Chris, maybe a point I should have made is that the 'pure' set
theories without ur-elements are purged of non-mathematical things but
also of mathematical things. They are *entirely* about sets. So it is
somewhat misleading to characterize them as being about mathematical
topics. Now, to be fair to John, he did say that they were *applied*
to mathematical topics, and in a sense this is correct: the whole
foundations-of-mathematics goal, as I said previously, was to build a
model of all of mathematics in pure set theory, in effect
reconstructing mathematics so that it is all about sets. But notice
how different this is from using set talk to refer to sets of
mathematical entities (like, say, the set of natural numbers; as
opposed to the set { { }, {{ }}, {{ }, {{ }}}, {{ }, {{ }}, {{ },
{{ }}}}, ...} ). That activity - using set language in doing
mathematics - is just as much ur-element dependent as using set
language to talk about non-mathematical things. (01)
Pat (02)
On Oct 10, 2009, at 5:22 AM, Chris Partridge wrote: (03)
> Pat,
>
>>> The fundamental distinction is between mathematical entities and
>>> physical entities:
>>>
>>> 1. The great developments in 20th century logic and set theory
>>> were applied to purely mathematical topics.
>>
>> John, you keep asserting this, and it is completely false. I have
>> given you citations from everyone from Frege through Russell to
>> Quine,
>> taking in Church, Carnap, Tarski and others along the way. They all
>> used as primary examples, sets of non-mathematical things. None of
>> the
>> published set theories make any stipulation that the entities in the
>> sets are 'mathematical' in nature.
>
> Firstly, to avoid any misunderstanding I wholly support ALL your
> comments
> (maybe a first :-) ). It would be great if we could put the
> confusion about
> sets and identity criteria behind us.
>
> However, there is a small technical point related to the item copied
> above
> that is illuminating - and we have discussed it before.
> So, in a sense, I am asking for confirmation from you - and (if we
> agree)
> making sure that others on the list are clear about this point.
>
> ZF and most other 'standard' set theories do not have URelements.
> There are
> non-standard (less used?) set theories around that do (e.g. ZFU).
> For the
> mathematical project, is was useful not to have to assume the
> existence of
> URelements (why should maths depend upon this?) So, technically, in
> ZF there
> are no "non-mathematical things".
>
> Regards,
> Chris Partridge
> Chief Ontologist
>
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>> -----Original Message-----
>> From: uom-ontology-std-bounces@xxxxxxxxxxxxxxxx [mailto:uom-ontology-
>> std-bounces@xxxxxxxxxxxxxxxx] On Behalf Of Pat Hayes
>> Sent: 09 October 2009 20:15
>> To: uom-ontology-std; John F. Sowa
>> Subject: Re: [uom-ontology-std] What is mass?
>>
>>
>> On Oct 9, 2009, at 3:14 AM, John F. Sowa wrote:
>>
>>> Dear Matthew,
>>>
>>> Some comments on your comments from different, but related notes:
>>>
>>> JFS>> Making a clear distinction between actual sets and
>>> hypothetical
>>>>> or imaginary sets would be useful.
>>>
>>> MW> In ISO 15926 we used the term "class" for this. An overloaded
>>>> term I know. But I'd rather not start inventing new ones.
>>>
>>> I would recommend an adjective. Instead of using two different
>> nouns,
>>> such as 'set' and 'class', it is much clearer to use one noun 'set'
>>> as the general term and add different adjectives to distinguish
>>> which kind of set you are talking about.
>>
>> I disagree. This adjectival usage suggests that there are kinds of
>> set, and that embodies the confusions that gave rise to this thread.
>> There are not different kinds of set. There are sets, which may be
>> sets containing different kinds of things. But they are all (just)
>> sets. A set of foodles is not a foodle set.
>>
>>>
>>> JFS>> But those sets [in possible worlds] are purely imaginary.
>>>
>>> MW> Lewis would disagree with you. He claims that are real
>>>> and not imaginary, hence modal realism.
>>>
>>> Lewis was using the term 'realism' in contrast with 'nominalism'.
>>> A strict nominalist, for example, would say that the so-called
>>> laws of nature are merely summaries of observations. But realists
>>> would say that any law that has been sufficiently tested, such as
>>> the Law of Gravity, is based on something real -- i.e., the law
>>> is an indication that there exists some *real* mechanism that is
>>> responsible for the observations.
>>>
>>> When people like Lewis say that there are such things as "real
>>> possibilities", they use examples such as the following:
>>>
>>> If I hold a pencil in my hand, I predict, with a degree of
>>> certainty that I would back with a wager of any amount you
>>> care to bet, that if I let go of the pencil, it will fall
>>> to the floor.
>>>
>>> Many people who are realists about physical laws would say that
>>> the possibility that the pencil would fall to the floor is real.
>>> In short, it is a "real possibility". I am willing to accept
>>> that terminology.
>>>
>>> However, saying that there are real possibilities does not imply
>>> that those possibilities are actualized. When I drop that pencil,
>>> the "mere possibility" becomes an actuality. Until then, its
>>> mode of existence is not the same as what non-philosophers would
>>> call "real". Despite my willingness to accept the term 'real
>>> possibility', I would still call an unactualized possibility
>>> "imaginary".
>>>
>>> If you prefer a different word, please suggest one. But it's
>>> important to recognize that real, but unactualized possibilities
>>> are not "really real" in the sense that ordinary people use.
>>
>> I fail to see what this discussion has to do with sets, however.
>>
>>>
>>> JFS>> Therefore, any use of set theory for representing physical
>>>>> entities must recognize that a set at one instant of time is not
>>>>> "exactly the same" set of supposedly "exactly the same" things
>>>>> at any other instant.
>>>
>>> MW> However, 4D neatly sidesteps this with states/stages/temporal
>>>> parts. The temporal part is unchanging and always a member of
>>>> the sets it is a member of. This is why 4D and set theory fit
>>>> well together.
>>>
>>> I'm sorry, but that is not a solution. Note that the person I
>>> quoted about the difference between the baby in the cradle and
>>> the grown man was Alfred North Whitehead. He had also developed
>>> one of the most detailed and elaborate 4D ontologies. In that
>>> quotation, he just happened to use a short illustration, but
>>> the point he was making is completely independent of 3D vs 4D
>>> ontologies. In fact, time is irrelevant to his argument.
>>
>> Not to the example he gave in the cited quotation. Do you (or ANW)
>> have any other examples?
>>
>>>
>>> For the record, I'll repeat the quotation before the example
>>> of the baby in the cradle:
>>>
>>> ANW> In logical reasoning, which proceeds by use of the variable,
>>>> there are always two tacit presuppositions -- one is that the
>>>> definite symbols of composition can retain the same meaning as the
>>>> reasoning elaborates novel compositions. The other presupposition
>>>> is that this self-identity of each variable can be preserved when
>> the
>>>> variable is replaced by some definite instance.
>>>
>>> The fundamental distinction is between mathematical entities and
>>> physical entities:
>>>
>>> 1. The great developments in 20th century logic and set theory
>>> were applied to purely mathematical topics.
>>
>> John, you keep asserting this, and it is completely false. I have
>> given you citations from everyone from Frege through Russell to
>> Quine,
>> taking in Church, Carnap, Tarski and others along the way. They all
>> used as primary examples, sets of non-mathematical things. None of
>> the
>> published set theories make any stipulation that the entities in the
>> sets are 'mathematical' in nature.
>>
>>> For those subjects,
>>> there is no need for epistemology or scientific methodology to
>>> determine whether two variables refer to "the same thing".
>>>
>>> 2. But things in the real world (and I mean "really real") are
>>> messy, and determining identity is nontrivial.
>>
>> The edge cases can be troublesome, but I don't agree that the problem
>> is a dire as you always claim it to be. Take persons, for example.
>> There really is almost never any doubt about identity of persons.
>> There are of course times where it is very hard to determine identity
>> in a particular case, because the necessary information is
>> unavailable, but the *actual criteria for identity* are never in
>> dispute. But this point about lack of information applies also to
>> 'mathematical' entities. What is the smallest number that can be
>> expressed as the sum of three perfect squares in three distinct ways?
>> I know that there must be one, but I have no idea how to find out
>> what
>> it is.
>>
>>> Just note
>>> the difficulties in criminal investigations and court trials.
>>> The same kinds of complexities arise in scientific experiments
>>> and engineering measurements.
>>>
>>> I agree that a 4D view is conceptually cleaner than a 3D view
>>> for many of the theoretical discussions. But it's essential
>>> to note that the only direct knowledge we have is of the
>>> immediate present. The past is known only from memory or
>>> from records whose reliability has to be determined. And
>>> the future is totally unknown.
>>>
>>> When dealing with the real world, you can't do ontology
>>> without getting into all the problems of epistemology.
>>
>> That is completely wrong, and a non-sequiteur. Not only CAN you do
>> ontology without getting into epistemology, you MUST do so. WIthout
>> stating what is is that you have knowledge about, the knowledge is
>> meaningless.
>>
>>>
>>> PH>> There are some cases that have been discussed in (at least)
>>>>> moral philosophy, involving identical twins, which at a very
>>>>> early stage of fetal development, in some cases, were one
>>>>> blastula. If life begins at conception, this is a difficulty.
>>>
>>> MW> What I like about 4D is that it allows this to be explained
>>> without
>>>> giving rules that forbid it as a matter of principle, but which
>>>> also
>>>> does not make or force a choice about which twin, or either or
>>>> neither or both split from which. It just allows the situation to
>>>> be described.
>>>
>>> No. The entropy gradient defines a time arrow through a 4D extent
>>> that
>>> clearly identifies before and after.
>>
>> Even if true, so what? That does not bear on Matthew's point.
>>
>> Pat
>>
>>> Theologians since Thomas Aquinas
>>> talked about 4D views, and they don't let you off the hook.
>>>
>>> John
>>>
>>>
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