[FOM] 802: Systematic f.o.m./1

[Harvey Friedman]

This is the first in a series of posting on Systematic Foundations of
Mathematics.

Here I want to reconstruct the essence of state of the art f.o.m. FROM
FIRST PRINCIPLES, starting from scratch - from zilch - from nothin -
except very ordinary common sense. .

Of course, this is all being done with the profound advantage of
HINDSIGHT. The approach taken here is based on the maximization of GII
= General Intellectual Interest.

The great hope for this endeavor is as follows.

1. Such a systematic reconstruction, which is focused on ideas and
novel presentation of results, and not proofs, will move f.o.m. to a
heightened level of ACCESSIBILITY to many communities in and around
mathematics. And even to some extent to the wider intellectual
community of scientists, engineers, social scientists, and humanists.

2. During the course of the reconstruction, many missed opportunities
for better results will be systematically uncovered. Some of these
opportunities will be realized.

3. The entire reconstruction will provide a model for foundations of
subjects other than mathematics - especially in terms of the lessons
learned, and how obstacles were overcome in the development of f.o.m.

WHERE DO WE START SYSTEMATIC F.O.M.?

I don't know about any kind of optimality, but a very good place to
start is to consider the most elemental aspects of the grammar of
mathematical assertions. This of course intersects strongly with the
general grammar of ordinary language assertions. But at some point,
treatment of the two grammars diverge greatly. One leads to Philosophy
of Language and aspects of Linguistics, and the other leads to an
essential component of f.o.m. By the way, expect more and deeper
integration of f.o.m. with PhilLang and Linguistics.

Now what is the most obvious grammatical phenomena in mathematical thinking?

Arguably the idea that we can combine two sentences by conjunction to
form a third sentence.

We now clearly see that there is meaning side to this operation and a
syntax side as well. On the meaning side, we have the familiar truth
table for A wedge B, which tells us the truth value of A wedge B
depending on the truth values of A,B.

On the syntax or "reasoning" side, we have a rule

A,B
________
A wedge B

But we also have the rules

A wedge B
_________
A

A wedge B
_________
B

We also have the multiple rule

A wedge B
_________
A,B

Now what we want to do is to pin down this tiny fragment of
propositional calculus in terms of syntax and semantics, and
completeness. This will serve a model of at least some aspects of what
we want to do when we go further and expand our horizons to take into
account what is going on with mathematical proofs.

I will continue with this in earnest in the next posting in the
series, where I will struggle to uncover at least something even here
of some interesting novelty.

************************************************************************
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 802nd 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
801: Big Foundational Issues/1

Harvey Friedman