[FOM] 801: Big Foundational Issues/1

[Harvey Friedman]

This is the first in a series of FOM postings devoted to a compilation
of and discussion of my perspective on the issues in the foundations
of mathematics of the highest GII = general intellectual interest.

As you know, I have focused on what may be regarded as Conventional
Line f.o.m. where I fully adhere to the great insights and
achievements of Aristotle, Frege, Cantor, Russell, Goedel, and others.
Here I include intuitionism and constructively and finitism and
ultrafinitism. This Conventional Line does not include, say,
categorical foundations or paraconsistency or any kind of "free form"
or "no foundations" or "pragmatism" or "naturalism". However, I intend
to cover these and other Non Conventional Lines as well. But there the
emphasis will be on fundamentals - criticisms, and forming new
initiatives to get these solidly off the ground,

So here is my first pass on a very rough preliminary initial list of
big f.o.m. issues.

1. In precisely what sense do we even have a foundation for
mathematics? I.e., what does ZFC really accomplish?
2. What are the prospects for a more powerful foundation for
mathematics? I.e., sensitive to a wider range of features of
mathematics or mathematical thinking?
3. What does it mean to say that mathematics is sound, or true, or
valid? What is or can be the nature of certainty in mathematics?
4. Why do we or should we believe that mathematics is in fact sound,
or true, or valid? What kind of evidence do we have, and what kind of
evidence can we expect to get, and what kind of evidence is possible
for us to get? I.e., the hierarchy of "isms" and mathematical theorems
that separate them, and consistency proofs.
5. In what sense is our current foundations for mathematics adequate
to "capture" our mathematical knowledge? In what senses do we have
completeness or incompleteness?
6. Is it possible to establish more strongly that our models of
computation are complete? I.e., novel proofs Church's Thesis.
7. In what sense is our f.o.m. systems canonical or forced on us or
inevitable? I.e., Strict Reverse Mathematics.
8. What is behind the profound mutual comparability of f.o.m. systems?
I.e., invariably the comparability of natural theories under
interpretability.
9. What regions of mathematical thought are subject to algorithmic
solvability? Is it hopeful that there is a universal algorithm that
decides all "natural" mathematical statements?

Well this is enough for a preliminary list. I will address each of
these, and many others, forming sub issues, and then addressing them.

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My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
This is the 801st in a series of self contained numbered
postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-799 can be found at
http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/

800: Beyond Perfectly Natural/6

Harvey Friedman