Re: Mathematical Logic, Podnieks'page ...
At 10:14 01/07/04 -0400, Hal Ruhl wrote: Re the discussion on mathematical realism etc. I ask for comments on whether or not definition that is the division of ALL in to two parts is a mathematical process. To me definition seems arbitrary but some definitions result in mathematical concepts such as the one I use which results in the concepts of incompleteness and inconsistency From this I can infer you are not following classical or more general standard logic where inconsistent theories are trivially complete in the sense that *all* propositions are provable (all the true one + all the false one!). This explains probably why it is hard to me to follow your post. I suggested to you (some years ago) to follow simpler paths, for pedagogical reasons. I read your posts but I have not yet a clue of what are your more primitive beliefs. You over-use (imo) analogies, which can be inspiring for some constructive path, but you don't seem to be able to realize the lack of clarity of your most interesting posts in that regards. I respect your willingness to try, and I hope my frankness will not discourage you. Bruno http://iridia.ulb.ac.be/~marchal/
Re: Mathematical Logic, Podnieks'page ...
Hi Bruno: The idea of my model is that the foundation system has two components one is inconsistent because it is complete - it contains all - and the other is incomplete - it is empty of all. These two components can not join but the incomplete one must attempt to do so - leading to the creation of metaverses. Hal At 10:36 AM 7/2/2004, you wrote: At 10:14 01/07/04 -0400, Hal Ruhl wrote: Re the discussion on mathematical realism etc. I ask for comments on whether or not definition that is the division of ALL in to two parts is a mathematical process. To me definition seems arbitrary but some definitions result in mathematical concepts such as the one I use which results in the concepts of incompleteness and inconsistency From this I can infer you are not following classical or more general standard logic where inconsistent theories are trivially complete in the sense that *all* propositions are provable (all the true one + all the false one!). This explains probably why it is hard to me to follow your post. I suggested to you (some years ago) to follow simpler paths, for pedagogical reasons. I read your posts but I have not yet a clue of what are your more primitive beliefs. You over-use (imo) analogies, which can be inspiring for some constructive path, but you don't seem to be able to realize the lack of clarity of your most interesting posts in that regards. I respect your willingness to try, and I hope my frankness will not discourage you. Bruno http://iridia.ulb.ac.be/~marchal/
Re: Mathematical Logic, Podnieks'page ...
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
Re: Mathematical Logic, Podnieks'page ...
Hi Bruno: By the way if some systems are complete and inconsistent will arithmetic be one of them? As I understand it there are no perfect fundamental theories. So if arithmetic ever becomes complete then it will be inconsistent. In the foundation system which I believe contains mathematics from the beginning arithmetic is complete so its inconsistent. Hal
Re: Mathematical Logic, Podnieks'page ...
Hi Hal, At 12:44 02/07/04 -0400, Hal Ruhl wrote: By the way if some systems are complete and inconsistent will arithmetic be one of them? As I understand it there are no perfect fundamental theories. So if arithmetic ever becomes complete then it will be inconsistent. Yes, if by arithmetic you mean an axiomatic system, or a formal theory, or a machine. No if by arithmetic you mean a set so big that you cannot define it in any formal theory, like the set of all true arithmetical sentences. That set cannot be defined in Peano arithmetic for exemple. Some logician use the word theory in that generalized sense, but it is misleading. Now the set of true sentence of arithmetic is that large sense is obviously consistent gieven that it contains only the true proposition! (but you cannot defined it mechanically). In the foundation system which I believe contains mathematics from the beginning arithmetic is complete so its inconsistent. No, because if it is complete, it will not be a mechanical or formal system. Only a theory will be inconsistent if both complete and enough rich. Not a model. To borrow Boolos title, I would like to say I get the feeling this list is missing the key road: Logic, logic and logic BTW an excellent introduction to elementary logic is the penguin book by Wilfried Hodges : http://www.amazon.co.uk/exec/obidos/ASIN/0141003146/qid=1088787942/sr=1-2/ref=sr_1_26_2/026-1716457-4246007 Only the first sentence of the book is false. (will say more on that book later ...) Bruno http://iridia.ulb.ac.be/~marchal/
Re: Mathematical Logic, Podnieks'page ...
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. I see what you mean, but we need *some* way of referring to specific (although perhaps imprecise) ideas or beliefs. I might feel comfortable defining Platonic essentialism as the belief that there exists a world of essences in which (say) the Ideal Horse exists, and all physical horses are imperfect copies of it, because I don't think this group already has multiple conflicting definitions of the term Platonic essentialism. However, this group definitely does have multiple conflicting definitions of the generic term Platonism, and people usually just assume their own definition when they hear the term. So someone asks someone else if they're a Platonist, and that person ends up answering a totally different question. Hi-larity ensues! 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 I agree, and in fact, avoiding the essentialist debate is exactly what I'm trying to do. My point is that every time we use the term Plantonism simply to refer to arithmetical realism, we run the risk of starting an essentialist debate (or a constructivist debate) that we didn't intend, because for many other people Platonism implies essentialism, or non-constructivism. -- Kory
Re: [InfoPhysics] Re: Mathematical Logic, Podnieks'page ...
At 03:09 PM 7/1/2004, Jim Whitescarver wrote: Platonist reasoning is the antithesis of constructionism. Thanks for the clarification. In this short discussion I've seen at least three conflicting ways that people use the term Platonism: 1. Platonism == Mathematical Realism. 2. Platonism == The belief in Ideal Horses, which real horses only approximate. 3. Platonism == Non-constructivism. -- Kory
Re: Mathematical Logic, Podnieks'page ...
Greetings Bruno, 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? Would it not be more to the point to askwhether I believe in an "ideal" computer, the affirmation of which might be construed as an essentialist view? If in factall "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 computerremained structurally complex enough to be potentially universal (Wolfram hassuggested "a bucket of rustynails" is, for instance!?!) than it could (would?) eventually re-self-organize and start running a new "routine". Cheers CMR- insert gratuitous quotation that implies my profundity here -
Re: Mathematical Logic, Podnieks'page ...
Kory Heath wrote: Thanks for the clarification. In this short discussion I've seen at least three conflicting ways that people use the term Platonism: 1. Platonism == Mathematical Realism. 2. Platonism == The belief in Ideal Horses, which real horses only approximate. 3. Platonism == Non-constructivism. Roger Penrose uses the word mathematical Platonism to describe his philosophy of math, which is clearer in that it obviously does not require believing in such a beast as the Ideal Horse. 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. Jesse
Re: Mathematical Logic, Podnieks'page ...
Just so my friend Jim's comments to Kory will have some context: From: Jim Whitescarver [EMAIL PROTECTED] Subject: Re: Re: Mathematical Logic, Podnieks'page ... Yes Kory, one needs to be explicit about what they mean by Platonist. I try to be explicit, by Platonic thinking, logic or reasoning I mean: 1. Platonic logic: law of excluded middle, a proposition may be true or false, there is no third alternative. Proof by induction is not questioned. Logical systems are necessarily incomplete. 2. Platonic existence: that which exists need not be constructible, infinities may be invoked at will and are attributed actuality. Platonist reasoning is the antithesis of constructionism. In constructionism you can have a set of points equal distance from one point but the set of all such points is considered imaginary, not real. You may have irrational numbers but only those generated by the countable set of algorithms exist. Others are random and cannot be constructed by any algorithm and therefore cannot exist. Jim Kory Heath wrote: At 09:19 AM 6/30/2004, Bruno Marchal wrote: Also, you said that your are not platonist. Could you tell me how you understand the proposition that the number seventeen is prime. (I want just be sure I understand your own philosophical hypothesis). CMR - insert gratuitous quotation that implies my profundity here -
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
