Model Questions for Logical Foundations of Mathematics
These questions were prepared by Matthias Baaz and Harvey Friedman in
accordance with Hyung Choi's questions' catalogue.
WHAT IS THE PHILOSOPHICAL BASIS FOR LOGIC?
How can we find a new and more convincing way to justify the axioms, or
substantial fragments of the axioms, of set theory? Are there good
arguments for their internal consistency that don't address their truth?
How can this be extended to large cardinal axioms? How can this be
extended to other kinds of extensions of the usual axioms for set theory?
How can we give clearer formulations of the main "isms" such as
Platonism, realism, formalism, constructivism, empiricism, nominalism,
conventionalism, etc., and formulate theorems that attack or defend these
positions? (This means understanding logic in Gödel's way: the
philosophical analysis is the most important instrument to deal with
mathematical reality, however, this instrument is effective for
mathematics and philosophy only if philosophical positions are connected
to mathematical theorems)
ARE THERE DIFFERENT LOGICAL SYSTEMS THAT ARE MUTUALLY INCOMPATIBLE?
How to deal with the conflict of deductive logic fundamental for
mathematics
and sciences, and abductive logic e.g. in legal decisions? (In legal
systems decisions have to be made with apriori limited resources. The
consistency of such decisions is ensured in an empirical way. From a
mathematical viewpoint all such systems are inconsistent, but there is no
doubt that they are effective)
Do different sciences relate to different possibly incompatible logics?
Is a recognizable failure of mathematization in many fields connected to
the unsuitability of classical logic? (The prime example is quantum logic
and von Neumann's Theorem of the impossibility to introduce hidden
variables.)
CAN LOGIC BE CONSIDERED EMPIRICAL? CAN LOGICAL PRINCIPLES BE DERIVED FROM
PHYSICAL PRINCIPLES?
How can we give a convincing, or more convincing, proof of Church's
Thesis?
What are the analogues of Church's Thesis for complexity classes such
as P, NP, PSPACE, etc.? How can we give a convincing, or more
convincing, proof of these forms of Church's Thesis?
Is classical logic a limit case of logics related to the physical world
similar to the limit status of the Euclidian space?
CAN CREATIVITY (OR CREATIVE PROCESSES) BE AUTOMATIZED?
How can we develop a proof theory of analogical reasoning?
How can we formalize the abstraction of general statements from "good"
examples? (Babylonian mathematicians communicated general facts by
examples. Even in contemporary mathematics there is a strong feeling, that
good examples comprise the essence of mathematical creativity.)
How can we classify the different levels of obviousness and triviality in
mathematics?
To what extent can we program computers to recognize these levels using
practical resources?
CAN LOGICAL SYSTEMS BE CONSTRUCTED DIFFERENTLY FOR DIFFERENT TYPES OF
WORLDS OR DIFFERENT WORLDS OF EXPERIENCE?
Should logic depend on the assumption of a common world of experience?
Intuitionistic logic corresponds to the cancellation of this assumption.
How can we give a convincing, or more convincing completeness proof for
intuitionistic logic that better explains the naturalness of the
usual axioms and rules?
Choice sequences are at the core of intuitionism. They are formulated as
a concept of real numbers, however they relate to all objects if
one considers objects as given by their stepbystep recognized
properties. Is intuitionism a better foundation than classical
mathematics for sciences, where general assumptions are problematic?
ARE THERE FUNDAMENTAL LIMITS OF LOGICS AND MATHEMATICS IN DESCRIBING THE
REAL WORLD?
In what sense and to what extent does reading a proof or interaction with
mathematicians provide absolute certainty about the result? In what sense
and to what extent does interaction with proof assistants provide
absolute certainty about the result?
Is there a general way to define what classification means in core
mathematics? E.g., the classification of finite Abelian groups, 2
manifolds, finite simple groups, etc. Can we show that in interesting
cases, in core mathematics, classifications do not exist?
WHAT ARE THE MOST FUNDAMENTAL ASSUMPTIONS IN MATHEMATICS AND LOGIC?
How can we deal with Brower's doubt in his early writings: "Are we sure,
that the repeated applications of formal rules do not change their
meaning?" Is syntax without elementary induction possible and reasonable?
How can we make logic less circular? How can we identify a core
circularity that cannot be removed?
WHAT IS A PROOF? WHAT IS THE RELATIONSHIP BETWEEN THE IDEAS OF PROOFS AND
TRUTH?
The most important but least understood concept of mathematics is the
concept of proof. When proofs are equal (i.e. which transformations are
invariant on proofs)? Under which circumstances can proofs be represented
by examples (as Babylonians did)? Are proofs objects independent of the
human mind as Erdös suggested?
What more do we know given the proof of a statement than simply knowing
that the statement is true? (This question by Georg Kreisel possibly
describes the essence of proof theory.)
IS MATHEMATICS CONSTRUCTED OR DISCOVERED?
If we assume that mathematics is constructed, a coherence of proof and
construction should exist. Do the Incompleteness Theorems lead to
nonconstructable objects and thereby refute this assumption? Are there
suitable transfinite constructions not sensitive to the theorems
mentioned?
Is it really possible to develop a method for discovery of new axioms in
set theory using Husserl's Phenomenology as claimed by Kurt Gödel?
HOW DO FINITENESS AND INFINITENESS INTERPLAY IN MATHEMATICS?
How can we develop recursion theory for the finite subsets of N? For the
subsets of {0,1,...,2^{1000}}? For the subsets of {0,1,...,1000}?
The ominimality concept asserts that every definable subset of the
(linearly ordered) domain is a finite union of intervals with endpoints
in the domain (with +infinity). There is a very limited supply of
ominimal structures in mathematics. Can we productively create a theory
based on the weakened condition that every n quantifier definable subset
is a finite
union of intervals with endpoints in the domain (with + infinity)?
(Tameness is a principal seminal theme in model theory. Generally
speaking, a structure is tame if and only if its definable relations are
"well behaved". The ominimality concept is a primary example of a
tameness notion that is very well studied. The definable subsets are
structured by finite means, the underlying (uncountable) infinite is
controlled.)
WHAT IS THE RELATIONSHIP BETWEEN RANDOMNESS AND INFINITE?
Can we integrate the probabilistic and recursion theoretic features of
random sequences? Is randomness a tool for better computing? Is randomness
useful for establishing formal concepts of learning and creativity?
Forcing was originally introduced as a technical tool by Paul Cohen in
order to obtain dramatic results about set theory. Today, it is used in
the formulation of interesting further results. How can we establish
that forcing is of fundamental intrinsic significance?
SHOULD CANTOR'S IDEA OF THE "COMPLETED INFINITY" BE CONSIDERED SELF
EVIDENT?
Large cardinals are considered to be "complete" for the projective
hierarchy of sets of reals. How can we state and prove this rigorously?
(Cantor's idea of the completed infinity cannot be considered self evident
as long the mathematical consequences are not obvious. The projective
hierarchy of reals is a prime example where there is the impression of
large cardinals being complete, although it is difficult to substantiate
this impression formally.)
The formal assumption of a completion within the theory leads to set
theories not compatible with ZF (ZermeloFraenkel set theory). The most
prominent set theory of this kind is Quine's NF(New Foundation). Can we
construct adequate semantics for NF and thereby prove its full
consistency?
