Thanks, Glenn. I appreciate your persistence in reading the whole article.
I think this is an important and timely article, mainly, I must admit,
because it is in my current area of research. I'm working on a different
thread than Mumford but my intention is very sympathetic to his. And I
think you might agree that his main points are that we are now in the
"age of stochasticity" and that we should now integrate "probability
thinking" into the foundations of mathematics.
To address your questions about the article, let me just suggest to
concentrate on his section 1, "Introduction", section 5, "Putting random
variables into the foundations" and section 7, "Thinking as Bayesian
inference". I think that unless one is a mathematician, etc. the other
sections can be skipped without too much loss. And even within those
sections, Mumford has pretty much segregated math/logic-speak from plain
English; and that one can usually skip the insider stuff when you want
to, and still get the significance of the article.
Grant
On 11/15/16 1:11 PM, glen ep ropella wrote:
Very cool article, Grant! Thanks. I started to get lost on page 11
with the meta-axioms that give the Bernoulli random variables. *8^(
It's interesting that the wikipedia page
(https://en.wikipedia.org/wiki/Continuum_hypothesis#Arguments_for_and_against_CH)
mentions Feferman's semi-intuitionistic ideas in the same context as
Freiling's argument against the CH.
But I was irritated by his maps from the traditional subdivisions of
math to the primitive elements of human experience. The geometry one
seems right to me. But either he didn't finish explaining the
referents of analysis, or I disagree. Analysis (to me, of course) is
all about _proximity_, the closeness of any bunch of things.
Differentiation being about the determination of a locality and
integration being about establishing totalities. Although it's obvious
(hindsight is 20/20) how to get to analysis from the calculus and from
forces. It doesn't strike me that forces (and acceleration and
oscillation) are the primitive human experiences referred to by
analysis, as a domain.
Also, I don't really agree with the map from algebra to recipes of
action. To me algebra is about the preservation of some ...
"substance" _through_ transformation. So, like with forces giving us
(well, Newton and Leibniz) a path into the calculus, the composition
of actions in algebra is a kind of side effect. The core of it (to
me, a non-mathematician!) is about the preservation of some quality
through equivalence (and equivalence classes).
Obviously, it would be silly for me to argue with Mumford on this sort
of thing. But I'm wondering whether you (or anyone on the list) see
these experience correlations more as he sees them?
As usual, I have no comment on the actual topic of the paper. 8^)
On 11/13/2016 10:21 AM, Grant Holland wrote:
http://www.stat.uchicago.edu/~lekheng/courses/191f09/mumford-AMS.pdf
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