The Current Mess in the Philosophy of Mathematics

by Forrest Dristy

 

 

Throughout most of the history of Western civilization geometry enjoyed the reputation of being the most certain and indubitable of all human knowledge. The knowledge of space itself, as codified in Euclid's Elements, was regarded as being perhaps the only human knowledge that was absolutely certain. Had not all the theorems of this subject been carefully deduced from postulates and axioms that all sane people accepted as self-evident truths? The challenge that faced all serious thinkers was to demonstrate a similar certainty for other kinds of subject matter, preferably by founding that subject matter directly on the bedrock of geometry. Thus when Spinoza and Descartes proved the existence of God, they used the methods of geometry. Similarly Newton used these methods in proving his theorems about the motion of terrestrial and celestial objects.

But in the Nineteenth Century, when it became clear that the geometry of Euclid was not the only possible consistent geometry, confusion entered_ the picture. The certitude of geometrical knowledge was called into question. What had been apparently a sure foundation was suddenly seen to be deficient, and mathematical philosophers had a full-blown crisis on their hands. What could be done to establish a sure foundation to replace the traditional one that was now seen to be wanting?

In this paper I hope to sketch briefly the story of what has been attempted and to show that even now, after more than a century of frantic effort, the crisis is unresolved. Moreover, I hope to show that this continuing crisis has caused among mathematicians, or at least among those mathematicians who wonder about the nature of mathematics, a kind of schizophrenia that would seem to be intolerable. Finally, at the end, I will try to draw from this story a helpful moral and relate it to certain passages from Divine Providence.

Why did the discovery of non-Euclidean geometries deliver such a devastating blow to the certitude of the theorems of geometry? It showed that there is no logical necessity to accept Euclid's postulates as true statements. Furthermore, there can be no empirical necessity for accepting them either. Any attempt to establish their validity empirically, say by measuring the angle-sum of a very large triangle, must involve some possible margin of error. One can never know with certainty that the empirically measured quantity does not differ from the predicted magnitude by an amount less than the possible error. Besides, the truths of geometry were supposed to be more certain than mere information gathered through the senses, so the very idea of having to justify the former by appeal to the latter would already imply a great loss of status for geometry.

As the Nineteenth Century progressed, traditional geometry encountered other difficulties besides the new rival geometries. Developments in analysis began to outstrip any intuitive basis that could have been provided by any of the geometries.

Mathematicians discovered, or devised, mathematical entities such as space-filling curves and nowhere differentiable continuous functions whose properties were simply more than geometry, of any version, could cope with.

The foundational failure of geometry led the leading mathematicians to seek another foundation for the huge and ever-growing superstructure of mathematics. One promising idea was to found mathematical analysis on arithmetic rather than geometry. Hence began the great program known as the “arithmetization of analysis” led by Dedekind and Weierstrass. All concepts and theorems were to be reduced ultimately to the simple laws of arithmetic, which, it was hoped, would provide a solid and contradiction-free foundation. As one step in the program, for example, the bewilderingly subtle system of real numbers had to be rigorously constructed from the system of positive integers, the so-called natural numbers.  In general the program met with much success. However, its development required the introduction of a new and abstract theory that came to be called set theory.

At first set theory, however abstract, seemed to be blessed with the beauty of simplicity. It seemed to be, in fact, a reformulation of the laws of logic. Was this theory itself, then, not the long-sought foundation for which mathematics had such desperate need? Frege thought that it was. He set to work with characteristic Teutonic thoroughness and after years of effort wrote a massive work rigorously deriving the basic concepts of mathematics from the intuitive properties of sets. But, alas, just as his work was ready for publication, he received from Russell a letter giving a clear, terse example which showed that set theory, as then understood, was self­contradictory and hence invalid as a basis for mathematical reasoning. Of course Frege was disappointed, to say the least, but to his credit he immediately acknowledged the seriousness of the difficulty. He replied to Russell that he was in the position of a builder who, just as he applied the finishing touches to his cherished building, saw its foundation collapse before his eyes.

The collapse of set theory of course had serious implications for the program of arithmetization, which was heavily dependent on the set concept.   By this time mathematics not only had no viable foundation, but it was becoming more and more sensitive to a certain embarrassing question: If mathematics was not derived from geometry, the laws of space, then what exactly was mathematics about? In other words, just what content, if any, did mathematics have? I shall refer to this question as the content question,

Now Russell was very impressed with Frege's work in spite of the set theoretic difficulty. Perhaps something like Frege's original goal could still be attained, not by using unreliable set theory but by deriving mathematics directly out of the rules of logic. Thus was established the logicist school of the philosophy of mathematics. The logicists would not only provide mathematics with a sound foundation, they would simultaneously give a definitive answer to the content question.

The content of mathematics, they would show, is nothing more nor less than the content of logic. Logic is related to mathematics as the child is to the adult: mathematics is logic grown up. Hence mathematics is, in its entirety, one vast tautology. Thus thought the logicists.

This position of the logicists offended the sensibilities of many prominent mathematicians. Did not every self-respecting mathematician have a deep, intuitive feeling of the reality of mathematical entities? Surely there was more to mathematics than mere empty logic. Probably even the logicists admitted to themselves, if not to others, that in reducing mathematics to a tautology they were paying a heavy price. But if that was the price required to buy for mathematics the certitude it was expect d to have, then they were prepared to pay it.

Thus Russell and his colleague Whitehead went to work on their Principia Mathematica, which was to show in agonizing detail how mathematics could be extracted from the rules of logic.

Unfortunately for their thesis, they had to incorporate into their “logic” some highly original, complicated, and unorthodox rules.  The resulting system from which they claim- to have built the basic concepts of mathematics is by no means mere logic in any reasonable sense. The passage of time seems to continually diminish the number of mathematicians who accept the logicist position.

Some of the mathematicians who especially abhorred the logicist thesis were the intuitionists, among whom the Dutch mathematician Brouwer was pre-eminent. To them mathematics did not at all consist of the logical manipulation of statements. They could not accept the assumption that all statements must be either true or false even when we have no way of determining which is the case. Thus there were certain rules of orthodox logic, notably the rule of the excluded middle, that they refused to use. To the intuitionists, mathematics had to be founded on the intuition of counting.

Their proofs came from constructing by this intuition structures which could be “seen” with the mind's eye. Their theorems were merely reports of the properties so seen. The intuitionist school, though small, is still active.  It claims to have a solid foundation for the work that it does, Unfortunately, its foundation seems to be a foundation for something other than orthodox mathematics. The acceptance of its position would require the abandonment of a great deal of what most mathematicians regard as valid and beautiful mathematics, Mathematicians in general are unwilling to do this.

In spite of Russell's paradox and several other set theoretic inconsistencies that were subsequently· brought to light, most mathematicians still regarded set theory as the best hope for providing a reasonable foundation. The strategy was to bring intuitive set theory under control, so to speak, by carefully axiomatizing it in such a way that the contra­ dictions would be avoided. Several such axiomatizations of set theory were worked out during the first half of the present century. Today they are regarded by most mathematicians as constituting the foundation of pure mathematics, at least so far as it has a foundation. The difficulty is this: since these axiomatic theories were contrived for the express purpose of circumventing known self-contradictions of set theory, how can we be sure that they do not contain, lurking somewhere deep within them, still other, as yet unknown, contradictions?

In an attempt to lay to rest all such doubts about possible insidious inconsistencies, David Hilbert devised the approach to mathematics known as formalism. His program was to reduce mathematics to a finite collection of symbols together with finitely many precise rules for combining these symbols into expressions and for manipulating these expressions to derive new admissible expressions. Thus, in formalism, the mathematical enterprise consists of rearranging meaningless symbols; so mathematics. becomes entirely a formal structure with no content at all. The expectation and the hope was that it could then be demonstrated that no contradiction could be possible. The formalists were prepared to pay the highest conceivable price to attain certainty. Their answer to the content question was that mathematics has no content at all – it is pure form. In the 1930s Hilbert and Bernays wrote their treatise Grundlagen der Mathematik, which was the formalist counterpart to Russell and Whitehead's Principia Mathematica. It was published in two volumes in 1934 and 1939, Although it almost seemed to achieve the desired goal it was actually not complete and their high expectations were unfulfilled.

In retrospect their efforts were doomed to failure by some amazing results that were obtained by a young Austrian mathematician, Kurt Gödel, already in 1931, Gödel showed that in a formal system that is rich enough to express the theorems of mathematics, or even the theorems of the natural number system, it is impossible to prove consistency by methods belonging to the system. Thus the formalist goal is unattainable, and so it seems that even the highest price that can be paid is insufficient to buy complete certainty for mathematics.

For a more detailed discussion of Gödel's theorems, see Cameron Pitcairn's paper Gödel's Results on the Completeness and Consistency of Mathematics given at last year's Colloquium. In this paper there is a quotation from Frank DeSua that I think is sufficiently novel and interesting to be repeated here. DeSua made (in The American Mathematical Monthly, 1956) this observation about the nature of mathematics. in the light of Gödel's work:

Suppose we loosely define a religion as any discipline whose foundations rest on an element of faith, irrespective of any element of reason which may be present. Quantum mechanics, for example, would be a religion under this definition. But mathematics would hold the unique position of being the only branch of theology possessing a rigorous demonstration of the fact that it should be so classified.

Although formalism did not achieve its ultimate goal, it still exerts an enormous influence over present day mathematics. This may be at least partly due to Hilbert's enormous prestige. At any rate, current pure mathematics built on axiomatic set theory is a highly formalized discipline.

Although it has not been proved to be consistent, most mathematicians feel comfortably safe from encountering contradictions in their work.

It is the unresolved content question that leads conscientious mathematicians into a state of discomfiting ambivalence. In his daily work it seems to the mathematician that he is dealing with reality; that mathematical structures have a real existence apart from the mathematician himself.

To this extent he is a Platonist. But when called upon to give an account of this mathematical reality, to explain it to someone else, the mathematician, aware of his inability to do so, falls back to the formalist position and admits to dealing only with formal systems void of content. Dieudonné, speaking for the consortium of mathematicians known as Bourbaki, has put it this way:

On foundations we believe in the reality of mathematics, but of course when philosophers attack us with their paradoxes we rush to hide behind formalism and say,  “Mathematics is just a combination of meaningless symbols,” and then we bring out Chapters 1 and 2 on set theory. Finally, we are left in peace to go back to our mathematics and do it as we have always done, with the feeling each mathematician has that he is working with something real. This sensation is probably an illusion, but it is very convenient. That is Bourbaki's attitude toward foundations,

So the mathematician is a split personality. He is a Platonist on weekdays while engaged in his ordinary work; but on weekends, or whenever he has time to engage in philosophical discussions, he becomes a formalist. We can almost describe the situation by paraphrasing a remark of desperation recently made by the government's haggard inflation fighter Alfred Kahn: Any mathematician who isn't schizophrenic these days just isn't thinking clearly.

In summary we can say that we have discussed two of the major issues of the philosophy of mathematics. One of these is the content question, which remains unanswered. The other is the quest for a sure, consistent foundation, which has ended in failure. It is from this rather unhappy story that I wish to draw a moral, Undoubtedly the story contains many worthy morals, but the one that seems clearest to me is this: that the human intellect, of and by itself, is unable to attain even the smallest shred of truth.

I think this moral is a significant confirmation of some important things that the Writings say about the faculty they refer to as human prudence. If we keep in mind this dictionary definition of prudence:  “the ability to regulate and discipline oneself through the exercise of the reason,” we see that the quest for a sure foundation for mathematics is pushing prudence to its limit. It is the attempt to so discipline the mind, through the rigorous exercise of reason, that one can attain by that means alone some absolutely certain bit of knowledge from which the verities of mathematics can be deduced.

But the Writings say that this kind of prudence is an illusion. The heading for Chapter X of Divine Providence, for example, reads in part, “There is no such thing as one's own prudence; there only appears to be and it should so appear.”

 Does this explain why mathematical entities seem so real to working mathematicians and why, nevertheless, they vanish into the meaningless symbols of formalism when one attempts to give an iron-clad demonstration of their consistency?

Commenting on the strength of the appearance of the reality of prudence and how difficult it is to convince someone of its illusory nature, D.P. 191 says,  “One who believes, on the strength of the appearance, that human prudence does all things, cannot be convinced except by reasons to be had from a more profound investigation and to be gathered from causes,”

The  “reasons to be gathered from causes” are enumerated in the next number, and the rest of Chapter X is devoted to an explanation of them. But nothing more is said about the  “reasons to be had from a-more profound investigation.” Could these be reasons obtained from results such as Gödel's?  Certainly these are from a more profound investigation than any that could have been done in the Eighteenth Century,

After explaining that there can be no thought except from some affection and that the thought is seen whereas the affection is only felt, Swedenborg writes (in D.P. 198):

For affection shows itself only in a certain enjoyment of thought and in pleasure over reasoning about it.

This pleasure and enjoyment make one with the thought in those who, from self-love or love of the world, believe in one's own prudence. The thought glides along in its enjoyment like a ship in a river current to which the skipper does not attend, attending only to the sails he spreads.

This passage readily brings to mind the vision of a working mathematician, swept along in the enjoyment of his very prudent reasonings, spreading ever more sail in the hope, perhaps, of landing on a result impressive enough to be published in a prestigious journal. Who could tell him that his own prudence is nothing, that his reasonings are vain and void of content?

In D.P. 293 some comments are made about the view of the angels on will and understanding in man, which is that there cannot be a grain of will or of prudence in man that is his own. One of the reasons given to support this view is that to think and will actually from one's own being is the Divine itself, and the Divine cannot be appropriated to anyone for then man would be God. The attempt to find some absolutely certain knowledge on which to build mathematics has been an attempt by man to think from his own being. Since the ability to do this is, in the nature of things, reserved to the Divine, there should be little wonder that it failed.

Still another way to look at the situation is suggested by D.P. 316, where human prudence is equated to the intellectual proprium in distinction to the volitional proprium or self-love. It is the nature of proprium, we are told, to strive to make everything its own. How well this teaching is illustrated by the story of the quest for mathematical certainty. We are told further that

All who are led by the Lord's divine providence are raised above the proprium and then see that all good and truth are from the Lord, indeed see that what in the human being is from the Lord is always the Lord’s and never man’s. He who believes otherwise is like one who has his master’s goods in his care and claims them himself or appropriates them. He is no steward but a thief.

From this point of view, the epic struggle to find a certain foundation for mathematics has been a dastardly attempt to chisel from the great monolith of truth a little chunk that we can call our very own. It has been an endeavor to steal from God a fragment of His altar and to glory in it as a triumph of our prudence--our intellectual proprium. Viewed in this way, the current mess in the philosophy of mathematics is not a disaster. On the contrary, it is a valuable lesson in humility and it is humility, not pride, that is instilled by all true learning.