Re: Mathematical Logic,
Dear Kory, an appeal to your open mind: in the question whether we discovered math or invented it..., many state that the first version is 'true'. Beside the fact that anybody's 'truth' is a first person decision, the fact that anything we may know (believe or find), is interpreted by the ways how our 'human' mind works - including comp and all kinds of computers, as we 'imagine' (interpret, even formulate) the thoughts. I find the above distinction illusorical. We may FIND math as existing 'before' we constructed it, or we may FIND math a most ingenious somersault of our thinking. To 'believe' that 17 is prime? of course, within the ways as we know (and formulate) the concept 'prime'. Axioms, conventions. With the ideas about 'quite' different universes why are we closed to the idea of 'quite' different mathematical thinking? We don't have to go to another universe: the Romans subtracted in their calendar (counting backwards from the 3 fixed dates in a month) like minus 1 = today, minus 2 = yesterday and so on. I wonder how would've done that Plato (before the invention of 0)? Our list-collegues think about math(s) in quite different concepts from the classic 'constructivist(?)' arithmetical equational thinking. how far can go a quite differently composed mind - maybe in an organizational thinking/observing system of a universe NOT based on space - time? What can be called 'mathematics'? (Theory(s) of Everything?) Vive le 'scientific agnosticism'! John Mikes (Bruno: am I still in your corner?) - Original Message - From: Kory Heath [EMAIL PROTECTED] To: [EMAIL PROTECTED] Sent: Friday, July 02, 2004 4:10 PM Subject: Re: Mathematical Logic, Podnieks'page ... 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. -- Kory
Re: Mathematical Logic, Podnieks'page ...
At 16:10 02/07/04 -0400, Kory Heath wrote: 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 http://iridia.ulb.ac.be/~marchal/
Re: Mathematical Logic, Podnieks'page ...
At 02:17 PM 7/2/2004, CMR wrote: Would it not be more to the point to ask whether I believe in an ideal computer No! It isn't more to the point. You may believe that all physical things are subject to entropy, and that therefore no physical computer could last forever, but you should still be able to talk about whether or not some program *would halt* if it were allowed to run forever. Look at the following program: 1: X = 1 2: X++ 3: if X 1 then halt 4: goto 2 This program would clearly never halt if it were run on an ideal computer - and we can recognize that fact even while believing in entropy and the physical impossibility of running a program forever, etc. So the question is, for every possible finite program, do you believe there's a fact of the matter about whether or not it would halt if we *were* able to run it forever? -- Kory
Re: Mathematical Logic, Podnieks'page ...
At 10:12 AM 7/3/2004, Bruno Marchal wrote: 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. Yes, but some confusions are so easy to avoid! Confusions will always appear in the middle of conversations, but I want them at least to be unexpected ones...! Anyway, I didn't mean to derail the conversation with my jargoning; I was just pointing out that whenever I see platonism in one of these conversations, I'm never sure what we're really talking about. -- Kory
