At 02:45 PM 7/2/2004, Jesse Mazer wrote:
As for the non-constructivism definition, is it possible to be a non-constructivist but not a mathematical realist? If not then these aren't really separate definitions.
It may be that all non-constructivists are mathematical realists, but some constructivists are mathematical realists as well (by my definition of "mathematical realism"). So "Platonism == mathematical realism" and "Platonism == non-constructivism" are two different statements. I can imagine a non-constructivist asking "Are you a Platonist?" (thinking "Do you accept the law of excluded middle?"), and a constructivist answering "Yes." (thinking, "yes, valid constructive proofs are valid whether or not any human knows them or believes them.") This miscommunication will lead to confusion later in their conversation.
True, but if we want to make sure no confusion will ever appear later in the conversation we will never start. So it is better to tackle confusion when they appear. You will tell me that CMR and me were in such state of confusion. I am not so sure. Well, I don't know, and to be clear, using the less confusing _expression_, I will avoid "platonism" and use "arithmetical realism" instead.
Please pardon me CMR but I will quote your answer, so as to be able to answer you and illustrate my point to Kory at the same time (without sending different cross-referent posts).
CMR wrote:
Would it not be more to the point to ask whether I believe in an "ideal" computer, the affirmation of which might be construed as an essentialist view? If in fact all "things" are subject to entropy, including quantum objects (http://www.maths.nott.ac.uk/personal/vpb/research/ent_com.html), then would not any "hardware" eventually degrade to a "halt"? I suppose if the decrepit computer remained structurally complex enough to be potentially universal (Wolfram has suggested "a bucket of rusty nails" is, for instance !?!) than it could (would?) eventually re-self-organize and start running a new "routine".
BM: OK. Here I see you postulate physical realism. But I am more sure of the non existence of a highest prime than of entropy or quanta. I don not postulate physical realism, but I postulate arithmetical realism.
To come back on Kory, Kory wrote also:
KORY:
1. Platonism == Mathematical Realism.
2. Platonism == The belief in Ideal Horses, which "real" horses only approximate.
3. Platonism == Non-constructivism.
So I propose we choose 1. By Godel's theorem 1 implies 3 (even for the intuitionist (= those who discard the excuded middle principle), but "non-constructivism" will acquire a different meaning, and I never refer to it so let us forget it.
So to be clear and simple I will always use the terme platonism in the sense
of Classical Arithmetical Realism (Classical = Boolean = admission of all classical tautologies excluded middle principle included (if I can say).
For 2, I would say that comp does not entail it, unless you define the ideal horse by the set of its digital approximation done at some level. Obviously "ideal computer" exist, any definition of something capable to emulate any turing machine will make the job, from c++ to universal unitary transformation in a Hilbert space. With Church thesis we can say the existence of an ideal computer can be proved in and by Peano Arithmetic, like PA can prove the inexistence of two numbers p and q such that (p/q)^2 is 2.
With comp the existence of the ideal computer entails the *appearance* of many relatively concrete computer which seems to obey quantum entropic decay ...
... I could suspect CMR of physical platonism and perhaps physical essentialism ;)
I must go now (saturday course!), but I want to say something about physical essentialism,
and Aristotle substantialism ...
Bruno
