Repairing the Number/Numeral Breach with the Restored Renaissance Counterpart Concept
by Robert H. Schmidt
The publication of Bonasoni's Algebra Geometrica is a good occasion to draw a distinction between the counterpart function of mathematical symbols, their imaging function and their signifying function. Phenomenologically speaking, all these kinds of symbols are different.
By COUNTERPART I mean a symbol that represents something else but is not meant to name it or point to it; instead, the counterpart exemplifies the object. When we are keeping score in a game, and make a single stroke on a piece of paper for each game won, one stroke exemplifies or "stands for" one game won. The counterpart need not resemble the object in its particularities. (I have introduced the word 'counterpart' because it is, on the one hand, something that stands against its object, and on the other hand, because it is counterparts that are most directly suited to becoming the medium for operational symbolism in which they become true counters.
By IMAGE I mean a symbol that represents something by being a likeness of it. Pictures and statues are obvious instances of images.
By SIGN I mean a symbol that names, points to, or intends something else, but need not resemble it in any way whatsoever. A road sign has such a signifying function.
There are also other kinds of symbols, such as emblems, symptoms, etc., but the above are the ones that immediately concern us here.
Many symbols may have more than one of the above functions, depending on how they are viewed. For example, the lines plotted on a nautical chart (or the elements of any other kind of model) might be considered a clear instance of images of pathways through the ocean, but they can also be used as counterparts to the actual courses they represent, particularly when we are involved in the chart and plotting the lines on it. And there are some cases that are truly difficult to analyze, and often times depend on our current scientific view of things. The nature of words is a good example of such a problem.
Signs and counterparts are in a certain sense opposites with respect to how they relate to their objects. It is the nature of counterparts to draw attention to themselves while it is the nature of signs to lead our attention away from themselves and toward the thing signified. In modern philosophical parlance, signs become transparent. Furthermore, it is the nature of counterparts to turn their objects "into" themselves (I use the preposition "into" here fully conscious of its ambiguity, for there is a sense in which the object becomes actually identified with its counterpart, and a sense in which the object only turns a certain side toward the counterpart; in my opinion, counterpart intentionality is at the root of all sympathetic magic.) It is the nature of signs to disclose their objects simply by pointing them out. Consider what happens when we dwell too much on the words as sounds (or as typographical entities). We lose the meaning of what is said entirely. On the other hand it is of critical importance to pay attention to the markings on the chart when plotting a course. Only at the end do we convert this into instructions for sailing. These common experiences are an indication that there is a legitimate distinction to be drawn between counterparts and signs.
Unfortunately, however, since the same symbol can exhibit more than one of these functions, they often get jumbled together in our ordinary mathematical work. For example, Algebra sometimes is referred to as the language of mathematics, and sometimes as a calculus. Historically, too, there can be a mixup. For example, a symbol that was originally a counterpart can be transformed into a written verbal abbreviation (understood to have a signifying function) through usage and familiarity, and later reinterpreted as a counterpart in a new way. As a matter of fact, such interplay has been one of the driving forces of deep conceptual revision in mathematics.
Counterparts & Images
Bonasoni's work is valuable in clarifying this distinction because he used geometrical figures as counterparts instead of images in his algebraic argumentation, and since this had not been done before, he was fairly explicit about what he was doing and why he was doing it.
Ordinary Geometry would use the figure as an image - a kind of sign. According to the Aristotelean tradition that Bonasoni endorses in his preface, Geometry abstracts from the sensible and material qualities of the drawn figure in order to obtain its proper object, which is the pure figure. It also stays in this mode while investigating the figure. Even geometrical construction itself is just another way of drawing attention to the figure in a certain way in order to disclose some aspect of the figure. In other words, the sensible figure should become transparent so that the pure figure can be seen with the mind's eye. The image has here a signifying function.
But Bonasoni thinks that Algebra deserves to be called a Mathesis Universa, a Universal Mathematical Discipline, which then would be a subject for the highest metaphysical science, the one that studies Being as Being. Such a discipline does not abstract from sensible and material quality as mathematics does. It dwells in the realm of the sensibles, finding there both its proper subject, and also its means of investigation.
Reading between the lines a little, Bonasoni seems to imply that Algebra (particularly his geometrical kind) can belong to the Universal Mathematical Science - which is a subject of metaphysical investigation - because it never abandons the senses. When it treats of geometrical figures, then, it does not view them as images of intelligibles, but revels in the fact that they are sensible, under the supposition that the ultimate geometrical evidence lies in the figure as a sensible entity. It does seem pretty obvious that, if two rectangles having an equal width are joined together, they make a rectangle having the same width and length equal to the sum of the lengths of the two rectangles themselves. And the actual manipulations of sensible figures by grouping and regrouping rectangles in the manner of Book II of Euclid's Elements, could serve as an intuitive guide through the vicarious comparisons of the pure geometrical figures in geometrical questions. This is close enough to what I mean by a counterpart.
Incidentally, in the constructions and demonstrations that follow the algebraic investigation of questions, Bonasoni appears to return to the normal classical mode of thinking that views the figure as an image. So here the sign as image is identical to the counterpart, with the usages isolated in the algebraic argument and the construction/demonstration.
I find an odd echo of this kind of thinking in Hilbert's statement that he was an intuitionist in metamathematics, but a formalist in ordinary mathematics.
Counterparts & Numerical Notation
Counterpart thinking also seems to dominate in the use of numerical symbols during the Renaissance. We should suspect this after reading Bonasoni, since he says explicitly at the beginning of this second book that the arbitrary line denoting the sought line is to be reckoned with in exactly the same way as the res is operated on in Numerical Algebra. So, from the fact that Bonasoni fashions his Geometrical Algebra on the pattern of the existing Numerical one, we should also expect to find a counterpart function in the symbols used by his contemporaries in their numerical algebra. It is their counterpart nature that would entitle them to be directly operated on. Furthermore, since these "algebraic numbers" are added to and subtracted from ordinary written numbers these latter would also seem to share a counterpart function in Algebra.
In fact, ordinary written numbers seem to be directly subjected to counterpart manipulation not only in Numerical Algebra but even in the ordinary practical arithmetics of the time. It was this observation that caused Johannes Buteo to apply a new name to the content of these practical arithmetics. In 1560 he published his Logistica, Quae et Arithmetica Vulgo Dicitur ... (a work that Bonasoni refers to). From the short history of calculation that Buteo prefaces to this work, it is apparent that he considers calculation with numerical symbols to be merely a refinement of pebble calculation. As the title indicates, he thought that the practical arithmetics of this time were misnamed-and, for the very reason that they taught such an art. Logistic was the art traditionally associated with sensible collections of things, not Arithmetic, which had to do with intelligible or pure collections. But whereas Proclus and certain other Greek writers apparently meant that Logistic was about sensible collections like a basket of apples, Buteo insisted on the name 'logistic' for the contemporary art of calculation because it was performed with, or on, sensible collections like the stones of a counting board, the fingers of the hand, or the symbols written on paper.
However Buteo did not now simply transfer Algebra from Arithmetic to Logistic. Instead he surmised that the Greeks and not the Arabs had been the true founders of the algebraic art, that this Algebra used a "kind of calculation wholly geometrical", that it was especially able to cope with irrationals, and that its foundations could be discerned in the second book of Euclid's Elements. Accordingly, he assigned Algebra to Mensuration, an art related to Geometry in the same way that Logistic was to Arithmetic, and he renamed it Quadrature.
The art of Quadrature could be borrowed by Logistic and accommodated to numerical problems, and this was Buteo's immediate concern in the third book of his Logistica.
Thus Bonasoni & Buteo each had to make the counterpart function of mathematical symbols explicit, although they were led to these considerations from totally opposite starting points.
Bonasoni was trying to introduce a counterpart role for a science which was the paradigm of pure signification. He wanted to use the figure drawn on paper in the same way as the sensible numerical counterpart was used in the Arithmetic and Algebra of his day - not so much as an image meant to signify the pure geometrical entity and thus direct attention away from itself, as a sensible counterpart to this geometrical entity, one that is directly subject to constructive operations with compass and straightedge.
What Buteo wanted to do was remind his contemporaries that there had been a pure arithmetical science - analogous to Geometry and surviving in the books of Euclid, Nicomachus etc. - that had been lost sight of primarily because they only thought in terms of counterparts in numerical matters.
Numerical Notation in Renaissance Arithmetic
But where is the counterpart nature to be found in numerical and algebraic symbols? In algebraic argumentation leading to an equation like '2 res plus 15 is equal to 1 census' (or their various Renaissance equivalents when, say, some Cossist symbol is used for 'res' and 'census'), we seem to be dealing with notation. How can we understand that two 'res' have been actually collected together into one group, and this group has then been aggregated with 15? To the extent that notation is a matter of signification, no matter whether the significand is a pure number, or some definite sensible multitude that we wish to study, it would not seem to be subject to counterpart manipulation.
It is somewhat ironic that we call the symbols used in reckoning "numerals." For these symbols were certainly counterparts as well as signs, and thus much more than mere names. But there is no question that written numbers are signs as well as counterparts. Renaissance writers are greatly concerned with the correct notation for numbers. How can the counterpart function and the signifying function coexist in the same symbol? This issue seems to be taken for granted by Buteo, and all his contemporaries. Here is how I think the two functions were combined.
In the Roman system of reckoning, 'I' was a counterpart to some other one thing, whether sensible or abstract; it is purely and simply a counterpart. 'X' was indeed a sign, but what it signified was a bundle of ten 'I's in that same place; it was worth that many 'I's. It is thus a special kind of sign that "stands for" the object it signifies, and is spatially coincident with it; in this case it is implicitly a counterpart. 'XXV' stood for a more highly articulated collection of the same kind.
Roman symbols were thus identical with sensible multitudes on paper, and to that extent they were directly subject to manipulations that were the vicarious comparisons of other multitudes, whether sensible or abstract-which is the true function of a counterpart. They did not signify or intend these other multitudes but represented them in a calculation. And so the individual figures that made up these symbols were more like tokens on a counting board or abacus than written words, while calculation consisted of grouping and regrouping the 'I's into similar bundles.
The Hindu-Arabic reckoning system was introduced into the West around 1200 C.E. The aspect of the notation that was new (it was positional) did not affect the way in which the symbols were understood to fulfill their function. For Leonardo of Pis a, '25' stood for a collection consisting of five single strokes of the pen together with two bundles containing ten strokes each. In this case it is almost as if the Hindu-Arabic figure stands for collections of the Roman figures themselves and so we have a higher order notation of the same kind. Later '25' sometimes meant one bundle of five strokes, and ten bundles of two strokes, but this is even more directly a version of the same kind of thinking.
So the Hindu-Arabic figures originally had a strong counterpart function as well as a signifying function. This is why mathematicians throughout the Renaissance consistently referred to the strings of Hindu-Arabic figures as numbers (instead of introducing a distinction between number and numeral), and felt themselves able to operate directly on them.
It is now easy to understand how the symbolism in the Numerical Algebras was also manipulable. An expression like '2 res' should be understood no differently than an expression like '2' or '2 units', except that the two elements of the collection would in this case not be individual strokes on the paper, or other bundles like 'X', but a certain bundle defined as the number of strokes necessary in order to be compared to an ordinary number in a certain way. Similarly, '1 census' might be the bundle of stokes required in order to get into the second rank of algebraic numbers, if '1 res' is the first rank. And so on.
A fully legitimate operational symbolism is our unclaimed legacy from this time. The Renaissance arithmetician could temporarily set aside the multitudes he was actually interested in and calculate with their counterparts, without having to deny that the notation has a signifying function as the modern formalist does. And where the modern semanticist would have it that the marks on paper signify a pure number and have no true counterpart status at all, the Renaissance arithmetician would say that these marks signify a sensible multitude in that very place and are counterparts to the pure numbers. They are thus signs of a very special kind and unlike the words in a language, are spatially coincident with the things they signify.
Counterparts and Literal Symbolism
Literal symbolism was introduced into Algebra with a similar kind of signification, which rendered it fully operational from the very beginning. Operating directly on literal signs only became problematical when mathematicians began to attribute a different motive to the use of letters - namely, that it was for the sake of generality - than the motives originally held by Viete and Descartes, the mathematicians who first endorsed letters.
Francois Viete is the mathematician who, with the publication of his Introduction to the Analytic Art in 1591, first introduced letters into Algebra as the proper counterparts for his 'new' Logistica Speciosa. Viete did this as part of his project to restore the ancient Greek art of Analysis & Synthesis, which he thought was backed up by a superior Algebra.
Despite the relentlessly numerical character of the algebraic arguments in the Arithmetic of Diophantus, Viete could not believe that the Greek Algebra had been numerical. Accomplished Renaissance mathematicians like Regiomontanus had sometimes failed to squeeze good geometrical constructions out of their numerical algebraic arguments - and this on problems far less difficult than the ones that the Greeks had solved routinely.
Viele came to the conclusion that, in comparison with the putative Greek Algebra, the Algebra of his contemporaries was being practised "impurely". By this he did not mean that mathematicians should avoid using a calculative art that operated in the sensible realm when engaged in a geometrical investigation, but that their calculative art should not be such as to force attention onto the multitude of unit measures in a magnitude (the matter of a number according to tradition) as the Numerical Algebra did. Rather it should draw attention (as the Greek art must have, in his opinion) to the formal aspects of a magnitude. This does not mean the particular shape that it possesses, but the formal character it possesses due to its relationship to certain reference magnitudes.
That is, instead of referring this magnitude to a unit measure, it should be referred to two or more magnitudes that are not equal but have some other relationship of size to one another, such as being greater or less, or proportional, etc. Then some area can be seen, dimensionally, as equal to the rectangle made under the greater and lesser, with these two magnitudes being present to one another in such a way as to result in another magnitude of a higher genus, a longitude & a latitude producing something twodimensional. Or this same area can be looked at as something homogeneous, whose parts are present to one another in such a way as to give a magnitude of the same genus, say the sum of the squares on the greater and the lesser. These are the two basic formal aspects under which Viete thinks that the Greeks saw their magnitudes when using their Algebra. We might almost say that the magnitudes were viewed as "functions" of the reference magnitudes.
Such a demand on the calculative art called for a new kind of counterpart to be used in calculations. Since Viete thinks of his restored calculative art as a Logistic, he presumably thinks of his counterparts as collections of discrete things. It is not certain whether Viete took this term from Buteo. His Logistica Speciosa could certainly qualify as a realization of Buteo's project of a geometrically pertinent Algebra, although it is quite different from Bonasoni's, which takes as its counterparts continuous quantities made up by fitting together sensible magnitudes like lines and rectangles.
The new counterparts, then, consisted not of collections of sensible dots or strokes regarded as all alike (as in Numerical Logistic), but rather of collections of unequal things that have a specified relationship of size to one another. A typical collection of this kind might be made by grouping the greater of two lines together with the lesser taken twice. Each of the two unequal lines has a role analogous to the unit in Arithmetic, although they cannot be reduced to each other; it is as if we had two entirely different kinds of units. They are the elements from which the collections are assembled.
Viete used letters as the elements in his collections perhaps because, when viewed purely as sounds, the formation of a syllable from two letters is analogous to the formation of a magnitude of a totally different kind from two others, while the formation of a word from syllables is analogous to the combination of homogeneous magnitudes into another magnitude of the same kind.
A letter might also signify a collection of these elementary letters in exactly the same way that the Roman numeral 'X' stands for a collection of ten 'I's, while the 'I' is itself the ultimate element in any numerical collection. In a sense, the letter bundles the collection all together in a convenient package that can be operated on during calculation. For example, 'B longitudo' could stand for the grouping made form two elementary sides 'A latus' and 'E latus' taken together.
More complex collections could be made by joining together two of these letter bundles with a '+' sign. But such aggregations were not the only kinds of collections that Viete was interested in, for they only correspond to the geometrical operation of joining two magnitudes together. Just as often we would want to consider taking one magnitude away from another. What would correspond to this is a collection of elements one of whose members is regarded as lacking. Such a collection could also be designated by a letter, or a more complex collection of this type could be formed by a '-' sign.
But here it is necessary to consider this difference collection to be an entirely different kind of collection from the aggregate collection. For, unlike Numerical Logistic, if the elementary letters in a difference collection are counterparts to a greater and a lesser magnitude, say, it will be impossible to re-express this same difference as any aggregate of these letters, no matter how many times the letters may be taken. This is a direct consequence of the fact that the elements are not all identical, so not every subtraction results in something which is itself an aggregate of units as in Arithmetic. So in Viete's Logistic, the difference collection has a status independent of the aggregate collection, and of equal rank with it. Similarly, there is a special kind of collection analogous to the result of multiplication in Arithmetic, and corresponding in Geometry to the formation of rectangles or parallelepipedal solids out of the sides that compose them. And again one analogous to division.
In deference to Greek precedent, Viete called these new kinds of counterparts species, as if to emphasize that he is interested in the way that magnitudes show themselves in a problem-that is, the role they have. This term is used by Diophantus to describe the unknown squares, cubes, etc., that are sought in arithmetical problems, and also the root of these "powers". ViNe simply extended this term to include the "forms" that are made from roots or sides that are not equal, such as rectangles etc. This is why he called his new art of calculation Logistica Speciosa.
Thus, letters were introduced into Algebra in accordance with the view of arithmetical notation that had been current throughout the Renaissance. This had nothing to do with achieving a greater generality of expression, The possibility of operating directly on them was tied up with the way they were thought to be formable into 'syllables' and 'words' or else to already signify such a sensible collection.
Although the introduction of letters into Algebra made a big impact, it does not seem that very many people understood what Viete meant by species. Unfortunately, Viete wrote very concisely, had few pupils, and died just when the collecting of his works had begun. So even in the first years after his death there was much disagreement as to just what function he meant his letters to serve. I don't think that anyone ever again viewed geometrical entities under this same formal aspect, or further pursued the different kinds of counterpart collections that arise by grouping dissimilar yet mutually interrelated elements.
Descartes had a different reason for using letters in his own Geometrical Algebra, but he too combined the counterpart function of these letters with their signifying function in the Renaissance manner.
Descartes thought that the Greeks had been on the right track with their idea of a Universal Mathematical Science, which he thought had concentrated on the relations between magnitudes, and he too thought that Greek Analysis had been an art in the service of this Universal Mathematical Science. Furthermore, Descartes also was convinced that Greek Analysis had been algebraic. Yet he believed that this Greek Algebra put too great a demand upon the imagination by confining itself entirely to figures in its counterpart manipulations (compare Bonasoni's Algebra, which in fact seems to possess many of the features that Descartes suspected the Greek art had): this made it hard for the intellect to consider the relations in a general way.
The Numerical Algebras that had preceded Descartes, even though they too were a step toward the Universal Science, erred in the opposite direction. They had the advantage of using notation in their calculations and so would not fatigue the imagination the way the Greek approach did. But they got so involved in a tangle of ciphers and notations to which nothing imaginable seemed to correspond that they ended up by confusing the intellect with too many special rules. In addition to this, numbers seem to introduce a distracting extraneous element into the univeral consideration of relations of size.
Descartes' central concern in his own Geometrical Algebra was to get the intellect and imagination working in accord. His plan was to combine the best elements of Greek Analysis (as he understood it) with the best elements of the Algebra of his time. So first of all he argued that extension would be the proper object for a general study of the relations of quantities- because the imagination was itself an extended body! Extended bodies would be used as counterparts in the comparison of any other quantities.
Descartes' expressed intention of correcting Greek Analysis with Algebra and vice versa is much more than a superficial remark. He means that he wanted to bring the two arts to the same level. This required him, on the one hand, to reconceptualize the relation between the shape of a body and its extension (which are separable only in thought) in accordance with the relation between numerical notation and what it denotes, so that the shape or figure is understood as a sign used to denote the extended body, which is itself the bearer of the counterpart function. On the other hand, he had to think of his literal algebraic notation and its extended object as inseparable, so that manipulations of the letters could be imagined to be manipulations of the extensions. When he wanted to study some particular relation he could employ his new figural "notation" and when he merely wanted to keep this relation in mind he could use his new literal "bodies". But he introduced a simple notation based on letters instead of the numerical and algebraic signs used before and he reduced the numerous figures used in Greek Analysis to the straight line and the rectangle. Finally, he showed how the arithmetical operations like addition and multiplication could be taken over into this Geometrical Algebra by introducing a unit measure, so that the product of two extended bodies would always be a fourth proportional and hence just another extended body.
In both of these ways, the sign and its counterpart significand are again coincident in the Renaissance manner. In Bonasoni's Algebra, however, the sensible counterpart magnitudes had indeed been designated by letters in the arguments themselves, but these letters were not in turn subject to counterpart manipulation. They were purely and simply signs used in the manner of verbal abbreviations to conveniently designate a certain element of the figure. Nor had the figure been detached from the sensible magnitude itself and thought of as notation. Thus Bonasoni had an algebraic symbolism without a corresponding algebraic notation.
For Descartes it was the counterpart function of his symbols that would allow him to disengage a problem from the particular subject matter and consider it in a general way. Isn't it in fact the very nature of a model to preserve the relations of what is modeled, but ignore the things that bear these relations? The construction of a model is not a matter of abstraction, for abstraction seems to be entirely an affair of signs and what they are supposed to intend.
And so for Descartes the letters stood for extended bodies-in this case sensible straight lines on paper. They did not signify other general or ideal magnitudes. And the relations that these counterparts can have are counterpart relations to the relations between any quantities whatsoever. And so these relations could be studied in a general way by means of these counterparts.
It is not clear whether Descartes knew of Viete's work when he did his own. In any case, it is different in that for him the object identified with the letters is an extended body (which makes it geometrical in a particularly important way), while for Viete it is a special kind of collection that ignores the figures of the individual elements in it and is only interested in the fact that they are different (which makes it more logistical or arithmetical).
Most of Descartes' thinking on the issue of signs & counterparts is packed into his little book Rules for the Direction of the Mind, a particularly difficult book to appreciate. It was not published until long after his death. By the time this book appeared it was customary to suppose that the main reason for using letters in Algebra was to achieve a certain level of generality. Indeed, both Viete's algebraic writings and this very book of Descartes were and have been read in that manner.
But if this had been the meaning of the literal symbolism, it would call into question the legitimacy of a literal operational symbolism, since the symbol cannot be identified with a definite sensible object. An ingenious attempt was made in this century to render general signs manipulable by making concepts, or 'second intentions,' the immediate significands of letters instead of things. This account was presented in context of an historical investigation called Greek Mathematical Thought and the Origin of Algebra, by Jacob Klein.
I would deny that the book has any historical validity, although, ironically, I think that the author invented a wholly new way of looking at symbols in his attempt to restore the philosophical ruminations of Viete and Descartes. He starts from the assumption that until the Renaissance symbols were pure signs, having as their objects things. He sees the role of Viete and Descartes as introducing, by means of a new kind of abstraction, signs that intend the general character of being a number, or the concept of number; they then understood these new symbols to be directly subject to manipulation almost as if they were primary mathematical objects themselves. It was only at this point that symbolism became operational. Klein states that arithmetical symbols first became operational in the work of Stevin, about the same time, and through similar reasonings.
I have argued that things took place in precisely the reverse fashion, namely, that the numerical symbols used in the West since 1200 C.E. were counterparts and thus fully operational from the outset; the sign was coincident with its counterpart significand. Then the introduction of manipulable letters into Algebra by Viete and Descartes was made possible because they were also to be coincident with their counterpart significands, which were fully definite sensible objects chosen so as to study traditional mathematical objects from new non-numerical points of view.
Perhaps a kind of justification of operational symbolism can be obtained in Klein's way, but it certainly takes the question of mathematical signification to its very frontier. It seems more natural to stay within the horizon of counterparts, and seek there for a suitable kind of signification.
This work by Klein is little known, however. The preferred modern view, which reduces symbols in a formal system to the level of signs that acquire meaning passively through a particular isomorphism, sidesteps the whole issue. The concept of a counterpartwhich holds the most natural way of accounting for operational use of letters - has virtually vanished. All this has rebounded back on the understanding of numerical notation, so that we have even lost the original concept of a numerical counterpart.
Numbers or Numerals
The modern understanding of a numeral as a name for a number effectively excludes the counterpart function of mathematical symbolism. This is causing us a lot of trouble in understanding the nature of mathematics, for we are even more inclined to think of other mathematical symbols (such as the letters in Algebra) in the same way.
I maintain that, as long as a numeral is understood to be merely a matter of nomenclature, and a number is assumed to be something totally distinct from what is written on paper, it is inevitable that there should be a stalemate between the formalists and the semanticists. For the formalists are forced to debate the issue of symbolism in terms of meaning, and so they can only insist that mathematical symbols acquire meaning passively through isomorphism. Yet the semanticists are led to the view that symbols actively signify mathematical entities as part of a mathematical language, and so cannot really account for the way we calculate with symbols. They must resort to making the significand something as questionable as a second intention in order to even begin to make them manipulable. So each of these camps is vulnerable to the attacks of the other.
The problem is that each of these views does account partially for the actual experience of the working mathematician. On the one hand, the mathematician has the feeling that his calculations are about something, which leads him to believe that he is writing some sort of language. On the other hand, he often puts aside the signification of the signs and simply calculates formally; at such times it would only slow him down to keep the signified entities in mind. The sharp distinction between number and numeral leaves him no choice but to endorse a position that accounts for one of these experiences, and to deny that the remaining experience has any real importance.
Is it any wonder, then, that the formalists take the next step of entirely reducing mathematics to an elaborate game of symbol shunting, denying that there is any special mathematical realm at all? Or that the semanticists, who hold that the true mathematical objects are different from the sensible marks on paper but signified by them, are plunged into a murky controversy as to whether the mathematical beings are creatures of thought or residents of some ideal realm?
The way that symbolism was regarded by mathematicians in the late Renaissance is an alternative to our two modern views. Serious consideration of this could go a long way toward breaking the grip of the number/numeral dichotomy.
Some Concluding Questions
Can the activity of operating directly on the mathematical symbolism of our time be legitmated by some extrapolation of this Renaissance view?
For example, is there really some sensible object that is implicitly signified by, and coincident with our notation, in the manner of Descartes?
Or is there some formal aspect under which the manipulations on letters can be regarded so as to make these letters themselves more clearly the sensible counterparts to the mathematical entities under consideration, in the manner of Viete?
In classical philosophy, questions of truth, being, signification, and intentionality are all interwined. (See a beautiful exposition of this in Heidegger's History of the Concept of Time.) In this editorial I have tried to show that, at least in mathematics since the Renaissance, the most natural and most intrinsic manner of relating mathematical symbols to their objects may not be with signification and intentionality, but rather as counterparts.
Where does this leave questions of truth and being in modern mathematics?