...
> He needs to know constructively (as in Mathematical intuitionism) what
> really happens and may be trusted like a normal person to generalize
> his experience UNLESS he is socialised to do things by rote &
> shibboleth.
Interesting that you again bring in something you know nothing about. In
this case intuitionism. Try once to read the works by Brouwer and
Well, I first encountered it in Evert W Beth and Stefan Korner. Your
foolish attack here is probably Nordic-nationalistic in the irritating
way of the Nordic male who like Heidegger prefers the bruder
schweigen, and like Adorno's attacker on a tram in 1920, hates young
whippersnappers who use High German, he being Lower than whale
scheissen himself.
Some sort of creepy lower middle class Dutch nationalism seems to be
afoot, since you seem offended that Brouwer's ideas are actually
understood in a different language.
From THE OXFORD HANDBOOK OF PHILOSOPHY OF MATHEMATICS AND LOGIC
(Shapiro, OUP 2005):
"...traditional *intuitionists*, such as L. E. J. Brouwer and Arend
Heyting held that mathematics has a subject matter [as opposed to
formalists]: mathematical objects, such as numbers, do exist. However,
Brouwer and Heyting insisted that these objects are mind-dependent.
Natural numbers and real numbers are mental constructions or are the
result of mental constructions. In mathematics, to exist is to be
constructed. Thus, Brouwer and Heyting are anti-realists in ontology,
denying the *objective* existence of mathematical objects. Some of
their writing seems to imply that each person constructs his own
mathematical realm. Communication between mathematicians consists in
exchanging notes about their own constructive activities. This would
make mathematics subjective. It is more common, however, for these
intuitionists, especially Brouwer, to hold that mathematics concerns
the *forms* of mental construction as such. This follows a Kantian
theme, reviving the thesis that mathematics is synthetic apriori."
"This perspective has consequences concerning the proper practice of
mathematics. Most notably, the intuitionist demurs from the law of the
excluded middle-(Av~A)-and other inferences based on it. According to
Brouwer and Heyting, these methodological principles are symptomatic
of faith in the transcendental existence of mathematical objects or
the transcendental truth of mathematical statements. For the
intuitionist, every mathematical assertion must correspond to a
construction."
This is an excellent definition of intuitionism, and there's a clear,
if to my knowledge undrawn, relationship between this and Dijkstra's
mode of thinking. In intuitionism, the only "reality" is mathematical
communication. In Dijkstra's view of programming, a program considered
as a text is a communication of an intent.
The rejection of the excluded middle is echoed in Dijkstra's belief
that testing can NEVER prove bug absence.
The deepest connection is with Kant (have you read Kant, dear Dik?)
Kant believed that noumenal reality (reality in itself) is unknowable
but can be approached as a limit through phenomena. Mathematics is a
form of intution in Kant in which "the medium is the message":
Euclid's theorems are at one and the same time forced upon us by the
way we see the world, and an ultimate reality: likewise, what makes
intuitionism different from formalism is that while the latter makes
mathematical realities into mere constructed games, intuitionism like
Kant says the the forms of perception HAVE NO ALTERNATIVE FORMS and
thus they are the ultimate reality. Other beings might "see" a world
with a different mathematical structure but this is less than an
unknowable noumenon, it's an only possible noumenon.
In Kant, we not only see space,
I SEE SPACE AND IT LOOKS LIKE NOTHING AND I WANT IT AROUND ME - Jenny
Holzer
we see objects arranged in it that never violate the laws of geometry
as we know it. Alternative beings perhaps dwelling amongst us but in a
different dimension (think Twilight Zone, dear Dik) might see
something other than "space", and/or the objects in their z-space
might arrange themselves according to a z-geometry, but we can only
speak of this, we cannot "visualize" it.
Which means that what the intuitionist discovers in proof by
construction are not assertions about the tools used, they are
"synthetic apriori" (informative but proved logically) assertions
about the way the world is, because in intuitionism, the world cannot
be considered apart from the way we apprehend it.
Which is probably why elegance mattered to Dijkstra: a program is just
one way of apprehending a truth in applied mathematics but it's also
part of that truth.
acknowledge that you are spouting nonsense. Intuitionism is about a way
to *prove* something, not about generalisations. Most mathematical
"Intuitionism is about a way to *prove* something" is incredibly poor
writing even if transliterated from Dutch:
* Why is it "about a way" and not "a way"?
* And are you seriously claiming that intuitionists wanted to prove
"something", at least one thing, or are you using the English idiom
that means a person with an inferiority complex, such as some Dutch
guy making a fool of himself, is out to "prove something"? Actually,
their programme was to prove "everything true" without using the
excluded middle.
generalisations are not valid in intuitionism because there is no
constructive proof.
> Actually, programmers who aren't socialized by corporate fear can
> intuit that the computer, like a geometrical diagram, has essential
> and accidental features, for the same reason intelligent students in
> geometry class don't worry about the thickness of the lines they have
> drawn using straightedge and compass in geometrical construction.
That is indeed the way trisection proofs are worked out. Sheesh, can you
not get anything right?
I don't know, but it sounds to me that you were not educated in
mathematics at all. Geometrical constructions use an unmarked
straightedge and a compass (a circle drawing tool, not a direction
finder):
http://mathworld.wolfram.com/GeometricConstruction.html.
If you insist on interrupting these conversations with news about
outdated computers that you happened to have worked on and complete
disinformation posted in order to participate in campaigns of personal
destruction...you're going to look the fool.