https://mindmatters.ai/2020/01/faith-is-...cal-tools/

EXCERPT (Daniel Andrés Díaz-Pachón): 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 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.

[...] 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.

[...] 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.” ... 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.

[...] 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. (MORE - details)

EXCERPT (Daniel Andrés Díaz-Pachón): 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 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.

[...] 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.

[...] 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.” ... 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.

[...] 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. (MORE - details)