Mathematics, Substance and Surmise: Views on the Meaning and Ontology of Mathematics (2015)
Let G be a group
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 København Ø, Denmark
Traditional philosophy of mathematics has dealt extensively with the question of the nature of mathematical objects such as number, point, and line. Considerations of this question have a great interest from a historical point of view, but they have become largely outdated in light of the development of modern mathematics. The prevalent view of mathematics in the 20th (and the early part of the 21st) century has been some variant of the so-called formalistic view, according to which mathematics is a study of axiomatic systems or structures. In such structures the objects have exactly the properties set out in the axioms and nothing else can be said about their nature. The question of the nature of mathematical objects has become a non-issue. Similarly, the structure itself is completely determined by its set of axioms and one can say nothing meaningful about its nature other than that. In particular, in the formalistic understanding of mathematics a structure exists if its axioms are consistent among each other (i.e., non-contradicting).
In this chapter we shall first outline this formalistic view of mathematics, then we shall follow the historical developments that led to it, and finally we shall critically analyze the formalistic view of mathematics, not from a purely philosophical point of view but from the view of the practice and history of mathematics. Instead of asking what can in principle exist and what can in principle be proved, we shall ask what is studied in practice, i.e., which mathematical structures are singled out by mathematicians and which statements in the structure are found interesting enough to be worthy of investigation. This question is not answered by the formalistic view of mathematics. In order to investigate the question one must study what drives mathematicians when they develop new mathematics. Here the history of mathematics points to intriguing problems as a particularly forceful driving force. Many (perhaps most) mathematical theories have been developed as an aid to solving problems both inside and outside of mathematics.
Aristotle considered mathematical objects as abstractions of phenomena in nature. However, many other generating processes such as generalization, rigorization, and axiomatization have played an important role in the development of the mathematical structures and theories that are currently being investigated. Thus, for a historian like me, it is tempting to replace an Aristotelian ontology with a richer historical narrative taking all the driving forces and generating processes into account. Although such a narrative may not explain the ontology of mathematical structures and objects it gives an account of how actually studied mathematical structures and objects came into being and to some degree also why they were singled out among the many potentially possible structures and objects.
1 Mathematical structures
A mathematical structure is a set of things called elements of this set, equipped with some relation or operation. It could be an order relation < or a binary operation that to two elements of the set assigns a third element of the set. An example of the latter is addition + which to two numbers a,b assigns a third number called a + b. The relation or operation must satisfy certain rules or axioms like the usual commutative law of addition: a + b = b + a. So a mathematical structure is a set with one or more relations or operations satisfying a number of axioms. One of the most studied structures is a group that is defined in Box 1. From the axioms mathematicians then deduce theorems, i.e., logical consequences of the axioms.
Box 1: Groups
In mathematics a group is a finite or infinite set of elements a,b,c,… and an operation denoted which to two elements in the group a,b attaches a third element denoted a b. A simple example of such an operation could be addition + on the set of integers. In order to be a group the operation must satisfy three axioms.
Axiom 1 called the associative law: for all elements a,b,c in the set,
In other words if one first finds b c and then operates on this element from the left with a, one gets the same element as when one first finds a b and then operates with c on the right.
Axiom 2: There exists an element called a neutral element e with the property that for all elements a of the set
In other words, nothing happens when one operates on either side with the neutral element.
Axom 3: For every element a in the set there exists a so-called inverse element denoted such that
In other words if one operates on an element on the left or the right with its inverse element one gets the neutral element.
In the example mentioned above of the integers with the operation +, the associative law holds, the neutral element is 0 and the inverse element of a is –a.
Another example is the set of displacements of a plane or of space where thea b means the displacement one gets by first using the displacement b and then the displacement a. Here the neutral element is the “displacement” that does not move anything, and the inverse of a particular displacement is the reverse displacement back to the original position.
Also the set of displacements of a plane or space, that move a certain figure onto itself form a group. For example displacements of a plane that map a given line onto itself make up a group. A reflection in the given line or a parallel displacement in the direction of the line are examples of such motions.
If a is any object the set consisting of this object can be made into a group by stipulating that a a = a. Of course, this is a trivial group, but it shows that any object whatsoever can play the role of an object in a group.
But in what sense do such structures exist, what kind of objects are their elements, and what kind of knowledge do the theorems represent? Let us start with the first question: How do mathematicians argue for the existence of a structure? The philosophically minded reader will probably be disappointed if he/she looks in a modern mathematics book for an answer. For example in an introductory text book in algebra one will meet a definition of a group and then the following theorems will typically start with the phrase: “Let G be a group.” In books about mathematical analysis theorems often begin with the words: “Let H be a Hilbert space,” etc. Such declarations may remind us of God’s creation of the world about which Genesis reports: “And God said: “let there be light and there was light.” Do mathematicians play God when they do mathematics? Do they think that they can create what they speak? In a sense they do. When they define a mathematical structure like a group or a Hilbert space, it is thereby created in a mathematical sense. In order for a mathematical structure to exist it is not required that one can point to an instantiation of it in the physical world; the only thing a mathematician will require of such a structure is that it is free of contradiction. That is, it must be impossible to find a statement in the theory that can be shown to be a consequence of the axioms and whose negation is also a consequence of the axioms. This seems to be a minimal requirement. In fact if there were such a statement in a structure, ordinary logic would allow one to prove that all statements in the structure be both true and false. Such a structure is clearly uninteresting.
But what is then the nature of such a mathematical structure and in particular what is the nature of the elements (objects) in the structure? A mathematician will in general refuse to answer this question, not because he does not know the answer or is philosophically naïve, but because he has a good reason to leave the question unanswered. For example it is a central characteristic of the definition of a group that we do not explain what the elements of the group are. In fact that allows us to deal with groups of many different types of objects at once. The elements can be numbers or transformations or matrices or vectors or .... In this way a proof about groups kills many birds with one stone.
Thus, saying that a structure is a group does not tell us anything about what kind of objects the elements are, nor the precise way the operation works. It only tells us that the operation must satisfy certain rules: the three axioms.
A modern formalistically oriented mathematician will have a similar view of the nature of more classical mathematical concepts such as number and geometry. Indeed one can define the ordinary real numbers as a set with two binary operations called addition and multiplication and an ordering which satisfies a long list of axioms. Similarly, plane Euclidean geometry can be defined as two sets of elements called points and lines, and a list of relations between them such as “lie on” (a point can be said to lie on a line) “between” (a point on a line can be said to be in between two other points on the line), etc. that satisfy an even longer list of axioms. In a way, these definitions create the real numbers and plane geometry with its points and lines. But they do not tell us anything about the ontology of a real number or a point and a line. They only tell us how their elements (numbers or lines and points, respectively) relate to each other.
In this way the whole question of the ontology of mathematical theories and objects have been side stepped. Of course one can ask if something in nature behaves like a particular mathematical system, but the answer cannot be found by mathematical means. For example it is a physical question to investigate if physical space can conveniently or correctly be described by the mathematical structure we call Euclidean geometry. And the answer does not affect the mathematical validity of Euclidean geometry.
This formalistic or structuralist view of mathematical theories and their objects became widely accepted around 1900. We shall now consider the historical development that led to it.
2 From idealized (or abstract) nature to arbitrary structures
It was the ancient Greek mathematicians and philosophers who developed the idea of mathematics as a deductive science in which one proves theorems from a system of axioms. Aristotle clearly explains the need for an axiomatic basis for a deductive science: Indeed, if you want to prove a theorem, you must base the proof on some prior knowledge, i.e., some theorems that you already have proved to be true. But a proof of these more basic theorems must be based on even more fundamental theorems, etc. In order not to end in an infinite regress, one must begin the deductive process with some statements (the axioms) that one assumes to be true. The other theorems of the theory must then be deduced from the axioms. This description of the role of the axioms in a mathematical theory is shared by a modern mathematician. However, the view of the nature of the axioms has changed (For a fuller survey of the historical developments analyzed in this paper see ).
For the ancient Greek mathematicians and philosophers the axioms were considered (evident) truths of the world. Not the crude physical world of phenomena that we sense, but an ideal or abstract world. For Plato the real world was the world of ideas, and it was this world that the axioms dealt with. For Aristotle they dealt with abstractions. For example, when a line in Euclid’s Elements (c. 300 BC) is defined as having no width and the first postulate (axiom) claims that one can draw a straight line between two points, this is of course not true of the physical world. No mathematician can draw a breadthless line. Similarly, the third postulate claims that it is possible to draw a circle with a given center and a given radius. But a circle is defined as a curve (line) of no width that lies equally far from one given point (the center). Of course we know from experience, that we cannot draw an exact circle. Many years of experience with a compass can teach us to draw very accurate circles, but they will never be perfect. However, that does not matter because the circles and the lines that the mathematician deals with are not the physically imperfectly drawn circles and lines. It is the ideal circles and lines (or the abstracted circles and lines). Mathematicians can prove things about the ideal or abstract circles by basing their arguments only on the definition of the circle and line and the axioms. For Plato this makes mathematics an important discipline for philosophers because it can teach them how to get insight into the world of ideas.
So, for the ancient Greeks the objects of mathematics were idealized or abstracted objects, and axioms were (self-evident) truths about them. Thus all consequences (theorems) deduced from them were also true, although not self-evident truths. This view of mathematics was generally accepted until the 19th century when it was challenged from several directions. We shall now turn to these challenges.
2.1 Symbolic algebra
The natural numbers 1,2,3,… have been handled by humans as far back as we have written sources. In fact writing began as a way to denote numbers. Archeologists have unearthed 20,000 years old bones with engraved marks that are usually interpreted as representing a number of something. One can question whether such representations can be called writing, but also one of the first real written languages the Mesopotamian cuneiform writing grew out of an earlier way to represent numbers.
Negative numbers can be traced back to about year zero when Chinese mathematicians represented numbers as bamboo sticks on a counting board. One color was used for the positive numbers and another color for the negative numbers. Special rules of calculation were invented for handling the new numbers, for example, the rule saying that negative times negative is positive. Other cultures like the Indians (c. 500 AD) developed similar ideas later but other equally advanced cultures, for example, the ancient Greeks, did not introduce negative numbers.
As late as the 18th century, where the negative numbers were freely used by leading mathematicians like Euler, other mathematicians voiced objections. The problem with negative numbers was not primarily how to work with them (although the rule for negative times negative was problematized) but rather the question of their nature or ontology. For the ancient Greeks a number represented a collection of units; the number of things so to speak. They also developed an advanced theory of continuous quantities such as line segments, but they did not consider such quantities as numbers. The medieval Arab and the early modern European mathematicians gradually erased the dichotomy between number and quantity by extending the notion of number to include all the positive real numbers. But they continued to consider them as a measure of a quantity, such as a length of a line or an area of a surface. Negative numbers are not numbers in this sense because they do not represent the number of something or the measure of a quantity.
There are other ways to interpret negative numbers: In ancient China buying -5 goats just meant selling 5 goats. Later (at least in the 17th century) a directional geometric interpretation was suggested: To advance -5 feet in a particular direction on a straight line simply means to advance 5 feet in the opposite direction.
Despite such interpretations, negative numbers were often shunned. For example in Medieval and Renaissance algebra (Arab as well as Latin) negative solutions to equations were in general left out, and for good reasons. The equations were often set up as a means to find the length of a line segment or an area of a figure, and it is of course unclear what a line of a negative length or a figure with a negative area could mean.
At the beginning of the 19th century, a group of Cambridge mathematicians called the Analytical Society tried to circumvent the problem of negative numbers by dividing algebra into an arithmetical and a symbolical algebra. In arithmetical algebra the objects are (positive) numbers and the operations + .-,⋅ and: are the usual ones. However, the operations cannot be used unlimited. For example a-b only makes sense if a > b. In symbolical algebra on the other hand, the objects are just symbols a,b,… and the operations have unlimited validity. Subtracting a larger symbol from a smaller symbol is allowed and the result is a negative symbol –c.
One of the members of the group, George Peacock, defined algebra as “the science which treats of the combinations of arbitrary signs and symbols by means of defined though arbitrary laws.” So, according to Peacock the nature of the objects were not important in algebra; only the laws of combination such as the commutative laws: a + b = b + a and ab = ba were essential. However, neither he nor his follower Augustus De Morgan made use of the arbitrariness of the laws of combination. They both assumed that they were the same as in arithmetical algebra. This changed in 1853 when William Rowan Hamilton constructed the so-called quaternions which are a kind of four dimensional number. And here one of the usual laws could no longer be maintained: For quaternions the commutative law of multiplication: ab = ba no longer hold true.
The undefined or symbolic nature of the objects and the arbitrariness of the laws (axioms) in 19th century British algebra was soon taken over to other areas of mathematics.
The development of the concept of a group was also an important ingredient in the structuralist turn in mathematics. The word group in the mathematical sense was introduced around 1830 by the young revolutionary French mathematician Evariste Galois in connection with his ground breaking investigations of the question: Which nth degree equations can be solved by a formula in which enters only the operations: +.-,⋅, : and the extraction of pth roots. Following the lead of Joseph-Louis Lagrange, Galois showed how this problem could be elucidated by studying those permutations of the roots of the equations that keep the expressions of the coefficients fixed. He pointed out that such a set of permutations have the property that if one first permute with one of the permutations and then with another, the combined permutation of the roots will be in the set. He called such a set of permutations a group.
Galois died in a duel when he was only 20 years old and so his ideas only became publically known after their publication in 1846. Gradually the idea of a group found its way into algebra books and it was noticed, that similar structures were found elsewhere in pure and applied mathematics . In fact as early as 1801 Carl Friedrich Gauss had studied transformations of so-called quadratic forms that exhibit a similar structure as Galois’ permutations, and around 1870 Camille Jordan, Felix Klein and Sophus Lie showed how the group concept could be used in connection with geometric transformations as well. In this way it became clear that the properties of groups did not depend on the nature of the elements in the group: they can be permutations of the roots of an equation, transformations of space, or numbers. The one distinguishing feature of the concept of a group is the axioms satisfied by the binary operation involved. Arthur Cayley (1854) was the first who explicitly wrote down an abstract set of axioms for a (finite) group similar, but not identical to those in Box 1. However, in the middle of the 19th century it was still unheard of to define a mathematical structure solely by its axioms, and the definition was ignored. It took another half century before the axiomatic introduction of groups and other algebraic structures such as rings and fields became commonplace. The first algebra book in which algebra was introduced from the outset as the study of algebraic structures was Van der Waerden’s Moderne Algebra (1930-31). Until then, algebra had been introduced as the study of equations.
2.3 Non-Euclidean Geometry
The philosophically deepest root of the structural view of axioms as arbitrary starting points of our deductions came from geometry . Until around 1800 geometry had been viewed as the paradigm of an exact mathematical theory. However, one particular axiom of the theory had been up for debate from the beginning, namely the so-called parallel postulate. From the other postulates, it can be shown that if two straight lines have a common normal they are parallel, in the Euclidean sense, i.e., they do not intersect. The parallel postulate says that if one of the lines is turned even the smallest angle around its point of intersection with the normal then the two lines will intersect. This postulate was considered far less self-evident than the other postulates. Ancient Greek, medieval Arab, and early modern European mathematicians tried to prove the postulate from the other postulates. If they had succeeded, one could have left out the parallel postulate from the list of postulates. It would have become a theorem. However, although many mathematicians believed they had found a proof, the majority of mathematicians were never convinced.
With the Italian Jesuit Giovanni Girolamo Saccheri (1733) and the Swiss mathematician Johann Heinrich Lambert (1766) the approach to the question changed. They tried to prove the parallel postulate indirectly. They assumed that the parallel postulate was invalid and tried to reach a contradiction. If the parallel postulate is false there are infinitely many lines through a given point parallel to a given line (not through the given point). From this assumption they deduced a lot of consequences and at the end they reached what at least Saccheri considered a contradiction.
Around 1820 three mathematicians Gauss in Göttingen, Nikolai Ivanovich Lobachevsky in Kazan, and Janos Bolyai in Hungary came to the conviction that there was no contradiction in Saccheri’s system and that therefore the parallel postulate could not be derived from the other postulates. Moreover, at least Gauss and Lobachevsky were convinced that physical space could very well turn out to be a space in which the parallel postulate does not hold. This kind of geometry was called non-Euclidean by Gauss. The three mathematicians derived many weird properties of such a space, for example: The angle sum of a triangle is less than 180° and it decreases when the area of the triangle grows. This means that there is an upper bound for the areas of triangles and that one cannot reduce the size of a figure without distorting it (an architect’s nightmare). But despite these unfamiliar properties the three mathematicians decided, that they were not absurd. Both Gauss and Lobachevsky tried if they could measure a discrepancy from 180° in the angle sum of a triangle in physical space (a large one for which such an effect would be measurable), but they could not detect a deviation from Euclidean space.
At first the investigations of non-Euclidean geometry were overlooked. There are many reasons for that:
Two of the discoverers, Lobachevsky and Bolyai were otherwise unknown mathematicians who worked far from Paris, the center of mathematics of the time. Gauss was the most famous mathematician of the time, but he chose not to publish his ideas.
The conclusions of Gauss, Lobachevsky, and Bolyai were in sharp contrast to the very influential philosophical ideas of Immanuel Kant. Kant had argued that the geometry of space is an a priori synthetic intuition, and the geometry he had in mind was of course Euclidean geometry. As an example of how we argue geometrically he had in fact shown how one can prove in an a priori way that the angle sum of a triangle is 180°. Gauss explicitly contradicted this point of view and claimed that the nature of physical space had to be determined empirically.
None of the three mathematicians could prove that there was no inconsistency in non-Euclidean geometry.
All this changed around 1870. In 1866 Gauss’ correspondence was published showing that the prince of mathematics (Gauss’s nickname) had entertained such unconventional ideas. This led to re-publications and translations of the works of Lobachevsky and Bolyai. And finally it was proved that non-Euclidean geometry was as consistent as Euclidean geometry. The first mathematician to give such a proof was the Italian mathematician Eugenio Beltrami (1868). He accomplished this by presenting what we would call a model of non-Euclidean geometry. He built his model on Gauss’ differential geometry of surfaces (1827). Gauss had introduced a notion of curvature of a surface, and Beltrami could show that the geometry on a surface with constant negative Gauss curvature was non-Euclidean, at least if one considers geodesics (locally shortest curves) as straight lines. Felix Klein (1872) discovered the same model from the point of view of projective geometry and 10 years later the leading French mathematician Henri Poincaré hit upon another model.
As pointed out most clearly by Poincaré, such models show that if there is a contradiction in non-Euclidean geometry, it would, in the model, appear as a contradiction in Euclidean geometry. Thus if Euclidean geometry is free of contradictions then so is non-Euclidean geometry.
This left mathematicians at the end of the 19th century in the following situation. There are two equally consistent geometries. In Euclidean geometry the parallel postulate is an axiom of the theory, in the other the negation of the parallel postulate is an axiom. How should one decide between them? One way to do that was to find out if the parallel postulate was true of physical space or not. However, mathematicians of the late 19th century were reluctant to use this way out. First, during the 19th century, mathematics had emancipated itself from physics. To be sure, physics was still a great source of inspiration for mathematicians, but to let the foundations of mathematics depend on physical measurements was more than many late 19th mathematicians would allow. Second, the measurement of the (non-) Euclidean nature of space turned out to be a non-trivial matter. Indeed, if space is non-Euclidean it differs very little from Euclidean space in the sense that angle sums in measurable triangles are very close to 180°. So it is very likely that measurements will not be able to reveal a discrepancy from Euclidean geometry. But of course that does not exclude that space is non-Euclidean, it only shows that the deviation from Euclidean geometry is below our accuracy of measurement. So, leaving the decision to the physicists would delay the decision indefinitely, perhaps infinitely.
Moreover, Poincaré pointed out that the nature of space cannot be investigated empirically in isolation independently of other physical theories of nature. He argued as follows: assume that some day one can measure, that a large enough triangle has in fact an angle sum less than 180°. In that case we have two alternatives: We can conclude that physical space is non-Euclidean, or we can conclude that the triangle was not a real triangle after all. For example if the measurement is conducted by optical sighting between the vertices of the triangle we might conclude that the light rays that compose the sides of the triangle are not straight lines. That would force us to revise the laws of physics (Maxwell’s laws) but we can uphold Euclidean geometry. For Poincaré this shows that the nature of physical space is a convention. He actually argued that since Euclidean geometry is simpler than non-Euclidean geometry we will always stick to the convention that physical space is Euclidean. In case of a discrepancy as the one explained above, we will always choose to revise the laws of physics rather than the more basic laws of geometry.
Poincaré’s prediction turned out to be wrong. In fact, in 1905 and 1917, based on new empirical evidence, Albert Einstein suggested that space (or rather space-time) is best modelled on a kind of non-Euclidean geometry.
But let us return to the situation in the late 19th century. Mathematicians had developed two equally consistent geometric theories and there was no good way to decide which one to choose. In that situation mathematicians decided not to choose. Both alternatives were left as proper mathematical theories. One could choose the parallel postulate as an axiom or one could choose its negation. It is up to the mathematician, and one choice is as good as another. Two different but equally valid geometries will result from the choices. This situation illustrates Georg Cantor’s often quoted dictum (1896): “The essence of mathematics lies in its freedom.” But of course this freedom does not allow the traditional view of axioms as self-evident truths. One cannot claim that both the parallel postulate and its negation are self-evident. Instead axioms began to be considered as arbitrary starting points for our deductions.
2.4 Hilbert’s Foundations of Geometry
Toward the end of the 19th century it was obvious that Euclid’s geometry had to be rewritten. To be sure, the millennium long debate concerning the parallel postulate had revealed, that this postulate is indeed necessary, if one wants a geometry as the one described by Euclid. But it had also been revealed that Euclid in his proofs used many geometric assumptions that were neither stated as axioms nor deducible from the explicitly formulated axioms in Euclid’s Elements. For example, already in the first theorem, where Euclid constructs an equilateral triangle, he postulates that two specific circles intersect each other despite the fact that no axiom guarantees the existence of an intersection point. For this and other reasons, several mathematicians tried to come up with complete axiom systems for geometry. The most famous of them was David Hilbert’s Grundlagen der Geometrie (1899) , . Hilbert needed 21 axioms instead of Euclid’s 5 in order to be able to deduce the usual Euclidean theorems. One of them was a slight reformulation of the parallel postulate.
Hilbert changed Euclid’s presentation in another and even more fundamental way namely with regard to the definitions. Euclid’s Elements begins with a definition of the basic objects of geometry.
· “A point is that which has no part”
· “A line is a breadthless length”
· “A straight line is a line which lies evenly with the points on itself.”
And so on. It is obvious that these definitions leave something to be desired. For example, one can ask: What are a width and a length, and in particular what does it mean that points lie evenly with other points? One could be tempted to answer the last question with: when they lie on a straight line; but in that case the definition is circular. We are here up against the same problem that we met when we discussed the beginning of the process of deduction. There we argued that we need to begin with axioms that are just stated without requiring proofs. When analyzing the problem of definitions we meet with same problem that was pointed out by Aristotle: If we want to define the meaning of a certain word as “straight line” we must do so in terms of other words. But in order to define these more basic words, we must use even more basic ones, etc. Thus, in order to begin the process of definition we must begin with words that we do not define.
Hilbert took the consequence of this analysis and left words as point, straight line, lie on, between, etc., as undefined. He pointed out that the meaning of the words indirectly appear from the axioms. Such implicit definitions were sharply criticized by the philosopher-mathematician Gottlob Frege. He pointed out to Hilbert that such definitions did not allow one to decide if a particular object is a point or a line. For example he asked Hilbert if his pocket watch could not be considered as a point if one followed Hilbert’s approach. Hilbert answered in the positive, but did not consider this as a problem. If one can find a set including the pocket watch that can play the role of points and another set that can play the role of lines, and a suitable interpretation of the words lie on, lie between, etc., then this is indeed a Euclidean geometry in Hilbert’s sense and the pocket watch is a point of the theory.
He had another reason to let the objects remain undefined. In projective geometry Joseph Diaz Gergonne (early 19th century France) and other mathematicians had discovered a property called duality. If a theorem about points and lines in a plane is true, then one can find another true theorem by interchanging the words point and line. For example it is true that through two different points there is precisely one line (this is an axiom both in Euclid and in Hilbert). By interchanging the words point and line we get the statement: Through two different lines there is exactly one point. As formulated here, the statement sounds a bit strange, but if we interpret “through” as “on” (which we can because the word “through” is undefined) the statement says that given two different lines there is exactly one intersection point. This still sounds wrong, because two parallel lines do not intersect in Euclidean geometry, but in projective geometry one has adjoined points at infinity that play the role of the missing intersection points. So in projective geometry the dual statement is indeed true.
In view of the duality of projective geometry it is irrelevant whether we consider points and lines as we use to do or whether the long thing we usually call a straight line is considered a point, and the dot we usually call a point is considered a line. In this way the question of the nature of the point and the straight line or any other object in a mathematical theory becomes meaningless.
Thus for Hilbert an axiomatic mathematical system is a system of arbitrary axioms from which one can deduce theorems about undefined objects. However he stressed that one need to check one thing: The consistency of the axioms. If the axioms are consistent then the system exists in Hilbert’s sense. Thus Hilbert reduced the problem of existence in mathematics to the problem of consistency.
Existence of objects in a theory is determined by the rules of existence laid down in the axioms. A particular geometric configuration exists in Euclidean geometry, if it is postulated in the axioms of geometry (for example one of Hilbert’s axioms posits that there are at least four points not lying in a plane) or can be deduced from the existential claims in the axioms. For example one can prove that there exists a square on a given line segment.
In 1901 the mathematician and philosopher Bertrand Russell described this new view of mathematics as follows:
“Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true. Both these points would belong to applied mathematics. We start, in pure mathematics, from certain rules of inference, by which we can infer that if one proposition is true, then so is some other proposition. These rules of inference constitute the major part of the principles of formal logic. We then take any hypothesis that seems amusing, and deduce its consequences. If our hypothesis is about anything, and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate”.
In fact, since Russell, this definition has been relativized even more. In Russel’s definition of mathematics the one absolute is the principles of formal logic. Since then mathematicians and logicians have debated what these laws are, and today there are several competing schools favoring different rules of logical inference (e.g., formalist and intuitionist logic).
Another unsettling discovery concerning mathematical structures emerged in the 1930s when Kurt Gödel proved that consistency of a (suitably forceful system) cannot be proved within the system itself. This has the unpleasant consequence, that we cannot be sure that our mathematical theories are consistent. This in a sense puts an end to the long search for absolute certainty in mathematics.
In this section we have given a historical account of a very important change in the philosophy of mathematics: the change from thinking about mathematics as an idealized or abstract physical reality to thinking about it as a study of mathematical structures whose elements are undefined and whose axioms are in principle arbitrary. This change is often considered as one of the main revolutions of mathematics.
3 How arbitrary are mathematical axioms in practice?
According to Russell one begins a mathematical theory by picking “any hypothesis that seems amusing” and then one starts deducing its consequences. In this view mathematics is just a game the rules of which are chosen entirely arbitrarily. At first this may seem to be in complete accordance with Hilbert’s ideas about axiomatic systems. However, in a lecture of 1919–20 about his axiomatic method Hilbert flatly disagreed with Russell emphasizing that “We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules.” The difference between the two great thinkers may be described as a difference between a philosopher and a working mathematician. From a philosophical point of view it may be defendable to view axioms as arbitrary points of departure of the mathematical deductive game. However, a mathematician like Hilbert would know, that in a mathematical sense the axioms are suggested by earlier mathematical developments. In fact Hilbert’s view is revealed in the term he often used to describe his method. He talked about axiomatization. Of course, in order to axiomatize a theory it is necessary that such a theory is already developed in a non-axiomatic way. As Hilbert pointed out (see ), mathematics is not build like a house where one starts by laying the foundations. Instead, a mathematical theory is often begun without proper foundation, and only when the lack of a proper foundation becomes urgent, it is laid. According to Hilbert, axiomatization is a way to provide a foundation for already rather mature mathematical theories. Hilbert axiomatized geometry, a discipline that had been around in an imperfectly axiomatized version for more than 2000 years. He did not pick his axioms of geometry just because he thought they were amusing, as Russell put it, but because they were needed to support the structure we call Euclidean geometry. In a sense the axioms can be said to be proof generated in the sense that they are chosen to allow certain deductions that we want to make in order to arrive at the important theorems of the theory such as Pythagoras’ theorem. In the same way Hilbert axiomatized the theory of arithmetic of the real numbers and other previously developed theories.
It is doubtful if any important mathematical theory has ever been developed as Russell suggests, by first choosing an amusing set of axioms and then deducing away. This raises an important question: What is an interesting mathematical theory? Or phrased differently which theories will a good mathematician consider worth studying? Hilbert tried to explain and analyze what a true mathematical system is, but he did not develop any way to distinguish interesting from uninteresting axiomatic systems. Considering how many strange axiom systems one could in principle investigate, it is striking how few are in fact studied by mathematicians. In a sense any game with precisely stated rules can be considered a mathematical system, but they are not studied by mathematicians. Chess and a few similar games have been studied from a mathematical point of view, but groups have been studied much more extensively.
Similarly, within an axiomatic system the theorems are usually not discovered simply by deducing away beginning with the axioms. Considering the huge amount of mostly uninteresting things one can deduce it is not surprising that mathematicians usually discover interesting theorems by formulating heuristic conjectures and subsequently trying to find a proof. But again one can ask: what distinguishes the important theorems from the unimportant ones.
4 What is important? An historical approach
There is no formal mathematical way to measure the degree of interest of a mathematical structure or of a theorem within a structure. Still among mathematicians, one will find wide agreement about the question, with some minor differences of opinion on the edges.
In order to approach the question we need to investigate the driving forces in mathematics. What makes mathematicians develop this theory rather than that; what makes them want to prove this theorem rather than that? One answer was given by Hilbert in a famous public address to the international Congress of Mathematicians in 1900. According to Hilbert problems are the driving force behind the development of mathematics. In the talk and the subsequent written account of it he formulated 23 problems whose solution he expected would develop mathematics in the 20th century. The problems mentioned by Hilbert were all problems formulated inside mathematics. But also problems outside mathematics have been powerful driving forces for the development of mathematics. In a sense such extra-mathematical problems constitute the origin of mathematics. Take the concept of number. This concept was surely developed as a means to deal with problems of size and of number of physical things or social questions as trade. But once it was established the number concept gave rise to questions of a more internal mathematical nature: How many prime numbers are there? (The answer: “infinitely many” was found by the Ancient Greeks). How many prime twins (i.e., primes which are two apart as 3 and 5) are there? (The answer is unknown). Similarly geometry clearly began as a way to deal with physical space, but it soon raised questions that have little bearing on practical applications for example: Can one prove the parallel postulate from the other axioms?
So problems in nature and society provided the background for the beginning of mathematics. And extra-mathematical problems have continued to influence the development of mathematics from antiquity till this day. In ancient Greece, music, astronomy, optics and statics provided stimulus to mathematics. These areas of application were followed in the Renaissance and the early modern period by kinematics and perspective drawing and in the 19th century by heat, electricity, and elasticity. Finally in the 20th century almost all aspects of science, technology, and economics and society were studied and managed by mathematical models. All through history these areas of application continued to pose new mathematical questions that necessitated the development of existing theories or the creation of new areas of mathematics.
The development of mathematics has taken place through a combination of theorem proving, problem solving, generalization, abstraction, rigorization, axiomatization, etc. But in the final end, what has determined the success of a theory, a structure or a theorem is its ability to be applied in reality or in other mathematical theories, or more generally to fit in a fruitful way into the web of already developed mathematics . For example the concept of a group was developed in order to solve equations and it has shown its worth because it fits into many other mathematical theories and helps solving problems in physics and chemistry. Similarly, Pythagoras’ theorem is continually being taught not only because it can be proved from the axioms of Euclidean geometry but also because it is useful both in practical applications and (often in a generalized higher dimensional version) in many parts of mathematics.
Thus, Russell may be philosophically justified when he claimed that mathematics is like a game whose rules or axioms are entirely arbitrary and that mathematical theorems are found by successive derivations from the axioms. However, this image does not capture the real driving forces behind the practice of mathematics, and it cannot explain why certain structures and theorems are deemed important and others are not. A historian on the other hand, can tell stories of their genesis, stories that can shed light on this question. Such a story will contain a story of repeated abstraction, which according to Aristotle reveals the ontology of mathematical objects, but it will be much richer. In addition to the process of abstraction it will operate with many other driving forces and processes, and it will ultimately be judged by the question of applicability within or outside of mathematics. The story will not be deterministic as an Aristotelian story of abstraction might suggest. Instead there will enter many contingencies and many influences from all areas of society and nature.
The modern formalist structuralist view of mathematics leaves very little to be said concerning the nature of mathematical objects. The whole point of the structuralist approach is that the objects are left undefined. Any object whatsoever can play the role of an element in a group, a point in Euclidean geometry or as the real number 708. The only thing that characterizes an element in a mathematical structure is how it relates to the other elements in the structure. This is laid down in the axioms. From a formalistic point of view there is nothing else one can say.
And what is the ontological status of a mathematical structure? According to the formalistic philosophy consistency is the only requirement of existence. If it is consistent the system exists as a mathematical structure. In this way modern mathematics has in a sense circumvented the question of ontology.
However, from the point of view of the practice and history of mathematics the formalist account of existence is not complete. History of mathematics can provide stories that elucidate the formalistic account in several respects. First the historian can tell the story of how the modern formalist, structuralist point of view came about. This is what I tried to do in the long middle section of this chapter. Second the historian can tell the story of how particular structures came to be. This story is always richer than the standard story told by Russell: The mathematician X found this or that set of axioms amusing and began to deduce theorems from them. In real life, the story mostly begins with a problem, either a problem outside mathematics (be it practical or theoretical) or a problem within mathematics. The story will then unfold through a messy series of new mathematical theorems and more importantly through a series of transformations of the theory through generalizations, abstractions, clarifications, axiomatizations, etc. On the way, new problems inside or outside the theory, will drive the development in different directions, until it reaches the intermediate stage that we call the present. And the story will always contain an essential human element.
The story will also reveal the origin of the objects that in the end turn up as undefined objects in the theory: The lines and points in geometry, the real numbers, the elements in a group. In a sense such stories can be considered an alternative ontology of the modern structures and their objects.
And at least the history of mathematics can explain how certain mathematical structures came to be studied intensely while the majority of possible structures are not considered at all. In this way history can reveal why certain structures, objects, and results exist, not in principle but in the sense that they are part of mathematical practice.
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D. Hilbert, Natur und mathematisches Erkennen, Ed. D. Rowe, Birkhäuser (1992)
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H. Wussing, The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory, Dover Publications (2007)