Talk:Hilbert 2nd problem

I created this article (obviously still under construction) partly in pursuit of my lifelong interest in the history and philosophy of mathematics.

I have only recently discovered the work of Gregory Chaitin. He and I are similar in age and, in addition, studied only minutes from one another along the Pennsylvania Railroad: he mathematics at CCNY in New York City, I philosophy at Rutgers University in New Brunswick, New Jersey.

Chaitin's interest in Hilbert's promotion of mathematical formalism and my own interest in Carnap's et. al. promotion of logical positivism are, I feel, mirror images of a single mind.

The subject has engaged the comments of many mathematicians and philosophers of mathematics since Hilbert's 1900 lecture. The title that Chaitin gave to his own 2000 lecture allows us to see his work as a homage to Hilbert's vision.

Feasible and infeasible

Encouraged by you, I've read that text by Chaitin, and like to share some impression. Here is my outrageous claim. Results of Goedel, Turing, Chaitin and probably some others are a wonderful proof of an intuitively evident point that a mathematical theory cannot prove "the truth, the whole truth, and nothing but the truth" about natural numbers. I do not know whether or not this was Hilbert's program, but if it really was, then it is a wonder that for Hilbert, the opposite was intuitively evident; and then we observe a dramatic change of our intuition, somewhat similar to the change from "evidently, a continuous function must be differentiable in most of the points" to "evidently, a generic continuous function is nowhere differentiable".

Once upon a time I was thinking hard about probabilistic cellular automata (there was a wonderful problem attacked by me, among others, but ultimately solved in an exciting work by Peter Gacs). And I got very pessimistic, for the following reason. Imagine a deterministic cellular automation powerful enough for having "computers" and "robots" among possible finite configurations. Introduce small randomness; being small it does not prevent these computers and robots from successful functioning during reasonably long time; being non-zero, the randomness introduces "mutations", it ensures that every possible computer/robot will emerge somewhere, sooner or later. Now ask a clever question about the asymptotic behavior of this automation on large time.

In order to answer such a question, you probably need to understand what is "the optimal civilization", a finite but growing combination of computers/robots most succesfully enlarging itself and fighting all other civilizations! Or alternatively you need to understand a sequence of civilizations that tends to optimality...

Why hope that such knowledge follows from reasonable axioms?

Yes, we mathematicians have an encouraging experience: when we want to prove or disprove something, sooner or later we can. But this only shows that our intuition is able to choose only feasible tasks.

Go to a wild, find a tiger that looks at a zebra, and ask a physicist to predict the result: who will succeed this time, the tiger or the sebra? No, the physicist cannot. He is successful in answering properly chosen questions. And the same applies to us mathematicians.

And now I wonder, which one of the following two scenarios is closer to the historical truth.

The first scenario. Hilbert believed that every question formulated in the formal arithmetic is "mathematically feasible". Thus, his intuition looks quite naive nowadays.

The second scenario. Hilbert's intention was more modest: to create a formal arithmetics that proves "the truth, and nothing but the truth" but not quite "the whole truth" about natural numbers. It was enough for him, if the theory solves all "mathematically feasible" questions.

Boris Tsirelson (talk) 11:03, 6 June 2015 (CEST)

Well, from the article (as of now) I understand that the answer to my question is "the first scenario". Boris Tsirelson (talk) 18:49, 9 June 2015 (CEST)

Yes, as you say, Hilbert, a mathematician of great renown, nevertheless entertained notions that we nowadays regard as naive.

Yet, as we say this, we smile and recall that few things in mathematics (even fewer in everyday life?) come into being fully formed and wholly mature -- as did Athena, who is said to have sprung full-blown from the head of Zeus!

It is enough to note that the subtitle of the ‘’E of M’’ article Set theory is naive to remind ourselves of the difficult journey from yesterday’s ‘’Unknown Unknowns’’ to tomorrow’s ‘’Known Knowns’’ -- I'm thinking here of U.S. Vice President Dick Cheney's famous remarks about these two notions. With respect to perplexing matters in mathematics (and in everyday life!), we repeatedly find ourselves somewhere in the midst of such a journey.

Hilbert was not the first mathematician to insist that some branch or other of mathematics needed to be placed on a firm(er) foundation, that mathematical concepts needed to be defined (more) rigourously, that mathematical assumptions needed to be stated (more) explicitly, and that more formal methods needed to replace intuition.

Hilbert must have known in 1900 what we know today, namely, that in all such efforts, the formalisms (e.g. definitions and axioms) that we develop inside mathematics are a way of expressing our regard for and paying our respect to the intuitive notions outside mathematics from which those formalisms have spring and upon which they are ultimately based.

William Hayes (talk) 17:22, 11 June 2015 (CEST) Best regards. It's always a pleasure to read and reflect on your comments.

The difference between mathematics and philosophy?

Thank you for your candid and very interesting reflection. Quite obviously you, yourself, have also been a student of philosophy!

No, never! :-) Though, one of the two my most influential teachers was a logician. [1] Boris Tsirelson (talk) 18:21, 8 June 2015 (CEST)

Thinking about this subject (Foundations of Math -- Indeed, are there any?) brought to mind a long-ago event in which one of my math profs at Rutgers, in a brief conversation, worked some mentoring magic in my undergraduate life.

Prof Dekker was, as I recall, a logician, though my studies with him were of algebra and topology. One afternoon, just before or just after a class (interesting that very important events happen on the periphery of and almost in spite of the plans we make for ourselves) I mentioned to him Nagel and Newman’s book on Godel’s incompleteness theorems, through which I was working on my own -- there was, at the time, no one in the philosophy department capable of helping me with that material. Prof Dekker made a bit of a face and then shared an anecdote from his own student past:

As an undergraduate, I had wrestled with the issue of academic direction: would I pursue philosophy or mathematics. The answer came to me as a revelation during a scholarly conference to which one one of my philosophy professors had kindly invited me.

A respected scholar rose and delivered a keynote address on a subject that was of great interest to me. Immediately afterwards, another scholar, equally respected, rose as a devil’s advocate to deliver some comments to the contrary of the keynote address. Finally, the keynote speaker rose once more and made a brief reply, which he began with the words, “But you have completely misunderstood me!”

Dekker paused for a moment, then looked at me and said the words that caused him to choose mathematics rather than philosophy:

You know, those words made the decision (between math and philosophy) for me. They would never have been said by a mathematician. We don’t ever misunderstand one another!

A bit of an exaggeration (?) and yet his words helped me with my own academic indecision, although in my case, the choice I made (for mathematics) was rendered moot by other, overarching events in my life. C’est la vie!

William Hayes (talk) 16:06, 8 June 2015 (CEST)

Feeling this article was growing unmanageably large, I today determined that it should deal only with the mathematics that preceded Hilbert's Problems Address, including (of course) his statement of the 2nd problem itself. Accordingly, I have revised the introductory remarks and excised headings for the sections no longer needed. I will deal with the subsequent history of the development of Hilbert's program itself in another article, Hilbert program, which I have also created today. Trust that this seems reasonable. Thanks for your occasional, watchful, and helpful edits. --William Hayes (talk) 17:43, 25 July 2015 (CEST)

Hilbert's Grundlagen

"6 undefined relations: being on, being in, being between, being congruent, being parallel, & being continuous" — I got puzzled with the last (being continuous). "Being on" is a relation between a point and a line, or a point and a plane, or a line and a plane, I guess. "Being parallel" - between two lines. "Being continuous" - between what? According to the text it should be some n−place relation; for which n? Boris Tsirelson (talk) 19:00, 19 August 2015 (CEST)

I have made (some) sense of Boyer's mention of the relation of "being continuous" as follows:

Consider Hilbert's 5 groups of axioms, grouped as follows: Incidence, Order, Congruence, Parallelism, and Continuity. Euclid Elements treats these as follows:

• Congruence and Parallelism are dealt with to some extent in axioms.
• Incidence is referred to (and therefore relied upon) in Euclid's 1st postulate that has to do with drawing a line between 2 points.
• Order and Continuity are not dealt with at all.

Hilbert's axioms of Order draw on Pasch's work in 1880s, to the extent that the 4th of Hilbert's axioms of Order is commonly called Pasch's Axiom.

In the various editions of the Grundlagen, Hilbert used different axioms for Continuity. In modern presentations (re-statements) of "Hilbert's Axioms," authors choose one or another of Hilbert's axiom sets for Continuity based on ... what ... personal preference? whim? ouija board consultation? I have no idea! Many authors have used two axioms (e.g. Archimedes and Line Completeness); at least one author has used a single, strong axiom of completeness (Dedekind's Axiom -- which implies Archimedes and Line Completeness).

In any case, the matter in Euclid's Elements that the axioms of Continuity seek to address is the intersection of lines and circles. It is this matter of two lines or two circles or one of each intersecting one another that makes sense of Boyer's talk of the relation of "being continuous".

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You ask: some n−place relation; for which n?

I took the "n" to be, very specifically, the $n$ (geometric) terms of Hilbert's axiom set $AX$.

Blanchette provides this more general example:

If $AX$ is the set $\{$There are at least two $points$; Every $point$ lies on at least two $lines$$\}$, then $R_{AX}$ is the relation that holds of any triple <$P$, $LO$, $L$> such that $P$ has at least two members, $L$ has at least two members, and $LO$ is a relation that holds between each member of $P$ and at least two members of $L$.

I am not persuaded that this example captures all of what Hilbert meant when he wrote the following in a letter to Frege:

If in speaking of my points I think of some system of things, e.g. the system: love, law, chimney-sweep … and then assume all my axioms as relations between these things, then my propositions, e.g. Pythagoras' theorem, are also valid for these things. In other words: any theory can always be applied to infinitely many systems of basic elements.

Hilbert's choices of "love, law, chimney-sweep" reminded me of Rudolph Carnap's own life-choices of "love, music, and logic" mentioned in a mathematical memoir.

- - - - -

Whenever I read the phrase "undefined relations" I think of the many modern families (of which mine is one) that are the product of multiple (re)marriages yielding step-grandkids (whom one of my friends dubs "gift-grandkids") and ever more compound-complex relations and sort-of descendants.

--William Hayes (talk) 18:43, 20 August 2015 (CEST)

All that is nice, about axiom(s) of continuity. No doubt, there are such axioms. I ask, what could be relation of continuity? Beween how many objects, and which kind of objects?

On WP I see 1+3+2=6 primitive relations, one ternary and 5 binary, and two axioms of continuity; and that looks logical (though I did not read Hilbert). Boris Tsirelson (talk) 22:31, 20 August 2015 (CEST)

If at all a continuity relation can be imagined, it is probably of infinite arity... Boris Tsirelson (talk) 12:50, 21 August 2015 (CEST)

As I understand them, the Axioms of Continuity (Axiom of Line Completeness and Axiom of Archimedes) are intended to address the matter of intersection that arises whenever Euclid deals with either of the following processes:

1. the actual intersection of 2 lines or 2 circles or one line and one circle
2. the implied intersection involved in the process of using one line segment to measure another line segment

Hilbert's Axioms of Continuity establish that lines and circles and line segments are sufficiently dense with points to establish that these processes do indeed determine points of intersection that both exist and also are unique. Euclid doesn't consider this matter at all.

In this sense, the property of continuity substantiates our speaking of both a relation of intersection (involving two lines or two circles or a line and a circle) and a relation of measurement (involving two line segments).

Either that or I am completely in the dark about what Boyer meant. :-))

As far as I know, Hilbert does not introduce new undefined relations if not quite necessary. Intersection of two lines is, by definition, a point that belongs to both lines. Only one undefined relation (one of the three containment relations) is used in this definition. No need to invent a new one.

I've revised Boyer's summary description of Hilbert's theory and added a comment to the footnote -- all for the better, I think. Thank you for your clarifying comments, which are both persistent and supportive.

Ah, yes, I see it written in Boyer. Now I should think more. Boris Tsirelson (talk) 22:19, 22 August 2015 (CEST)

Interesting discussion of in-finite arity relations. The notion of validating infinitely long sentences generated by axiom schemas of first-order logic using truth tables of infinite size is confounding (and amusing) enough for me. https://en.wikipedia.org/wiki/First-order_logic#Infinitary_logics ?? --William Hayes (talk) 20:27, 21 August 2015 (CEST)

Probably, Hilbert would be disturbed, too. Metamathematics was created as a finitary theory that avoids infinity in order to be more reliable than mathematics. But now other options are tried, too. Clearly, infinitary logic can exists only within the set theory (or another very strong, and therefore not so reliable) mathematical theory. Hardly related to Hilbert's ideas. Boris Tsirelson (talk) 21:18, 21 August 2015 (CEST)