Re: Mathematical Logic, Podnieks'page ...

From: Bruno Marchal <>
Date: Fri, 02 Jul 2004 18:45:59 +0200

At 03:21 01/07/04 -0400, Kory Heath wrote:

>At 03:25 PM 6/30/2004, CMR wrote (quoting

>>"Mathematical realism holds that mathematical entities exist independently
>>of the human mind. Thus humans do not invent mathematics, but rather
>>discover it, and any other intelligent beings in the universe would
>>presumably do the same. The term Platonism is used because such a view is
>>seen to parallel Plato's belief in a "heaven of ideas", an unchanging
>>ultimate reality that the everday world can only imperfectly approximate.
>This is a perfect example of what I'm complaining about. The quote implies
>that the term "Platonism" can be used as just another term for
>"mathematical realism", but then it immediately provides a definition that
>goes beyond simple mathematical realism. The belief that mathematical
>entities exist independently of the human mind - that humans discover
>mathematics rather than invent it - does not automatically entail the
>belief that there's a "heaven of ideas" containing (say) the Essence of
>Horseness which everyday horses only imperfectly approximate. These two
>ideas are logically distinct, and it seems sensible to call them by two
>different names. I prefer "mathematical realism" and "essentialism", or
>maybe "Platonic essentialism". I'd prefer not to use the term "Platonism"
>all by itself, but if I had to use it, I'd use it to refer to "Platonic
>essentialism", not "mathematical realism".

Perhaps you could say more on "Platonic essentialism", but I would have
attributed the beginning
of Essentialism to the Aristotle reading of Plato. Plato is too vague on
these question imo. Aristotle essentialism is much more clear especially
through the development of modal logic (Aristotle's invention). But it is a
complex problem which I find premature.
Quine criticized the use of quantifier in modal logic because, he argues,
this would reintroduce essentialism in the scientific field. Comp is
vaccinated in that respect because the modal logic G and G* have quantifier
entirely defined by their arithmetical interpretations, so that there is a
clear non essentialist view of them, and at the same time, it explains why
some form of essentialism is just inevitable once we listen to the (sound)
machine's point of view.
Note that in my these I have not use the Gq and Gq* (G and G* first order
Ruth Barcan Marcus wrote a book on that
Quantifier-in-modal-logic/essentialism question. See for a nice link with
Now I agree with you, let us avoid the use of the term "platonism" (only
mathematicians use it for (mathematical) realism. Note that I avoid it most
of the time, but I could defend it's use as well, giving that Pythagore and
Plato have appreciate it so much. With comp, note, there is a sense to say
that not only the "almost-one-horse lives" in Platonia, but all possible
apparently concrete one too.
But that is probably a good reason to avoid the terme "platonism" before
being sure everyone grasp that aspect of comp.
Sometimes I define an arithmetical realist as someone who believes in all
the the propositions of the form (A or not A) with A an arithmetical
proposition. That's enough for my use of the term. G. Boolos make a case
that there is no notion of "alternative world" without the use of the (A or
not A) exclude middle propositions. I have order his book "logic, logic
and logic" and don't know yet his argument, which I find a priori
astonishing giving that you can do (and people does that) intuitionistic
modal logic (that is manage a notion of possible world without the exclude
middle principle).
To finish, Kory, I would avoid the term "essentialist" giving that its
modern philosophical use is more precise than our admittedly rather
imprecise use of it. It is better not to use the word more precisely than
the way we are using them ....
This reminds me one of my favorite replies by Bruno in the (not so well
known) "Sylvie and Bruno" by Lewis Carroll. By memory:
There was a herd of sheeps near Bruno who was talking with the Professor
somewhere in the country, and Bruno said "oh, look there is about 1004
sheeps there in the field". The Professor told him that he should not say
"about 1004" but "about 1000" giving that "about" is in contradiction with
the precise use of "4". Bruno replied that he was absolutely sure about the
four, seeing them near here, and that he was using the "about" concerning
the use of "1000" giving that he could hardly be sure of that!
Since, I am used to call that error (suspected by the Professor in Bruno's
exclamation), the 1004 error:
It is the error consisting of using words in a way more precise than the
way you are using them.
Not all jargon are 1004 errors, but 1004 errors lead always in the limit
toward jargon.
Kory, I am not pretending that your are "jargoning" but I would like to
avoid the risk of pointing to the essentialist debate too early, especially
without the modal logical tools. But I will try to avoid "platonism", and
this should indeed make me and CMR more alike with respect to the ontology.
This is equivalent to say yes in the test for "platonism" given in the
Podnieks page.
CMR, do you believe that a running program (on an ideal computer) will
stop, or will not stop?

Received on Fri Jul 02 2004 - 12:42:48 PDT

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