6
I - Sets and Functions
mathematicians who use it daily while restricting themselves to the imagining
of the simplest geometrical and physical images, on the other hand, from the
"formalised" point of view of logicians who, by systematically applying rules
of argument, and methods of construction stated once and for all, construct
everything from nothing "to allow thought to rise above the void by leaning
on the void" as wrote a physicist studying Newton's cosmology7. We shall
confine ourself to expounding an intermediate point of view very succinctly,
without using logicians' language, but nevertheless not letting the reader
believe that intuitively "evident" results require no justification. The use of
ordinary language can make propositions seem intuitive, though professional
logicians will consider them by no means evident, and computers will not
understand for the simple reason that they lack intuitions: to make oneself
understood by a stupid though disciplined machine one must explain oneself
in a perfectly correct fashion, even if one only wants to receive information 9
on the Silicon Valley Paedophiles Brotherhood through the Internet. We shall
give some very summary details on the language of logicians later.
7 Loup Verlet, La malle de Newton (Gallimard, 1993), p. 292. Highly recommended
reading.
8 Although the invention of computer science has presented several inconveniences
to mankind - ask ordinary Iraqis what they think of the computers on a cruise
missile -, it offers a great advantage to mathematical pedagogy: a proof is totally
correct if a suitably programmed computer could understand it. This point of
view does not eject the role of intuition from mathematics; it simply shows that
one must not confuse an intuition with an argument.
9 For computer scientists, "information" is a sequence of the digits 0 and 1. It is
interesting to note that before becoming a "cultural rubbish bin" , as the Chinese
call it, the Net was first conceived as a military system with the purpose of
assuring American telecommunications in the event of nuclear war, thanks to the
total interconnection of the given bases and centres of calculation or command;
it was soon linked to laboratories and university departments highly delighted
to profit from it (and to profit the military through their expertise) after which
the military provided themselves with a system for their exclusive use while the
"civil" system passed into the control of the National Science Foundation, from
which it has finally been removed.
There is an enormous American literature on the history of computer science; the following titles, by historians who cite their sources, are particularly
recommended: Kenneth Flamm, Creating the Computer: Government, Industry,
and High Technology (Brookings Institution, 1988), Paul Edwards, The Closed
World. Computers and the Politics of Discourse in Cold War America (MIT
Press, 1996), Arthur L. Norberg and Judy O'Neill, Transforming Computing
Technology. Information Processing for the Pentagon 1962-1986 (The Johns Hopkins UP, 1996), very dry, Thomas P. Hughes, Rescuing Prometheus (Pantheon
Books, 1998) which also treats other similar subjects (missiles, SAGE, systems
analysis, etc.), Janet Abbate, Inventing the Internet (MIT Press, 1999), from a
Ph.D. thesis, Kent Redmond and Thomas M. Smith, Prom Whirlwind to Mitre.
The R& D Story of the SAGE Air Defense Computer (MIT Press, 2000), where
one can see to what extent the development of the gigantic SAGE net for the air
defence of the American continent influenced the progress of computer science
and electronics in the 1950s.
I - Sets and Functions
mathematicians who use it daily while restricting themselves to the imagining
of the simplest geometrical and physical images, on the other hand, from the
"formalised" point of view of logicians who, by systematically applying rules
of argument, and methods of construction stated once and for all, construct
everything from nothing "to allow thought to rise above the void by leaning
on the void" as wrote a physicist studying Newton's cosmology7. We shall
confine ourself to expounding an intermediate point of view very succinctly,
without using logicians' language, but nevertheless not letting the reader
believe that intuitively "evident" results require no justification. The use of
ordinary language can make propositions seem intuitive, though professional
logicians will consider them by no means evident, and computers will not
understand for the simple reason that they lack intuitions: to make oneself
understood by a stupid though disciplined machine one must explain oneself
in a perfectly correct fashion, even if one only wants to receive information 9
on the Silicon Valley Paedophiles Brotherhood through the Internet. We shall
give some very summary details on the language of logicians later.
7 Loup Verlet, La malle de Newton (Gallimard, 1993), p. 292. Highly recommended
reading.
8 Although the invention of computer science has presented several inconveniences
to mankind - ask ordinary Iraqis what they think of the computers on a cruise
missile -, it offers a great advantage to mathematical pedagogy: a proof is totally
correct if a suitably programmed computer could understand it. This point of
view does not eject the role of intuition from mathematics; it simply shows that
one must not confuse an intuition with an argument.
9 For computer scientists, "information" is a sequence of the digits 0 and 1. It is
interesting to note that before becoming a "cultural rubbish bin" , as the Chinese
call it, the Net was first conceived as a military system with the purpose of
assuring American telecommunications in the event of nuclear war, thanks to the
total interconnection of the given bases and centres of calculation or command;
it was soon linked to laboratories and university departments highly delighted
to profit from it (and to profit the military through their expertise) after which
the military provided themselves with a system for their exclusive use while the
"civil" system passed into the control of the National Science Foundation, from
which it has finally been removed.
There is an enormous American literature on the history of computer science; the following titles, by historians who cite their sources, are particularly
recommended: Kenneth Flamm, Creating the Computer: Government, Industry,
and High Technology (Brookings Institution, 1988), Paul Edwards, The Closed
World. Computers and the Politics of Discourse in Cold War America (MIT
Press, 1996), Arthur L. Norberg and Judy O'Neill, Transforming Computing
Technology. Information Processing for the Pentagon 1962-1986 (The Johns Hopkins UP, 1996), very dry, Thomas P. Hughes, Rescuing Prometheus (Pantheon
Books, 1998) which also treats other similar subjects (missiles, SAGE, systems
analysis, etc.), Janet Abbate, Inventing the Internet (MIT Press, 1999), from a
Ph.D. thesis, Kent Redmond and Thomas M. Smith, Prom Whirlwind to Mitre.
The R& D Story of the SAGE Air Defense Computer (MIT Press, 2000), where
one can see to what extent the development of the gigantic SAGE net for the air
defence of the American continent influenced the progress of computer science
and electronics in the 1950s.