairs,* or *beer mugs*! They would become undefined terms, whose properties as well as the relations between them would be determined by using only the stated axioms. In the *formalisation* of a mathematical theory its objects appear as 'meaningless signs' expressed in a very precise, formal logical language whose syntax (or grammar) expresses the rules for combining individual signs into longer strings, or 'sentences' of the language.

In such a formalisation the proof of a mathematical theorem comprises a series of valid steps—derived from the axioms specific to that area of mathematics and expressed in terms of the symbolic language, while obeying the logical axioms and rules of of inference—that transform one such sentence into another. In this way one can ensure that no hidden assumptions or unstated logical principles have crept into the deductive process. The relationships between the dierent sentences representing mathematical statements are demonstrated explicitly and completely

It then becomes possible to ask specific questions about the nature of and relationships between dierent strings of symbols. Among these is to ask whether it is possible, within the system, to construct a sentence, say *,* such that and its negation *¬* are both consequences of the underlying set of axioms. The sentence might, for example, be expressed within the formal language through the formula '0=0'*.* Its negation would be expressed as the formula '0 6= 0'. The question is whether both can be deduced from the underlying axiom system.

A proof that such a situation cannot occur within the system would verify that the formalised system is *absolutely consistent,* that is, free from inherent contradiction. The objective of Hilbert's program was to achieve this aim for each area of mathematics by what he called *finitary reasoning*, and so rebut a key element of Brouwer's critique of classical methods.

To illustrate Hilbert's approach we consider the case of *arithmetic*, as codified by the Peano axioms. To express these precisely requires a suitable and well-defined logical language. In mathematical logic, a *first-order language L* for an axiom system *S* would satisfy five requirements, only one of which is specific to *S.* The general requirements for *L* are a (denumerable) list of *variables* (*x, y, z, ...*) and three types of logical symbols: *connectives* (symbols for 'not' ( *¬*), 'and '(\_), 'or' (^) and 'implies' (!)), *equality* (=) and the *quantifiers* 'for all' (8) and 'there exist' (9). The symbols specific to *S* are its *undefined terms*: For Peano arithmetic these are the notions of 'distinguished' element (0)*,* 'immediate successor' (*s*)*,* and the operations of addition (+) and multiplication (⇥)*.*

The Peano axioms (expressed in this formal language), together with logical axioms and rules of inference, are used to derive the *theorems* of *S,* i.e. the statements that can be proved using just the axioms and rules of inference. No meaning is attached to these statements—as noted in [**33**], a page *'of "meaningless marks" of such a formalised mathematics does not assert anything'.*

Hilbert's programme, as well as his conception of what constituted *finitary reasoning*, were to evolve substantially throughout the 1920s.4 His goal, however, remained to find, for any mathematical theory, a way of proving, entirely *within* the formal language, that it is impossible to arrive at a contradiction when using only the (finitely many) given axioms specific to the theory as well as the logical axioms.

If this could be achieved, Hilbert argued, he would finally have laid the ghost of Kronecker to rest. He would also have demonstrated that, by accepting consistency as the standard of truth for a mathematical system, its theorems could be seen to have solid foundations and thus could safely be used and explored further, without subjecting their proofs to Brouwer's more demanding criteria.

In its grand ambition, as well as in its careful attention to detail, Hilbert's programme is perhaps reminiscent of Frege's and Russell's attempts–in their dierent ways—to subsume mathematics under formal logic. And while *Principia* provided most of the new tools needed for Hilbert's formalisation of arithmetic and the rules of inference in logic, it was the methodology of this process which laid the groundwork for Gödel's investigations. The question whether there is a proof of consistency for a given axiom system is a statement that is not phrased *within* the logical language constructed to that system – it is a statement *about* the system itself, a *meta-mathematical* question.

**4.2. Decidability and consistency.** This is reminiscent of the diculty that underlies *Richard's paradox,* which we discussed at the beginning of this chapter. In fact, Gödel confirmed that his arguments were to some extent modelled on reasoning suggested by this paradox, namely that its statement *appeared* to be about decimal expansions, but was actually concerned with the properties of the set *E* of real numbers definable in a finite number of words. By listing the various combinations of letters one can attach a number to each such combination, representing its position in the sequence.

4A detailed description of the philosophical implications of Hilbert's evolving use of the term *finitary* is given in the Stanford were to evolve throughout the 1920s, were to evolve throughout the 1920s, of Philosophy under the heading 'Hilbert's Program' – the details are too technical to include here. In **[43]** a definition of 'finitary', given by the French formalist *Jacques Herbrandt*, is reproduced as follows: 'By a finitary argument we understand an argument satisfying the following conditions: In it we never consider anything but a given finite number of objects and of functions; these functions are well defined, their definition allowing the computation of their values in a univocal way; we never state that an object exists without giving the means of constructing it; we never consider the totality of all the objects *x* of an infinite collection; and when we say that an argument (or a theorem) is true for all these *x*, we mean that, for each *x* taken by itself, it is possible to repeat the general argument in question, which should be considered to be merely the prototype of these particular arguments.'

We apply the same idea to all *definitions* that define a property of numbers (e.g '*k* is prime'). The list of these definitions (expressible in English, say) is countable, so they may be written as (*k*)*k*1*,* for example. We can check whether any given number has the property expressed by *k* (e.g if *k*(*n*) expresses '*n* is prime', then *k*(71) holds, and so does *¬k*(65)*,* since 71 is prime, while 65 is not). The formal equivalent of the statement *'k does not have the property designated by the defining expression with which k is correlated in the serially ordered list of definitions'* then becomes *¬k*(*k*)*.* Following [**33**], let us call such a number *Richardian.* But now *'n is Richardian'* is also a definition in the list, hence it is allocated a number, say *m*. Then the question: *is m Richardian?* is self-contradictory: we now have a natural number *m* such that for all *k, m*(*k*) holds if and only if *¬k*(*k*) holds, and therefore *m*(*m*) holds if and only if *¬m*(*m*) holds.

Working in the formal system *S* representing Peano arithmetic, Gödel (using an adapted version of the system developed in *Principia*) was similarly able to 'mirror' certain meta-mathematical statements by statements *within* the language of *S.* By assigning a unique *Gödel number* to each elementary sign in the formal language, to each formula and to each proof (which is a finite sequence of formulas), he was able to analyse the structure of the formal language with great precision and formulate 'mirror images' *within* the language of *S* of various statements *about* the language—a fairly non-technical acount can be found in [**33**]**.**

In his *First Incompleteness Theorem*, Gödel showed that, *if S is consistent, there is a statement G (a Gödel statement) in this language such that neither G nor its negation ¬G (not-G) can be proved within S.*

In other words, the theorem shows that *S* is an *incomplete* system.

Gödel's encoding of all statements and proofs in *S* by natural numbers (expressed as products of powers of their prime factors in ascending order) enabled him to assign a Gödel number *g* to the statement *G*, expressed in the language of *S*, that mirrors the (meta-mathematical) statement *'This statement is not provable'*. 5 The statement *G* is codified by its Gödel number *g,* so we can read this claim as '*The statement with Gödel number g is not provable'.* Gödel next showed that if *G* were provable in *S,* then the same would hold for its negation *¬G*, which would mean that *S* is inconsistent, contradicting the hypothesis of the theorem.

The first theorem therefore says that the statement *G* is (formally) *undecidable in S*. 6

5Note the close similarity to the *'who shaves the barber?'* version of Russell's paradox (see Footnote 17 in **Chapter 9**) or the so-called *Liar paradox*, stating *'This statement is false*'.

6However, the statement G is true. It makes the (meta-mathematical) assertion: 'there is no proof of G within S', which is a true statement, since we have just seen that G is undecidable in S. Thus: provability of a statement within a formal system and truth are not the same thing!

Adding either *G* or its negation as a new axiom to *S* will not improve matters. Gödel's numbering technique works for any axiom system that (like *S*) is rich enough for arithmetic while remaining *computable*—so that its axioms can be recognised by a computer. Adding 'Gödel statements' to the axioms one by one does not destroy this property. In other words, *S* is *essentially incomplete.*

Gödel's *Second Incompleteness Theorem* followed from the first. Informally, it states:

*If S is consistent, then that fact cannot be proved within S.*

To justify this claim, Gödel first codes the statement *'S is consistent'* by *Consis*(*S*)*,* while the statement '*G* is not provable' is coded by *G* itself. The First Incompleteness Theorem states that if *S* is consistent then *G* is not provable in *S.* Coding this statement in *S* as *Consis*(*S*) ! *G,* the proof of the first theorem can be mirrored in *S* to show that *Consis*(*S*) ! *G* is provable in *S,* contradicting the first theorem. Therefore the consistency of *S cannot* be proved within *S.*

This conclusion was a hammer blow to Hilbert, showing that the original objective of his proof theory programme is unachievable. If there were a finitary demonstration of the consistency of *S*, it should be possible to formulate it as a theorem of *S,* which would mean, in turn, that *Consis*(*S*)*,* and therefore *G,* would be provable in *S,* contradicting the first incompleteness theorem!

While this does not exclude the possibility of finding a 'finitary' proof (in Hilbert's sense) that *cannot* be mirrored in the formal language of *Principia* (which is essentially the system we called *S*) but which shows the consistency of arithmetic, no such proof has yet been found. Work by various of Hilbert's former co-workers and others has produced partial results, but his original goal has eectively been abandoned as hopeless.7

Thus the optimistic claim that *'every mathematical problem can be solved'* and Hilbert's stirring epitaph: *'Wir müssen wissen. Wir werden wissen.'* ('We must know. We will know.') have, for the present at least, had to be replaced by somewhat more modest objectives. Gödel's results are a salutary reminder that the axiomatic method itself – at least as understood today – has severe limitations.

In 1938/9 Gödel lectured in the USA, but, showing little interest in politics, he returned to Vienna despite Nazi Germany's takeover of Austria in 1938. He did not succeed in obtaining a paid position—possibly because he had many Jewish friends. He was even mistaken as Jewish and attacked on

7In 1936, *Gerhard Gentzen* proved the consistency of *S*, imposing a linear ordering on statements in *S*. However, this went far beyond finitary statements: he needed not only the ordinals of Cantor's class (I), but ordinals up to a certain *infinite* ordinal ("0).

the street. He obtained a US visa in 1940, having to travel to the US via Russia and Japan, and settled at the Institute for Advanced Study in Princeton, where he remained until his death in 1978—and where he formed a close friendship with *Albert Einstein.*

Soon after arriving in Princeton he published *Consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory,* now regarded as a classic. He defined, within ZF, a smaller collection of sets (which he called *constructible*) in which the axioms of ZF, as well as the Axiom of Choice (AC) and the Generalised Continuum Hypothesis (GCH), are true. Hence, if the *negation* of AC could be proved in ZF, that would probably also hold within the constructible universe, and would make ZF inconsistent.8 The same would apply with GCH. So: if ZF is consistent, then so is the axiom system ZF+AC+GCH—no *new* inconsistencies can arise by adding AC and/or GCH to ZF.

In 1963, *Paul Cohen* (1934-2007), used a revolutionary technique (*forcing*) to show that, provided ZF is consistent, it also remains so if we add the *negations* of AC and/or GCH to it instead. Hence, if ZF is consistent, then neither AC nor GCH can be *proved* within ZF. Combining this with Gödel's result, therefore, AC and GCH are *independent* of ZF, i.e. neither provable nor disprovable.

In this more modest sense, and despite eectively ending Hilbert's hopes of proving absolute consistency, Gödel was instrumental in rescuing aspects of 'Cantor's paradise' that Hilbert had sought to preserve, as quoted at the beginning of this section.

**4.3. von Neumann's verdict.** John von Neumann summarised the impact of intuitionism and Gödel's theorems as follows in his essay *'The Mathematician'*, published in *'The Works of the Mind'* (vol.1, pp. 180-196, University of Chicago Press, 1947):

*It is dicult to overestimate the significance of these events. In the third decade of the twentieth century two mathematicians—both of them of the first magnitude, and as deeply and fully conscious of what mathematics is, or is for, or is about, as anybody could be—actually proposed that the concept of mathematical rigour, of what constitutes an exact proof, should be changed! The developments which followed are equally worth noting.*

*1. Only very few mathematicians were willing to accept the new, exigent standards for their own daily use. Very many, however, admitted that Weyl and Brouwer were prima facie right, but they themselves continued to trespass, that is, to do their own mathematics in the old, "easy" fashion—probably in the hope that somebody else, at some other time, might find the answer to the intuitionistic critique and thereby justify them a posteriori.*

8 For a discussion of what is meant by *true* in this context, see article on 'The Axiom of Choice' in the *Stanford Encyclopedia of Philosophy*, http://www.plato.stanford.edu.

*2. Hilbert came forward with the following ingenious idea to justify "classical"' (i.e. pre-intuitionistic) mathematics: Even in the intuitionistic system it is possible to give a rigorous account of how classical mathematics operates, that is, one can describe how the classical system works, although one cannot justify its workings. It might therefore be possible to demonstrate intuitionistically that classical procedures can never lead into contradictions—into conflicts with each other. It was clear that such a proof would be very dicult, but there were certain indications how it might be attempted. Had this scheme worked, it would have provided a most remarkable justification of classical mathematics on the basis of the opposing intuitionistic system itself! At least, this interpretation would have been legitimate in a system of the philosophy of mathematics which most mathematicians were willing to accept.*

*3. After about a decade of attempts to carry out this program, Gödel produced a most remarkable result. This result cannot be stated absolutely precisely without several clauses and caveats which are too technical to be formulated here. Its essential import, however, was this: If a system of mathematics does not lead into contradiction, then this fact cannot be demonstrated with the procedures of that system. Gödel's proof satisfied the strictest criterion of mathematical rigour—the intuitionistic one. Its influence on Hilbert's program is somewhat controversial, for reasons which again are too technical for this occasion. My personal opinion, which is shared by many others, is, that Gödel has shown that Hilbert's program is essentially hopeless.*

*4. The main hope of a justification of classical mathematics—in the sense of Hilbert or of Brouwer and Weyl—being gone, most mathematicians decided to use that system anyway. After all, classical mathematics was producing results which were both elegant and useful, and, even though one could never again be absolutely certain of its reliability, it stood on at least as sound a foundation as, for example, the existence of the electron. Hence, if one was willing to accept the sciences, one might as well accept the classical system of mathematics. Such views turned out to be acceptable even to some of the original protagonists of the intuitionistic system. At present the controversy about the "foundations" is certainly not closed, but it seems most unlikely that the classical system should be abandoned by any but a small minority.*

*I have told the story of this controversy in such detail, because I think that it constitutes the best caution against taking the immovable rigour of mathematics too much for granted. This happened in our own lifetime, and I know myself how humiliatingly easily my own views regarding the absolute mathematical truth changed during this episode, and how they changed three times in succession!*

Even while the foundational debates were raging in the 1910s and 1920s, the great majority of mathematicians paid only occasional attention to these matters, and got on with their particular research projects, while paying due respect to the eorts of those grappling with dicult foundational questions. Following the shock of Gödel's 'negative' results, this tendency became more marked, while the pace of progress in many areas of mathematics increased dramatically, with new areas developing so quickly that no single researcher after Poincaré and Hilbert could truly be regarded as a 'universalist' with contributions and interest stretching across the entire spectrum of the subject. The emergence of computing has shifted some attention back towards the discrete and to the rapid development of recursive techniques as well as of rules of inference, including probabilistic techniques and multi-valued logics, but their impact on the fundamental questions around the nature of 'mathematical truth' is by no means clear at present.

#### **5. A logician's revenge?**

While the controversy over Cantor's transfinite cardinals raged in the period 1880-1900, there was nevertheless near-unanimity among mathematicians that, following the construction of arithmetical models of the continuum R*,* the 'infinitely small' could no longer lay any claim to legitimacy in the foundations of the Calculus. Among the few notable dissenting views, those expressed by *Paul du Bois-Reymond* (1831-1889), especially in his *Die Allgemeine Funktionentheorie*(General Function Theory), were more strikingly and elaborately articulated than most (see [**2**]).

This work mixes a certain amount of mysticism and metaphysics with mathematical analysis in building contrasting pictures of the continuum that foreshadow later debates. In a dialogue between an 'idealist' and an 'empiricist', various conceptions of the continuum are analysed, but no final conclusions are reached. However, in an earlier paper du Bois-Reymond had developed a theory of infinitesimals, arguing that if infinite numbers were to be regarded as legitimate, so should their inverses, the infinitesimals.

Although du Bois-Reymond was not regarded as a mathematician of the first rank, he later made claims of priority in the use of the second diagonal method, which is universally attributed to Cantor. His more famous brother *Emil*, a physiologist who later became a philosopher, was somewhat more influential in late nineteenth-century society. Emil's writings focused on limits to our knowledge of nature, arguing that there were 'transcendental' questions that were unsolvable in principle. His statement *'Ignoramus et ignorabimus'* ('we are ignorant and we shall remain ignorant') became something of a rallying cry among groups of university students at the time. In 1900, in his Paris address, Hilbert responded regally: *'We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus'.*

Cantor was also dismissive of infinitesimals in analysis. In the long, discursive, introduction to his *Grundlagen* paper in 1883 he addressed the question whether his sequence of transfinite cardinal numbers might help in the search for a consistent grounding of a theory of infinitesimals. Could these be regarded as 'finite' numbers which were distinct from rational or irrational numbers, but would lie between such numbers, or which, similarly, could be inserted between algebraic and transcendental numbers?

His answer was unequivocal: the theory of well-ordered sets would show such attempts, by various authors at the time, to be fallacious, resting on the one hand on confusion between the actual and potential infinite, and on dubious reasoning on the other. The description (by some philosophers) of the potential infinite as 'bad' infinity was unjustified, since it had frequently proved itself useful in mathematics and the natural sciences. To his knowledge, *all* uses of the infinite in real analysis could relate only to the potential infinite. He argued that any attempt to 'force' the infinitely small into the guise of an actual infinite would be purposeless. Even if such objects existed, or could be defined successfully, they could not stand in any immediate relation to normal quantities that were *becoming* infinitely small.

Yet the use of infinitesimals, if only as *façon de parler* used in textbooks, did not die out as Cantor may have wished. There continued to be serious attempts to construct number systems that included infinitesimals, notably in 1907 by the Austrian mathematician *Hans Hahn* (1879-1934), who constructed a non-Archimedean ordered field containing non-zero elements *x* such that, for all *n* in N*,* 0 *< |x| <* 1 *n .* But there did not, as yet, seem to be a consistent way of exploiting such objects within real analysis.

This situation remained more or less intact for half a century. In 1958 a partial theory of infinitesimals was published by the German mathematicians *Curt Schmieden* (1905-1991) and *Detlef Laugwitz* (1932-2000). Their construction did not arouse much initial interest, as the number system they produced was not an ordered field. It was soon overshadowed by the publication in 1961 of an article by the logician *Abraham Robinson* (1918-1974), followed in 1966 by his ground-breaking volume *Non-Standard Analysis,* [**39**].

Robinson employed novel techniques from logic and model theory to construct an ordered field (necessarily non-Archimedean, of course) ⇤R – nowadays called the *hyperreals –* which extends the real number system R to include both infinite and infinitesimal numbers and is linked to real numbers in a way that makes applications to Real Analysis possible. This correspondence is based on a completely new concept, the *Transfer Principle,* which states (informally) that exactly the same formal statements (within symbolic logic, as described earlier) will hold in both R and ⇤R. This may seem contradictory at first sight, since ⇤R contains infinitesimal elements, while R does not—the resolution of this conundrum lies in the precise description of the formal statements that are to be included in the Principle. In this sense the Transfer Principle might be regarded as a modern version of Leibniz' Continuity Principle. *Plus ça change!*

Versions of the hyperreals can be constructed in a variety of ways, not all equivalent. We will assume as *given* an ordered field that we denote by ⇤R*,* in which R may be regarded as an embedded subset (much as Q was taken to be embedded in R in **Chapter 7**). This extension extends to subsets of R as well as to real functions (i.e. between real numbers): *A* ⇢ R has a counterpart ⇤*A* in ⇤R such that *A* = ⇤*A* \ R, and a function *f*:*A* 7! *B* extends to ⇤*f*: ⇤*A* 7! ⇤*B*, so that *f* is the restriction of ⇤*f* to *A* and the graph of ⇤*f* is the extension of the graph of *f.*

The algebraic operations (+*,* ⇥) and order relation (*<*) similarly extend to ⇤R , as does the absolute value (*|.|*) of a real number—for ease of notation, we will use these symbols also in ⇤R, without the asterisk.

In ⇤R we can distinguish between three types of elements: *x* 2⇤ R is

(i) *infinitesimal* if *|x| <* " for all " *>* 0 belonging to R,

(ii) *finite* if *|x| < r* for some *r* in R,

(iii) *infinite* (equivalently, *not* finite) if *|x| > r* for all *r* in R.

(Note that infinitesimals are also finite.)

A version of the Transfer Principle that ensures that all *first-order* properties of subsets and functions extend to ⇤R is proved in Chapter 1 of **[1].** In this setting, *completeness,* the key feature of R*,* does *not* extend to ⇤R*.* For example, the set of all *finite* elements of ⇤R cannot have a least upper bound in ⇤R: any upper bound of this set would be an infinite element, *s* say, hence *s* 1 would still be an upper bound. This illustrates the need to take care when using the Transfer Principle.

For hyperreals *x, y* we write *x* ⇡ *y* if their dierence is infinitesimal, and say they are *infinitely close*. Clearly, a sum or product of infinitesimals is again infinitesimal, as are the inverse of an infinite element and the product of a finite with an infinitesimal element of ⇤R*.* So if *x* ⇡ *y* and *u* ⇡ *v* then *x* + *y* ⇡ *u* + *v.* If *x, u* are finite, then *xv* ⇡ *uv.*

Sums and products of finite elements are finite, as is the sum of a finite and an infinitesimal element. The inverse of a non-infinitesimal finite element is also finite. While the sum of two infinite elements of the same sign, and the product of any two infinite elements, are again infinite, products of infinite numbers with infinitesimals can belong to *any* of the three classes.

The following theorem ensures that any finite hyperreal is infinitely close to a unique real number.

#### **Standard Part Theorem**

*If x is a finite hyperreal there is a unique real number r* ⇡ *x. So there is a unique infinitesimal such that x* = *r* + *.*

*Proof:* The set *A* = *{a* 2 R : *a x}* is bounded as a subset of R*,* as *x* is finite. So *r* = sup *A* exists in R*.* For any real " *>* 0*, r*" is not an upper bound of *A,* so we can find *a* 2 *A* with *r* " *< a x,* and so *r x <* "*.* As *r* is an upper bound of *A, r*+" is not in *A,* hence *x<r*+"*,* so that *xr <* "*.* We have shown that *|x r| <* " for all " *>* 0 in R , hence = *x r* is infinitesimal. The uniqueness claim is obvious, as 0 is the only infinitesimal in R*.*

We write *r* = *st*(*x*) and call it the *standard part* of *x.* For finite elements, the standard part of their sum (product) is the sum (product) of their standard parts. If *x y* in ⇤R then *st*(*x*) *st*(*y*) in R (but not conversely). For finite *x* in ⇤R*\*R we see that = *x st*(*x*) 6= 0 is infinitesimal, and 1 is infinite. For infinite *x, x*1 is a non-zero infinitesimal.

These preparations allow us to characterise convergence of sequences and continuity of functions very simply:

(a) for a real sequence (*sn*) and *a* 2 R, lim*n*!1 *sn* = *a* is equivalent to: ⇤*sK* ⇡ *a* for all infinite *K* in ⇤N;

(b) if *f* is defined on *I* = (*a, b*) in R*,* then *f* is continuous at *c* 2 *I*

if and only if ⇤*f*(*x*) ⇡⇤ *f*(*c*) for all *x* ⇡ *c.*

Both claims are proved by an appeal to the Transfer Principle, i.e. constructing a sentence in first-order logic that can be transferred. We only consider the second case (which is very close to Bolzano's definition of continuity):

Suppose lim*x*!*c f*(*x*) = *f*(*c*)*,* fix any hyperreal *z* ⇡ *c* and " *>* 0*.* We need to show that *|* ⇤*f*(*z*) ⇤ *f*(*c*)*| <* "*.*

We know that there is *>* 0 such that the following (first-order) sentence holds in R:

*for all x, |x c| < implies |f*(*x*) *f*(*c*)*| <* "*.*

By Transfer, the corresponding sentence holds in ⇤R:

*for all X, |X c| < implies |* ⇤*f*(*X*) ⇤ *f*(*c*)*| <* "*.*

Taking *X* = *z* we see that ⇤*f*(*z*) ⇡⇤ *f*(*c*) whenever *z* ⇡ *c.*

For the converse implication, assume that *|* ⇤*f*(*z*) ⇤ *f*(*c*)*|* ⇡ 0

whenever *z* ⇡ *c,* and suppose that " *>* 0 is given.

Taking any infinitesimal *Y >* 0 in ⇤R, we have:

*there exists Y* 2 ⇤R *such that for all X* 2⇤ R *, |X c| < Y*

*implies |* ⇤*f*(*X*) ⇤ *f*(*c*)*| <* "*.*

By Transfer, the corresponding sentence holds in R :

*there exists y* 2 R *such that for all x* 2 R *, |x c| < y*

*implies |f*(*x*) *f*(*c*)*| <* "*.*

We can take any such *y* as the required *>* 0*,* hence lim*x*!*c f*(*x*) = *f*(*c*)*.*

This elementary illustration should suce to show that basic facts of Real Analysis can be recovered in an intuitively attractive manner—see, e.g. [**1**] for an account of this and a variety of advanced applications. In *MM* we outline the construction of a version of ⇤R reminiscent of Cantor's model for the reals.

## **Epilogue**

I will leave matters there. Other extensions of the real number system have been proposed—see [**28**] for a striking example. By starting with counting and linking this to notions of the 'number line', we have seen how mathematicians have extended the number concept progressively over the centuries. Infinitesimals, it seems, are not as easily banished into the outer darkness as Weierstrass and others had supposed—but they had to change their clothes significantly in order to appear more respectable!

That said, what David Hume, who so clearly abhorred 'horn angles' (see the end of **Chapter 5**), might have thought about the infinitely many dierent orders of infinitesimals now made possible, must remain an open question. And it is easy to imagine how Kronecker (and, no doubt, Brouwer) might have reacted to the hyperreals. We might counter their concerns with Hadamard's confident assertion (quoted in Section 1) that *'essential progress in mathematics'* results from including notions which, for earlier generations, *'were "outside mathematics" because it was impossible to define them'.* And, after all, not even Plato's Olympian edicts stopped Archimedes from employing *neusis* constructions – nor, indeed, infinitesimal slices!

The present mathematical community has largely taken hyperreals in its stride, while seldom showing great interest in the details. One reason is the existence of a meta-theorem that maintains (roughly speaking) that any result which can be proved by nonstandard methods also has a 'standard' proof–which may well be rather longer, however! In this sense, the practitioners of nonstandard analysis appear to be closer to the current 'mainstream' than is the group at the opposite end of the spectrum, the constructivists, who not only reject infinitesimals, but also restrict the real numbers they accept to the numbers (essentially) definable in finitely many words.

Neither of these opposite poles has attracted more than a fairly small minority of practitioners to date. In both cases the 'entry fee' to participation, having to learn radically new techniques and adopt unfamiliar perspectives, may seem quite high to many researchers, trained as they usually are in techniques and subject matter still dominated by the groundwork laid in the late nineteenth century. Whether and how this may change only time will tell.

Let us therefore leave the last word to the venerable Sir Francis Bacon.

*Etiam capillus unus habet umbram suam. (The smallest hair casts a shadow.)* Sir Francis Bacon, *Ornamenta Rationalia, or, Elegant Sentences*, 1625.

## **Bibliography**

#### 260 BIBLIOGRAPHY

## **Name Index**

Abel, Niels, 63 al-Kashi, 48 al-Khwarizmi, 47 al-Uqlidisi, 48 Alcuin of York, 50 Alexander the Great, 41 Apollonius of Perga, 68 Archimedes, 5, 74 Argand, Jean-Robert, 100 Aristotle, 20, 31 Baire, René-Louis, 237 Barrow, Isaac, 96 Beltrami, Eugenio, 199 Berkeley, George, 128 Bernoulli, Jacob, 93, 125 Bernstein, Felix, 224 Berry, GG, 235 Bessel, Friedrich Wilhelm, 198 Bishop, Errett, 244 Boethius, 50, 51, 198 Bolyai, Janos, 197 Bolzano, Bernhard, 143 Bombelli, Rafael, 64 Boole, George, 103 Borel, Emile, 237 Bradwardine, Thomas, 115 Brahe, Tycho, 68 Briggs, Henry, 82 Brouwer, LEJ, 243 Burali-Forti, Cesare, 229 Cantor, Georg, 211

Cardano, Girolamo, 58 Carnot, Lazare, 135

Cauchy, Augustin-Louis, 131, 135 Cavalieri, Bonaventura, 93, 117 Charlemagne, 50 Chuquet, Nicolas, 2, 57 Cipolla, Michele, 239 Cohen, Paul, 249 d'Alembert, Jean le Rond, 105, 132 de Morgan, Augustus, 103 del Ferro, Scipione, 58 Descartes, René, 78, 85 Dinostratus, 187 Diophantus, 41, 50 du Bois-Raymond, Paul, 251 Einstein, Albert, 199, 244 Eratosthenes, 112 Euclid, 21 Eudoxus, 70 Euler, Leonhard, 124 Fermat, Pierre de, 43, 85 Ferrari, Lodovico, 60, 62 Fior, Antonio, 59 Fontana, Niccolo (Tartaglia), 59 Fraenkel, Abraham, 234 Galilei, Galileo, 68, 115 Galois, Evariste, 63 Gauss, Carl Friedrich, 100, 198 Gerbert d'Aurillac, 51 Goedel, Kurt, 244, 246–248 Graham, Ronald, 4

#### 262 NAME INDEX

Hadamard, Jacques, 237 Hahn, Hans, 252 Halley, Edmund, 68, 128 Hamilton, William Rowan, 101 Heine, Eduard, 144, 177 Heraclitus, 69 Hilbert, David, 199, 237, 251 Hippias, 72 Hippocrates of Chios, 71, 187 Hume, David, 129, 257 Huygens, Christiaan, 118 Kasner, Edward, 4 Kepler, Johannes, 68, 117 Knuth, Donald, 5 Lacroix, Sylvestre Francois, 136 Lagrange, Joseph-Louis, 133 Laplace, Pierre Simon, 136 Laugwitz, Detlef, 252 Lebesgue, Henri, 237 Legendre, Adrien-Marie, 160, 186, 197 Leibniz, Gottfried Wilhelm, 85, 117 Lejeune Dirichlet, Peter Gustav, 144 Leonardo of Pisa (Fibonacci), 51 Lhulier, Simon, 134 Liouville, Joseph, 188 Lobachevsky, Nikolai, 197 Maclaurin, Colin, 129 Menaechmus, 72 Meray, Charles, 176 Mersenne, Marin, 86 Minkowski, Hermann, 199 Napier, John, 80 Neugebauer, Otto, 14 Newton, Isaac, 96, 97, 120, 121, 123, 124 Nicholas of Cusa, 116 Nicomachus, 51 Nieuwentijt, Bernard, 132

Omar Khayyam, 50, 58 Pacioli, Luca, 57 Pappus of Alexandria, 90 Parmenides, 69, 108 Peacock, George, 103 Peano, Giuseppe, 200 Philolaus, 20, 23 Plato, 20, 34, 69 Playfair, John, 196 Poinsot, Louis, 136 Proclus, 25, 196 Pythagoras of Samos, 20 Recorde, Robert, 57 Richard, Jules, 234 Riemann, Bernhard, 199 Robinson, Abraham, 252 Rolle, Michel, 147 Rudolph, Christo, 57 Runi, Paolo, 63 Russell, Bertrand, 165, 193, 229 Schmieden, Curt, 252 Schroeder, Ernst, 224 SierpiÒski, Wac≥aw, 239 Skolem, Thoralf, 234 Stevin, Simon, 79 Stifel, Michael, 77 Theaetetus, 36

Theodorus of Cyrene, 34 Torricelli, Evangelista, 93

van Schooten, Frans, 95 Viéte, Francois, 77 von Neumann, John, 203, 249

Wallis, John, 95 Wantzel, Pierre, 186 Weierstrass, Karl, 80, 131 Wessel, Caspar, 100 Weyl, Hermann, 235, 244 Whitehead, Alfred North, 231

Zeno, 108 Zermelo, Ernst, 223, 234

## **Index**

abacus, 52 aleph, 223 algorithm, 52 antiderivative, 148 approximable by rationals, 189 Archimedean property, 170 Archimedes 'Sand Reckoner', 5 Conoids and Spheroids, 113 infinitesimal slices, 112 law of the lever, 112 neusis construction, 74 On Sphere and Cylinder, 170 The Method, 114 Argand diagram, 100 Aristotle 'Metaphysics', 21 'Physics', 110 'Problems', 20 'Topics', 34 the excluded middle, 31, 43, 243 axiom of Choice, 236 of denumerable choice, 237 axiom system Zermelo-Fraenkel (ZF), 234 axiomatic method model for the natural numbers, 204 model of hyperbolic geometry, 199 primitive terms, 195

Babylon Seleucid period, 15 Berkeley The Analyst, 128 Cantor second diagonal argument, 226 Cardano Ars Magna, 58, 60 formula, 62 cardinality, 223 Cartesian product, 164 Cauchy Cours d'Analyse, 137 sequence, 141 clay tablet cuneiform, 14 Plimpton 322, 25 Yale7289, 17 closed, 169 coecients binomial, 121 common factor, 31, 39 conic section, 58 ellipse, 67 hyperbola, 68 parabola, 68 construction neusis, 69 constructivists French school of, 237 continuity, 110 Cauchy's definition, 142

principle of, 117, 120 continuum, 110, 136 hypothesis, 228 coordinate system polar, 93 rectangular, 87 Coss tradition, 57 decimal point, 80 derivative of a function, 147 Descartes cogito ergo sum, 86 coordinate geometry, 86 Discours de la methode, 86 folium of, 92 geometric curves, 92 La Geometrie, 86, 89 mechanical curves, 93 rule of signs, 98 Rules for the Direction of the Mind, 79 dierentiation, 117 Diophantus Arithmetica, 41 indeterminate problems, 42 Eddington's number, 10 equation cubic, 59 linear, 18 quadratic, 18 quartic, 62 quintic, 63 Euclid Elements, 21, 57, 67, 96, 137 Book V, 34 Book X, 39, 170 Books VIII-IX, 23 common notions, 195 postulates, 196 Lemma, 161 Sectio Canonis, 22 Eudoxus method of exhaustion, 111

expansion decimal, 180 eventually periodic, 181 truncation of decimal, 183 Fibonacci Liber Abaci, 51, 56 sequence, 53 Frege Basic Law V, 230 Foundations of Arithmetic, 230 predicate calculus, 230 rules of inference, 230 function, 125 derived, 134 dierentiable at a point, 147 Fundamental Theorem of Algebra, 105 of Arithmetic, 36 of the Calculus, 119, 149 Gaussian plane, 100 geometric progression, 81 geometry hyperbolic, 197 googol, 4 Hilbert problems, 191 Hilbert programme, 241 consistency of arithmetic, 242 finitary methods, 245 formalism, 242 proof theory, 242 Holy Roman Empire, 50 horn angle, 130 House of Wisdom, 47 induction, 201 inequality triangle, 141 infinite actual, 108, 217 divisibility, 110 potential, 108 infinitesimal, 117 Cauchy's formulation, 138, 139

method, 135 numbers, 251 infinity proper and improper (Cantor), 220 integral Cauchy's definition, 148 integration, 117 intuitionism, 243 Kant a priori knowledge, 198 Critique of Pure Reason, 198 Lagrange Mecanique Analytique, 133 Theorie des Fonctions analytiques, 133 Lebesgue measure countable additivity, 240 Leibniz characteristica universalis, 232 on imaginary quantities, 99 Library of Alexandria, 41 limit, 132 of a sequence, 140 of a variable (Cauchy), 139 Liouville constant, 190 locus of points, 90 logarithm Briggsian, 82 Napierian, 81 logicism, 230 long scale, 2 maestri d'abbaco, 56 magnitudes, 23, 110 apotome, 40 binomial, 40 commensurable, 39 in square only, 40 expressible, 40 in proportion, 23, 33

inexpressible, 40 major and minor lines, 40 medial, 40 means geometric, 22 multiples, 23, 110 muqabala, 49 myriad, 7 neutral element, 103 Newton De Analysi, 122 fluent, 121 fluxion, 121 Prime Ratio, 124 Principia Mathematica, 124 Ultimate Ratio, 124 Universal Arithmetick, 96 number algebraic, 215 cardinal, 219 complex, 64, 154 imaginary unit, 65 composite, 158 constructible, 186 dekad, 29 factors of a, 158 Fibonacci, 54 integer, 153 irrational, 11, 153, 184 multiple, 26 negative, 11 Newton's definition, 97 oblong, 27 octillion, 2 prime, 157 rational, 11, 153 real, 154 square, 26 tetrad, 29 triangular, 26 unit, 26 number line, 93 number system decimal, 2, 77

266 INDEX

Hamilton's approach to, 101 rational, 168 real, 94 sexagesimal, 14 number theory, 29 diophantine analysis, 43 numbers 'numeri ficti', 64 algebraic, 185 commensurable, 32 greatest common divisor (gcd) of two, 160 hyperreal, 252 incommensurable, 32, 57 infinite, 125 irrational, 78 natural, 153 negative square roots of, 64 real, 94, 95 relatively prime, 161 transcendental, 186 transfinite, 217 comparability of, 223 generation principles, 220 limitation principle, 220 ordinal, 221 transfinite number class, 221 one-one correspondence, 212 order of magnitude, 3 order relation, 154 on the integers, 166 on the rationals, 169 total, 167 transitive, 166 trichotomy, 167 order type, 224 paradox Burali-Forti, 229 Cantor, 228 Richard, 235 Russell, 231

paradoxes of the infinite, 115 Zeno, 108 Peano Arithmetices principia, 201 axioms, 201 Plato 'Theaetetus', 34 'Timaeus', 23 Platonic solids, 40 Theory of Forms, 69 power series, 123 prime factorisation, 158 uniqueness of, 162 proof by contradiction, 31 Pythagorean arithmetic, 21 pebble arithmetic, 26 gnomon, 26 triple, 25, 28, 29 Pythagoreans neo-Pythagoreans, 21 Philolaus, 20, 23 Theodorus, 34 quadratrix of Hippias, 73, 187 ratio Euclid's definition, 23 golden, 54 in music, 21 diatonic scale, 23 tetachord, 23 whole tone, 22 mean and extreme, 54 regular pentagon, 55 relation equivalence classes of a, 166 scientific method, 86 scientific notation, 3 sequence finite, 212 fundamental, 177

infinite, 213 null, 179 series divergent, 140 Fourier, 212 partial sum of a, 140 representation of a function, 134 sum of a, 140, 190 set countably infinite (denumerable), 213 derived, 212 infinite, 213 of first species (Cantor), 212 proper subset of a, 153 second species, 212 simply ordered (Cantor), 224 subset of a, 153 theory, 152 union, 153 well-ordered, 224 sets equipotent, 219 short scale, 2 Stevin De Thiende, 79 L'arithmetique, 80 successor, 201 tangent, 118, 121

telescoping sum, 118 Theodorus lesson, 36, 54 Theorem Pythagoras, 24 Theaetetus', 36 theorem binomial, 121 Bolzano-Weierstrass, 145 extreme value, 145 Gauss (roots of polynomials), 104 incompleteness, 244 intermediate value, 145 mean value, 147 Rolle, 147 Schroeder-Bernstein, 224 three famous problems doubling the cube, 70 trisecting the angle, 72 Transfer Principle, 252 trigonometry, 74 variable, 139 Viete Canon mathematicus, 77 In artem analyticam isagoge, 78 Wallis Algebra, 98 Arithmetica Infinitorum, 95

infinite product formula, 95

# Making up Numbers

## EKKEHARD KOPP A History of Invention in Mathematics

*Making up Numbers: A History of Invent on in Mathemat cs* off ers a detailed but accessible account of a wide range of mathema� cal ideas. Star� ng with elementary concepts, it leads the reader towards aspects of current mathema� cal research.

Ekkehard Kopp adopts a chronological framework to demonstrate that changes in our understanding of numbers have o� en relied on the breaking of long-held conven� ons, making way for new inven� ons that provide greater clarity and widen mathema� cal horizons. Viewed from this historical perspec� ve, mathema� cal abstrac� on emerges as neither mysterious nor immutable, but as a con� ngent, developing human ac� vity.

Chapters are organised thema� cally to cover: wri� ng and solving equa� ons, geometric construc� on, coordinates and complex numbers, a� tudes to the use of 'infi nity' in mathema� cs, number systems, and evolving views of the role of axioms. The narra� ve moves from Pythagorean insistence on posi� ve mul� ples to gradual acceptance of nega� ve, irra� onal and complex numbers as essen� al tools in quan� ta� ve analysis.

*Making up Numbers* will be of great interest to undergraduate and A-level students of mathema� cs, as well as secondary school teachers of the subject. By virtue of its detailed treatment of mathema� cal ideas, it will be of value to anyone seeking to learn more about the development of the subject.

As with all Open Book publica� ons, this en� re book is available to read for free on the publisher's website. Printed and digital edi� ons, together with supplementary digital material, can also be found at www.openbookpublishers.com

*Cover images from Wikimedia Commons. For image details see capt ons in the book. Cover Design by Anna Gat* .

**OBP**

 **EKKEHARD KOPP**

MAKING

UP NUMBERS

**Making up Numbers**

**A History of Invention in Mathematics**

**EKKEHARD KOPP**