“A man can live three weeks without food, three days without water, and three minutes without air, but he cannot live three seconds without hope” is a popular saying.  The American sociologist Lewis Mumford (1895-1990) wrote in The Conduct of Life (1951):
 
“Without food man can survive for barely thirty days; without water for little more than three days; without air hardly for more than three minutes: but without hope he might destroy himself in an even shorter time.”
 
In 1953, the saying was attributed to “internationally renowned surgeon,” but it’s not known who this is. The saying has continued to be used since its 1950s popularization.
 
 
Google Books
The Conduct of Life
By Lewis Mumford
New York, NY: Harcourt, Brace and Co.
1951
Pg. 30:
Without food man can survive for barely thirty days; without water for little more than three days; without air hardly for more than three minutes: but without hope he might destroy himself in an even shorter time.
 
Google Books
Living God’s Way
By Reuben K. Youngdahl
Minneapolis, MN: T.S. Denison
1953
Pg. 88:
An internationally renowned surgeon once said, “A man can live three weeks without food, three days without water, and three minutes without air, but he cannot live three seconds without hope.”
 
Google Books
Positive Judaism
By Richard C. Hertz
Detroit, MI: Temple Beth El
1955
Pg. 43:
One thing we know for sure: you can’t live without hope. A person can live three weeks without food, three days without water, but not three seconds without hope.
 
8 October 1964, Trenton (NJ) Evening Times, “Employer Visisted Handicapped On Area Frontier of Space,” pg. 11, col. 9:
“It has been said that man can live 30 days without food, five days without water, five minutes without air, but not one second without hope.”
     
Google News Archive
27 June 1971, Rome (GA) News-Tribune, “Billy Beale’s Washington letter,” pb. 4-D, col. 4:
A touch of the old school gallantry was the windup of Paul Rogers’ toast. “Man can live 30 days without food,” he said, “Five days without water and five minutes without air; but not one minute without hope.”
   
Google Books
The Terminal Generation
By Hal Lindsey with Carole C Carlson
Charlotte, NC: Commission Press
1976
Pg. ?:
Man can live about 40 days without food, about 3 days without water, about 8 minutes without air . . . but only about 1 second without HOPE.
 
Google Books
50 Days of Hope:
Daily Inspiration for Your Journey Through Cancer
By Lynn Eib
Carol Stream, IL: Tyndale House Publishers
2012
Pg. 203:
In fact, there’s an old proverb that says you can live three weeks without food, three days without water, three minutes without air, but only three seconds without hope.
     
OA Online
WELLMAN: The new normal
Posted: Wednesday, April 17, 2013 9:06 pm
By Jared C. Wellman
(...)
An old adage says that a man can live 40 days without food, three days without water, and eight minutes without air, but not a moment without hope.