Hilbert program

a history of Hilbert’s program for the foundations of mathematics, initiated by his statement of the 2nd problem in his Problems Address, Paris, 1900 -- see Hilbert problems

In his 1990 lecture to the International Congress of Mathematicians in Paris, David Hilbert presented a list of open problems in mathematics. He expressed the 2nd of these problems, known variously as the compatibility of the arithmetical axioms and the consistency of arithmetic, as follows:

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. Upon closer consideration the question arises: Whether, in any way, certain statements of single axioms depend upon one another, and whether the axioms may not therefore contain certain parts in common, which must be isolated if one wishes to arrive at a system of axioms that shall be altogether independent of one another.

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 definite number of logical steps based upon them can never lead to contradictory results.

Hilbert’s 2nd problem arose from a principle that had only recently emerged in his thought, namely, that “mathematical existence is nothing other than consistency.”

In the decades that followed his lecture, Hilbert made this 2nd problem more explicit by developing “a formal system of explicit assumptions” (see Axiom and Axiomatic method) upon which he intended to base the methods of mathematical reasoning. He then stipulated that any such system must be shown to have these characteristics:

1. the assumptions should be "independent" of one another (see Independence)
2. the assumptions should be “consistent” (free of contradictions) (see Consistency)
3. the assumptions should be “complete” (represents all the truths of mathematics) (see Completeness)
4. there should be a procedure for deciding whether any statement expressed using the system is true or not (see Decision problem and Undecidability)

Hilbert's 2nd problem is said by some to have been solved, albeit in a negative sense, by K. Gödel (see Hilbert problems and Gödel incompleteness theorem).

And yet, in his 2000 Distinguished Lecture to the Carnegie Mellon University School of Computer Science, Gregory Chaitin began his remarks as follows:

I’d like to make the outrageous claim, that has a little bit of truth, that actually all of this that’s happening now with the computer taking over the world, the digitalization of our society, of information in human society, you could say in a way is the result of a philosophical question that was raised by David Hilbert at the beginning of the century.

The philosophical question to which Chaitin was referring is the surmise at the heart of Hilbert’s 2nd problem. The title Chaitin gave to his lecture was “A Century of Controversy Over the Foundations of Mathematics.”

The question for us today is this:

How are we to view this century-old-and-more controversy?

“There can be no other way,” we are told, “than from our own position of understanding and sophistication…. [W]e have to try to appreciate the difference between our viewpoint and that of mathematicians centuries ago.” This article attempts to assist our appreciation of that difference.

For a history of the mathematics preceding and relevant to Hilbert’s statement of the 2nd problem, which initiated his program, see the article Hilbert 2nd problem.

Two caveats about Hilbert’s 2nd problem

The history of Hilbert’s program that follows will benefit from a brief preliminary mention of two reservations:

1. Intuition in mathematics
2. Hilbert’s 2nd problem vs. Hilbert’s program

Intuition in mathematics

The vision of a mathematics free of intuition was at the core of the 19th century program known as the Arithmetization of analysis.

Hilbert, too, envisioned a mathematics developed on a foundation “independently of any need for intuition.” His vision was rooted in his 1890s work developing an axiomatic theory of geometry. In Hilbert’s view, the theory of any area of mathematics, if developed rigourously, would be as follows:

• it would be developed independently of any need for intuition
• it would clarify logical relationships between basic concepts and axioms.

There was a deep irony, ably expressed as follows, in this vision of an intuition-free mathematics,:

At the time of Hilbert’s Problems Address, there was no mathematical formalization of Algorithm (or indeed of computational device, computational procedure, or computable function.) There was only an intuitive notion, spoken of as follows:

• a specified discrete process that follows a finite and fixed set of rules
• it (deterministicly) follows the same steps with the same input
• . . . a “mechanical procedure” (Hilbert’s own statement)

Hilbert himself did succeed in stating the crucially important Decision Problem precisely, but not until the late 1920s.

Precise notions of computation (and indirectly of algorithm) and the enabling concept of the Turing Machine needed to await the mid-1930s work of Turing and Post.

In other words, at the time of Hilbert's address, the notion of a rigorously developed mathematics utterly free of intuition was itself a thoroughly intuitive notion, not perhaps of the mathematical, but rather of the meta-mathematical sort – a distinction that, at the time, had also not yet been developed.

Hilbert’s 2nd problem vs. Hilbert’s program

From time to time, questions are raised about the connection between Hilbert’s 2nd problem and his program for the foundations of mathematics. Here, for example, are excerpts from a recent (2011) question:

Is Hilbert's second problem about the real numbers or the natural numbers?

In his famous "23 problems" speech, Hilbert gave his second problem as follows:

The axioms of arithmetic are essentially nothing else than the known rules of calculation, with the addition of the axiom of continuity. I recently collected them and in so doing replaced the axiom of continuity by two simpler axioms, namely, the well-known axiom of Archimedes, and a new axiom essentially as follows: that numbers form a system of things which is capable of no further extension, as long as all the other axioms hold (axiom of completeness). I am convinced that it must be possible to find a direct proof for the compatibility of the arithmetical axioms, by means of a careful study and suitable modification of the known methods of reasoning in the theory of irrational numbers.

Now, I'm not sure what he's referring to in the "recently" but it might be his paper "On the concept of number" published also at 1900. In this paper Hilbert gives an axiomatic system for the real numbers (with order). . . .

So, what was Hilbert's 2nd problem about? Is it correct to interpret it as a question about Peano arithmetic? Is it correct to claim that Gödel's theorem had a major impact on the question? Or is it a confusion between Hilbert's program and the 2nd question?

asked -- Gadi A Oct 25 '11 at 11:16

Here are excerpts from the answer to the question provided on the site:

The universal understanding is that a positive solution to Hilbert's second problem requires a convincing proof of the consistency of some adequate set of axioms for the natural numbers. The history of [Hilbert’s 2nd] problem is [... as follows]:

Hilbert provided such an axiomatization in, but it became clear very quickly that the consistency of analysis faced significant difficulties, .... Hilbert thus realized that a direct consistency proof of analysis, i.e., one not based on reduction to another theory, was needed. He proposed the problem of finding such a proof as the second of his 23 mathematical problems in his address to the International Congress of Mathematicians in 1900 and presented a sketch of such a proof in his Heidelberg talk (1905).

Note that the term "analysis" in that article is the traditional term for the theory of natural numbers and sets of natural numbers, which is now called second-order arithmetic. In the first decades of the 20th century, the study of formal logic and model theory was still in its infancy, and many basic facts which we now take for granted were not known to researchers in that era. In particular, Hilbert would have had no reason to expect that the theory of the real numbers as a field would behave differently from the theory of second order arithmetic.

answered -- Carl Mummert Oct 25 '11 at 12:46

Primary sources

• Hilbert, D. (1900). “Über den Zahlbegriff,” Jahresbericht der Deutschen, Mathematiker-Vereinigung 8, 180–184. (English translation in Ewald, W. (1996). “On the concept of number,” From Kant to Hilbert: A source book in the foundations of mathematics, vol. 2, Oxford University Press.

• Hilbert, D. (1902). "Mathematical problems," Bull. Amer. Math. Soc. , 8 pp. 437–479, MR1557926 Zbl 33.0976.07, (Reprint: ‘’Mathematical Developments Arising from Hilbert Problems’’, edited by Felix Brouder, American Mathematical Society, 1976), URL: http://aleph0.clarku.edu/~djoyce/hilbert/problems.html, Accessed: 2015/06/03.

References

• Bertossi, L. (2008). “From Hilbert to Turing and beyond . . .,” Talk at University of Concepcion, Dec. 2008, URL: http://people.scs.carleton.ca/~bertossi/talks/hilbTur08.pdf, Accessed: 1025/06/01.

• Calude, C.S. and Chaitin, G.J. (1999). “Mathematics / Randomness everywhere, 22 July 1999,” Nature, Vol. 400, News and Views, pp. 319-320, URL: https://www.cs.auckland.ac.nz/~chaitin/nature.html, Accessed: 2015/06/19.

• Chaitin, G. (2000). “A Century of Controversy Over the Foundations of Mathematics,“ Journal Complexity -- Special Issue: Limits in mathematics and physics, Vol. 5, No. 5, May-June 2000, pp. 12-21, (Originally published in Finite Versus Infinite: Contributions to an Eternal Dilemma, Calude, C. S.; Paun, G. (eds.); Springer-Verlag, London, 2000, pp. 75–100), URL: http://www-personal.umich.edu/~twod/sof/assignments/chaitin.pdf Accessed 2015/05/30.

• Ferreirós, J. (1996). “Hilbert, Logicism, and Mathematical Existence,” URL: http://personal.us.es/josef/HLME.pdf, Accessed: 2015/06/11.

• Math Stack Exchange. (2011). “Is Hilbert’s second problem about the real numbers or the natural numbers?” Math Stack Exchange, URL: http://math.stackexchange.com/questions/75705/is-hilberts-second-problem-about-the-real-numbers-or-the-natural-numbers, Accessed: 2015/05/31.

• O’Connor, J J. and Robertson, E F. (1997) “An Overview of the History of Mathematics”, MacTutor History of Mathematics archive, URL: http://www-history.mcs.st-andrews.ac.uk/HistTopics/History_overview.html Accessed: 2015/06/18.

• Pon, S. (2003). “Hilbert’s Second Problem: Foundations of Arithmetic,” Undergraduate paper for Math 163: History of Mathematics, U.C. San Diego, URL: https://www.math.ucsd.edu/programs/undergraduate/history_of_math_resource/history_papers/math_history_05.pdf, Accessed: 2015/06/09.

• Zach, R. (2015). “Hilbert’s Program,” The Stanford Encyclopedia of Philosophy, (Summer 2015 Edition), URL: http://plato.stanford.edu/archives/sum2015/entries/hilbert-program/, Accessed: 2015/06/01.