[FOM] Wittgenstein

[Harvey Friedman]

Reply to Buckner 4/24/03  8:09PM, and Buckner 4/25/03  10:04PM

How would LW prove the following results:

There are more straights than flushes in poker?
The most common distribution in bridge is 4-4-3-2?

>Harvey:
>
>
>*DID LW WRITE ANYTHING THAT CAN AT LEAST BE REASONABLY INTERPRETED AS
>BEING SIGNIFICANT FOR THE FOUNDATIONS OF MATHEMATICS? IF SO, EXACTLY
>WHAT?*
>
>In Tractatus 6.02 he develops a formalistic theory of number in about half a
>page.  John Mayberry (in his book, discussed in other postings) describes
>this as a "farrago".

6.02 *by itself* does not amount to anything at all - *by itself*, is 
not distinguishable by me from a bad student's notes from a bad 
lecture by a bad logician. If you are serious, you ought to present a 
theory of numbers associated with LW in a meaningful way, here on the 
FOM.

>
>My main source is the end of "Philosophical Grammar", which is about half a
>book on set theory, Skolem's Theorem, recursion and other matters.  Happy to
>quote, but Harvey asked for no quotes.  W.'s view is that set theory is
>fundamentally a mistake, which if true has clear significance for f.o.m.

I am not interested in more quotes unless their meaning is 
sufficiently clear to move the discussion forward. I am, however, 
much more interested in perceptively interpreted quotes.

>
>>is LW a major force in nominalism and related views?
>
>"Nominalism" is strictly speaking a medieval school of philosophy.  Let's
>say "existential conservative".  Then yes, very much so.  But, as with
>politics, there are huge varieties.
>
>As to W. a major force, it's more a legacy.  However, Ray Monk is writing a
>book on f.o.m.

But it is at least claimed that LW's hugely negative views are backed 
up with some unusual finesse that makes them potentially valuable. 
That's what I would like the FOM list to get to the bottom of.

>
>>Roughly speaking, as I understand it, this [nominalism] is the view that
>only
>>objects which have been named are legitimate items of discourse.
>>Quantifiers should range only over objects which have been named.
>
>Not really.  Shorten it to "Quantifiers should range only over objects".

You are jumping the gun. The conventional wisdom is that many 
mathematical objects exist that do not have names, and that this is 
an essential feature for the smooth development of contemporary 
mathematics.

>At
>the heart of all Wittgensteinian thought is the idea of the _category
>error_, subsequently popularised by Ryle. This is to read what is in fact a
>distinction between linguistic categories, as though it were a distinction
>between things.  There is in human nature (and particularly in philosophers
>nature) a mistaken, but inevitable tendency to"look for a substance whenever
>one sees a substantive".  (For example, to look in the meaning of "the
>number of unicorns is zero" for something named by "the number of unicorns",
>"zero" &c).

Is the idea that mathematicians make category errors of the kind you 
are talking about? If so, give the simplest example of where 
mathematicians make such category errors.

I asked you this kind of question before and never received a 
response: are you saying that mathematicians are doing something 
wrong?

Is the following a category error?

"We let <a1,a2,...,an> denote the list of items a1...,an of length n. 
In particular, consider the list <5,7,5,<5,7>,3>".

OBSERVE: In the above, <5,7> is a list (of length 2) which is also an 
item in the list <5,7,5,<5,7>,3> of length 5.

What if anything in the above discussion denotes, represents, refers 
to, etc., an object?

>Imagine we are given ALL the irrational numbers EXCEPT pi.  How exactly
>would this fact manifest itself?  And what would change if pi were added?

If pi were missing, then the circles of unit radius would not have an area.
>
>If I search hard enough, I can find some rational number whose expansion
>appears to correspond to that of pi.  Of course, for any such rational
>series, there is a point which the expansion must diverge.  But then I can
>find another one that agrees with the expansion of pi "still further".

Yes, every real number can be approximated by a rational number 
arbitrarily closely.

>
>So if we now add pi, what exactly changes?  At what point is pi "now really
>needed"?  At EVERY point, it has a companion (in the set from which it was
>omitted) agreeing with it from the beginning up to that point.  So how is pi
>actually necessary?

It is necessary to support the idea that any reasonable region has an area.

>
>Doesn't that prove that pi is not "the extension of an infinite decimal
>fraction", that it's just a rule or algorithm of some sort?

It doesn't *prove* anything. A charitable interpretation of what you 
are suggesting is that we don't need to have areas represented by 
single entities. Rather, maybe we can get away with only 
approximations.

This idea has been around for a very long time and has merit. It is 
generally called such things as: the finitistic interpretation of 
mathematics.

All mathematical objects under this scheme are to be finite.

Actually, the posting of mine, #164, is more radical. There I propose 
a strict upper bound on the available mathematical objects, and there 
are only a finite number of them, and that number is very simply 
expressed in ordinary arithmetic notation.

It has never been worked out clearly enough to see what the 
difficulties are. One major difficulty is that it appears to be 
perhaps unworkably complicated.

We do know that some mathematical theorems that make clear sense from 
this standpoint cannot be proved without invoking mathematics in 
which strong forms of completed infinite totalities are used in a 
demonstrably essential way.

>
>
>["Set theory is wrong because it apparently presupposes a symbolism which
>doesn't exist instead of one that does exist (is alone possible).  It builds
>on a fictitious symbolism, therefore on nonsense."

What about the list notation I wrote about in this posting, above? 
I.e., <5,7,<5,7>,3>?

If this by now completely standard finite list notation is admitted, 
then one trivially has finite set notation.

>
>"Mathematics is ridden through and through with the pernicious idioms of set
>theory.  *One* example of this is the way people speak of the line as
>composed of points.  A line is a law and isn't composed of anything at all".
>(Wittgenstein, Remarks, XV § 173)
>

Of course, LW didn't publish serious mathematics, so we don't 
actually see how he would actually avoid the set theoretic 
interpretation of mathematics, which is extremely effective and 
extremely convenient. So he had the luxury of perhaps saying these 
things as provocative jokes. He may well be laughing in his grave 
that people actually try to take such comments at face value!

Nevertheless, I don't care if it is all a hoax.

It is important to prove that the removal of completed infinite 
totalities and abstract entities of various kinds cannot be achieved 
in various senses, and can be achieved in various senses. I do this 
sort of thing for a living.