# Gauss's views on pure mathematics

According to Wikipedia's entry on Gauss:

"Though Gauss had been up to that point supported by the stipend from the Duke, he doubted the security of this arrangement, and also did not believe pure mathematics to be important enough to deserve support"

As a mathematician, I find, rather obviously this opinion to be very shocking (especially coming from Gauss himself).

Wikipedia does not give any sources related to the quote above. Is there any reference, or evidence for Gauss holding such a position?

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Even speaking of the "pure vs. applied" distinction seems rather ahistorical to me... –  Yemon Choi Apr 22 '12 at 22:56
The wikipedia entry has other errors, it says "his 1799 doctorate in absentia" whereas the doctorate was actually in mathematics. –  Will Jagy Apr 23 '12 at 0:30
If one could get a doctorate for work in absentia, many more people would be doctors! ;P –  Mariano Suárez-Alvarez Apr 23 '12 at 0:39
Gauss attended the University of Goettingen, but his doctorate was conferred later by the University of Helmstedt. That is what the reference to a doctorate "in absentia" is about. –  Michael Renardy Apr 23 '12 at 1:45
@Michael, I like my version better. –  Will Jagy Apr 23 '12 at 2:20

Quotation from Gauss:

"...the greatest thing is purely mathematical thinking: this is worth much more than the application of mathematics."

In conversation in 1854, a few months before his death, that was. In Gauss, Titan of Science by G. Waldo Dunnington, p. 303.

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In his letter to Zimmermann (March 12, 1797; Werke X), Gauss wrote

May God give the noble Duke a long life, and what may Science expect from him since he deems a work that is only a little interesting not unworthy of his support; how much I wish that I could present a work that is more profitable for society [serving the public good] or more excellent.

Gauss was talking about the Duke's support for printing his Disquisitiones Arithmeticae.

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I believe that Gauss may have drawn the distinction not between pure and applied mathematics (that distinction only appeared at the end of his career), but between mathematics and the sciences, and, indeed, Gauss was quite prominent in the sciences (astronomy, electromagnetism -- consider "degaussing"), and engineering (he and Weber constructed one of the first (if not THE first) telegraph. A lot of Gauss' mathematics was inspired by applications, so he was never an "art for art's sake" kind of mathematician. I am guessing that the extent of his contributions to mathematics is explained by the depth of his understanding.

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Did he really view, say, his Disquisitiones are something not for art's sake? –  Mariano Suárez-Alvarez Apr 23 '12 at 0:40
I haven't spoken to him lately, so I don't know what he thought. When I was writing my answer, Disq. Arith. did come to mind as a possible counterexample, but I think that in pre computer age, arithmetic was a practical pursuit (by understanding more structure, you could maybe perform computations faster, solve equations more efficiently, etc, etc). Just a guess. –  Igor Rivin Apr 23 '12 at 1:09

I consider that the following lines of Hardy might help the OP to form a definite idea as to the accuracy of such an asseveration:

"... I must deal with a misconception. It is sometimes suggested that pure mathematicians glory in the uselessness of their work*, and make it a boast that it has no practical applications. The imputation is usually based on an incautious saying attributed to Gauss, to the effect that, if mathematics is the queen of the sciences, then the theory of numbers is, because of its supreme uselessness, the queen of mathematics—I have never been able to find an exact quotation. I am sure that Gauss’s saying (if indeed it be his) has been rather crudely misinterpreted. If the theory of numbers could be employed for any practical and obviously honourable purpose, if it could be turned directly to the furtherance of human happiness or the relief of human suffering, as physiology and even chemistry can, then surely neither Gauss nor any other mathematician would have been so foolish as to decry or regret such applications. But science works for evil as well as for good (and particularly, of course, in time of war); and both Gauss and less mathematicians may be justified in rejoicing that there is one science at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean.

*I have been accused of taking this view myself. I once said that ‘a science is said to be useful if its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life’, and this sentence, written in 1915, has been quoted (for or against me) several times. It was of course a conscious rhetorical flourish, though one perhaps excusable at the time when it was written. "

P.S. a) The emphasis is mine. b) The excerpt comes from the last paragraph of section 21 of Hardy's Apology.

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What Gauss said is typical old-style Ciceronian rhetoric, and I don't suppose anyone at the time would have taken it very seriously. –  Charles Matthews Apr 23 '12 at 8:28
Indeed, he (CFG) may have been taunting his critics (see the other answers). –  Igor Rivin Apr 23 '12 at 14:35

An interesting observation about Gauss, the Duke and his neighbors:

When the Duke of Brunswick (Braunschweig) , Carl Wilkelm Ferdinand, has increased the Gauss stipend from 158 Thalers per year to 400 Thalers per year, the reaction of some residents of Brunswik (Braunschweig) was very critical.

"The residents of Brunswick (Braunschweig) were not the only one critical of a pure mathematician. A crank wrote to the editor of the Monthly Correspondence, complaining that if certainly would be nicer if our scientists did something useful instead of all this theoretical nonsense"

Ref. M.B.W. Tent, The Prince of Mathematics, Carl Friedrich Gauss, A.K. Peters, Ltd., 2006, pp. 129.

Obs.: 400 Thalers over 60 pounds (from: W.H. Bruford, Germany in the Eighteenth Century: The Social Background of the Literary Revival, pp. 278)

Gauss to his father: "... Mathematics is the foundation of science, and my challenge is to explore it in important ways. ... I have not asked the Duke for his support. He has offered it to me because he believes in my work. He believes that it is important." M.B.W. Tent, pp. 119.

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Surely that playing with billions, diverting $.0001$ percent into its pocket and eventually loosing them is considered much more useful. –  Roland Bacher Apr 23 '12 at 13:53
I wonder how that number (400 thalers) compared to, e.g., a shopkeeper's income... –  Igor Rivin Apr 23 '12 at 14:25
Do any of these authors give a reference for this letter? –  Franz Lemmermeyer Jun 13 '12 at 9:22
@Lemmermeyer: Tent's book has no references... In Preface, The narrative of Gauss' life is based on stories Gauss told about himself and letters and descriptions that have come down to us. The vignettes and conversations are based as closely as possible on reports of what actually happened. No references... –  Papiro Jun 13 '12 at 10:38

Is it possible that the author means to make this refer to Gauss' reason for "doubt[ing] the security of this arrangement"?

Maybe Gauss got the feeling that the Duke "did not believe pure mathematics to be important", but an unskilled writer failed to express that clearly.

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If the Duke was not an idiot, he understood that having the likes of Gauss under his wing was good for his prestige (sort of like the space program, but cheaper). Since he did pay Gauss for several decades, he (the duke) presumably was a reasonably wise man. I always wondered if he (the duke again) is the same guy crushed by Morphy, or whether that was his son. –  Igor Rivin Apr 23 '12 at 1:13
I don't doubt that the Duke believed pure mathematics important. But I thought maybe Gauss might have misread him. Link-following leads to a letter from Gauss' grandson to Cajori <gausschildren.org/genwiki/index.php?title=Letter:GAUSS,Charles_Henr‌​y_to_Florian_Cajori-_1898-12-21> that says Gauss didn't want his children to lower the family name by entering mathematics. –  user1241 Apr 23 '12 at 1:17
Yes, but the sentiment behind that quote was more along the lines of "No one remembers Shakespeare's children." It's better to include the full quote. "He - that is Grandfather - did not want any of his sons to attempt mathematics for he said he did not think any of them would surpass him and he did not want the name lowered. Probably he felt the same in a measure of any other line of scientific study. " –  stankewicz Apr 24 '12 at 15:11
@Igor: According to wikipedia, it was his grandson who played the chess game. –  Zavosh Jun 13 '12 at 3:01