[FOM] Why Bother?

Harvey Friedman friedman at math.ohio-state.edu
Sat Oct 18 22:53:47 EDT 2003

Reply to Slater. Tennant (10/16/03 10:33PM) has already commented on the
technical aspects of some of Slater's recent posting(s); among these are
Slater's unexpected use of long discarded conventions concerning the
subscripts of quantified variables. I now want to comment on some
nontechnical aspects.

On 10/16/03 7:52 PM, "Hartley Slater" <slaterbh at cyllene.uwa.edu.au> wrote:

> I grant you the latter is a very, *very* interesting issue, but only
> for this reason: it reflects on how few people want to get to grips
> with substantive issues.

What substantive issues is Slater raising in his recent postings? I haven't
seen any substantive issue raised by Slater in the foundations of

>It was quite remarkable how many people
> responded about this, but were not man enough to tackle my assertion
> 'the theory of number has been mis-represented for 100 years'.  ... people
> to have turned to some trivial pursuit, to avoid the anxiety which
> realisation of the dimension of the real issue would bring.

What real issue in the foundations of mathematics?

Which of the following "theories" have been "mis-repesented for 100 years"?

theory of heat
theory of color
theory of light
theory of sound
theory of force
theory of time
theory of space
theory of metal
theory of surfaces
theory of fire

> Where, by contrast, in von Neumann's work, does he say anything like
> this, or employ formulas of the form '(nx)Fx', where 'n' is one of
> his constructs?  

Von Neumann was concerned with the foundations of mathematics, and not
philosophy of ordinary language.

>Calling something 'a cardinal number' or 'an ordinal
> number' does not guarantee they are such.

Von Neumann made this definitions within set theory and class theory, and
proved that they have many properties, including various existence theorems.
An example of such is the emptyset. This guarantees that there are such.

>Where in *any* of 
> twentieth century FOM are there words with the sense of 'first',
> 'second', 'third' etc. (remember them?)

I sometimes use those words in technical contexts. It is usually preferable
to using symbols like 1,2,3 for the reader, who may need a break from
looking at a lot of symbols. Other times it is necessary for clarity and
exactness to use symbols.

> Certainly von Neumann's
> 'ordinals' do not have this sense,

"sense" makes no sense.

>so all you need, it seems, is some
> ungrounded assertion that von Neumann's chosen sets are ordinals

This is provable in set/class theory. I.e., it is provable that the emptyset
is an ordinal, that {emptyset} is an ordinal, that if alpha is an ordinal
then so is alpha union {alpha}.

> everybody thinks they have a foundation for the theory of ordinal
> numbers.  

The theory of ordinal numbers is explicit within set/class theory and so any
foundational issue is, prima facie, answered by the foundations of set/class

One may of course want something more subtle such as: the development of
some concept of "ordinal numbers" independent of set/class theory, or the

>Alas: "You can make anybody believe anything, so long as
> they are clever enough" (Tom Stoppard).

"You can make anybody disbelieve anything, so long as they are clever

On 10/16/03 9:26 PM, "Hartley Slater" <slaterbh at cyllene.uwa.edu.au> wrote:

Avron wrote::

>>  The only question is: why are they
>> bothering the people on this list who are really interested in the
>> foundations

I also have the same question:

why are they bothering the people on this list who are really interested in
the foundations of MATHEMATICS???

I hereby propose an answer.

***They actually think that the "issues" that they raise have something to
do with the foundations of mathematics!***

This is only part of the reason that they are "bothering the people on this
list who are really interested in the foundations of mathematics".
Discussion of other reasons may not be suitable for FOM.

> There is clearly some conflict in Avron's mind if he can be
> interested in the Foundations of Mathematics but not attach much
> importance to what Natural Numbers are.

I attach no importance whatsoever to any discussion of "what Natural Numbers
are" that is pointless, unproductive, with no new ideas, that does not even
remotely take into account what is well known concerning the matter for many
many decades. To make matters worse, even after the relevant well known
information has been spelled out, the discussion just continues without any
incorporation or even acknowledgement. This constitutes a drag on the FOM
list, and should not continue.

That doesn't mean that nobody in the world can make very interesting
postings concerning "what Natural Numbers are" with plenty of FOM content.

It just means that IF one is going to make postings about matters like

what is a Natural Number?
what is a Real Number?
what is a thing?
what does it mean to exist?
what is an object?
what is an idea?
what does 'and' mean?
what does 'not' mean?
what is truth?
what is a set?
what is a thing?
what does 'what is' mean?
what does 'is' means?

THEN it had better have some glimmer of an insight, or idea, or else I think
it does damage to the list. Also, any repeated postings had better take into
account fundamental well known information, particularly when made

>The underlying belief
> is that there is no right answer, but more to the point is the reason
> why many people want there to be some freedom in the matter, since it
> is that which relates to why there is a need to 'bother' people about
> these issues. 

I still don't understand your explanation as to why you are bothering to
bother people on the FOM about 'what is a Natural Number'.

>The danger is that the mathematics people like to do
> might be just recreational.

I have seen your use of the word "recreational" on the FOM before, but I had
thought that it must have been a typographical error of some sort. I have no
idea what you are talking about.
> Maybe Aaron has tenure, or a private income, so he does not have to
> justify what he likes to do to anybody,

Why don't you "justify" why you "like" to bother people on the FOM about
'what is a Natural Number'? It is because you "have tenure, or a private
income, and do not have to justify what you like to do to anybody"?

>Von Neumann's answer is wrong (remember the word 'wrong'?).

To my knowledge, von Neumann never addressed the issue 'what is a Natural
Number' in any sense that you seem to be interested in. So von Neumann make
no errors in this regard.

Harvey Friedman

More information about the FOM mailing list