Hilbert program

"a history of Hilbert’s program for the foundations of mathematics, initiated by his Problems Address given in Paris, 1900" -- see Hilbert problems

In his 1900 lecture to the International Congress of Mathematicians in Paris, Hilbert proposed that an axiomatic treatment of any field of mathematics required the demonstration of the independence, the completeness, and the consistency of its axioms. More specifically with respect to geometry, he noted that the consistency (as he put it, the “compatibility”) of the axioms of geometry could be proved by providing an interpretation of the system in the real plane. In some sense, then, the consistency of geometry could be “reduced to” (proved indirectly as a result of proving directly) the consistency of analysis:

In geometry, the proof of the compatibility of the axioms can be effected by constructing a suitable field of numbers, such that analogous relations between the numbers of this field correspond to the geometrical axioms. Any contradiction in the deductions from the geometrical axioms must thereupon be recognizable in the arithmetic of this field of numbers. In this way the desired proof for the compatibility of the geometrical axioms is made to depend upon the theorem of the compatibility of the arithmetical axioms.

What was needed, in other words, was “a direct consistency proof of analysis, i.e., one not based on reduction to another theory.” This was the challenge of Hilbert’s 2nd problem and the hope of Hilbert’s program that proceeded from it.

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.”

Hilbert’s address was preceded by and founded on 30 years of efforts to construct rigourously the whole of mathematics, which involved the development of the following -- see Hilbert 2nd problem:

• the algebra of logic -- Boole/De Morgan/Peirce
• naive set theory -- Cantor
• the predicate calculus -- Peirce/Frege
• the axioms of arithmetic -- Frege/Dedekind/Peano
• transfinite arithmetic -- Cantor
• the axioms of geometry -- Pasch/Hilbert

Going forward from his 1900 Problems Address, Hilbert’s program sought to “pull together into a unified whole” these developments, together with abstract axiomatics and mathematical physics. His views in this regard, “exerted an enormous influence on the mathematics of the twentieth century.”

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 this:

“A Century of Controversy Over the Foundations of Mathematics.”

The question for us today is this: How are we to view this century-and-more-old 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.

Some caveats about Hilbert’s address

Hilbert’s address provided what was essentially his view of what the new century could and, he hoped, would bring:

• his talk embodied a very personal vision of mathematics and science
• his list of problems reflected what he saw on the mathematical horizon

An understanding of Hilbert’s program benefits from a brief preliminary mention of these issues:

1. Intuition in mathematics
2. Hilbert’s 2nd problem vs. Hilbert’s program
3. Infinite sets and the continuum

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 and response:

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 response 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

This exchange highlights an interesting assumption of Hilbert’s program, at least in its earliest stage. Since the “axioms of arithmetic” to which he referred in the statement of his 2nd problem as having been “recently collected” were indeed those axioms of ordered real numbers that he “presented by him in “Über den Zahlbegriff” several months prior to this talk,” then clearly it was Hilbert’s expectation that the axioms for “the theory of irrational numbers” would provide a basis for establishing a “compatible” set of axioms for the arithmetic of natural numbers.

Infinite sets and the continuum

As the first problem in his list, Hilbert chose Cantor’s continuum hypothesis. He pursued the subject of the continuum in his discussion of the 2nd problem, stating clearly his belief that proving the consistency of arithmetic using his “recently collected” axioms not only would prove the consistency of Euclidean geometry, but also could provide a proof for the very existence of the continuum of real numbers, including the higher Cantorian cardinals and ordinals:

The concept of the continuum or even that of the system of all functions exists, then, in exactly the same sense as the system of integral, rational numbers, for example, or as Cantor's higher classes of numbers and cardinal numbers.

In saying this, Hilbert relied on his hopes for the axiomatic method to draw a powerful connection between the two problems. He was pointing forward towards his own personal vision of the mathematical horizon, “unaware of the difficulties involved in realizing this point of view, and, more generally, [with] no precise idea of what an elaborate theory of systems of axioms would involve.”

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.

• Corry, L. (2006). “The Origin of Hilbert’s Axiomatic Method,” Jürgen Renn et al (eds.). The Genesis of General Relativity, Vol. 4 Theories of Gravitation in the Twilight of Classical Physics: The Promise of Mathematics and the Dream of a Unified Theory, Springer (2006), pp. 139-236. URL: http://www.tau.ac.il/~corry/publications/articles/pdf/Hilbert%20Kluwer.pdf, Accessed: 2015/08/16.

• Ewald, W B. (2002) Review of Grattan-Guinness, I. (2000) The Search for Mathematical Roots, 1870-1940, Princeton University Press. Bull. (New Series) Amer. Math. Soc., Vol. 40, No. 1, pp. 125–129.

• 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.