The Problems of Understanding Mathematics
In an outstanding book by P. J. Davis and R. Hersh, The Mathematical Experience, there is a small chapter devoted to the crisis of understanding mathematics. Alas, this fragment focuses only on the presentation of the difficulties the students have with understanding a certain proof and some common-sense attempts to explain them, whereas it totally ignores the very notion of understanding. The only statement characterizing this notion is the remark that understanding is connected with effort.
The problem of understanding mathematics requires, in my opinion, a short presentation of a more general issue, that is the issue of understanding as such. I will treat understanding as a kind of indirect cognition, determined by the perception of the relations between the objects of various order (y becomes comprehensible for x as a part of the relation xRy, in which y is an object of a different order intentionally grasped by x). As it can be seen, I neglect here the problem of understanding another human being, although it is usually achieved through understanding the phenomenally accessible human behaviours, i.e. linguistic or extralinguistic creations.
It seems that the Polish philosopher Izydora Dąmbska grasped the problem of understanding accurately and concisely, stating that this kind of cognition is characterized by the following factors:
The essence of this kind of cognition, which we call understanding, decides about the hypothetical and constructive character of propositions based on understanding.
I would like to stress some points of the above statement which I am going to develop with reference to mathematics. Thus, in my opinion, understanding is: firstly-some kind of cognition, secondly-indirect cognition, thirdly-it consists in grasping the sense of what is to be understood, fourthly-it manifests itself in the ability to apply that, what we understand in a new situation.
Let's start our considerations from a trivial, perhaps, statement that the people who have some contact with mathematics, for example who learn it, try not only to remember the mathematical text but also to understand it. It suggests that the very memorization of the 'rules' or mechanical assimilation of the definitions or the formulations of the theorems does not mean that somebody knows mathematics. I think that it is possible to get acquainted in this way with some branch of mathematics (at great expenditure of work and time) and to give the right answers to the questions, asked in connection with this fragment of mathematics. However we won't say about somebody who has achieved it that he knows this branch of mathematics. It seems that the 'true', deeper contact with mathematics cannot be done without understanding, so in this case, understanding is a peculiar method, refering especially to mathematics, of acquiring knowledge. Great mathematicians, however, stress that while inventing or creating mathematics, intuition, illumination, direct intellectual insight into the subject matter of mathematics plays a big role. One of the outstanding Polish mathematicians, participating in Gödel's seminar in Vienna, characterized anecdotically his greatness, stating that Gödel had such knowledge of the problems he dealt with as if he was in a direct telephone contact with God. Those, who are occupied with mathematics but are not endowed with the special 'God's spark', must use an indirect way.
Let's try to explain briefly this thesis. It seems that in mathematics, the only way of directing attention to what we want to understand, is a mathematical text-written, spoken or 'thought', in other words, contact with a mathematical text is a necessary condition for a contact with mathematics to occur. Hence, if understanding is a form of getting acquainted with mathematics, it is also an indirect cognition in the sense that it is achieved through a text which plays the role of 'direct data'. However, the problem arises whether understanding does not simply refer to the text. This suggestion must be rejected on closer examination, because the text is only to introduce us into the domain of mathematics, into the subject matter of this science. Grasping-on the basis of the text-the sense necessary to develop the whole domain allows us to detach ourselves completely from the text. We can, as if, ignore the text of the mathematical message for the benefit of the domain, which this text renders accessible, because through understanding, we can discover the structure of the subject domain, no matter if the further part of the text continues its sense or not. Thanks to that, we can find mistakes (if they exist) in mathematical or logical texts, however, it's unimaginable to point to an extralogical error (excluding grammatical, orthographic or factual ones), refering to the plot in a literary text. Besides, using a mathematical text, we can acquire the knowledge not only about that, what is contained explicite in it; for example we can examine other properties of the investigated objects, formulate new hypotheses and try to prove them. It suggests that understanding mathematics refers to the subject domain and not to the mathematical text.
First, however, we have to understand a mathematical text in the sense, in which it appears in the following statement: I understand a text in a foreign language. When we say: I understand a foreign language text, we state the fact of having the knowledge about a given language, allowing us to translate it, better or worse, into the mother tongue. The understanding of the mother tongue seems obvious and is linked with the knowledge of both the syntax and the semantics of this language, achieved in the process of learning this language. John R. Searle, giving up complex analyses, even explains his notion of understanding through the reference to the intuition of understanding a language-in his case English. In my opinion, the matter isn't so obvious, because we can read a text in the mother tongue, which is 'comprehensible' for us, and still not understand it. It can happen, when the text concerns a domain which is totally unfamiliar to us, or when the text exceeds our intellectual capabilities. We can also 'understand' a text in a foreign language and, at the same time, not understand it.
Coming back to a mathematical text, it should mainly be comprehensible in such a sense as a foreign language text. A mathematical text consists of the signs, which denote some notions which are specific for mathematics. Such a text will be a nonsensical sequence of graphic signs, in the absence of knowledge or interpretation of what these symbols denote, what they point to. The basis of this denotation doesn't matter at all, it is unimportant if it is a consequence of some convention, or if it comes from some similarity between what is given and what the data point to. To put it shortly, a mathematician should have a knowledge pertaining to the meaning of the physical signs occuring in the text, since this knowledge plays the role of the initial data and it allows him to reach the contents of the text.
The next step is the understanding of the contents, which pertains to the subject domain, grasped in some categories. Let's notice that, even if we know which signs represent the given notions, the understanding of a given problem can be wrong. The reason, why it is so, can be the insufficient knowledge as regards the applied notions: 1) we don't know what they mean, what content they have, 2) the relations and links between the notions are not obvious for us, or 3) we don't know wide enough background, i.e. the theoretical context of the considered problem. Each of the enumerated cases contributes to the fact, that the problem we are dealing with, has an insufficiently precise sense. Enlightening it or discovering it or even giving some sense to it, is just the process of understanding. My intentions can be briefly described by the thesis: to understand means to discover the sense.
It seems that, already at the common-sense level of cognition, we should treat understanding functionally-as the process of acquiring sense. A given notion acquires sense by being confronted with other notions and being considered in various possible contexts, in which it can be used. Similar, although not so simple, is the situation at the scientific level of cognizance. A greater complexity is the consequence of the fact that, in this case, understanding is a multi-level process, taking place simultaneously on many planes. These planes represent the different, as to the accepted point of view, characteristics of a given object of understanding. It's not the question of the depth of the analyses, but of the acquistion process of various complementary descriptions of a given object. The basic plane, on which the process of understanding takes place, is the plane which I propose to call the theoretical plane of understanding. The considerations, carried out in its framework, are a synchronic analysis of the object of understanding in its 'natural' surroundings, within the theory or domain, of which it is an element. Other planes of considerations, essential for the issue of understanding, are: historical, methodological and philosophical. In each of these three planes, we can differentiate three basic levels, at which the object of understanding can acquire sense and thus can be understood: 1) the level of understanding the meanings of the terms and notions; 2) the level of understanding structures of the object of understanding (which consists in fixing the sense of the sequences of notions and terms), 3) the level of understanding the 'role' of the object of understanding (i.e. grasping the sense of the object of understanding in the context of greater entity, i.e. description of the so-called background of the problem and the structure of this background).
I think that, as regards the components of the mathematical theories such as: notions, definitions, theorems, proofs and the whole theories, the process of their understanding follows the scheme presented below. Let's note, however, that bringing the notion of understanding to only one of the enumerated aspects doesn't render the whole richness of intuition we expect from it.
Let's try to exemplify the above considerations by analysing the widely known mathematical formula: eip=-1
The idea is to reconstruct, how, in this case, the process of understanding this formula will look like. According with what we stated before, the first level of understanding concerns fixing the sense of the notions and terms, occuring in the object of understanding. We won't understand our example if we don't know, what the particular symbols used in this formula mean. Enumerating them in the order they appear, they are:
e-the so-called Napierian number, the base of natural logarithms; it is an irrational number and even non-algebraic; we define it as the limit of the sequence
(1 + 1/n)n at n® ¥; the number e approximately equals 2,71828182...
i-a new kind of a number, unfortunately called 'imaginary unit'; it was introduced as the root of the equation x2 = -1 which doesn't have a solution in the set of real numbers; it is the number which allows us to define the complex numbers; sometimes wrongly described as the 'root of-1'.
p-a mathematical constant, describing the ratio between the length of the circle to the length of its diameter; this number is irrational and it isn't a root of any equation with integral coefficients, p-approximately equals 3,141582...
=-a sign used to denote the equality of the expressions on its both sides
-1-a number unknown to the Greeks which ( together with other negative numbers) was accepted in Europe only in the XVIII th century; Chinese mathematical treatises mentioned negative numbers already in the second century BC, and Hindu mathematicians used zero and negative numbers in VI-VII centuries AD.
Let's notice that, in the considered formula, there appear 'uncommon' numbers, perhaps only -1 is more often met in common use and has more or less sensible interpretation (as a debt or a lack of something). On the other hand, the number i doesn't resemble any known number, for example, it is neither bigger nor smaller than zero, which is an unusual property for the notion of a number in the everyday meaning. The numbers e or p are strange in that they are, in fact, the result of the infinite operations-of approaching the limit and summing up infinite series. So, to understand fully our example, we need to know the fragments of several branches of mathematics, among others geometry, trigonometry, algebra, the theory of limits, the theory of the infinite series. Considering the given formula on the theoretical plane of understanding, we state that, behind the modest looking symbols applied in the formula and the short characteristics of the notions represented by them, there are hidden important mathematical theories (which constitute their natural context), while numbers e and p appear unexpectedly in various branches of mathematics, acquiring the status of fundamental mathematical constants. In this context, the previously considered formula is a significant example, confirming the methodological and subject matter unity of mathematics.
The next step in understanding the above formula-our object of understanding-is a transition to understanding the structure which, as we postulated, consists in grasping the sense of the sequence of the notions. Otherwise we know, that placing the number on the right side and at the top of a number means the operation of raising to a power. By analogy, the left side of the formula should be understood as number e raised to the power ip. The expression ip is the complex number: 0 + ip, the real part of which equals 0, and the imaginary part equals p. However, in this article, we don't have to worry about the operation of involution, because Leonhard Euler proved that:
eix = cosx + isin x,
where x is an arbitrary real number. For x = p we get:
eip = cosp + isinp = -1 + 0 = -1,
but, of course, first we have to know, what is hidden behind the symbols sin and cos. The notation of the whole considered formula should be understood as: e raised to the complex power ip equals -1.
And so, these four different kinds of numbers, which in mathematics appeared completely indepedently and in various periods of time, were combined into the formula which is one of the most famous equations in mathematics. It is a totally unexpected and unforseeable result, although it is the consequence of the definition of the natural number (and its further extensions onto the negative, rational, real and complex numbers).
The above considerations constitute only the most general scheme of the process of gradual understanding of some elements of higher mathematics. I am afraid that, if we confined ourselves only to that, our understanding would be only superficial. At this moment, the process of understanding doesn't stop. There still remains the third level of understanding, at which we have to take into account the sense of this formula in a wider context as well. As I mentioned earlier, understanding is a multi-plane process. To characterize exhaustively the structure of the 'background' of the object of understanding, we also have to consider, apart from the theoretical plane of understanding, the historical, methodological and philosophical planes.
It is obvious that the deep understanding of the notions, without investigating their genesis, is impossible. The sense of the particular elements, that the object of understanding, which we are interested in, consists of, manifests itself only after taking into account diachronicity, because they have their 'human' history. There is no place here for the detailed investigation into the history of mathematics, however, if we wanted to observe the development of these notions before they were finally combined in the above formula, we would have to get acquinted with a significant part of history of mathematics.
Next, let's mention the methodological plane. Having to do with such different examples of numbers,we should try to understand the connections between these kinds of numbers in the hope that such investigation will throw light on the very notion of a number. For example, we would have to interpret methodologically the stages in the development of the notion of a number and explain, why the true history of the notion of the number: filiation N-Q+-R+-R-C (where N denotes the natural numbers, Q+-positive rational numbers, R+-positive real numbers, R+-real numbers with zero and negative numbers, C-complex numbers) is different from the commonly accepted set theory interpretation of the range of the notion of the number, i. e.: N Ì Z Ì Q Ì R Ì C.
From the point of view of methodology, filiation N-Z-Q-R-C, although it does not agree with the factual order of history, can be considered as the model of the development of the notion of the number. The mechanism of this development reveals itself in the situation, in which there occurs the specific shortage of the objects in relation to the operations which can be carried out on the known objects. For instance, in ancient times, natural numbers (considered to be the only true numbers) were added and, as I suspect, substracted, but the latter operation was not always feasible. Not in every case did the natural numbers enable the substraction. We can say that, at some moment, there appeared 'an operational pressure', which brought the notion of a negative number into existence. It happened by postulating some mathematical objects, which met the operational needs. Such beings were firstly treated as quasi-objects, and only later, usually after finding the suitable interpretation, they were treated as fully accepted mathematical objects. Similarly, we can interpret the issue of the emergence of fractions-first treated as substandard 'faulty' numbers, irrational numbers-treated as non-rational, or complex numbers-first called imaginary ones. Let's notice that methodological analyses deepen essentially the understanding of the general notion of the number and so they contribute to its creative generalization ( which has already occured in the conceptions: of the numbers called quaternions, extrafinite numbers or in the matrix calculus.)
In order to explicate more profoundly the significance of the methodological analyses for the process of understanding, let's recall the situation, which occured in the thirties of the XXth century, in connection with the publication of the famous Gödel's theorem. It is known that Gödel first presented his breakthrough discovery at the conference on epistemology, organized in 1930 in Königsburg. To understand this theorem, it is necessary to have a thorough knowledge and understanding of such notions as: consistency of the system, Richard's paradox, formalization, arithmetization, Gödel's numeration, Gödel's formula (not to mention here the basic logical categories). Besides, it is necessary to grasp the structure of this theorem's demonstration. It means that, as in every case of the process of understanding, very important is the theoretical plane. However, we can trace this proof step-by-step and, as if, 'understand' the idea of this proof, but at the same time, we may comprehend neither its significance nor its consequences. We may draw such conclusions from the lack of any response to the theorem on the part of other participants of the conference. The analysis of Carnap's speech delivered at this conference indicates that he didn't realize the breakthrough character of the discovery. Also Hans Reichenbach in his report on the conference placed in Die Naturwissenschaften didn't even mention Gödel's pronouncement. We shouldn't suspect these great scientists of inability to understand the demonstration procedure proposed by Gödel, therefore I put forward the thesis, they didn't know enough about the background of the problem to understand fully the contents of Gödel's theorem. It cannot be understood profoundly without understanding the historical background of the problems, which led to the formulation of the problem of consistency of mathematics and the restrictions of methodological type, connected with the question of accepted methods and rules of demonstration. It is linked with the situation which occured in mathematics after Hilbert's presentation, in 1900, of his famous 'Programme', in which he presented 23 mathematical problems. Nowadays, as we have a higher methodological awareness and a deeper historical knowledge, we can understand this theorem better than (at the beginning) even the famous philosophers and mathematicians, Gödel's contemporaries.
Finally, let's try to signal how, in my opinion, the problem of understanding mathematics should be perceived in the philosophical plane. First, I will present briefly the basic and fundamental, in my view, philosophical problem connected with mathematics. Philosophy is not, perhaps, a science sensu stricto, but for me, it is a kind of knowledge about reality, about that what exists in some way. Thus, the question of the role and place of mathematics in acquiring knowledge about the world of nature-is the most typically philosophical approach to mathematics. The view that mathematics is an esoteric art for art's sake, shared by some mathematicians not a long time ago, should be considered a specific ideology of 'mathematical class'. It is well known that mathematics is an essential element of the contemporary natural sciences. Contemporarily, mathematics, as a whole, is commonly considered to have sense if it is useful. This usefulness shouldn't be comprehended too simply, as its purely specialized applications.Mathematics is, then, in some sense, also a science about the 'tools of cognition'-categories used in empirical sciences. Therefore the explication of the effectiveness of mathematics in the process of acquiring scientific knowledge would allow a better understanding of the 'background' of mathematics, which, in turn, would enable us to be moderate in our admiration for mathematical knowledge. Besides, this point of view is the only one, as I think, in the light of which we could investigate-properly, undogmatically and non-arbitrarily-other 'classical' problems of philosophy of mathematics such as: the nature of mathematical objects, the way they exist and the issue of admissible methods that can be used to get acquainted with them.
It follows from this article that teaching mathematics shouldn't consist only in inculcating abstract formulas and conducting formalized considerations. We can't learn mathematics without its thorough understanding. My postulate is that, in the process of teaching mathematics, we should take into account both the history and philosophy (with methodology) of mathematics, since neglecting them makes the understanding of mathematics superficial and incomplete.
1. Philip J. Davis & Reuben Hersh, The Mathematical Experience, Birkhäuser Boston, 1981.
2. Izydora Dąmbska, W sprawie pojęcia rozumienia, in: Ruch Filozoficzny 4, 1958.
3. John R.Searle, Minds, Brains and Programs, in: Behavioral and Brain Sciences 3, Cambridge University Press 1980, p.417-424.
4. Danuta Gierulanka, Zagadnienie swoistości poznania matematycznego, Warszawa 1962.
5. Roger Penrose, The Emperor's New Mind, Oxsford University Press 1989.
6. Andrzej Lubomirski, O uogólnieniu w matematyce, Wrocław 1983.