[FOM] Wittgenstein?

[Harvey Friedman]

   Reply to Steiner, 4/24/03  5:03PM.

>  To ask whether Wittgenstein's LATER philosophy of mathematics has any
>significance for FOM, in the sense that it can inspire research in the
>field, may seem unfair--LW expresses hostility to the whole project of fom,
>such as that of Russell and that of Hilbert.

As I said before, a sufficiently well argued and/or clever argument 
against foundations of mathematics might very well count as a 
contribution to f.o.m. because of what has to be developed in order 
to fully defend against the attack.

Is there anybody on the FOM list prepared to outline the LW attack or 
any other attack interactively?

A long time ago on the list, there were some specialists in model 
theory and some sympathetic people in core mathematics who made some 
effort to do this. They got interested in doing this because of the 
suggestion by myself and some others that major work in f.o.m. had 
greater general intellectual interest than major work in core 
mathematics, and therefore there was a serious undervaluation of 
f.o.m. in the mathematics community.

There were also some people who defended the categorical approach to 
f.o.m. as not only a viable alternative to standard f.o.m., but 
generally superior.

>I also read LW as arguing that
>Goedel's theorems have no significance for the philosophy of mathematics--a
>disastrous error even from LW's own point of view, as I understand it (cf.
>my remarks on Juliet Floyd's views in Philosophia Mathematica, vol. 9,
>257-279).

It would be interesting if Floyd or her supporters on this issue 
would argue this interactively on the FOM email list.

At least, it would be interesting to see at least a brief synopsis of 
the Floyd/Steiner exchange right here in front of us on the FOM list.

E.g., there might be a new imaginative defense against this that some 
subscriber might think of, and/or some new some way to strengthen the 
attack.

>
>     Yet I believe that the kind of research that Harvey does (or some of it,
>including "reverse mathematics"), and the motivations for it which he has
>expressed from time to time on the fom list, are compatible with LW's
>"descriptive" point of view (and even if LW would not have agreed to this
>proposition, he should have) concerning mathematics.

In fact, I discovered "reverse mathematics" in the 60's just by 
pondering very carefully, the extremely harsh private attacks against 
f.o.m. It is commonplace that the vast majority of mathematicians 
think privately that working with formal systems is no more 
interesting or important than the contemplation of your own navel for 
about 10 years without interruption  - largely "because" you 
"generally make up any axioms you want and play with them with no 
constraints or purposes".

So these commonplace ignorant, prejudicial, and inane banalities can 
be viewed as being a real practical force in the development of 
f.o.m. 

>(LW's insistence that
>set theory, to be mathematics, must be relevant to applications
>(mathematical and physical), is, as I understand it, exactly Harvey's view,
>except that Harvey and other fom people have demonstrated such
>connections--LW's prejudice to the contrary is, I feel, the same prejudice
>that "core mathematicians" have about set theory, so LW should not be
>singled out for blame here.)

Again, I have benefited from the commonplace view that "how can such 
abstract contentless gibberish have anything to do with real math - 
or even real thinking?"

>
>     Let me give an example of this.  Wittgenstein, in a face to face
>exchange with Turing (who attended his classes in 1939), argues that the
>obsession with consistency on the part of logicians (Hilbert) is a
>superstition.  Hidden consistencies he argues can never affect the
>application of mathematics, and when an inconsistency crops up no engineer
>would argue that "anything follows from this."  Turing, on the other hand,
>argued that "bridges could fall down" if we apply an inconsistent system,
>because there are ways of arguing for any conclusion q in an inconsistent
>system without actually going through the step "p and not-p".  E.g., by
>arguing from not-p to if p then q in one proof, and then proving p in
>another.

I have thought from time to time about the idea that one could 
possibly productively protect an inconsistent system from itself. 
i.e, one might be able to talk about the consistent part, so that one 
can still distinguish between the status of sentences and their 
negation.

However, I have not been able to do something productive with this 
idea, and the literature that I know a little about in this direction 
does not get to the heart of the matter.

I know that there are several subscribers who at least implicitly 
think that this is productively handled by various forms of relevance 
and minimal logic, etc. But I have not found it convincing (yet), and 
it would be interesting to see them argue interactively on the FOM 
email list.

Of course, I did an easy form of what you are talking about, 
connected with size. I.e., one can easily work with inconsistent 
systems all of whose inconsistencies are huge. I proved the finite 
Godel theorem, which was later taken up by Pudlak and other proof 
theorists who also work on the edge of complexity theory.

>
>     Now, in fact, "core" mathematicians inform me that the basic mathematics
>used in quantum field theory, the Feynman Path Integral, has presently no
>consistent formalization--unlike, say, the Dirac delta function, or the
>Newtonian/Leibnizian differential calculus.  Feynman himself believed that
>his formalism was in fact INconsistent (i.e. that there is no way to give it
>a consistent foundation).  Nevertheless, physicists know how to extract the
>most amazingly accurate "numbers" from this formalism, much as Euler was
>able to extract mathematicial information from manipulating divergent
>series.

The Euler matter was later given a satisfactory rigorous treatment, 
as i understand it.

So what happens when such things are handled properly? One has a 
major increase in

THE QUALITY OF KNOWLEDGE.

This is not the highest priority of physicists.

>
>     I would be grateful to the fom community if they could give me
>references to literature, or create the literature themselves, on degrees of
>inconsistency, "hidden" vs. "manifest" inconsistency, etc.  I'm not sure
>myself how to formulate the problems mathematically, but an elementary
>question raises itself: in the same way that Harvey has discussed the
>completeness of set theory when limited to proofs of a certain magnitude or
>"length" one could obviously ask the same question concerning inconsistency
>(since the question is so obvious, there must already be literature on
>this).

Other than the size of inconsistencies, I am not familiar with 
anything in which I have confidence. E.g., look at the Handbook of 
Proof Theory, ed. Sam Buss. .

So, subscribers, let us know what you find interesting along these lines.

>
>     A deeper question would be--is there any formal analogue to the notion
>that a physicist knows how to "work around" an inconsistency (in the same
>way that Euler did with divergent series), knowing where it is and where it
>is not possible to draw conclusions (so that bridges don't fall)?

It should be noted that, arguably, the only interesting serious 
modern case of where an inconsistency was found in a sensible formal 
system for f.o.m. purposes was Kunen's inconsistency proof of VB + 
AxC + "there is a nontrivial elementary embedding from V into V".

Perhaps a better way of saying it is Kunen's inconsistency proof of 
ZFC + "there is a nontrivial elementary embedding from a limit rank 
into itself".

The inconsistency is by no means simply any kind of rehashing of the 
usual diagonal arguments.

During the brief period when this was in the air - just a few years - 
people were somewhat queasy about it - although Reinhardt proposed it 
and it is called Reinhardt's axiom.

They carefully studied things that are directly weaker, and these 
weaker ones are now principal features of modern set theory. People 
have become somewhat confident of their consistency. You can view 
this as some sort of perhaps weak example of the sort of thing you 
are asking for.

Also, the weaker

VB + "there is a nontrivial elementary embedding from V into V"
ZF + "there is a nontrivial elementary embedding from a limit rank into itself"

are still not known to be consistent or inconsistent, and there is 
probably no strong feeling on the part of the set theory community.

>
>     More generally, we can ask concerning the whole question of rigor in
>"core" mathematics.  Physicists tend to pooh pooh rigor in mathematics as
>unnecessary in physics--they sometimes claim that rigor in a proof is
>necessary only to rule out "monstrous" counterexamples never seen in nature.


When they say this, they are insensitive to

THE QUALITY OF KNOWLEDGE.

F.o.m. people like me revel in this aspect of intellectual life.

>Physicsts, for example, have no problem in considering a magnet of, say, 3+i
>dimensions, as a formal trick to be able to apply complex analysis (analytic
>continuation) to thermodynamics.  The physicists' attitude is similar to
>LW's fulminations about the invasion of mathematics by formal logic.  It
>would be ironic if the "logicians" were able to explain the efficacy of
>nonrigorous mathematics as they give an account of "correct" mathematics.

In a sense, we already do this to some extent. In order to have such 
efficacy, it must be the case that if someone performs such 
nonrigorous mathematics in a somewhat new way, then they are not 
going to get a different conclusion. Or if someone performs such 
nonrigorous mathematics they are not going to get demonstrably false 
conclusions. When we formalize and prove consistency (or relative 
consistency), we get such as Corollaries.

>
>     From the historic point of view I can testify that LW's problem of the
>falling bridges has provoked at least one logician, Saul Kripke, to study
>the problem.  He thinks LW is wrong and Turing is right, but in any case a
>study of the problem would be fascinating.
>
It should be easy to set up some math based on inconsistent 
principles which gives a nice looking answer that can be used to 
build a bridge which later collapses, and wouldn't collapse if the 
answer that was used was different. In a sense, this happens whenever 
a buggy computer program is used in connection with correct 
mathematics. The mathematics that is wrong is "the computer program 
is correct". You probably don't think that I have answered your 
question, so I invite you to say more about just what you are looking 
for.