[FOM] vagueness in mathematics? -- why so little?

Dustin Wehr wehr at cs.toronto.edu
Wed Feb 22 11:24:59 EST 2017

Stewart Shapiro wrote:
> >> Mathematics goes to great lengths to avoid any kind of vagueness or
> >> indeterminacy. In what sense has it succeeded or not succeeded?
> >> Doesn't vagueness enter in to almost every other subject?

I understand why we do this, mostly for quality control, but I
question whether it's how things should remain. Only a  small portion
of vague concepts get sorted out in a satisfactory way in mathematics.
The rest remain in the realm of philosophy, law, policy, etc,
practically untouched by a large class of geniuses whose primary
preoccupations are truth and rigour. I suspect it's a great loss.

It's not because mathematics and formal logic do not apply. I think to
some of you that is obvious, and to the rest, I'll point to my
dissertation, "Rigorous Deductive Argumentation for Socially Relevant
Issues" (arxiv.org/abs/1502.02272). Here is the abstract:

"The most important problems for society are describable only in vague
terms, dependent on subjective positions, and missing highly relevant
data. This thesis is intended to revive and further develop the view
that giving non-trivial, rigorous deductive arguments concerning such
problems -without eliminating the complications of vagueness,
subjectiveness, and uncertainty- is, though very difficult, not
problematic in principle, does not require the invention of new logics
--classical first-order logic will do-- and is something that more
mathematically-inclined people should be pursuing. The framework of
interpreted formal proofs is presented for formalizing and criticizing
rigorous deductive arguments about vague, subjective, and uncertain
issues, and its adequacy is supported largely by a number of major
examples. This thesis also documents progress towards a web system for
collaboratively authoring and criticizing such arguments, which is the
ultimate goal of this project."

Note the "without eliminating" part, which is part of what
distinguishes what I'm talking about from the rigorous deductive
reasoning of game theory, for example. Another part is the focus on
work by "importance" in the everyday, outside-of-mathematics sense, as
opposed to interestingness, novelty, etc. Of course, there are loads
of applied statisticians who use that criteria, but they either (1)
don't reason deductively, resulting in arguments with badly obscured
assumptions that can be arbitrarily hard to criticize by outsiders,
however patient and talented; or (2) like in game theory, they begin
with a reductionist phase to eliminate vagueness, but that phase ends
up being the most disputable part of the argument.

This continues to be my life's work, now under the name of a nonprofit
rather than an academic institution, and these days I call it Formal
Deductive Argumentation. Besides my phd supervisors Alasdair Urquhart
and Steve Cook, many colleagues who encourage me but are too busy
publishing papers in their particular niche to help, and friends,
mostly lawyers, without backgrounds in math who want to help but
currently can only help in limited ways, it's been pretty lonely. My
cofounder and I are currently working on writing formal proofs about a
likely case of wrongful conviction, with help from the convicted man
and his lawyer (I have another one in my dissertation, though it's
been improved since then. It was assisted by Innocence Canada, and
they got the man freed without needing any help from me; I'd be happy
to share the improved version with anyone interested, after some
polishing). Please contact me if you're interested in any of this.

A QUESTION for the list: why do you think proof-focused mathematicians
do not engage in such work, even if only in their free time?

I think it's because it introduces another kind of complexity that is
simply unpleasant to the kinds of people who are drawn to
proof-focused math (including me). And we don't have a reward system
to compensate for that, since often such work will have a bad ratio of
size to interestingness/novelty/impressiveness, making it fail the
standards for publication in venues that publish proofs. Leibniz said
something to that effect, in this (translated) quote about the lack of
interest in his imagined Characteristica Universalis: "The true reason
for this straying from the portal of knowledge is, I believe, that
principles usually seem dry and not very attractive and are therefore
dismissed with a mere taste."

That said, I did learn and use Sturm's Theorem recently to prove a
claim that I left unproved in my dissertation (in the proof about the
now-resolved wrongful conviction case). And generally I KNOW I
would've produced more interest by now if only I was a more talented
several times I've temporarily abandoned a possibly-promising argument
because of some difficult pure math problem that came up, and then for
one reason or another never returned to it.

-Dustin Wehr

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