Pure Mathematics and the Human Condition: Mathematics as a Respite from the World
Roger G. Olson
Associate Professor of Mathematics
Saint Josephís College
Lilly Project on Faith and Reason: 1600-Present
December 8, 1997
It is God's privilege to conceal things, but the kings' pride is to research them. (Proverbs 25:2)
Introduction and Editorial Comments
This paper is a reflection of my personal feelings about mathematics. By mathematics I mean pure (or abstract) mathematics unless otherwise stated. It is largely self-conscious but draws upon the opinions of genuine mathematicians, in particular G.H. Hardy, whose A Mathematicianís Apology is heavily quoted. At the outset let me say that not all persons trained in pure mathematics agree with me, but I believe these to be a small minority. I could not find examples of mathematicians claiming that pure mathematics influences the human condition via its impact on Faith and Reason.
First, Iíll recount at little about my own intellectual journey. I am not now, nor was I ever a verbal person. Were I forced to make a living using my writing talent I would now be permanently at the mercy of the vestiges of the Great Society. On the other hand I donít have the mathematical creativity of the kind possessed by the great pure mathematicians cited in this paper. I do appreciate their work and can fairly well distinguish inspired from prosaic work, at least in the branches of mathematics closest to my training. Trying not to be too maudlin, I oft times think of myself as a weak analog of Antonio Salieri who in the semi-historical play Amadeus is condemned to behold and understand the great genius of Mozart but is never allowed, even after much supplication to God, to attain an equivalent talent.
The standard American mathematics curriculum of thirty years ago left little room for creativity or those sorties into free thought that so frightened me in those days. I felt safe in a world where questions always had answers, and I was succored by this confidence into developing the basic skills necessary to learn real mathematics. The basic mathematics of elementary and high schools valued correct and complete answers. This built a foundation that allowed me to undertake abstract math when the time was right. [Aside: I believe this to be the correct pedagogical approach still. There is a movement in American mathematics education today to replace the standard K-12 math curriculum by a more "exploratory" approach. I strongly disagree with this method that discards the fundamental concepts, skills, and drills. California opponents of this new approach (often called "whole-math") have an Internet site that is replete with information: http://ourworld.compuserve.com/homepages/mathman.]
I wasnít ready for abstract mathematics until my mid-twenties. There were two reasons for this. First, previously I lacked the ambition to undertake the rigors of a careful study. Secondly, I was simply intimidated by upper-level mathematics, and, possessing a rather weak self image, I simply avoided it. The college mathematics (calculus and differential equations) I had before this time was computational and required an ever increasing "bag of tricks". This study is appropriate for a budding physicist or engineer but is intimidating to all but the most motivated.
I fell into abstract mathematics by accident. In 1982 I enrolled in a course in Linear Algebra, ostensibly to help me become more proficient with a linear programming software system at work. Fortunately, Dr. Choi took a pure, rather than computational, approach to the course. I saw the axiomatic method for the first time and fell in love with it. I took one course per semester for the next three years, then entered graduate school in 1985 at the age of thirty.
The middle of this paper will explore a sequence of pure mathematicians from approximately 1600 to the early twentieth century. In particular I will look at what may be inferred of their world views and how or if these influenced their mathematics. I will spend a longer time on G.H. Hardy, since he wrote a work that allowed outsiders a view at the soul of a pure mathematician. I will start with an explanation of what pure mathematics is, and I will close with a brief comment on the need for consistent standards for the teaching of basic math skills.
What is Pure Mathematics?
Pure mathematics differs from science, in which I include applied mathematics, in its method of reasoning. The scientific method uses inductive reasoning, i.e., inferring or inducing a general result by observation of specific examples. The researcher makes an educated guess, or hypothesis, then makes observations and, if all these fit the hypothesis, he/she may infer that the educated guess is correct or "proven". This is most assuredly not the same thing that a mathematician means by "proof". Further observations, especially after, e.g., the development of new technology, may refute the hypothesis, in which case the scientist must either scrap or modify the explanation.
Pure mathematics uses deductive reasoning. It is very much like playing a game such as Chess. There are the game pieces and a set of rules. All the moves in the game have to be consistent with the rules and involve only the game pieces. Analogously, in deductive reasoning, the rules of the game are the axioms. These are statements that are assumed without proof. E.g., in the "game" of Euclidean Geometry one axiom is given a line and a point not on that line there exists exactly one line through the point that is parallel to the given line. The "game pieces" are called undefined terms and objects. Again, using Euclidean Geometry, point and line are undefined objects and on, as in point A is on line L, is an undefined term.
Pure mathematics, or more correctly, any particular branch of it (such as Euclidean Geometry, Number Theory, or Algebraic Topology), exists in an abstract world. The undefined terms and objects may be able to be modeled by objects in the real world, but their abstract nature remains unchanged. A streak of chalk dust on a blackboard may be used to represent a line, but it is no more a line than a painting of a person actually is that person. At times this abstract world impinges on the real one, and applied mathematics results.
Deductive reasoning takes the axioms and undefined terms and objects and uses rules of inference to prove statements of the "if-then" kind. An example of a fundamental rule of inference is modus ponens: that from (p implies q) and (p) one may infer q. A proved statement is called a theorem or a lemma, depending upon how "serious" the result. [Aside: Mathematicians actually say a theorem is proved rather than proven in order to distinguish this concept from either the scientific or the commonplace sense of "proof".]
When is a proved result "serious"? This is really the crux of difference between pure and applied mathematics (or between pure math and science or pure math and engineering). Here I draw heavily from G.H. Hardyís Apology [Har]. "Serious" does not mean "important" in any practical sense, e.g., a serious result will probably not lead to a faster computer chip or reduce world hunger. A serious result certainly has generality (it will lead to further mathematics), it will be recognized as an achievement by the mathematical community, and usually has depth (requires new techniques or clever application of existing ones).
I couldnít find an example of a serious result that was proved in the period 1600-present and yet has a proof that is accessible to a general audience. The ancient Greeks had a few that fit this bill. For example, Euclidís proof that there is no largest prime number, which uses the technique of proof by contradiction (reductio ad absurdum) is fairly straightforward:
Suppose there is a largest prime P. Consider N = (2x3x5x7x11x...xP) + 1. Here we are taking the product of all prime numbers and then adding 1. What kind of number is N? Clearly, it isnít divisible by 2, 3, 5, 7, ..., P because these would yield a remainder of 1 upon division. By assumption though, it canít be prime, thus has a prime divisor, which must be larger than P. This contradicts the initial assumption. Thus, no such prime number P exists.
This result is "serious" in that it has implications for the foundations of arithmetic. The Fundamental Theorem of Arithmetic states that every integer can be uniquely factored into primes. Euclidís Theorem above is reassuring in that it concludes that there is an infinity of raw material for arithmetic.
An example of a non-serious yet interesting result, whose proof is "beyond the scope of this paper", is the following tidbit from Number Theory:
8712 and 9801 are the only four digit numbers that are integral multiples of their reversals, i.e., 8712 = 4 X 2178 and 9801 = 9 X 1089
According to Hardy, "This is an odd fact, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in it which appeals much to a mathematician."
Some History: The 17th Century
The line between pure and applied mathematics in 17th century Europe is quite blurred. I believe that this blurring is a good thing, and in fact spurred the development of both disciplines. It is clear that modern science got its start in this century, especially with the celestial mathematics of Kepler (1571-1630) and Galileo (1564-1642). According to the mathematics historian and expositor Morris Kline [Kli1], "The secret of the success of modern science was the selection of a new goal for scientific activity. This new goal, set by Galileo and pursued by his successors, is that of obtaining quantitative descriptions of scientific phenomena independently of any physical explanations."
This method differed sharply from that of the ancient Greeks, as well as the Scholastics of Medieval Europe. The Greeks spent their time attempting to explain why physical phenomena occur. For example, Aristotle tried to explain why a ball released from a elevated position falls back to earth. Some of these explanations were absolutely obtuse, e.g., Plato explained why the earth maintained its fixed position in the center of the universe (this is of course not true), "... a thing in equilibrium in the middle of any uniform substance will not have cause to incline more or less in any direction". St. Thomas Aquinas had an equally opaque explanation of motion, "... that it is the act of that which is in potentiality and seeks to actualize itself".
[It is my opinion that pure mathematics is not sufficient alone to describe physical reality. One can determine relationships among pieces of the self-contained world of an axiomatic system, but it is as impossible to obtain information outside its walls as it is for light to escape the Schwarzschild radius of a black hole. Even though I love abstract math, I fully realize that the 17th century invasion of the mathematical world by the shock troops of the physical world was necessary to "kick start" pure mathematics.]
Galileo described the motion of the dropped ball quantitatively. He observed that the distance, d, the ball falls from its point of release increased with elapsed time, t. The relationship between these two variables was determined by repeated observations to be d=16t2. Of course, the obvious modern answer to why the ball falls is "gravity", a fundamental force in nature that has not been fully "explained" today.
Newton (1643-1727) described laws of motion and, together with Leibniz (1646-1716), were the Fathers of the Calculus. Their work was essentially empirical. The formal proofs and foundation of the Calculus would not be provided until the 19th century with the works of Cauchy, Weierstrass, et.al. However, its basic concepts of functions and instaneous rates of change, which Newton and Leibniz applied mainly to the motion of objects, were abstracted in the next two centuries by brilliant scientists/mathematics who were getting more and more formal in their work. I found interesting the following quote from Vera Sanford [San, p.328], an erstwhile instructor at the undergraduate institution I attended:
The history of the calculus is remarkable for the way in which methods were developed in connection with practical problems, so that many cases were met by use of this branch of the subject before the validity of the work had been established. Few parts of mathematics are associated with so many incidents of human interest, and more than any other branch of the subject, the history of the calculus shows the influence of critics who, although not necessarily in sympathy with the subject itself, were in the end its actual benefactors.
Newton believed in a "mechanomorphic" universe. God had created the cosmos and established certain laws that applied throughout it. Newton wished to know how the Almighty had fashioned the universe but was not presumptuous enough to believe he could determine the Creatorís purpose. He was quite satisfied with mathematically describing those laws. Here is a quote of Newton from the Principia (after [Kli2]):
...But to derive two or three general principles of motion from phenomena, and afterwards to tell us how the properties and actions of all corporeal things follow from those manifest principles, would be a very great step in philosophy though the causes of those principles were not yet discovered. (Underscoring is mine.)
Newton, Leibniz, and Galileo did not purport to know, at least via mathematics, Godís purpose for humanity. Pascal (1623-1662), however, attempted to apply his work on probability to the world of Faith and Reason in his fascinating but refutable "wager". He claimed to prove, very much in the sense of deductive reasoning, that belief in God is rational with the following argument:
If God does not exist, one will lose nothing by believing in him, while if he does exist, one will lose everything by not believing.
As a Christian, this argument is very compelling to me. However, as a mathematician, I realize that this statement is outside of the language of any mathematical axiom system. Its logic takes as an axiom, for example, something like "if God exists then He will punish (or at least not reward) non-belief in Him while rewarding (eternally?) belief..."
More History: The 18th Century
The mid-late 18th century witnessed some brilliant mathematical work. Pure mathematics began to separate from the natural sciences and reestablish itself in its axiomatic glory. Leonhard Euler (1707-1783) may have been the greatest mathematician of all time (usually itís considered a toss-up between him and Gauss (1777-1855)). For Euler the axiomatic world of pure mathematics and the world of Faith and Reason sometimes blurred, as it had for Pascal.
The following is a famous anecdote (related by [Gui]) of an encounter of Euler, who was a Christian, with the encyclopedist and atheist Denis Diderot:
Euler, it seems, accepted an invitation to meet Diderot, who at the time was in attendance at the royal court of the Russian Czar. On the day of his arrival, Euler strode up to Diderot and proclaimed: "Monsieur, (a+bn)/n = x, donc Dieu existe, repondez!" In the past the French scholar had eloquently and forcefully refuted many a clear philosophical argument for Godís existence, but at the moment, at a loss to comprehend the meaning of the mathematical equation, Diderot was intimidated into silence.
Euler was probably trying to embarass Diderot with this exchange. Surely Euler knew the axiomatic method well enough to understand this was not a valid argument. What is hard to understand was that Diderot fell for this ploy. Iíll offer a possible explanation, which will also bring to light some of 18th century reason.
As mentioned before, while in the 17th century the physical world impinged upon pure mathematics, in the 18th one could argue the reverse occurred. Some of the great thinkers of the day attempted to apply the axiomatic method to the physical world, even in aspects of human behavior. "The search for axiomatic truths upon which the science of human behavior was to constructed took on the appearance of a gold rush." [Kli1]. Locke, Hume, Berkeley all wrote treatises on this theme.
An example of an axiom of human nature is "all men are created equal". There is obviously no reason to find fault with such a statement. However, it is not the same as a mathematical axiom in that, for example, "man" is not an undefined term, but has biological, social, and commonplace meanings. Any conclusion drawn from such a system would be at best a model of aspects of humanity.
The jurist and philosopher Jeremy Bentham (1748-1832) attempted not only a rational and deductive system of human ethics, but tried also to make it quantitative. He assigned numerical values to a list of pleasures and pains based on objective factors and "sensibilities". The "value" of any act could be computed, and courses of conduct could be determined by comparing values of chosen acts. The philosopher David Hartley (1705-1759) carried this idea from the sublime to the ridiculous with his summary of moral and religious truth in the formula W=F2/L, where W is the love of the world, F is the fear of God, and L is the love of God. Presumably, as a person grows older, L increases without bound, so that W decreases and approaches zero.
It is easy to see that, in the spirit of the "science of human nature", Diderot could have been flummoxed by Eulerís attack. Fortunately, thinkers of the 19th century mostly abandoned this type of reasoning, and a proper of separation of pure mathematics, science, and the study of human nature was attained.
The Addition of Rigor: The 19th Century
Karl Gauss (1777-1855), a child prodigy who became the greatest mathematician of the 19th century, and perhaps of all time, gave evidence that the gedanken of his age had changed in the following succinct statement [New]:
There are problems to whose solution I would attach an infinitely greater importance than to those of mathematics, for example touching ethics, or our relation to God, or concerning our destiny and our future; but their solution lies wholly beyond us and completely outside the province of science.
Axiomatic (pure) mathematics assumed its rightful place in this century. The great mathematicians concentrated on establishing the fundamentals (Grundlagen) of well-known methods, most notably the calculus. Cauchy (1789-1857) and Weierstrass (1815-1897) added rigor to the calculus with the concepts of limits and the insistence on proof for all their results. Gauss developed a new type of calculus, called differential geometry, that gave the tools necessary for the physicists of 20th century to formulate theories of space, time, the infinitesimal, and the unimaginable hugeness of the cosmos.
One of the most significant accomplishments in the 19th century was a rigorous axiomatic description of the infinite. Georg Cantor (1845-1918) initiated the discipline of Set Theory and used its language to build orders upon orders of infinity. Such stuff now is the motherís milk of budding pure mathematicians. A shibboleth for a would-be mathematician is how he/she feels upon first encountering the concept of transfinite cardinal numbers and, in particular Cantorís remarkable proof that the continuum (the number of real numbers) is a higher order of infinity than the integers (a countable infinity). [A realization for me personally, upon taking a graduate course in mathematical logic, was that these orders of infinity do not refer literally to the vastness of the universe, nor to God or Eternity. They are "merely" axiomatic constructs. However, I could relate this to my Christian faith in the sense that these constructs helped me "model" God and Eternity in my mind.]
Pure mathematics and applied mathematics (mostly physics) were nearly completely divorced by the end of the 19th century. By that time pure mathematics was quite close in its deductive nature to the that of the ancient Greeks, although the axiomatic systems and their development were much more complicated.
A Mathematicianís Apology: Mathematics as a Respite
A mathematician, like a painter or a poet, is a maker of patterns...The mathematician's patterns, like the painter's or the poet's, must be beautiful. [Har]
Godfrey Harold Hardy (1877-1947), long-time dean of British mathematics, may not have been the greatest pure mathematician of the 20th century, but he certainly wrote the most eloquently about being a pure mathematician. Actually, his Mathematicianís Apology is a rather short oeuvre of fewer than one hundred pages, but I know of no other work that displays the soul of a mathematician. It is a vivid description of how a mathematician thinks, and how mathematics is a pleasure.
I will quote freely from Hardy to develop the point that pure mathematics, while it doesnít directly affect faith and the human condition, provides a respite from the physical and social world. I assert now that this respite is in fact as beautiful as any created by the most talented artist, poet, or novelist, although it is only accessible to those who have made the effort to climb the hill to reach its door.
On the subject of the "significance" of pure mathematics Hardy writes:
What we do may be small, but it has a certain character of permanence; and to have produced anything of the slightest permanent interest, whether it be a copy of verses or a geometrical theorem, is to have done something utterly beyond the powers of the vast majority of men.
In these days of conflict between ancient and modern studies, there must surely be something to be said for a study which did not begin with Pythagoras, and will not end with Einstein, but is the oldest and the youngest of all.
Hardy discusses what he believes are the three most important motives which may lead a person to do research, mathematical, scientific, social, or otherwise: intellectual curiosity, professional pride, and ambition. By professional pride he includes the avoidance of shame in doing work not worthy of oneís talent. By ambition he means a desire for reputation and position, and perhaps even power or money. He conspicuously omits the motive of improving the human condition. In fact,
It may be fine to feel, when you have done your work, that you have added to the happiness or alleviated the sufferings of others, but that will not be why you did it. So if a mathematician, or a chemist, or even a physiologist, were to tell me that the driving force in his work had been to benefit humanity, then I should not believe him (nor should I think the better of him if I did). His dominant motives have been those which I have stated, and in which, surely, there is nothing of which any decent man need be ashamed.
If intellectual curiosity, professional pride, and ambition are the dominant incentives to research, then assuredly no one has a fairer chance of gratifying them than a mathematician. His subject is the most curious of all - there is none in which truth plays such odd pranks. It has the most elaborate and the most fascinating technique, and gives unrivaled openings for the display of sheer professional skill. Finally, as history proves abundantly, mathematical achievement, whatever its intrinsic worth, is the most enduring of all.
To support the point of the last sentence, Hardy uses the example of ancient Greek mathematics:
Oriental mathematics may be an interesting curiosity, but Greek mathematics is the real thing. The Greeks first spoke a language which modern mathematicians can understandÖ So Greek mathematics is Ďpermanentí, more permanent even than Greek literature. Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. ĎImmortalityí may be a silly word, but probably a mathematician has the best chance of whatever it may mean.
Finally, on mathematics as a respite Hardy states:
For mathematics is, of all arts and sciences, the most austere and the most remote, and the mathematician should be of all men the one who can most easily take refuge where, as Bertrand Russell says, Ďone at least of our nobler impulses can best escape from the dreary exile of the actual worldí
Truly pure Mathematicians, unlike scientists or accountants, are not professionally concerned with the practical application of their discipline to the everyday workings of the Real World. In fact, an ennui about the lack of practicality seems to be the distinguishing characteristic between Pure and Applied Mathematicians. Conversely, neither is the Puristís personal Weltanschauung of any seeming influence on her or his work.
Tobias Dantzig states the conclusion better that I could [Dan]:
Between the philosopher's [and the scientist's] attitude toward the issue of reality and that of the mathematician there is this essential difference: for the philosopher [or scientific researcher] the issue is paramount; the mathematician's love for reality is purely platonic. The mathematician is only too willing to admit that he is dealing exclusively with acts of the mind. To be sure, he is aware that the ingenious artifices which form his stock in trade had their genesis in the sense impressions which he identifies with crude reality, and he is not surprised to find that at times these artifices fit quite neatly the reality in which they were born. But this neatness the mathematician refuses to recognize as a criterion of his achievement: the value of the beings which spring from his creative imagination shall not be measured by the scope of their application to physical reality. No! Mathematical achievement shall be measured by standards which are peculiar to mathematics. These standards are independent of the crude reality of our senses. They are: freedom from logical contradictions, the generality of the laws governing the created form, the kinship which exists between this new form and those that have preceded it. The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit.
Before we take to sea we walk on land, Before we create we must understand...[Har]
Hardyís statement points out the necessity for the thorough founding of a student in the basics of mathematics before creativity is possible. It is clear that basic math skills are necessary to function in a modern society, so that substitution of anything less ultimately hurts the culture, as well as more immediately the individual. However, a further tragedy of a weakening of elementary mathematics pedagogical standards is the barring of even more people from the gate of the world of pure mathematics.
[Dan] Dantzig, Tobias, "The bequest of the Greeks", Scribner, New York, 1954
[Fer] Ferguson, Kitty, "The Fire in the Equations: Science, Religion & the Search for God", Wm. B. Eerdmans Publishing Co., 1994
[Gui] Guillen, Michael, "Bridges to Infinity", Jeremy P. Tarcher, Inc., Los Angeles, 1983, 204 pages
[Har] Hardy, G.H., "A Mathematicianís Apology", Cambridge University Press, 1967, 153 pages, forward by C.P. Snow
[Kin] King, Jerry P., "The Art of Mathematics", Plenum Press, 1992, 313 pages
[Kli1] Kline, Morris, "Mathematics in Western Culture", Oxford University Press, 1953, 484 pages
[Kli2] Kline, M., "Mathematics and the Search for Knowledge", Oxford University Press, 1985, 257 pages
[Kli3] Kline, M., "Mathematical Thought from Ancient to Modern Times", Oxford University Press, 1972, 1238 pages
[New] J. R. Newman (ed.), The World of Mathematics, Simon and Schuster, New York 1956
[San] Sanford, Vera, "A Short History of Mathematics", Houghton-Mifflin Co., 1958
[Sch] "Stephen Hawking, The Big Bang, and God" (by Henry Schaefer, a quantum chemist)
The essay quotes a number of famous scientists. For example,
``Toward the end of Schroedinger's career he made this statement, "I am very astonished that the scientific picture of the real world around me is very deficient. It gives us a lot of factual information, puts all of our experience in a magnificently consistent order but it is ghastly silent about all and sundry that is really near to our heart, that really matters to us." Schroedinger believed that science has limits; it knows nothing of beautiful and ugly, good or bad, God and eternity.''