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All over the web one can find statements to the effect that:

"One must be able to say at all times--instead of points, straight lines, and planes--tables, chairs, and beer mugs"

There are many variations, some in quotes (lots of variations here) and some not, all paraphrases of the same thing.

But I can't seem to find any kind of original source on-line. I can't seem to find anything like it in Die Grundlagen der Geometrie (suitably using German words).

Does anyone know where this first occurred in Hilbert's writings (if at all!)? And if made up, who did it?

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He can't possibly have said that. For instance, every line contains at least one point, but, sadly, there have been times in my life when my table has not contained a beer mug. –  Ben Crowell Aug 9 '11 at 23:07
@Ben: Good table. –  joriki Aug 10 '11 at 4:10
Exec summary: not really an urban legend, only a tiny bit of hearsay but Hilbert didn't state it intentionally in his published works. Thanks so much all for all the scholarship, amazing (yes, probably just a google search, but it doing it is something)...my next question will be 'who invented the variable?'. –  Mitch Aug 10 '11 at 17:39

3 Answers 3

up vote 12 down vote accepted

Funny enough, this is not the first time that I've seen this question come up. It is reportedly from a conversation that Hilbert had with Blumenthal at a train station in Berlin, on his way back to Königsberg.

Grattan-Guiness included this in his book The Search for Mathematical Roots on page 208. It is also supposedly in Blumenthal's Lebensgeschichte, which has several bits on Hilbert, on pages 402-403 (published in 1935 by Blumenthal himself).

A quick search also revealed That there is some sort of reference in the collection of Hilbert's Papers Gesammelte Abhandlungen, in 3 volumes, published in 1934. I don't know who edited this collection, but most every source I have heard believes that this came from a conversation with Blumenthal in Berlin. Or that Blumenthal made it up because it was convenient. One or the other.

Edit: (T.B.) The relevant section 4.7.2 on pages 208–209 from Grattan-Guiness's book (taken from Google Books):

Grattan-Guinness, p.208i Grattan-Guinness, p.208ii Grattan-Guinness, p.209i Grattan-Guinness, p.209ii

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Excellent...is the Grattan-Guinness mention on-line? –  Mitch Aug 10 '11 at 15:39
I took the liberty of including the relevant passage from Google Books. If you have qualms about copyright issues, feel free to roll back. @Mitch: see edit. –  t.b. Aug 10 '11 at 16:56
The second paragraph is the most relevant...I really don't know how to evaluate its last line though: "This famous result is normally misunderstood and Hilbert may not have thought it though at the time". How -is- it supposed to be understood, and in what way was it misunderstood, and which way did Hilbert think of it? I thought it was a statement that labels for types are arbitrary and that the axioms provide the meaning. Is that the misunderstanding? –  Mitch Aug 10 '11 at 17:49
@Theo Thank you Theo - I didn't even think to look at Google Books. Very nice. –  mixedmath Aug 10 '11 at 20:27

In Otto Blumenthal's biography of Hilbert, included as Lebensgeschichte on pages 398–429 in David Hilbert, Gesammelte mathematische Abhandlungen, Springer 1935 we find on pages 402f the following passage. Note: Link goes to the freely accessible version from the Göttinger Digitalisierungszentrum—the entire collected works of Hilbert are easily available from that link.

Diese Entwicklung scheint schon sehr früh eingesetzt zu haben. Sicher wissen wir erst, daß ein starker Anstoß von einem Vortrag ausging, den H. Wiener 1891 auf der Naturforscher-Versammlung in Halle über „Grundlagen und Aufbau der Geometrie“ hielt1. In diesem Vortrag stellt Wiener mit völliger Klarheit die Forderung auf, daß man die für die Punkte und Geraden der Ebene und die Operationen des Verbindens und Schneidens geltenden Tatsachen aus solchen Grundsätzen müsse ableiten können, deren Aussagen nur diese Elemente und Operationen enthalten, so daß „man aus diesen eine abstrakte Wissenschaft aufbauen kann, die von den Axiomen der Geometrie unabhängig ist“ Als ein vollständiges System solcher Grundsätze findet Wiener den Desargues und den speziellen Pascal (Pappus) und macht auch einige Angaben über das gegenseitige Verhältnis der beiden Sätze. Diese Ausführungen packten Hilbert, der im vorhergehenden Semester Projektive Geometrie gelesen hatte, so, daß er gleich auf der Rückreise den Fragen nachging. In einem Berliner Wartesaal diskutierte er mit zwei Geometern (wenn ich nicht irre, A. Schoenflies und E. Kötter) über die Axiomatik der Geometrie und gab seiner Auffassung das ihm eigentümliche scharfe Gepräge durch den Ausspruch: „Man muß jederzeit an Stelle von „Punkte, Geraden, Ebenen“ „Tische, Stühle, Bierseidel“ sagen können“. Seine Einstellung, daß das anschauliche Substrat der geometrischen Begriffe mathematisch belanglos sei und nur ihre Verknüpfung durch die Axiome in Betracht komme, war also damals bereits fertig. Im April 1893 schreibt er an Minkowski: „Ich habe mich jetzt in die Nichteuklidische Geometrie hineingearbeitet, da ich im nächsten Semester darüber zu lesen gedenke“. Die Vorlesung ist im Sommer 1894 gehalten worden. Ihre Frucht ist der (schon oben erwähnte) Brief an Klein „Über die gerade Linie als kürzeste Verbindung zweier Punkte“2, in dem, wohl unter dem Einfluß Minkowskischer Ideen, Geometrien betrachtet werden, deren Punkte das Innere eines konvexen Körpers erfüllen (so wie in Kleins Realisierung der Lobatschefskyschen Geometrie das Innere einer Kugel), und gezeigt wird, daß bei Definition der Entfernung durch den Logarithmus des Doppelverhültnisses mit den unendlich fernen Punkten die Dreiecksungleichung gilt. Historisch von Bedeutung ist, daß in dieser Arbeit die Axiome der Verknüpfung und Anordnung und das Archimedische Axiom vorangestellt werden, und zwar im wesentlichen in derselben Formulierung wie in den „Grundlagen“, die Anordnungsaxiome unter ausdrücklicher Berufung auf M. Pasch.

1 Jber. dtsch. Math.–Ver. Bd. 1, (1892) S. 45–48.
2 Grundlagen der Geometrie, 7. Auflage, Leipzig und Berlin: B. G. Teubner 1930, Anhang I oder Math. Ann. Bd. 46, (1895) S. 91–96.

The relevant sentence „Man muß jederzeit an Stelle von „Punkte, Geraden, Ebenen“ „Tische, Stühle, Bierseidel“ sagen können“. is towards the middle of the text. Your quote seems to be a quite accurate translation.

This doesn't appear to be written in any of the texts I have available electronically by Hilbert himself and mixedmath's answer seems to be rather faithful to what Blumenthal writes, so I won't elaborate and only point out the rather peculiar Bierseidel which I know from Austria and Bavaria but which strikes me as a strange word to choose by a man of prussian descent in a train station in Berlin.

Also of interest is Blumenthal's first biographical note on Hilbert: O. Blumenthal, David Hilbert, Die Naturwissenschaften, Volume 10, Issue 4, Jan. 1922, pp.67-72 (where there is no talk of beer mugs, however).

Added: I should have pointed out that Hilbert wrote the preface of his collected works. The end of the one of the third volume is displayed in this picture:

Excerpt preface

While I don't know what one is allowed to conclude from this alone, one might interpret the first sentence displayed here combined with the fact that it was printed in his collected works as an endorsement of the Lebensgeschichte.

On page 208 of Constance Reid's Hilbert one finds the following passage supporting this view:

Excerpt from Reid's biography

Of course, all this does not prove the factual veracity of this small episode but there are worse urban legends and less substantiated tales in the lore of mathematics.

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Excellent answer..thanks for the scholarship. I imagined it would have been in Reid's but it looks like all she does is -use- that phrase rather than ascribing any kind of quote. With respect to your addenda, are you trying to say that Blumenthal was most likely not making it up himself but reporting what he thought Hilbert had said? –  Mitch Aug 10 '11 at 15:47
I'm not sure what to make of that excerpt from Reid, honestly. I interpreted Reid's report on that note as sort of indirect speech (indeed, I think that H. must have written something of that kind, otherwise that line doesn't make much sense to me, as H. was to live about 7 more years). Concerning the train-station incident: It must be a second hand report because Blumenthal was born in Frankfurt and remained there until his Abitur in 1894. He first saw Hilbert in 1895 when the latter arrived in Göttingen from his position in Königsberg (see the section starting on p.399 of the Lebenslauf). –  t.b. Aug 10 '11 at 16:35
Where was I getting at? It seems to me a bit too much to conclude that Hilbert endorses that small and rather trivial part of this lengthy Lebenslauf, but he must have been happy with the overall account. If that story were a blatant falsehood or to his disliking, he would have objected, I believe. I can't substantiate this more at the moment. Here's an online archive containing an interesting biography of Blumenthal by Felsch, by the way but the Biography of Hilbert is only mentioned in passing. Most sources there are not available online. –  t.b. Aug 10 '11 at 16:39
Theo: +1 for the link to the Gesammelte Abhandlungen and for the last sentence of your post. To (mis)quote a line from The man who shot Liberty Valance: "This is Hilbert, Sir. When the legend becomes fact, print the legend." –  Did Aug 13 '11 at 0:22
@Didier: Thanks for reminding me of that movie! (btw. do you know where that widespread Leone quote: "it was the only film where Ford learned about something called pessimism." comes from?). I was going after this in the hope of finding something a bit more interesting than the original question---my main motivation for plowing the archives for something silly... This time I found Felsch's notes on Blumenthal's life (pdf available under footnote 2). I don't know if you read German but if you do, you should have a look at them. –  t.b. Aug 13 '11 at 1:06

A similar statement about love, law and chimney sweeps is in a letter from Hilbert to Frege:

"Wenn ich unter meinen Punkten irgendwelche Systeme von Dingen, z.B. das System: Liebe, Gesetz, Schornsteinfeger ..., denke und dann nur meine sämmtlichen Axiome als Beziehungen zwischen diesen Dingen annehme, so gelten meine Sätze, z.B. auch der Pythagoras auch von diesen Dingen." ("If I subsume under my points arbitrary systems of things, e.g. the system: love, law, chimney sweep ..., and then just assume all my axioms as relationships among these things, then my theorems, e.g. also the Pythagorean theorem, are true of these things, too.") (Gottlob Freges Briefwechsel mit D. Hilbert , E. Husserl , B. Russell sowie ausgewählte Einzelbriefe Freges, Felix Meiner Verlag, 1980, p. 13)

You can find some more references on both quotes here, and this passage draws an interesting connection to Hermann Wiener, whose talk Hilbert had apparently just heard when he made the tables/chairs/beer mugs statement. Let me know if you need help with the German.

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Excellent find...the letter is dated Dec 1899. Of course, none of this was published (and so generally accepted as from Hilbert until the 30's) –  Mitch Aug 10 '11 at 15:37

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