[FOM] Wittgenstein

[Dean Buckner]

>Harvey:

>[Tractatus] 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.

V. funny.  That's why I quoted Mayberry saying it was a "farrago" (which
means a "hodge-podge" - a mess).

Harvey:

>>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.

Dean:
>Not really.  Shorten it to "Quantifiers should range only over objects".

Harvey:
>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.

No, on one version of nominalism, quantifiers should range only over
objects, period.  Nominalism and common sense are comfortable with the idea
that there are things that have no (proper) names.  Any planets apart from
the usual ones, have no names.  But they could be named.  Or in the Harvey
standard way of putting things, could be possibly potentially referred to
denoted represented by.

Harvey:
>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?

Well fancy that.  I must have responded via FOM.  Well, once again:

(1)  Rendering "7 is a prime number" as the relation

 7 R {x: x is a prime number}

where something that is not a referring phrase "is a prime number", and
which does not contain any referring phrase, is analysed by a referring
phrase (namely the expression consisting of the curly brackets).

(2)  Rendering what is grammatically (i.e.logically) a sentence by an
expression that is grammatically a referring phrase, namely rendering "7 is
a prime number" as

 is_a_prime_number(7)

Prof. Hartley Slater has (as I understand) discussed the error underlying
this move at great length on FOM..

Harvey:

>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?

Well, <a1,a2,...,an> is itself a list, and cannot itself denote or signify
anything, unless it's a list of names.  Did you mean

 We let "<a1,a2,...,an>" denote the list of items <a1,a2,...,an>

I know (judging from offline correspondence) that some mathematicians regard
quotation marks as a tiresome triviality, but they're useful to indicate
when we are talking about one sort of thing (a list) and when another (the
name of a list).

So, what if anything in the above discussion denotes, represents, refers
to, etc., an object?  Very simple, we pick out anything in the discourse
which is a referring expression, surround it by quotation marks, then add
the expression "refers to", then append the expression itself.  Thus

 " "<a1,a2,...,an>" " refers to the expression "<a1,a2,...,an>"

That is because the first occurrence of the list expression is itself within
quote-marks.  To talk about this, we need two sets of quote marks.  What
else?

 "the list of items <a1,a2,...,an>" refers to the list of items
<a1,a2,...,an>
 "<5,7>" refers to the list of length 2
 " the list <5,7,5,<5,7>,3>" refers to the list <5,7,5,<5,7>,3>

and so on.  Anything which is a referring phrase, refers to an object.  What
does not refer?  Expresssions like

 "is an item in the list <5,7,5,<5,7>,3>"

are not conventionally held to refer, because they are fundamentally
predicative in nature.  They are (as medieval nominalists used to say)
"syncategorematic": they contribute to the meaning of expressions
(sentences), without themselves having a meaning.


Dean:
>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?

Harvey:
>If pi were missing, then the circles of unit radius would not have an area.

That's a beautiful answer, and correct.  I see no difficulty in there being
objects to which infinitely many things stand in a given relation.  For
example, regarding the number pi as the sum of every x such that x is a
value of the decimal expansion of pi.

 3  + 1/10 + 4/100 + 1/1000 .

What's wrong with the statement that pi is the sum total of every number in
this series?

But I DO see a difficulty by contrast in the idea of a set as an object that
bears a relation to all objects of a certain kind, in virtue of their being
objects of that kind.  Something that bears a semantical, not a mathematical
relation to its parts.  For, on a strictly nominalist view, we can only
quantify over single objects, or finite pluralities of objects.  Thus, to
express the fact that there are infinitely many F's, we must say that no
(finite) set of F's constiture all the F's there are.  "Constat quod omne
continuum habet plures partes, et non tot finitas numero quin plures, et
omnes suas partes actualiter et simul habet, igitur omne continuum simul et
actualiter habet partes infinitas."  - "Every continuum has further parts,
and not so many parts finite in number that there are not further parts, and
has all its parts actually and simultaneously, and therefore every continuum
has simultaneously and actually infinitely many parts"

Wittgenstein wrote " The number word 'four' in "there are four things" has a
quite different function from the word 'infinitely' in "there are infinitely
many things".  We mistakenly treat the word "infinitely many" as if it were
a number word like "four", because in ordinary language both are given as
answers to the question "how many?"

Dean:
>>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".

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

Is that what I'm saying here?  I am saying, every digit in the expansion of
pi whatsoever, without remainder, with nothing left over, corresponds to
some rational number.

Dean:
>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?

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

I meant, if pi is construed as the "set" of all the digits in its expansion,
then we can do without this notion.  We cannot of course do without any of
the digits in the expansion of pi.  Every one is essential in contributing
to the mathematical sum, the mathematical totality.  That is what
distinguishes pi from any rational number.  Pi is the only number that
stands in a certain relation to EVERY member of the sequence.

Harvey:
>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.

No, exact amounts.  Every number in the expansion of pi represents some
approximation.  No approximation stands in such a relation to every member
of the sequence.

> [Blah blah]


Wittgenstein

>["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."

Harvey:
>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.

Of course.  Why would I deny it?  "A cardinal number is an internal property
of a list.  Philosophical Grammar  § 19  p. 332."  Lists are essentially
finite objects.  And they are objects, too.  We can express the idea of
infinity only by using predicates, which do not signify objects.  So, we can
say "ther are infinitely many F's".  But this is not the same as saying "the
set of F's is infinitely large (which we cannot grammatically say at all).
When we say there are infinitely many F's, we are not saying of ANY THING
that it has a certain number.

W:
>"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)
>

Harvey:
>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!

I take his comments at face value.  As W. had no sense of humour whatever, I
take it he was not joking.  Very often he says things that sound like jokes,
of course.

>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.

This requires that "completed infinite totality" is a coherent notion.
Conceived as a mathematical sum, no difficulty.  Conceived as the reference
of a predicate that has an infinite extension, it is incoherent.  Predicates
have no reference.

I have no problem with abstract objects, either: for example humour, set
theory, existence.  Anything that looks like a noun phrase conceivably
refers to something.  The problem occurs when we take something that is
demonstrably not a referring expression, and impose the logic of a referring
expression upon it.  For example

 "a is B" = "a satisfies the referent of  "is B""

The first problem this gives rise to is Bradley's regress.  If the empty
space between the name and the predicate signifies the "satifisfies"
relation, and "satisfies the referent of  "is B"" is iteself a predicate,
then we can have a regress.  Second, what about

         "a does not satisfy a"?


Why can't we do good mathematics, without getting muddled up in poor
semantics?


Dean
London
ENGLAND

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