At 03:21 01/07/04 -0400, Kory Heath wrote:
At 03:25 PM 6/30/2004, CMR wrote (quoting www.fact-index.com):
"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 extension).
Ruth Barcan Marcus wrote a book on that Quantifier-in-modal-logic/essentialism question. See
http://www.fordham.edu/gsas/phil/klima/ESSENCE.HTM for a nice link with references.
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?
Bruno
