Afterwards, of all things, which God the Creator of the whole world, brought forth by his incomprehensible wisdom and goodness, on earth, were not found more excellent than the Soul or the reasoning ingenuity of man: so we ought to do our utmost best that we share this God given gift for the examination of the truth, and what knowledge depends upon all its perfection, continually to be used and practiced.
Among all the exercises now understood, which ones could serve here, and none were found as progressive as the Mathematics or Mathematical Exercises, taken so as to exceed and surpass all others in better knowledge and wisdom. For some consider that the very senses which deceive us many times, have no place in these exercises and have no understanding whatsoever with that occupation, and how the principle of these very well known things takes us in short to the knowledge of men's hidden secrets to bring forth and thus understand the earthly conduct as in Heaven; so it is easy to decide that for some reason it should not be considered unjust to use the old Counting and Measuring Art in all other sciences and arts.
Thus the Leiden professor Franciscus van Schooten wrote in his book Mathematical Exercises around 1650. These words may, in the first place, show his great reverence for mathematics, but this quote also tells us of a certain conception of the essence of mathematics and the task of the mathematician. The use of reason, logical reasoning, allows us to deduce propositions from some very well-known facts, which one certainly believes to be true and certain. Nowadays not everyone will share this view without doubt, even if his love for the profession equals that of the aforementioned professor. The views on the foundations of mathematics and the certainty of its statements have since undergone significant changes and the difference of opinion that is encountered in this field has increased rather than decreased. This difference of opinion is even so great that where there is agreement in the principles in technical issues in mathematics, the disagreement appears to be the principles in issues concerning its foundations. The "better knowledge and wisdom", which according to van Schooten characterizes mathematics, seems to be completely lost here.
You know that in mathematics new theorems are always derived from old ones. The question has been heard several times: "Will this never end?", probably out of concern for the fate that the practitioners would have to wait in this case. However, they have not shared these concerns, but have always been interested, not for a possible end, but precisely for the beginning, the principles and for the rules, the methods used in mathematical reasoning, in short for the source of the certainty of mathematical statements. They were always interested in the pillars on which the entire building rests, because what value will this have if some or only one of these pillars prove to be defective. But, you may wonder, is this really a problem for the mathematician? These issues affect the essence of mathematics. This is a question of what mathematics is. In the following, we want to take a closer look at some of the views on this, where we will mainly limit ourselves to geometry.
The question of what geometry is, used to be more unanimous in the past than it was in the last century. It was agreed that geometry, always referring to three-dimensional Euclidean geometry, is the science that has the objects of space as its subject. That does not mean that its knowledge was acquired by measuring figures, on the contrary, because, as Van Schooten put it: "the very senses will often deceive us." Already the Greeks felt the need to derive the theorems of geometry by logical reasoning from some very simple facts, called axioms, which in turn were derived from experience. Of course, people were aware that the statements of geometry could only be realized in approximate ways, but because of the great degree of accuracy, people gradually began to see more in these statements than empirical truths.
In the theory Kant develops about geometry, the statements of geometry are therefore seen as exact truths. According to Kant, the basic properties of geometry, in this case the axioms of Euclid, would be immediately intuitively clear to us and would have their origin in the human mind. This conception of priority of space is not contrary to the aforementioned objective of geometry. Only through this present concept of space can the experiences become conscious. Consequently, Euclidean geometry will also guide the world of experience. Kant's conception has been able to maintain itself for a long time, until the confidence in his point of view was shaken by the discovery of non-Euclidean geometry.
We have already seen that the statements of geometry were obtained by logical reasoning from some axioms. It goes without saying that these axioms - that is, the facts that one accepts without proof - are required to express very simple and well-known things, that it is those statements of geometry that require the least proof. Now, among the axioms and postulates of Euclid, there is one that does not meet these requirements, since it is vague. This so-called parallel postulate, formulated for the flat surface, comes down to this, that one can draw from a point not on a straight line, one and only one line, which does not cut the former line. Already among the oldest interpreters of the writings of Euclid, we find that they believe that this postulate is not of itself evident enough to be used as an axiom. Many mathematicians have therefore tried to prove this postulate on the basis of the other axioms of Euclid. One of the methods for this was to start from the assumption that several lines are possible through a point not on a given straight line that lies on a flat plane and do not intersect this given straight line. It was hoped that this assumption in combination with the other axioms of Euclid would lead to a contradiction. However, the result was completely different from the expectation. Contradictions did not occur and in this way Lobachevsky, Boyai and Gauss succeeded in building up a complete geometry independently of each other, a so-called non-Euclidean geometry. This geometry does not stop there. We owe to Riemann the creation of a second non-Euclidean geometry, in which parallel lines do not occur at all. In this geometry, two straight lines located on a flat surface always have one intersection. The surprising thing is that several geometries can coexist.
This great discovery from the beginning of the 19th century has brought a radical change in the conception of geometry. Kant's position is difficult to maintain in the long term, especially when it later emerges that Euclidean space is not the most suitable for describing some experiences. The question now arises as to what significance should be assigned to the statements of geometry. Since the three geometries now available are based on three different systems of axioms, it is obvious that one starts to consider the significance of an axiom. An axiom was previously a statement, the validity of which was self-evident. But two mutually exclusive axioms cannot both speak for themselves. The question of certainty in geometry is again being raised, for how can one apparently base mutually exclusive geometries on the same experience or the same intuition?
This altered attitude, which therefore arises with respect to geometry, is closely related to the axiomatic method, which has been largely implemented by Hilbert. In short, this method comes down to this. We imagine three different systems of things, which we call points, lines, and planes. According to Hilbert, we need not imagine these things. Moreover, we think of certain relations between these objects, which we can indicate with words such as "lie on", "between", "parallel", etc., or also with symbols, which has the advantage that, as a result of which the disruptive influence of language is more easily avoided. The precise descriptions of these relationships are the axioms of geometry, from which the theorems of geometry are now derived, using logical reasoning as the only tool.
In contrast to the older views, when choosing the axioms it is no longer necessary to be guided by facts from the world of experience, but we are relatively free in this choice. Only a few requirements are necessary. Hilbert demands from his axioms, first of all, that they are not contradictory and, secondly, that they are mutually independent, the first being essential. The system is considered worthless if this requirement of non-conflict is not met, i.e. if it is possible to deduce from the axioms both the correctness and the incorrectness of the same statement. The geometer now finds himself faced with the problem of proving that the axioms he has chosen cannot indeed lead to a contradiction, a problem which the Kantian approach does not know at all, since for it the contradiction of the basic propositions of (Euclidean) geometry is simply unthinkable. Now the certainty of geometry must be proven by giving a proof of non-conflict. This question of non-conflict arises for the first time in mathematics after the discovery of non-Euclidean geometry. How do we know that the axioms of this geometry or even those of Euclidean geometry are free of contradictions? It is not enough to point out that contradictions have not yet occurred, we must show they cannot occur.
Around 1870 Klein discovered that it is possible to make a Euclidean model of non-Euclidean geometry. By this it is meant that it is possible to add to each concept and every relation from non-Euclidean geometry according to a certain rule a concept and a relation from Euclidean geometry, such that the non-Euclidean axioms correspond to certain Euclidean theorems. Every operation and every proof in non-Euclidean geometry corresponds to an operation and proof based on the Euclidean axioms. If non-Euclidean geometry were to lead to a conflict, it immediately follows that the same must be said of Euclidean geometry. Conversely, one can also construct a model of Euclidean geometry in non-Euclidean geometry. The same order of certainty therefore belongs to each of the geometric arts. They are not mutually exclusive, but mutually inclusive, geometric arts.
As a result of this consideration, confidence in non-Euclidean geometry may have increased, but the question asked has not been definitively answered. It remains to be shown that one of the geometric techniques, e.g. Euclidean geometry, is free from contradictions. People often resort to analysis and algebra for this problem. The connection between geometry on the one hand and algebra and analysis on the other is found and applied in analytical geometry. Herein, if we confine ourselves to the flat plane, a point is represented by a pair of numbers (x, y), a line by a linear equation in x and y, etc. Each geometric theorem now corresponds to an algebraic one, e.g. the axiom pronouncing that two distinct points determine one line, the algebraic theorem formulated in geometric language, it states that a system of two linear and homogeneous equations in three unknowns, of which the coefficient matrix has the rank of two, always has one solution.
As Hilbert points out, the entire Euclidean geometry is realized in this way in the doctrine of real numbers. And he rightly concludes from this that a contradiction cannot occur in Euclidean geometry if the doctrine of real numbers is free of contradiction. Do we, however, have to see in the doctrine of real numbers the logical foundation of the various geometries? Some take this view on the basis of the considerations just mentioned. Yet this conclusion is not right. Although geometry can be reduced to analysis and algebra, the reverse is no less the case. Every operation with real numbers corresponds to certain constructions in geometry. What this correspondence demonstrates is therefore that the order of certainty in geometry is the same as that in analysis. Geometry and the doctrine of real numbers are two different models of the same abstract system. It cannot give us an answer to the question whether the doctrine of real numbers should be the logical foundation for geometry or, conversely, consider geometry as the foundation of the doctrine of real numbers. Some even lean towards the latter, because in their opinion geometry is primary and the concept of a real number originated from geometry.
Here one then often appeals to Dedekind, one of the founders of the theory of irrational numbers. In his work "Stetigkeit und irrationale Zahlen" he remarks: "If one wants to translate all the properties of the straight line arithmetically, the rational numbers do not suffice; it is necessary to complete the structure already created by the rational numbers, by adding other numbers so that the system of numbers gets the same continuity as the straight line." It would then appear from this that the original concept is not the real numbers, but rather the arbitrary point of a segment, that is, the geometric concept. However correct these considerations may be, they do not teach us anything about the logical dependence of the concepts considered. After all, the way in which the various mathematical concepts originated in the course of time does not necessarily have anything to do with their logical dependence.
But let us return to the question concerning contradictions in the geometric system. Are we sure that the doctrine of real numbers, in which we have depicted our geometric system, is free of contradictions? It has been believed that it was possible to base the doctrine of real numbers on the system of natural numbers. Assuming this view to be correct, the natural numbers could be regarded as the foundation of all mathematics, including geometry, and the question of the basis for the certainty of arithmetic would remain.
Attempts have been made to answer this question in various ways. So they have tried to base arithmetic on logic. If such an attempt, inter alia, undertaken by Frege and Russell, were successful, this would mean that mathematics should be regarded as a part of logic, insofar as it is based at least on the natural numbers. It would therefore have no foundation other than logic and, as it were, logically emerged from non-existence. Apart from those who went too far with this consequence, there were some serious objections to Frege's view through the discovery of some paradoxes. The first one to remind Frege that his system can give rise to a paradox was Russell. However, he does not leave it at this. He also shows how to avoid these antinomies, namely by being more careful with the concept of a set. But the distrust of logical reasoning once aroused and is not easily deflected. Russell can't give either certainty that no paradoxes can occur in his system. Poincaré says of this: "Have we perhaps enclosed, by the fence which we have erected around the sheep of the doctrine of sets, the unnoticed wolf?"
Thus it is increasingly the opinion that mathematics cannot be reduced to logic, that mathematics cannot be completely built from the construction of logical systems. In mathematics, however, logical reasoning is again used, so that a separate structure is no longer eligible. Attempts are therefore made to establish logic and mathematics in common.
The method used here is the aforementioned axiomatic method, but now for the whole of mathematics and logic. We have already made the observation that, according to Hilbert and his school, the axioms in mathematics are interdependent concepts, to which no other content belongs than that attributed to them by said relationships. It is therefore preferable to use a symbol. The same is now done for logic. The meaning of the logical concepts is abandoned; the signs entered in logic are symbols without content. However, there will be some relations between these symbols, the axioms of logic. Mathematics thus formalized can be seen as a game. A statement is a certain configuration of symbols. It is provable if it can be deduced from the starting position with the help of the rules of the game. The core problem can now again be demonstrated that in the system built up in this way no contradiction can occur, that is to say that it is impossible for two claims that are mutually exclusive to be both proven according to the rules. People do not seem to want to interfere with systems that do not have mutually exclusive claims, for which this problem does not exist.
Did the solution to this problem work? No more answers can be given to this question, since it was not correctly stated. The point is, after all, which submissions to a proof are considered admissible. It would be obvious to limit oneself to those that can be formulated in the system under consideration, but Gödel has proved that proof is impossible under this limitation. Proof of the non-contradiction must always be used in the form of deliberations that cannot be formulated in the system under consideration.
With Hilbert, therefore, in addition to formalized mathematics, another mathematical system occurs, a meta-mathematics, which is used to base the former. The objects with which meta-mathematics is concerned are the proofs in formalized mathematics. Its statements are judgments about the aforementioned rules and drawing configurations. In this meta-mathematics, Hilbert allows reasoning based on observation, in contrast to the purely mechanical rules of mathematics. In this way, Hilbert and his followers have succeeded in proving the lack of contradiction of various systems.
Has Hilbert now achieved the goal he imagined, namely, as he puts it, nothing less than to banish the general doubt about the certainty of mathematical reasoning? We doubt it. In his meta-mathematics, he uses reasoning from classical mathematics, albeit with great limitations, but the doubt concerns the certainty of the statements of this mathematics. Proof of non-conflict can only reduce the correctness of certain arguments to the correctness of other more confident arguments. The difficulties do not disappear, they are moved. However, Hilbert's attempts have been very enlightening. In meta-mathematics, it is often about how much, or rather, with how few observations one can achieve the set goal. This work is also highly valued outside the circle of formalists. It is important not only for those who see the formal structure of mathematics as necessary, but also for those who find formal considerations useful or desirable and for whom axiomatics do not serve as a basis for mathematics.
We now return to geometry. We have considered the problem of the foundations of arithmetic for so long as a result of the question of certainty in geometry. It has been found that this problem cannot be solved without outside help. The study of non-conflict does not therefore lead to the conviction that geometry must find its foundation in the doctrine of real numbers. It has not even brought us to the understanding that the possibility of reducing the axioms of geometry to that of analysis gives the statements of geometry a higher degree of accuracy.
But from what does geometry derive its certainty? For the first practitioners of geometry, experiment, experience, were the source of certainty for its statements; for the Kantian, the origin of geometry resided in the human mind, in the intuition of space and time, which means that the statements of geometry have absolute value. Attempting to see logic as the foundation of mathematics did not provide a satisfactory solution. What now? Do we go back to Kant's view or to a certain empiricism? Indeed, both views can be found in modified form. According to some, the study of the world around us has produced mathematics and logic. The basic concepts of all branches of mathematics, such as point, line, plane, integer, sum, etc. would have been derived from experience, with some having experience only looking at what, by observing the physical world comes to us, but others do not want to exclude a certain experience from the field of thought, a mental experience. Mathematics can also not be practiced without diagrams and figures, so without relying on concrete representations. Constantly mathematics now has to justify its existence through experimental verification of the predictions it makes. What has been said here in general for mathematics, naturally also applies in particular to geometry, by which in this view one could understand that part of mathematics that deals with those objects which correspond to figures in physical space.
We also already became acquainted with the current notion, in which the mathematical concepts are regarded as free creations of the human spirit and no ties with contemplation or anything of that nature are recognized. If one takes this formalistic standpoint, a seclusion of a certain part of mathematics under the name of geometry can only be accepted if it is defined with the axioms of the systems under consideration alone and not with the help of anything located outside of this system. We could, for example, divide mathematics into mathematics of the countable and mathematics of the non-countable and use geometry for the latter. Or we could refer to geometry as the mathematics of the objects point, straight line, plane, etc. with certain interrelations, which can lead to the known geometric systems. But why would we do this, since nothing prevents the formalist from putting other relationships first, which completely change the character of the concepts of point, line, etc. It is precisely this freedom of choice in the choice of relationships defining the objects that gives a separation from a certain part of mathematics under the name geometry something unreal.
An entirely different aspect is given to this matter, if it is assumed that the basic concepts used are bound to our intuition, so that they are immediately clear to the thinking mind, a position that shows a great deal of kinship with Kant's conception. The mathematical relations then certainly have a sense. This view is already found in Poincaré. When he, dealing with the foundations of geometry, introduces the concept of a point, he tries to show how it is possible that we know this concept. He provides a psychological explanation for this.
Of course, there is still a difference of opinion as to which concepts one can accept as immediately obvious. Those who recognize the intuitive concept of space will construct part of the mathematical system of geometry, in which the concepts and relationships are derived from this space intuition. In the literature one finds again and again attempts to accurately define the field of geometry. However, these often give either a description of what is considered to be the goal of geometry, or a more detailed description of the method that one would have to follow to reach that goal, e.g. Klein's statement, which describes geometry as the invariant theory of a transformation group. Needless to say, such designations are highly dependent on the development of mathematics. Time and time again, different descriptions are used when the older ones no longer satisfy. This inconsistency has led Veblen and Whitehead to say that a certain branch of mathematics is called geometry, because a sufficient number of competent people give it this name on the basis of feeling and tradition.
What is the case anyway? With these classifications, the majority is guided by the usefulness or non-usability of the mathematical system under consideration for describing displacements, movements, etc. in the physical world of experience. That part of mathematics, which one can use or can use for this purpose, is simply called geometry and has tried to describe it. In this classification, therefore, attention is paid to what mathematics has to offer to the world of contemplation, in contrast to the aforementioned description, which is based on what mathematics has derived from intuition, thus from contemplation in a broad sense. Space intuition does not give us one, but several geometric systems. Which of these systems is used to express the physical laws in the world of experience is a matter of taste, convenience and also habit.
What concepts and relationships are they now that are derived from the intuitive concept of space? They will be the most fundamental axioms of geometry, and the axioms of the continuum can be regarded as such. In their investigations, the geometries generally assume a topological space, namely a continuum. For example, Lie first defines the space as a "Zahlenmannigfaltigkeit", i.e. a set of points that can be represented by a number of numbers, called the coordinates of the point. The topological nature of this space is then guaranteed because Lie contends a certain geometry defines the motion group through a continuous transformation group, the axioms of geometry are divided into two groups: the axioms of the continuum still come from the movement, but the latter are also topologically invariant, i.e. they can be characterized topologically. The axioms of Euclidean geometry are not tied to a rigid model of straight line and flat plane, and if a model of Euclidean space is continuously deformed, it remains Euclidean, provided that the deformed lines are still called lines in the deformed planes. What makes our sketches in geometry different than other topological images?
The axioms of the continuum, to which a central position is assigned on this basis, are the ordering axioms and the axioms of continuity. For the one-dimensional continuum, for example, the first-mentioned axioms define the concepts of "lying between" and orientation, while the axiom of continuity provides us with the concept of limit, of the concept of geometric point, intuitive notion of space, or if you want to be given by time-intuition, and not be based on anything else, so this intuition is not only based on the description of geometry, it is also the source for the certainty of its statements.
The axioms of the continuum cannot be regarded as the only pillar of mathematics, not even geometry. The second pillar is the sequence of natural numbers, which is also considered to have been given to us by intuition. Together they carry the whole of mathematics, as well as the doctrine of whole numbers, arithmetic, and that of the continuum, of geometry: According to this view, mathematical concepts have a dual origin. A part groups around whole numbers, the others around the continuum.
Some people object to the idea that the geometric continuum is immediately given by intuition of the physical continuum, e.g. the colour spectrum has a structure which differs essentially from that of the mathematical continuum, since it is not possible to speak of individual colours in the spectrum, whereas separate points of the mathematical continuum are referred to as if it were composed of separate points. On the other hand, it can be noted that intuition cannot be identified with experiment.
But this has not proved to be the only objection. Criticism has just been made from different sides, not least from the supporters of an intuitionistic view of mathematics. The aforementioned view is not shared at all by those mathematicians, who are commonly referred to as intuitionists, among whom our compatriots Brouwer and Heyting occupy a prominent place. According to their view, mathematics is identical with the exact part of our thinking. It is not based on any other science, not even philosophy or logic. It is pointless to use propositions from these sciences as evidence in mathematics, since mathematical conceptions are already necessary for formulating these propositions. There is no other source for mathematics other than intuition, which clearly presents the permissible concepts to us. That which can be deduced constructively from this intuition is mathematics. In contrast to the view just outlined above, this intuition would, however, be limited to the sequence of natural numbers and does not include the continuum. According to the intuitionists, mathematics as a whole consists of constructively working with the sequence of natural numbers. The doctrine of the continuum, as classical mathematics knows it, is therefore not entitled to the name mathematics. Brouwer did succeed in constructing a kind of continuum in a constructive way, which has a lot in common with the classical continuum.
The consequences of these basic ideas for mathematics are of a radical nature. They require a completely new structure. For the intuitionist, there is no separate geometry at all, he denies any independent existence of geometry. An axiomatic geometry based on concepts such as point, line and plane with certain relationships between them, as Hilbert has developed, remains without meaning and content as long as an analytical model cannot be given. Because only analytical geometry, insofar as it is based on arithmetic, has a mathematical meaning. If one considers the development of geometry, then according to this view it has been in a pre-mathematical stage for a very long time. Is it surprising that it is difficult for many to sacrifice such large parts of their structure to these ideas? Almost all mathematicians agree that there are good reasons to mistrust classical proofs and concepts in mathematics on certain points, but not everyone is inclined to draw the line between permissible and unacceptable concepts and conclusions, where the intuitionist puts it.
Thus we see that there is by no means any unity about the views on geometry and the source of its certainty. Will we in the future see a unity in this area that we are used to seeing in the more technical issues of mathematics? Probably not. The problems have a completely different character. Many a question is of a philosophical nature. But this will not prevent the mathematician from dealing with the question of the certainty of its statements, not only from an urge for truth, but also because the development of mathematics is closely related to the problems of its foundations.