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Faith Is the Most Fundamental of the Mathematical Tools

An early twentieth century clash of giants showed that even mathematics depends on some unprovable assumptions

It was August 1900 in Paris. David Hilbert (1862–1943), one of the best-known mathematicians of his time (right), posed a list of twenty-three open problems.

The impact was huge; much of the mathematical research of the dawning century was consumed by Hilbert’s problems. Nobel Prize winners, Fields medalists, and winners of other prestigious awards were among those who worked to solve them. Some of them (the Riemann hypothesis, for instance) remain unsolved. Large sums of money are offered for a successful solution.

Nineteen hundred was felt to be a significant year. The Dark Ages were past; the Enlightenment had come. The Scientific Revolution had brought progress. God was dead, now the Superman (Friedrich Nietzsche’s Übermensch) lived. The universe, with its infinite history, did not require a God. Darwin had proposed a mechanism through which all biological species have merely emerged.

The twentieth century was shaping up to be promising, the beginning of a new age in which man would take his destined position, far from the noise of all those meaningless myths. Reason should be able to explain all things. Each event should have a natural explanation for its occurrence. Every proposition should be subject to a logical explanation to verify its truth value. If every new year brought the happiness and hope of a new beginning, how much more must a new century bring! And how much more must the twentieth century, the first truly Modern century, promise! No wonder prominent mathematicians tackled the problems with such fervor.

In a way, their optimism was understandable, if not justified by events. Not even Hilbert could escape the enthusiasm of the times. Two of his twenty-three problems, the second and the sixth, reflected the modern aspiration to subject everything to human reason. The second problem aimed to prove that the axioms of arithmetic were consistent —that is, the axioms of the natural numbers do not lead to any contradictions. The sixth problem aimed to axiomatize physics, particularly probability and mechanics.

The sixth problem conveys Hilbert’s modern heart: physics should be subjected to cold reason; even chance must submit to reason! Mathematics, the most rigorous way of knowing, should extend itself beyond abstraction to dominate chance and physical reality.

He put the matter this way when he posed the second problem:

When we are engaged in investigating the foundations of a science, we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science. The axioms so set up are at the same time the definitions of those elementary ideas; and no statement within the realm of the science whose foundation we are testing is held to be correct unless it can be derived from those axioms by means of a finite number of logical steps…

But above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to the axioms: To prove that they are not contradictory, that is, that a finite number of logical steps based upon them can never lead to contradictory results.

David Hilbert, “Mathematical Problems: Lecture delivered before the International Congress of Mathematicians at Paris in 1900″ at Göttinger Nachrichten, (1900)

Hilbert was a modern man, no doubt about it. He wanted all of scientific knowledge to be obtained from basic axioms by means of a finite number of logical steps. His goal was an extension of his particular dream for mathematics, the eponymous Hilbert’s program—to establish a consistent and complete finite number of axioms as a foundation of all mathematical theories. The goal was of cardinal importance to him. On his gravestone at Göttingen, you will find inscribed the words:

We must know. We will know. (Wir müssen wissen Wir werden wissen)

That epitaph on the gravestone (image by Kassandro, CC-BY-SA-3.0) was his response to the Latin maxim ignoramus et ignorabimus (“we do not know, we shall not know”), a dictum of the German physiologist Emil du Bois-Reymond from a speech given at the Prussian Academy of Sciences in which du Bois-Reymond argued that there were questions that neither science nor philosophy could aspire to answer.

Seen from the perspective of the age (what C. S. Lewis has called the “climate of opinion”), Hilbert’s aspiration was understandable. The two World Wars had not happened yet; science had not been used to create biological weapons; no one knew that the twentieth century would become the bloodiest in history; progress and industrialization had not caused widely noticed environmental issues; the Left had not founded its Gulag and the Right had not built its Auschwitz.

These events (and some others) overthrew modern aspirations in the way the rolling stone in the vision of Daniel broke the statue with feet of clay into pieces. (Dan 2:34) And in all of these events, the problem was easily singled out —the human being. It is impossible to make a superman out of a man. Enlightened modernity, blinded by pride, failed to see what all religions, even the oldest and the false ones, have seen so clearly—that man is wicked and the intention of his thoughts is only evil continuously, that from the sole of his foot even to the head there is nothing sound in him, that man’s heart is deceitful more than any other thing. (See Jer. 17:9) In brief, the problem of man is nothing other than himself.

Thus, the practical problem of modernity turned out to be man himself—and it was devastating. But the conceptual problem was still to come and it was equally devastating to modern aspirations.

Enter Gödel

On Monday, September 8, 1930, Hilbert opened the annual meeting of the Society of German Scientists and Physicians in Königsberg with a famous discourse called “Logic and the knowledge of nature.” He ended with these words:

For the mathematician there is no Ignorabimus, and, in my opinion, not at all for natural science either…

The true reason why [no-one] has succeeded in finding an unsolvable problem is, in my opinion, that there is no unsolvable problem. In contrast to the foolish Ignorabimus, our credo avers: We must know, We shall know.

In one of those ironies of history, during the three days prior to the conference opened by Hilbert’s speech, a joint conference called Epistemology of the Exact Sciences also took place in Königsberg. On Saturday, September 6, in a twenty-minute talk, Kurt Gödel (1906–1978) presented his incompleteness theorems. On Sunday 7, at the roundtable closing the event, Gödel announced that it was possible to give examples of mathematical propositions that could not be proven in a formal mathematical axiomatic system even though they were true.

The result was shattering. Gödel showed the limitations of any formal axiomatic system in modeling basic arithmetic. He showed that no axiomatic system could be complete and consistent at the same time.

What does it mean for an axiomatic system to be complete? It means that, using the axioms given, it is possible to prove all of the propositions concerning the system. What does it mean for the axiomatic system to be consistent? It means that its propositions do not contradict themselves. In other words, the system is complete if (using the axioms) all proposition in the system can be proven either true or false. The system is consistent if (using the axioms) no proposition in the system can be proved simultaneously true and false.

In simple terms, Gödel’s first incompleteness theorem says that no consistent formal axiomatic system is complete. That is, if the system does not have propositions that are true and false simultaneously, there are other propositions that cannot be proven either true or false. Moreover, such propositions are known to be true but they cannot be proven using the system axioms. There are true propositions of the system that cannot be proven as such, using the axioms of the system.

Gödel’s second incompleteness theorem is more stringent. It says that no consistent axiomatic system can prove its own consistency. In the end, his theorem entails that we cannot know whether a system is consistent or not; we can only assume that it is.

Implications for Hilbert’s program

Let’s recall a portion of Hilbert’s statement of his second problem: [N]o statement within the realm of the science whose foundation we are testing is held to be correct unless it can be derived from those axioms by means of a finite number of logical steps.

Hilbert knew the difference between science and mathematics, of course. So this introduction to his second problem actually fits well to his sixth problem—to axiomatize science. In this regard, his sixth problem is more ambitious than the second one because it purports to translate to science—beyond mathematics—what mathematics should be doing… at least in Hilbert’s mind. But inasmuch as Hilbert was broadening his concepts to take in science as well as mathematics, it was of particular importance that his statement be true of mathematics. The word “science” should be replaceable by the word “mathematics”: [N]o statement within the realm of the mathematics whose foundation we are testing is held to be correct unless it can be derived from those axioms by means of a finite number of logical steps.

But Gödel’s first incompleteness theorem voids such a statement. There are indeed true mathematical propositions that cannot be derived from a finite number of axioms through a finite number of logical steps. Mathematics, our best way of knowing, the one we consider the most certain, is, in the most optimistic case, incomplete!

But even this is not the end of the matter. Returning to Hilbert’s presentation of his second problem, note what he says in his second paragraph:

Above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to the axioms: To prove that they are not contradictory, that is, that a finite number of logical steps based upon them can never lead to contradictory results.

Well, Gödel’s second incompleteness theorem destroys this statement too. Because it proves the opposite: no consistent formal axiomatic system can prove its own consistency. If Hilbert’s program is the Titanic, Gödel’s incompleteness theorems are the iceberg that sunk it.

Moreover, Gödel’s first incompleteness theorem throws Comte’s positivism into the trash and it does the same with today’s “scientism.” There are indeed true statements that are beyond mathematics and science.

The faith of a rationalist

Gödel’s second incompleteness theorem is a source of hopelessness to a rationalist viewpoint. If no consistent formal system can prove its own consistency, the consequences are devastating for whomever has placed his trust in human reason.

Why? Because provided the system is consistent, we cannot know it is; and if it is not, who cares? The highest we can reach is to assume (which is much weaker than to know) that the system is consistent and to work under such assumption. But we cannot prove it; that is impossible!

In the end, the most formal exercise in knowledge is an act of faith. The mathematician is forced to believe, absent all mathematical support, that what he is doing has any meaning whatsoever.

The logician is forced to believe, absent all logical support, that what he is doing has any meaning whatsoever.

Some critics might point out that there are ways to prove the consistency of a system, provided we subsume it into a more comprehensive one. That is true. In such a case, the consistency of the inner system would be proved from the standpoint of the outer system. But a new application of Gödel’s second incompleteness theorem tells us that this bigger system cannot prove its own consistency. That is, to prove the consistency of the first system requires a new step of faith in the bigger one. Moreover, because the consistency of the first system depends on the consistency of the second one—which cannot be proved— there is more at stake if we accept the consistency of the second one. And suppose there is a third system which comprehends the second one and proving that it is consistent. Faith is all the more necessary if we are to believe that the third system is also consistent. In such a system, faith does not disappear. It only compounds, making itself bigger and more relevant in order to sustain all that it is supporting.

In the end, we do not know whether the edifice we are building will be consistent; we do not have the least idea. We just hope it will be, and we must believe it will be in order to continue doing mathematics. Faith is the most fundamental of the mathematical tools.

The question is not whether we have faith, the question is what is the object of our faith. It is the rationality of mathematics what is at stake here, its meaning. But we cannot appeal to mathematics to prove its meaning. Thus, Platonic reality, given its existence, does depend on a bigger and more comprehensive reality, one beyond what is reasonable, one that is the Reason itself.

The ambition to know all things is nothing more than a statement on a gravestone.

Postscript on Christian apologetics

Even though for years I have enjoyed applying analytical philosophy to Christian apologetics, these and other considerations have led me to question that approach. At this point, I don’t see that it creates a clear advantage. Instead, I see it as a concession to the unbeliever in order to lead him to question his own faith and place it instead in Christ.

It is sad to see that many a Christian apologist has placed his faith in logic, not in the Logos. At the end of the day, logic does not prove anything because it is grounded in unprovable propositions. It is impossible to use Aristotelian logic to prove Aristotelian logic. It begs the question; to accept it requires faith. Axioms are undemonstrable by definition and, as theory develops, they become less and less intuitive. To accept them requires faith. Similarly, the consistency of any formal axiomatic system cannot be proven, to accept it requires faith. All of our knowledge is sustained by faith. All of it.

Sustaining faith in reason, besides making for a cheap faith, constitutes an unacceptable abdication to rationalism because reason and logic cannot sustain anything. They cannot even support themselves. Moreover, in order for faith and reason to have a foundation, not merely from an epistemological viewpoint but also from an ontological one, there must be something that sustains it —a First Sustainer undergirding them all.

There is no logic without a Logos. Faith’s only task is to accept that such a Logos does exist. The opposite is despair, meaninglessness. With this in mind, John 1:1-4 14, and Colossians 1:15-17 are illuminated by a wonderful light.

Daniel Andrés Díaz-Pachón

Contributor, the Bradley Center's Mind Matters
Daniel Andrés Díaz-Pachón is Research Assistant Professor at the Division of Biostatistics at the University of Miami, where he also was a postdoc. He has a PhD in Probability from the University of São Paulo, Brazil, and a bachelor's in Mathematical Statistics from Universidad Nacional de Colombia, in Bogotá. His research is focused on some kind of intersection between probability, statistics, information theory, and machine learning. He has a deep interest in Christian apologetics and Christian life, and frequently lectures and writes on the relation of faith and science.

Faith Is the Most Fundamental of the Mathematical Tools