Of Hilbert and Gödel

It was the year 1900; August, to be more accurate, when in Paris, David Hilbert, one of the best-known mathematicians of his time, posed a list of twenty-three open problems. The impact was huge, so much so that much of the mathematical research of the dawning century was consumed by these problems, the most important ones for Hilbert. Nobel Prize winners, Fields medalists, and other winners of prestigious awards were among those who work to solve them. Some of them (the Riemann hypothesis, for instance) are so hard that they are still open, and large sums of money are offered to whomever is able to solve them.

It was the year 1900. The Enlightenment had come, the Dark Ages were past. The Scientific Revolution brought progress. God was dead, now the Superman (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 as the most promising one. It would be the beginning of a new age in which man will take the position he was destined to take, 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 subjected to a logical explanation to verify its truth value. If every new year brings with itself 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 through and through, promise! No wonder such a fervor.

In a way the optimism was at least understandable, if not justified. 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 of subjecting everything to human reason. The second problem was to prove that the axioms of arithmetic were consistent —that the axioms of the natural numbers did not lead to any contradictions. The sixth problem was 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.

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. To the point that the his gravestone at Göttingen has these words inscribed (in German):

We must know.
we will know.

Hilbert epitafio
Image by Kassandro, CC-BY-SA-3.0

The epitaph on the gravestone was his response to the Latin maxim ignoramus et ignorabimus (“we do not know, we shall not know”), a dictum pronounced by the German Physiologist Emil du Bois-Reymond in a speech to 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 called ‘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 will become the bloodiest in history; progress and industrialization had not caused widely noticed environmental issues; the Left had not produced its Gulag and the Right had not built its Auschwitz.

These events (and some others) overthrew the modern ideal in the way the rolling stone in the vision of Daniel broke the statue with feet of clay into pieces. 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. In brief, that the problem of man is no other than himself.

Thus, the practical problem of Modernity was man himself, and it was devastating. But the conceptual problem was still to come and it was equally devastating to the modern aspirations.

Enter Gödel

It was September 8 of 1930, Monday, when Hilbert opened the anual 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, also in Königsberg, during the three previous days 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-minutes talk, Kurt Gödel 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 the 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; and 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 stronger. It says that no consistent axiomatic system can prove its own consistency. In the end, his second 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 enunciation 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. That is, 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, he says this 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: it shows that 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 trashcan and it does the same with today’s “scientism”. There are indeed true statements that are beyond mathematics and science.

Faith

Gödel’s second incompleteness theorem is really strong, overwhelming, and even a source of hopelessness from 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, it 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 that there are ways to prove the consistency of a system. For instance, provided we subsume it in a more comprehensive one. It 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 faith step in the bigger one. Moreover, since 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. Now suppose there is a third system which comprehends the second one and proving that the it is consistent. Faith is all the more necessary if we are to believe that the third system is also consistent. 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 is going to 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 pretense to know all things is nothing more than a statement on a gravestone. 

Postscript in Christian apologetics

Even though for years I enjoyed applying analytic philosophy to Christian apologetics, this and other considerations have led me to question that approach. At this point, I don’t see that it shows anything definite. Rather, 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. Minerva’s worshippers, rather than Christ’s. 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 indemonstrable 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 only from an epistemological viewpoint but from an ontological one, there must be a 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. 

***

In the beginning was the Logos,
and the Logos was with God,
and the Logos was God.
He was in the beginning with God.
All things were made through him,
and without him was not any thing made
that was made.
In him was life,

and the life was the light of men…
And the Logos became flesh
and dwelt among us, 
and we have seen his glory,
glory as of the only Son from the Father,
full of grace and truth.
John 1:1-4, 14

He is the image of the invisible God,
the firstborn of all creation.

For by him all things were created,
in heaven and on earth,
visible and invisible,
whether thrones or dominions
or rulers or authorities
—all things were created through him
and for him. 
And he is before all things,
and in him all things hold together. 

Colossians 1:15-17

Categorías:English

1 respuesta

  1. I want to thank to Denyse O’Leary for her amazing review of my translation into English of this article. It also appeared at Mind Matters: https://mindmatters.ai/2020/01/faith-is-the-most-fundamental-of-the-mathematical-tools/

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