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student resistance to the evaluation of  @xmath0 as the real number  @xmath1 ( henceforth referred to as the _ unital evaluation _ ) has been widely discussed in the mathematics education literature . it has been suggested that the source of such resistance lies in a psychological predisposition in favor of thinking of  @xmath0 as a process , or iterated procedure , rather than the final outcome , see for instance d.  tall s papers @xcite , @xcite , @xcite ( see also @xcite for another approach ) . we propose an alternative model to explain such resistance , in the framework of non - standard analysis . from this point of view , the resistance is directed against an unspoken and unacknowledged application of the standard part function `` st '' ( see section  [ ten ] , item [ 103 ] ) , namely the stripping away of a ghost of an infinitesimal , to echo george berkeley  @xcite , implicit in unital evaluation :    @xmath5{9tail.ps}}}\end{pmatrix } = 1.\ ] ]    what is the significance of such fine ( indeed , infinitesimal ) districtions between @xmath0 and @xmath1 ? as this text is addressed to a wide audience , it may be worth recalling that _ dividing _ by the difference @xmath6 will only be possible if the latter is nonzero , a matter of immediate import for studying rates of change and calculating derivatives ( see section  [ zoom ] ) . the hyperreal approach to a certain extent vindicates the student resistance to education professionals toeing the standard line on unital evaluation : so long as the number system has not been specified , the students hunch that @xmath0 can fall infinitesimally short of @xmath1 , can be justified in a mathematically rigorous fashion .    in section  [ two ] , based on a formula for a geometric sum , we summarize the standard argument in favor of unital evaluation . in section  [ three ] , we summarize the standard resistance to the latter . in section  [ four ] , we construct a hyperreal decimal in @xmath7 , represented by a string of @xmath4-infinitely many repeated 9s , and identify its lightstone representation . in section  [ five ] , we square the strict inequality with standard reality , by means of the standard part function . in section [ six ] , we represent the hyperreal graphically by means of an infinite - magnification microscope , already exploited for pedagogical purposes by keisler @xcite and tall @xcite . in section  [ zoom ] , we exploit the hyperreal to calculate @xmath8 .    in section [ eight ] , we develop an applied - mathematical model of a hypercalculator so as to explain a familiar phenomenon of a calculator returning a string of 9s in place of an integer . in section [ nine ] , we examine the cognitive concept of _ generic limit _ of cornu and tall in mathematics education , in relation to a hyperreal approach to limits . in section [ flat ] , we set the situation in perspective , with e. a. abbott . section  [ ten ] is a technical appendix containing basic material on non - standard calculus . the historical section  [ history ] contains an examination of the views of courant , lakatos , and e. bishop . section  [ proposal ] contains a 10-step proposal concerning the problem of unital evaluation . evaluating the formula @xmath9 at  @xmath10 , we obtain @xmath11 or alternatively    @xmath12    multiplying by  @xmath13 , we obtain @xmath14 for every  @xmath15 . as  @xmath16 increases without bound , the formula @xmath17 becomes @xmath18 or does it ? when i tried this one on my teenage daughter , she remained unconvinced . she felt that the number  @xmath0 is smaller than  @xmath1 . after all , just look at it ! there is something missing before you reach  @xmath1 . i then proceeded to give a number of arguments . apologetic mumbo - jumbo about the alleged `` non - unicity of decimal representation '' fell on deaf ears . the one that seemed to work best was the following variety of the old - fashioned `` because i told you so '' argument : factor out a  @xmath19 : @xmath20 to obtain @xmath21 and `` everybody knows '' that the number  @xmath22 is exactly `` a third '' _ on the nose_. q.e.d . this worked for a few minutes , but then the validity of was called into question , as well . then i finally broke down . in abraham robinson s theory of hyperreal analysis @xcite , there is a notion of an infinite hyperinteger ( see section  [ ten ] , item  [ 108 ] ) . h.  jerome keisler @xcite took to denoting such an entity by the symbol @xmath23 possibly because its inverse is an infinitesimal  @xmath24 typically appearing in the denominator of the familiar definition of derivative ( it most decidedly does _ not _ stand for `` howard '' ) . taking infinitely many terms in formula is interpreted as replacing  @xmath15 by an infinite hyperinteger @xmath25 the transfer principle ( see section  [ ten ] , item  [ 101 ] ) applied to then yields @xmath26 where the infinitesimal quantity  @xmath27 is nonzero : @xmath28 therefore we obtain the strict nonstandard inequality @xmath29 and my teenager was right all along . note that hyperreal extended decimal expansions were treated by a. lightstone in @xcite . in his notation , the hyperreal appearing in the left - hand - side of   appears as the extended decimal @xmath30 with the hat `` @xmath31 '' indicating the  @xmath4-th decimal place , where the _ last _ repeated digit  @xmath3 occurs . we have employed the underbrace notation as in rather than lightstone s semicolon , as it parallels the finite case more closely , and seems more intuitive . an alternative construction of a strict inequality @xmath32 may be found in @xcite , however in a number system which is not a field . to obtain a real number in place of the hyperreal @xmath34 , we apply the standard part function `` @xmath35 '' ( see section  [ ten ] , item  [ 103 ] ) : @xmath36 to elaborate further , one could make the following remark . even in standard analysis , the expression  @xmath0 is only shorthand for the * limit * of the finite expression when the standard integer  @xmath16 increases without bound . from the hyperreal viewpoint , `` taking the limit '' is accomplished in two steps :    1 . evaluating the expression at an infinite hyperinteger  @xmath4 , and then 2 .   taking the standard part . the two steps form the non - standard definition of limit ( see section  [ ten ] , item  [ 1010 ] ) . now the first step ( evaluating at  @xmath4 ) produces a hyperreal number dependent on  @xmath4 ( in all cases it will be strictly less than  @xmath1 ) . the second step will strip away the infinitesimal part and produce the standard real number  @xmath1 in the _ cluster _ ( see section  [ ten ] , item  [ 104 ] ) of points infinitely close to it . [ 51b ] the existence of more than one infinite hyperreal is not only a requirement to have a field , but is actually extremely useful . for example , using the natural hypperreal extension  @xmath37 of  @xmath38 ( see section  [ ten ] , item  [ 101 ] ) , it is possible to write down a pointwise definition of uniform continuity of a function  @xmath38 ( see below ) . such a definition considerably reduces the quantifier complexity of the standard definition .    to elaborate , note that the standard definition of uniform continuity of a real function  @xmath38 can be said to be _ global _ rather than _ local _ ( i.e. pointwise ) , in the sense that , unlike ordinary continuity , uniform continuity can not be defined as a pointwise property of  @xmath38 . meanwhile , in the framework of robinson s theory , it is possible to give a definition of uniform continuity of the real function  @xmath38 in terms of its natural hyperreal extension , denoted @xmath39 in such a way that the definition is local , in the sense of depending only on each pointwise cluster ( see section  [ ten ] , item  [ 104 ] ) in the domain of  @xmath37 . thus ,  @xmath38 is uniformly continuous on  @xmath40 if the following condition is satisfied : @xmath41 here  @xmath42 stands for the relation of being infinitely close . the condition must be satisfied at the infinite hyperreals  @xmath43 , in addition to the finite ones . this addition is what distinguishes uniform continuity from ordinary continuity . the symbol `` @xmath44 '' is employed in standard real analysis to define a formal completion of the real line  @xmath40 , namely @xmath45 ( sometimes a formal point  `` @xmath46 '' is added , as well ) . such a formal device is helpful in simplifying the statements of certain theorems ( which would otherwise have a number of subcases ) . the symbol is used in a _ different _ sense in topology and projective geometry , where the thom compactification  @xmath47 of  @xmath40 is a circle : @xmath48 we have refrained from using the symbol `` @xmath44 '' to denote an infinite hyperreal , as in @xmath49 even though the symbol `` @xmath44 '' does convey the idea of the infinite more effectively than the symbol `` @xmath4 '' that we have used . the reason is so as to avoid the risk of creating a false impression of the uniqueness of an infinite point ( as in or above ) , in the field  @xmath50 ( see section  [ ten ] , item  [ 102 ] ) . we represent the hyperreal @xmath51 visually by means of an infinite - resolution microscope already exploited for pedagogical purposes by keisler @xcite . the hyperreal  @xmath34 appearing in the diagram of figure  [ micro ] illustrates graphically the strict hyperreal inequality @xmath52 where , as we already mentioned , the symbol  @xmath4 is exploited to denote a fixed infinite robinson hyperinteger ( see section  [ ten ] , item  [ 108 ] ) . to calculate the slope of the tangent line to the curve @xmath53 at @xmath54 , we first compute the ratio @xmath55 where we have deleted the underbrace @xmath56 and also shortened the symbol  @xmath0 to @xmath57 , so as to lighten the notation .     at @xmath54,height=288 ] therefore the slope is @xmath58 note that @xmath59 in lightstone s notation , where the digit `` 1 '' appears in the @xmath60-decimal place , as illustrated in figure  [ slope ] . everyone who has ever held an electronic calculator is familiar with the curious phenomenon of it sometimes returning the value @xmath61 in place of the expected  @xmath62 . for instance , a calculator programmed to apply newton s method to find the zero of a function , may return the  @xmath63 value as the unique zero of the function @xmath64 developing a model to account for such a phenomenon is complicated by the variety of the degree of precision displayed , as well as the greater precision typically available internally than that displayed on the lcd . to simplify matters , we will consider an idealized model , called a _ hypercalculator _ , of a theoretical calculator that applies newton s method precisely  @xmath4 times , where  @xmath4 is a fixed infinite hyperinteger , as discussed in the previous section . let  @xmath38 be a concave increasing differentiable function with domain an open interval  @xmath65 and vanishing at its midpoint . then the hypercalculator applied to  @xmath38 will return a hyperreal decimal  @xmath0 with an initial segment consisting of an unbounded number of repeated  @xmath3 s . assume for simplicity that  @xmath66 . we have @xmath67 by the mean value theorem , there is a point  @xmath68 such that  @xmath69 where  @xmath70 , or @xmath71 since  @xmath38 is concave , its derivative  @xmath72 is decreasing , hence @xmath73 thus  @xmath74 . inductively , the point  @xmath75 satisfies  @xmath76 for all  @xmath16 . by the transfer principle ( see section  [ ten ] , item  [ 101 ] ) , the hyperreal  @xmath77 satisfies a strict inequality @xmath78 as well ( cf .  ) . hence the hypercalculator returns a value strictly smaller than  @xmath1 yet infinitely close to  @xmath1 , proving the theorem . the precise meaning of the finite expression @xmath79 is that the repeated digit 9 occurs precisely  @xmath16 times . the standard non - terminating decimal @xmath80 as it is traditionally written , is said to have an unbounded number of repeated digits 9 , but the expression  infinitely many  9 s " is only a figure of speech , as  infinity " is not a number in standard analysis , in the sense that , whenever a precise meaning is attributed to the phrase  infinitely many  9 s " , it is almost invariably in terms of _ d. tall writes in @xcite as follows :    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ [ ... ] the infinite decimal  @xmath81 is intended to signal the _ limit _ of the sequence  @xmath82  @xmath83  @xmath84 , ... which is @xmath1 , but in practice it is ofter imagined as a limiting process which never quite reaches  @xmath1 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _    tall @xcite describes a concept in cognitive theory he calls a _ generic limit concept _ in the following terms :    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ [ ... ] if a quantity repeatedly gets smaller and smaller and smaller without ever being zero , then the limiting object is naturally conceptualised as an extremely small quantity that is not zero ( cornu @xcite ) . infinitesimal quantities are natural products of the human imagination derived through combining potentially infinite repetition and the recognition of its repeating properties . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _    ( see @xcite for the related notion of _ procept _ ) . a technical realisation of the cognitive concept of _ generic limit _ is achieved in the hyperreal line ( see section  [ ten ] , item  [ 102 ] ) as follows . given an infinite hyperinteger  @xmath4 , consider the hyperreal repeated decimal where the repeated digit  @xmath3 occurs precisely  @xmath4 times , in the sense routinely used in non - standard calculus , for example , when one partitions a compact interval into  @xmath4 parts in the proof of the extreme value theorem ( see section  [ ten ] , item  [ 109 ] ) . such a number can be denoted suggestively by  @xmath34 with an underbrace indicating that the digit  @xmath3 occurs  @xmath4 times , resulting in a strict inequality  @xmath85 ( with the underbrace indicating that we are not referring to the standard real ) . in lightstone s notation , this hyperreal would be expressed by the hyperdecimal @xmath86 the last digit 9 occurring in the  @xmath4-th position . such a hyperreal appears to be the mathematical counterpart of the cognitive concept of _ generic limit_. as far as standard limits are concerned , given the sequence  @xmath87 , @xmath88 ,  @xmath89 , etc . , from the hyperreal viewpoint we have @xmath90 where  st " is the standard part function which  strips away " the infinitesimal part , and outputs the standard real in the cluster of the hyperreal  @xmath91 ( see section  [ ten ] , item  [ 104 ] ) . what may be bothering the students is this unacknowledged application of the standard part function , resulting in a loss of an infinitesimal . we have , in fact , been looking at the problem `` from above '' , in the context of non - standard analysis . perhaps a helpful parallel is provided by the famous animated film _ flatland _ ( cf . @xcite ) , where the two  dimensional creatures are unable to conceive of what we think of as the sphere in 3-space , due to their dimension limitation . similarly , one can conceive of the difficulty in the understanding of the unital evaluation of  @xmath0 , as due to the limitation of the standard real vision . the notion of an infinitesimal appeals to intuition ( see r.  courant s comment in section  [ history ] ) and would not go away inspite of what is , by now , over a century of  @xmath92-ideology ( see e.  bishop s comment in section  [ history ] ) . highschool students are exposed to the thorny  @xmath0 issue before they are exposed to any rigorous notion of a _ real number_. they are not aware of fine differences between rational numbers , algebraic numbers , real numbers , hyperreal numbers . a related point is make by keisler in his textbook @xcite , when he points out that `` we have no way of knowing what the line in physical space looks like '' . most students ( perhaps all ) initially believe that the mysterious number with `` infinitely many '' repeated digits 9 falls short of the value  @xmath1 . if an education professional claims that the students are making a mistake , might he in fact be making a pedagogical error ? in this section we present some illustrative terms and facts from non - standard calculus @xcite . the relation of being infinitely close is denoted by the symbol  @xmath42 . thus ,  @xmath93 if and only if  @xmath94 is infinitesimal .      the _ extension principle _ of non - standard calculus states that every real function  @xmath38 has a hyperreal extension , denoted  @xmath37 and called the natural extension of  @xmath38 . the _ transfer principle _ of non - standard calculus asserts that every real statement true for  @xmath38 , is true also for  @xmath37 . for example , if  @xmath95 for every real  @xmath43 in its domain  @xmath96 , then  @xmath97 for every hyperreal  @xmath43 in its domain  @xmath98 . note that if the interval  @xmath96 is unbounded , then  @xmath98 necessarily contains infinite hyperreals . we will typically drop the star  @xmath99 so as not to overburden the notation . internal set is the key tool in formulating the transfer principle , which concerns the logical relation between the properties of the real numbers  @xmath40 , and the properties of a larger field denoted @xmath100 called the _ hyperreal line_. the field  @xmath50 includes , in particular , infinitesimal (  infinitely small " ) numbers , providing a rigorous mathematical realisation of a project initiated by leibniz . roughly speaking , the idea is to express analysis over  @xmath40 in a suitable language of mathematical logic , and then point out that this language applies equally well to  @xmath50 . this turns out to be possible because at the set - theoretic level , the propositions in such a language are interpreted to apply only to internal sets rather than to all sets . note that the term  language " is used in a loose sense in the above . a more precise term is _ theory in first - order logic_. internal sets include natural extension of standard sets . the standard part function st " is the key ingredient in abraham robinson s resolution of the paradox of leibniz s definition of the derivative as the ratio of two infinitesimals @xmath101 the standard part function associates to a finite hyperreal number  @xmath43 , the standard real  @xmath102 infinitely close to it , so that we can write @xmath103 in other words , `` st '' strips away the infinitesimal part to produce the standard real in the cluster . the standard part function st " is not defined by an internal set ( see item  [ 102 ] above ) in robinson s theory . each standard real is accompanied by a cluster of hyperreals infinitely close to it . the standard part function collapses the entire cluster back to the standard real contained in it . the cluster of the real number  @xmath104 consists precisely of all the infinitesimals . every infinite hyperreal decomposes as a triple sum @xmath105 where  @xmath4 is a hyperinteger ,  @xmath106 is a real number in  @xmath7 , and @xmath107 is infinitesimal . varying  @xmath107 over all infinitesimals , one obtains the cluster of  @xmath108 .      to define the real derivative of a real function  @xmath38 in this approach , one no longer needs an infinite limiting process as in standard calculus . instead , one sets @xmath109 where  @xmath107 is infinitesimal , yielding the standard real number in the cluster of the hyperreal argument of `` st '' ( the derivative exists if and only if the value   is independent of the choice of the infinitesimal ) . the addition of `` st '' to the formula resolves the centuries - old paradox famously criticized by george berkeley @xcite ( in terms of the _ ghosts of departed quantities _ , cf . * chapter  6 ) ) , and provides a rigorous basis for the calculus . a function  @xmath38 is continuous at  @xmath43 if the following condition is satisfied :  @xmath110 implies  @xmath111 . a function  @xmath38 is uniformly continuous on  @xmath96 if the following condition is satisfied :    * standard : for every  @xmath112 there exists a  @xmath113 such that for all  @xmath114 and for all  @xmath115 , if  @xmath116 then @xmath117 . * non - standard : for all  @xmath118 , if  @xmath93 then  @xmath119 .    see remark  [ 51b ] for a more detailed discussion .      a hyperreal number  @xmath4 equal to its own integer part @xmath120\ ] ] is called a hyperinteger ( here the integer part function is the natural extension of the real one ) . the elements of the complement @xmath121 are called infinite hyperintegers . let  @xmath4 be an infinite hyperinteger . the interval  @xmath2 $ ] has a natural hyperreal extension . consider its partition into  @xmath4 subintervals of equal length  @xmath122 , with partition points  @xmath123 as  @xmath124 runs from  @xmath104 to  @xmath4 . note that in the standard setting , with  @xmath16 in place of  @xmath4 , a point with the maximal value of  @xmath38 can always be chosen among the  @xmath125 partition points  @xmath126 , by induction . hence , by the transfer principle , there is a hyperinteger  @xmath127 such that  @xmath128 and @xmath129 consider the real point @xmath130 an arbitrary real point  @xmath43 lies in a suitable sub - interval of the partition , namely  @xmath131 $ ] , so that  @xmath132 . applying `` st '' to the inequality , we obtain by continuity of  @xmath38 that  @xmath133 , for all real  @xmath43 , proving  @xmath68 to be a maximum of  @xmath38 ( see @xcite ) .      we have  @xmath134 if and only if whenever the difference  @xmath135 is infinitesimal , the difference  @xmath136 is infinitesimal , as well , or in formulas : if  @xmath137 then  @xmath138 .    given a sequence of real numbers  @xmath139 , if  @xmath140 we say  @xmath141 is the limit of the sequence and write  @xmath142 if the following condition is satisfied : @xmath143 ( here the extension principle is used to define  @xmath144 for every infinite value of the index ) . this definition has no quantifier alternations . the standard  @xmath92-definition of limit , on the other hand , does have quantifier alternations : @xmath145      given a real decimal @xmath146 consider the sequence  @xmath147 ,  @xmath148 ,  @xmath149 , etc . then by definition , @xmath150 for every infinite  @xmath4 . now if  @xmath151 is a non - terminating decimal , then one obtains a strict inequality  @xmath152 by transfer from @xmath153 . in particular , @xmath154 where the hat  @xmath155 indicates the  @xmath4-th lightstone decimal place . the standard interpretation of the symbol  @xmath0 as  @xmath1 is necessitated by notational uniformity : the symbol  @xmath156 in every case corresponds to the limit of the sequence of terminating decimals  @xmath157 . alternatively , the ellipsis in  @xmath0 could be interpreted as alluding to an infinity of nonzero digits specified by a choice of an infinite hyperinteger  @xmath158 . the resulting  @xmath4-infinite extended decimal string of  @xmath3s corresponds to an infinitesimally diminished hyperreal value  . such an interpretation is perhaps more in line with the naive initial intuition persistently reported by teachers . prior to robinson , mathematicians thought of infinitesimals in terms of  naive befogging " and  vague mystical ideas " . thus , richard courant @xcite wrote as follows :    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ we must , however , guard ourselves against thinking of  @xmath159 as an  infinitely small quantity " or infinitesimal " , or of the integral as the  sum of an infinite number of infinitesimally small quantities " . such a conception would be devoid of any clear meaning ; it is only a naive befogging of what we have previously carried out with precision . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _    and again on page 101 :    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ we have no right to suppose that first  @xmath160 goes through something like a limiting process and reaches a value which is infinitesimally small but still not 0 , so that  @xmath160 and  @xmath161 are replaced by  infinitely small quantities " or  infinitesimals " @xmath159 and  @xmath162 , and that the quotient of these quantities is then formed . such a conception of the derivative is incompatible with the clarity of ideas demanded in mathematics ; in fact , it is entirely meaningless . for a great many simple - minded people it undoubtedly has a certain charm , the charm of mystery which is always associated with the word  infinite " ; and in the early days of the differential calculus even leibnitz [ sic ] himself was capable of combining these vague mystical ideas with a thoroughly clear understanding of the limiting process . it is true that this fog which hung round the foundations of the new science did not prevent leibnitz [ sic ] or his great successors from finding the right path . but this does not release us from the * duty of avoiding every such hazy idea * [ emphasis added  mk ] in our building - up of the differential and integral calculus . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _    needless to say , courant s criticism was not without merit _ at the time of its writing _ ( a quarter century prior to robinson s discovery ) . i.  lakatos @xcite wrote in 66 as follows :    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ robinson s work ... offers a rational reconstruction of the discredited infinitesimal theory which satisfies modern requirements of rigour and which is no weaker than weierstrass s theory . this reconstruction makes infinitesimal theory an almost respectable ancestor of a fully fledged , powerful modern theory , lifts it from the status of pre - scientific gibberish , and renews interest in its partly forgotten , partly falsified history . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _    not everyone was persuaded . a decade later , courant s _ duty of avoiding every such hazy idea _ was taken up under a constructivist banner by e.  bishop . in his essay @xcite cast in the form of an imaginary dialog between brouwer and hilbert , e. bishop anchors his foundational stance in a species of mathematical constructivism . thus , bishop s opposition to robinson s infinitesimals , expressed in a bristling review  @xcite of keisler s textbook  @xcite , was to be expected ( and in fact was anticipated by editor halmos ) . in a memorable parenthetical remark , bishop writes @xcite :    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ although it seems to be futile , i always tell my calculus students that mathematics is not esoteric : it is common sense . ( even the notorious  @xmath163-definition of limit is * common sense * [ emphasis added  mk ] , and moreover it is central to the important practical problems of approximation and estimation . ) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _    bishop is referring , of course , to the type of _ common - sense _ definition reproduced in  , which he favors over keisler s definition in terms of robinson s hyperreals . bishop expressed his views about non - standard analysis and its use in teaching in a brief paragraph toward the end of his essay  crisis in contemporary mathematics " @xcite . after discussing hilbert s formalist program he writes :    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ a more recent attempt at mathematics by formal finesse is non - standard analysis . i gather that it has met with some degree of success , whether at the expense of giving significantly less meaningful proofs i do not know . my interest in non - standard analysis is that attempts are being made to introduce it into calculus courses . it is difficult to believe that * debasement of meaning * [ emphasis added  mk ] could be carried so far . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _    bishop s view of the introduction of non - standard analysis in the classroom as no less than a  debasement of meaning " , was duly noted by j.  dauben  @xcite .    to illlustrate how bishop anchors his foundational stance in a species of mathematical constructivism , note that in @xcite , he writes :    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ to my mind , it is a major defect of our profession that we refuse to distinguish [ ... ] between integers that are computable and those that are not [ ... ] the distinction between computable and non - computable , or constructive and non - constructive is the source of the most famous disputes in the philosophy of mathematics ... _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _    on page 511 , bishop defines a principle ( lpo ) as the statement that    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ it is possible to search `` a sequence of integers to see whether they all vanish '' , _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _    and goes on to characterize the dependence on the lpo as a procedure both brouwer and bishop himself reject . s.  feferman @xcite explains the principle as follows :    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ bishop criticized both non - constructive classical mathematics and intuitionism . he called non - constructive mathematics  a scandal " , particularly because of its  deficiency in numerical meaning " . what he simply meant was that if you say something exists you ought to be able to produce it , and if you say there is a function which does something on the natural numbers then you ought to be able to produce a machine which calculates it out at each number . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _    the constructivist objections to lpo are similar to objections to the law of excluded middle , and to proofs by contradiction ( for example , the traditional argument for the irrationality of @xmath164 is a proof by contradiction , but using classical continued fraction estimates , it is easy to rewrite it in a numerically meaningful fashion , acceptable from the constructivist viewpoint ) .    given that a typical construction of robinson s infinitesimals ( see keisler @xcite ) certainly does rely on lpo , bishop s opposition to such infinitesimals , expressed in a bristling review @xcite of keisler s textbook , was to be expected . non - standard calculus in the classroom was analyzed in the chicago study by k.  sullivan @xcite . sullivan showed that students following the non - standard calculus course were better able to interpret the sense of the mathematical formalism of calculus than a control group following a standard syllabus . such a conclusion was also noted by m.  artigue  @xcite . in the matter of teaching decimal notation , we would like to obtain some reaction from educators to the following proposal concerning the problem of the unital evaluation of  @xmath0 . a student may ask :    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ what does the teacher mean to happen exactly after _ nine , nine , nine _ when he writes _ dot , dot , dot _ ? _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _    how is a teacher to handle such a question ? experience shows that toeing the standard line on the unital evaluation of  @xmath165 possesses a high - frustration factor in the classroom . rather than baffling the student with such a categorical claim , a teacher could proceed by presenting the following ten points , based on the material outlined in section  [ ten ] :    1 . the reals are not , as the rationals are not , the maximal number system ; 2 .   there exist larger number systems , containing infinitesimals ; 3 . in such larger systems , the interval  @xmath2 $ ] contains many numbers infinitely close to  @xmath1 ; 4 .   in a particular larger system called the hyperreal numbers , there is a generalized notion of decimal expansion for such numbers , starting in each case with an unbounded number of digits  `` @xmath3 '' ; 5 . all such numbers therefore have an arguable claim to the notation  `` @xmath0 '' which is patently ambiguous ( the meaning of the ellipsis  `` @xmath166 '' requires disambiguation ) ; 6 . all but _ one _ of them are strictly smaller than  @xmath1 ; 7 . the _ convention _ adopted by most professional mathematicians is to interpret the symbol `` @xmath0 '' as referring to the _ largest _ such number , namely  @xmath1 itself ; 8 . thus , the students intuition that  @xmath0 falls just short of  @xmath1 can be justified in a mathematically rigorous fashion ; 9 . the said extended number system is mostly relevant in infinitesimal calculus ( also known as differential and integral calculus ) ; 10 . if you would like to learn more about the hyperreals , go to your teacher so he can give you further references . a goal of our , admittedly non - standard , analysis is both to educate and to heal . the latter part involves placing balm upon the bewilderment of myriad students of decimal notation , frustrated by the reluctance of their education professionals to yield as much as an infinitesimal iota in their evaluation of the symbol  @xmath167 , or to acknowledge the ambiguity of an ellipsis . we are grateful to david ebin and david tall for a careful reading of an earlier version of the manuscript , and for making numerous helpful suggestions . abbott , edwin a. : the annotated flatland . a romance of many dimensions . with an introduction and notes by ian stewart . reprint of the 2002 original [ perseus publ . , cambridge ] . basic books , new york , 2008 . bishop , e. : review : h. jerome keisler , elementary calculus , _ bull . * 83 * ( 77 ) , 205 - 208 . cornu , b. : limits , pp .  153 - 166 , in advanced mathematical thinking . edited by david tall . mathematics education library , 11 . kluwer academic publishers group , dordrecht , 1991 . courant , r. : differential and integral calculus . i. translated from the german by e. j. mcshane . reprint of the second edition ( 1937 ) . wiley classics library . a wiley - interscience publication . john wiley & sons , inc . , new york , 1988 . dauben , j. : appendix ( 1992 ) : revolutions revisited . 7282 in revolutions in mathematics . edited and with an introduction by donald gillies . oxford science publications . the clarendon press , oxford university press , new york , 1992 . lakatos , imre : cauchy and the continuum : the significance of nonstandard analysis for the history and philosophy of mathematics . intelligencer 1 ( 1978 ) , no . 3 , 151161 ( originally published in 66 ) .          robinson , abraham : non - standard analysis . reprint of the second ( 1974 ) edition . with a foreword by wilhelmus a. j. luxemburg . princeton landmarks in mathematics . princeton university press , princeton , nj , 1996 . stewart , i. : from here to infinity . a retitled and revised edition of the problems of mathematics [ oxford univ . press , new york , 1992 ] . with a foreword by james joseph sylvester . the clarendon press , oxford university press , new york , 1996 . tall , d. : the psychology of advanced mathematical thinking , in advanced mathematical thinking . edited by david tall . mathematics education library , 11 . kluwer academic publishers group , dordrecht , 1991 .
is  @xmath0 equal to  @xmath1 ? 
 a.  lightstone s decimal expansions yield an infinity of numbers in  @xmath2 $ ] whose expansion starts with an unbounded number of repeated digits `` @xmath3 '' . 
 we present some non - standard thoughts on the ambiguity of the ellipsis , modeling the cognitive concept of _ generic limit _ of b.  cornu and d.  tall . 
 a choice of a non - standard hyperinteger @xmath4 specifies an @xmath4-infinite extended decimal string of  @xmath3s , corresponding to an infinitesimally diminished hyperreal value  . in our model , the student resistance to the unital evaluation of @xmath0 is directed against an unspoken and unacknowledged application of the standard part function , namely the stripping away of a ghost of an infinitesimal , to echo george berkeley . 
 so long as the number system has not been specified , the students hunch that @xmath0 can fall infinitesimally short of @xmath1 , can be justified in a mathematically rigorous fashion .    supported by the israel science foundation ( grants no .  84/03 and 1294/06 ) and the bsf ( grant 2006393 ) ]