Quote of the Day
From: Harvey
Friedman ...
Sent: Monday, January 17, 2005 4:09 AM
Subject: [FOM]
Atlanta Meeting
...
I asked the panel members whether they were interested in
this line of investigation: no simple axiom settling the
continuum hypothesis.
Woodin responded by saying that, overwhelmingly, he
really wanted to know whether the continuum hypothesis is
true or false. He is far more interested in pursuing
that, as he is now, than any considerations of
simplicity, which for him, was a side issue.
Martin responded by saying that the projective
determinacy experience showed that one could have axioms
with simple statements, but with very complicated
explanations as to why they are correct.
I did not have an opportunity to respond to Martin's
statement - I would have said that by the standards of
axioms for set theory, projective determinacy is NOT
simple. It is far more complicated than any accepted
axiom for set theory.
I did ask Cohen specifically to comment on whether
simplicity (of a new axiom to settle the continuum
hypothesis) was important for him. Cohen responded by
saying that such an axiom, for him, must be simple.
Let me end here with something concrete. In my papers in
Fund. Math., and in J. Math. Logic, it is proved that all
3 quantifier sentences in set theory (epsilon,=) are
decided in a weak fragment of ZF, and there is a 5
quantifier sentence that is not decided in ZFC (it is
equivalent to a large
cardinal axiom over ZFC). All of the axioms of ZF are an
at most four quantifier sentence and an at most five
quantifier axiom scheme. It has been shown that AxC over
ZF is equivalent to a five quantifier sentence (see Notre
Dame Journal, not me). Show that over ZFC, any
equivalence of the
continuum hypothesis requires a lot more quantifiers.
Show that over ZFC, any statement consistent with ZFC
that settles the continuum hypothesis, requires a lot
more quantifiers.
...
Full text at http://www.cs.nyu.edu/pipermail/fom/2005-January/008756.html
Previous quotes
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My Main Theses
I define mathematical theories as
stable self-contained systems of reasoning, and formal
theories - as mathematical models of such systems. Working
with stable self-contained models mathematicians have
learned to draw a maximum of conclusions from a minimum
of premises. This is why mathematical modeling is so
efficient.
For me, Goedel's results are the
crucial evidence that stable self-contained
systems of reasoning cannot be perfect (just
because they are stable and self-contained). Such systems
are either very restricted in power (i.e. they cannot
express the notion of natural numbers with induction
principle), or they are powerful enough, yet then they
lead inevitably either to contradictions, or to
undecidable propositions.
For humans, Platonist thinking is the best way of
working with stable self-contained systems. Thus, a
correct philosophical position of a mathematician should
be: a) Platonism - on working days -
when I'm doing mathematics (otherwise, my
"doing" will be inefficient), b) Advanced
Formalism - on weekends - when I'm thinking
"about" mathematics (otherwise, I will end up
in mysticism). (The initial version of this aphorism is
due to Reuben
Hersh).
Next step
The idea that stable self-contained system of basic
principles is the distinctive feature of mathematical
theories, can be regarded only as the first step in
discovering the nature of mathematics. Without the next
step, we would end up by representing mathematics as an unordered
heap of mathematical theories!
In fact, mathematics is a complicated system
of interrelated theories each representing some
significant mathematical structure (natural numbers, real
numbers, sets, groups, fields, algebras, all kinds of
spaces, graphs, categories, computability, all kinds of
logic, etc.).
Thus, we should think of mathematics as a
"two-dimensional" activity. Most of a
mathematician's working time is spent along the first
dimension (working in a fixed mathematical theory, on a
fixed mathematical structure), but, sometimes, he/she
needs also moving along the second dimension (changing
his/her theories/structures or, inventing new ones).
Do we need more than this, to understand the nature of
mathematics?
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Whether your own philosophy of mathematics
is Platonism, or not,
can be easily determined by using the following test.
Let us consider the twin prime number sequence:
(3, 5), (5, 7), (11, 13), (17, 19),
(29, 31), (41, 43), (59, 61), (71,73), (101, 103),
(107,109),
(137, 139), (149, 151), (179, 181), (191, 193), ...,
(1787, 1789), ..., (1871, 1873), ...,
(1931, 1933), (1949, 1951), (1997, 1999), (2027, 2029),
...
It is believed that there are infinitely many twin
pairs (the famous twin prime conjecture),
yet the problem remains unsolved up to day. Suppose,
someone has proved that the twin prime conjecture is
unprovable in set theory. Do you believe that, still, the
twin prime conjecture possesses an "objective truth
value"? Imagine, you are moving along the natural
number system:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
12, 13, 14, 15, 16, 17, 18, 19, 20, 21, ...
And you meet twin pairs in it from time to time: (3,
5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59,
61), (71,73), ... It seems there are only two
possibilities:
a) We meet the last pair and after that moving forward
we do not meet any twin pairs (i.e. the twin prime
conjecture is false),
b) Twin pairs appear over and again (i.e. the twin
prime conjecture is true).
It seems impossible to imagine a third possibility...
If you think so, you are, in fact, a Platonist. You
are used to treat natural number system as a specific
"world", very like the world of your everyday
practice. You are used to think that any sufficiently
definite assertion about things in this world must be
either true or false. And, if you regard natural number
system as a specific "world", you cannot
imagine a third possibility: maybe, the twin prime
conjecture is neither true nor false. Still, such a
possibility would not surprise you if you would realize
that the natural number system contains not only some
information about real things of human practice, it also
contains many elements of fantasy. Why do you think that
such a fantastic "world" (a kind of Disneyland)
should be completely perfect? To continue click here.
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