[FOM] Theory of Actual Proofs
Harvey Friedman
friedman at math.ohio-state.edu
Tue Jun 17 23:49:49 EDT 2003
Reply to Tennant 7:57PM 6/17/03.
>On Tue, 17 Jun 2003, Harvey Friedman wrote:
>
>> My proposal of the condition to study is this. Every assertion made
>> in the first proof has a trivial proof from the set of assertions
>> made in the second proof, and every assertion made in the second
>> proof has a trivial proof from the set of assertions made in the
>> first proof.
>
>This presupposes---does it not?---that we have a way of identifying what
>counts as an "assertion" within a proof. With a Frege-Hilbert style of
>proof, this is unproblematic: every line or sentence in such a proof
>counts as an assertion. But with a natural-deduction style of proof, not
>every sentence-at-a-node is really an assertion. There are, for example,
>assumptions made for reductio ad absurdum; or for conditional proof; or
>for existential elimination; or for proof by cases. The person
>constructing the proof would not, in general, be committed to asserting
>any of these---nor, for that matter, many of the sentences following from
>them within the proof.
>
>So my first question is: For one whose preferred style of proof is natural
>deduction, how are we to understand and apply the condition quoted above?
I cut my posting of 2PM 6/17/03 off before I could get to such
issues, issues that I am fully aware of.
1. My proposal concerns actual mathematical proofs. In nontrivial
actual mathematical proofs, there are all kinds of assertions that
are made in the course of that proof. Many of these assertions are in
the form of quoting previously obtained results. E.g., typically a
nontrivial Theorem is preceded by a number of Lemmas. The proofs of
each Lemma may typically quote earlier Lemmas, as well as earlier
Theorems. The proof of the Theorem itself will generally quote
earlier Lemmas as well as other previously obtained results.
2. I have not seriously explored the consequences of my proposal in
the context of various formalisms. But actual mathematics rules, not
toy examples in logically convenient formalisms.
3. The way model trees come in to the way I want to formalize
mathematics is as follows. In order to give a proof of a Theorem, one
first imports a handful of previously obtained results, or perhaps
instantiations of previously obtained results, including
instantiations of axioms of ZFC, (instantiations of free variables
that are implicitly quantified). The importation should be sufficient
to logically imply the Theorem being proved. Then the model tree must
be finite. However, the model tree might be complex. One requirement
on mathematical text with proofs is that these imports are spelled
out, and ALL model trees are required to be simple. In particular,
they are very feasibly computer obtainable. In other words, in this
approach, ALL PROOFS ARE TRIVIAL LOCALLY. The nontrivial steps are
the importations, and the constructions of suitable Lemmas.
4. A typical objection to this approach is that one might want to
argue by reductio on a creative hypothesis. I.e., suppose we want to
prove Theorem T. We argue inside the proof that suppose A holds. Then
T follows. And suppose A fails. Then T follows. We have all done this
kind of proofs. But then it would be required that we create two
Lemmas:
LEMMA 1. Suppose A. Then T.
LEMMA 2. Suppose notA. Then T.
And then we have:
THEOREM. T.
with a completely trivial proof by importing Lemmas 1 and 2.
5. Another typical objection is that the proof of the Theorem may run
into an existential quantifier, where the term is not right in front
of your face. There are two ways I am thinking of handling this.
The first is to maintain, come what must, the idea that every proof
is trivial. So what I would do here is to again force one to break
this up into Lemmas. The nontrivial construction (i.e., term) will
actually appear in the statement of the Lemmas. Or, if the theorem is
itself existential, or AE, then state a stronger theorem with the
extra information that the term works. Then there would be again a
trivial model tree because the term is right in front of your face.
I guess I would prefer, at first, the all out attempt to make every
proof a trivial model tree. Nontrivial proofs by cases or reductio on
a statement, have the cases or statement appear in the Lemmas.
Nontrivial constructions; i.e., terms, would actually appear also in
the statement of the Lemmas.
6. One might call my model of mathematical practice, Administratively
Broken Down Proof Development. Everything is broken down,
administratively, into bite sizes that have trivial model trees.
7. Above and beyond all, this must be done so that it is entirely
readable, and flows perfectly mathematically. Everything nontrivial
is above board. What is suppressed is completely routine trivial
model trees.
8. In fact, it may even be more readable than technical Journal
articles! Of course, it is extremely time consuming to prepare. You
would have to know the ins and outs of elementary things far better
than people see any reason to know them. You would have to know them,
say, at the level a professional pianist knows the Piano Concertos
he plays with Orchestra. But it is not necessary to have this extreme
knowledge in real time, as a pianist must have.
9. But can be apply such ideas to some standard formal systems?
Again, this is an interesting, but secondary question, as the real
question is how we should model mathematical proofs.
10. Let's take a Hilbert style proof system for propositional logic
first. Let us say that we have two proofs of the same propositional
formula, with no hypotheses. We can ask about the set of lines in the
proof, all of which are valid formulas.
11. One immediate question that comes to mind is this. Does every
tautology have a Hilbert style proof where every line except the last
is a trivial tautology?
12. Also, consider a sequent calculus. Here every line of the proof
is a sequent, even if it is in tree form. Every sequent is obviously
a formula, by conjuncting the left side, disjuncting the right side,
and using implication for the sequent sign. So my proposal makes some
sense in the context of sequent calculus.
13. Even in natural deduction, at every stage in a natural deduction
proof, there is a set of working assumptions, and a statement being
asserted. This is also a formula, by taking the conjunction of the
working assumptions and arrowing the statement being asserted.
I'll stop here.
Harvey Friedman
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