[FOM] Math Depth/Maddy

Harvey Friedman hmflogic at gmail.com
Wed Aug 13 03:13:24 EDT 2014

Dear Pen,

Thanks for the info (conference on mathematical depth). I'm hoping to spend
some more time on those videos, but in any case I will be quite interested
to look at the resulting papers.

Recall that I was pointing to the blackboxing of the notion of "good
mathematics". Of course, as you clearly know, this is not the same as
depth, though for most mathematicians, depth is a substantial component.

Here are my thoughts.

1. I have never heard mathematicians talk directly about what good
mathematics is in any kind of generality. Normally, they talk about the
relative merits of some particular developments in some particular areas,
without delving into any general considerations or criteria.

2. In talking separately to different kinds of mathematicians over the
years, it is obvious that there is a huge amount of disagreement about how
to evaluate mathematics, what it's purpose it, what it means, and so forth.

3. Just about the only thing that there is widespread agreement among
mathematicians is the following.

i. If the problem has resisted solution for a very long time, and it is
known that some mathematicians with very strong reputations worked on it
and failed to solve it, then mathematicians will generally regard the
solution as extremely good mathematics. There are exceptions to this, and
the main exception, which will sometimes split the opinion, is whether or
not the solution uses considerable machinery. This is considered an extreme
plus over it being solved by extremely clever special methods. There is
rationale for this, mainly that if big machines are used, then that
promises further solutions to further problems more than an extremely
clever special method. However, this kind of attitude is somewhat
bothersome to me because it illustrates how far the mathematicians
generally are from evaluated the mathematics on the basis of information
content. To bring in an obvious example, I couldn't care less in my
evaluation of Goedel's Second Incompleteness Theorem whether the proof was
very easy, easy, fairly hard, hard, or extremely hard. Or for that matter
whether the proof was very deep, deep, fairly deep, deep, or extremely
deep. Information content is all I personally care about, but making that
coherent across mathematics seems to require something like 4 below.

ii. There is a major premium paid for interactions between areas of
mathematics - particularly if the interaction is unexpected.

iii. There is increasingly a premium paid to problems of a concrete nature,
particularly if there is some finite computerizable component. There has
long been a premium paid for stuff of clear geometric meaning.

4. Here is what I think mathematics needs most, in terms of a practical
project. It needs a thorough systematic foundational exposition. I think I
know how to go about this, generally speaking, and would love to find the
time I don't have to do a chunk of this. But I haven't seen, nor have I
done, a foundational exposition of even mathematical logic - which has
largely veered away from its foundational roots.

5. For me, I take mathematics to be both a tool and an object of study. I
am a foundationalist who uses and studies mathematics. But I am still
interested in the dynamics of the mathematics community, even if I find
many features of it rather unattractive intellectually.

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