jrk@information-systems.east-anglia.ac.uk (Richard Kennaway CMP RA) (01/23/91)
In article <1991Jan21.130344.19720@sics.se> torkel@sics.se (Torkel Franzen) writes: >Anyway, the >"statistical" argument for the consistency of elementary arithmetic or >ZFC is a very poor one. Agreed. But (IMHO) it's the only one we've got. > On the other hand, there is nothing mysterious about the claim that >e.g. first order arithmetic is obviously consistent. There is no need >to speak poetically about intuition: the theory is obviously consistent >because its axioms are obviously true. The mere occurrence of the word >"true" in an observation is often, it seems, taken to imply that >the observation must be metaphysical or otherwise deplorable. To say >that the axioms are true, however, is to say no more than what is said >in asserting the recursion equations for addition and multiplication, >and the principle of mathematical induction. Now if somebody does in fact >find these assertions obscure or doubtful or false, he will have no reason to >believe that first order arithmetic is consistent. It is mere ritual >however, to speak of the consistency of arithmetic as dubious without >attempting to establish just what is obscure, false, or doubtful about >the axioms. If one does in fact have serious doubts concerning the >consistency of arithmetic, one must have serious doubts concerning the >validity of practically every theorem of mathematics. Dealing with this is wandering away from the original subject, and out of comp.ai.philosophy for that matter. But I'm not sure if I can cross-post (broken news software), so I won't attempt to move the thread to another group. To speak of the consistency of arithmetic as obvious is just as much a ritual (i.e. non-productive activity making no practical difference to anything). The axioms seem intuitive; no inconsistency has been found; they underpin "the validity of practically every theorem of mathematics"; so we might as well use them, pro tem. No-one needs to believe in them to use them. If you like, I have non-serious doubts. But they're serious enough to torpedo, for me, at its outset, the argument that we are more powerful than machines because we can intuit the truth of various statements, including the consistency of arithmetic. I do not regard myself as able to intuit the consistency of arithmetic any better than a machine fed with knowledge about the history of mathematics. As for finding the axioms obscure, doubtful, or false...well, I don't suppose I do. But here is an analogy. I am sitting on a chair, at a desk, on which is a computer. These facts seem to me as clear, obvious, and true as the axioms of arithmetic or of the various flavours of ZF (all of which can be true simultaneously, of different concepts of "set"). Yet when I ask myself how I create from my senses the notions of "this chair", "this desk", etc., I see that I have not the faintest idea. And neither does anyone else, it would seem, or it wouldn't be so difficult to make a robot that can pick parts out of a jumbled bucket. (Ah, comp.ai relevance!) More than once, I have had the experience - surely not unique to me - of seeing something, and then realising that it is something else entirely; thereafter seeing it as such. My fundamental uncertainty regarding the physical senses applies just as much to my awareness of my own thoughts. The axioms of arithmetic are only obvious to me as long as I don't think too closely about them. When I do, I cannot find any reason to assume that someone will not eventually prove from those axioms that 0=1, only reason enough not to be concerned about the possibility in practice. It only makes a difference in philosophical arguments like these - I continue to sit on "this chair", typing on "this keyboard", expounding "these thoughts". I just don't really believe in any of it. -- Richard Kennaway SYS, University of East Anglia, Norwich, U.K. Internet: jrk@sys.uea.ac.uk uucp: ...mcsun!ukc!uea-sys!jrk "As I was sitting in my chair I knew the seat just wasn't there Nor legs, nor back, but I just sat Ignoring little things like that."
torkel@sics.se (Torkel Franzen) (01/23/91)
In article <5794.9101222235@s4.sys.uea.ac.uk> jrk@information-systems. east-anglia.ac.uk (Richard Kennaway CMP RA) writes: >If you like, I have non-serious doubts. But they're serious enough to >torpedo, for me, at its outset, the argument that we are more powerful >than machines because we can intuit the truth of various statements, >including the consistency of arithmetic. What is this talk about 'intuiting truth'? It means nothing to me. My claim is that arithmetic is obviusly consistent because there is a trivial consistency proof for it: the axioms are true, the rules of inference are truth-preserving, hence all theorems are true, hence no contradiction is a theorem. Now I am not claiming that it is impossible to reject this proof. Intuitionistist will reject it because of the use of classical logic in elementary arithmetic; finitists will reject it because of its use of quantification over an infinite totality. So these people do in fact take the view that the axioms are obscure, doubtful, or false (at least coupled with the use of classical logic). What is mere ritual is to reject this particular proof while not raising any comparable hullabaloo about mathematical proofs in general. As I tried to emphasize: if the proof appears insufficiently mathematical, consider the fact that it is easily formalizable in elementary analysis. And if we say of one particular arithmetical theorem of elementary analysis that the only reason we have for believing it to be true is that no counterexample has been found (and thus that its proof proves nothing at all), we are indulging in a very peculiar ritual unless we go on to consider just which proofs of elementary analysis, if any, prove anything at all.
jrk@information-systems.east-anglia.ac.uk (Richard Kennaway CMP RA) (01/23/91)
In article <1991Jan23.045926.17528@sics.se> torkel@sics.se (Torkel Franzen) writes: > What is this talk about 'intuiting truth'? Talk like this: > the axioms are true Why do you believe the axioms are true? Formalising the proof in elementary analysis, besides the objections you mention of certain schools of philosophy, won't tell you that arithmetic is consistent unless you believe that the axioms of elementary analysis are true. To get truth out of a mathematical argument, you - obviously - have to supply truth yourself at some point. > What is mere ritual is to reject this particular proof while not raising > any comparable hullabaloo about mathematical proofs in general. But I am raising a comparable hullabaloo - that is to say, hardly any at all. I just use mathematics, I don't concern myself with whether it's true. "Truth" in mathematics is not a metaphysical or deplorable concept, just an unnecessary one. -- Richard Kennaway SYS, University of East Anglia, Norwich, U.K. Internet: jrk@sys.uea.ac.uk uucp: ...mcsun!ukc!uea-sys!jrk
torkel@sics.se (Torkel Franzen) (01/24/91)
In article <7129.9101231301@s4.sys.uea.ac.uk> jrk@information-systems.east- anglia.ac.uk (Richard Kennaway CMP RA) writes: >But I am raising a comparable hullabaloo - that is to say, hardly any at >all. I just use mathematics, I don't concern myself with whether it's true. >"Truth" in mathematics is not a metaphysical or deplorable concept, just an >unnecessary one. This doesn't even begin to touch on the question I raised. You were saying, of a particular theorem of elementary analysis of the form "every natural number has property P", with P a primitive recursive predicate, that we have no reason whatever to believe that every natural number has property P, except that no counterexample has been found. So my question is, would you extend this to every theorem of analysis? If it were proved in elementary analysis that the equation x^n+y^n=z^n has no non-trivial solution for n>2, would you proclaim to the net that we still have no reason whatever to believe that this is so, except that no counterexample has been found? I rather suspect not. And as long as you prefer to ignore this issue, you have said nothing of consequence regarding consistency.
jrk@information-systems.east-anglia.ac.uk (Richard Kennaway CMP RA) (01/25/91)
In article <1991Jan23.180914.8116@sics.se> torkel@sics.se (Torkel Franzen) writes: > This doesn't even begin to touch on the question I raised. You were >saying, of a particular theorem of elementary analysis of the form >"every natural number has property P", with P a primitive recursive >predicate, that we have no reason whatever to believe that every >natural number has property P, except that no counterexample has been >found. So my question is, would you extend this to every theorem of >analysis? If it were proved in elementary analysis that the equation >x^n+y^n=z^n has no non-trivial solution for n>2, would you proclaim to >the net that we still have no reason whatever to believe that this is >so, except that no counterexample has been found? I rather suspect >not. You suspect rightly. I fail to see the relevance of the fact that both FLT and the consistency of arithmetic can be expressed as statements of the given form. You might equally well have imputed to me the claim that every long-undecided conjecture, of whatever syntactic form, may be accepted as true. The difference between "arithmetic is consistent" and FLT is that there has been vastly greater opportunity for an inconsistency to show up in arithmetic than a counterexample to FLT. The effort expended on the latter is a tiny fraction of that expended, not necessarily on the metamathematical study of arithmetic itself, but on ordinary, everyday mathematics that uses it and can be formalised in it. On the whole, I suspect that arithmetic is consistent; I have no opinion regarding FLT. But looking back on this thread, I believe you have misinterpreted the thrust of my comments on consistency. I was primarily rejecting the argument to consistency from intuition: "the axioms are obviously true and the inference rules obviously truth-preserving". I meant only to remark in passing that to justify using arithmetic, the fact that it has worked so far is enough reason for not worrying. Weak as it might be, I see is no other. I also notice that you have generally refrained from expressing personal opinions about the consistency of various systems. You may feel that personal tastes are not relevant to the discussion; but I would be interested in knowing what they are anyway. Do you have opinions on the consistency of arithmetic, ZFC, and ZFC+UM? If so, what are they, and on what grounds do you hold them? -- Richard Kennaway SYS, University of East Anglia, Norwich, U.K. Internet: jrk@sys.uea.ac.uk uucp: ...mcsun!ukc!uea-sys!jrk Relevance-to-newsgroup detector now registering approx. 0.0...
torkel@sics.se (Torkel Franzen) (01/25/91)
In article <11656.9101241836@s4.sys.uea.ac.uk> jrk@information-systems. east-anglia.ac.uk (Richard Kennaway CMP RA) writes: >You suspect rightly. I fail to see the relevance of the fact that both >FLT and the consistency of arithmetic can be expressed as statements of >the given form. Your remarks are unrelated to what I wrote. I didn't say that you should ascribe equal 'statistical' weight to FLT and the consistency of arithmetic. I was assuming that FLT turned out to be provable in elementary analysis. My question is, what proofs in elementary analysis, if any, do you regard as proving anything? If FLT turned out to be provable in analysis, would you still say that we have no reason for believing it to be true (i.e. that the equation x^n+y^n=z^n has no non-trivial solutions for n>2) except that no counterexample has been found - as you do in the case of the theorem "elementary arithmetic is consistent"? Perhaps I had better say this one more time? The 'hullabaloo' you raised in the case of one particular theorem of elementary analysis consisted in claiming that its proof proves nothing; that we still have no reason for believing its conclusion to be true (i.e. arithmetic to be consistent) except that no counterexample has been found. The question is, would you raise a corresponding hullabaloo in the case of any and all proofs of elementary analysis? And if not, what are the grounds for this distinction? Again, I am not saying that good answers to such questions can't be given. But no good answer - so I claim - will give any special status to those theorems of classical mathematics which have the form "T is consistent". Hence my initial remarks on the merely ritual character of much talk about consistency.
jrk@information-systems.east-anglia.ac.uk (Richard Kennaway CMP RA) (01/29/91)
In article <1991Jan25.075046.4464@sics.se> torkel@sics.se (Torkel Franzen) writes: >My question is, what proofs in elementary >analysis, if any, do you regard as proving anything? All of them - provided elementary analysis is consistent. >If FLT turned out >to be provable in analysis, would you still say that we have no reason >for believing it to be true (i.e. that the equation x^n+y^n=z^n has >no non-trivial solutions for n>2) except that no counterexample has >been found - as you do in the case of the theorem "elementary >arithmetic is consistent"? The reason - at least, my reason - for believing FLT under such circumstances is not the "statistical" evidence of no counterexample to FLT having been found, but of no counterexample to the consistency of elementary analysis having been found. ... >Hence my initial remarks on the merely ritual character of much talk about >consistency. Quite so. Consistency is no more worth remarking on than that the earth turns. One does not usually bother to preface a mathematical proof with "provided arithmetic (or analysis, ZF, etc.) is consistent". It begins to matter, when people start claiming (as in the discussion that this subthread arose from) that arithmetic "really" is true, rather than merely exhibiting a certain syntactic object (viz. a proof of consistency of arithmetic in elementary analysis). There is no problem with the latter. The former is - to me - unclear and unnecessary. -- Richard Kennaway SYS, University of East Anglia, Norwich, U.K. Internet: jrk@sys.uea.ac.uk uucp: ...mcsun!ukc!uea-sys!jrk
torkel@sics.se (Torkel Franzen) (01/29/91)
In article <17928.9101281842@s4.sys.uea.ac.uk> jrk@information-systems. east-anglia.ac.uk (Richard Kennaway CMP RA) writes: >>My question is, what proofs in elementary >>analysis, if any, do you regard as proving anything? >All of them - provided elementary analysis is consistent. First let me note that this only makes any obvious sense if what you mean is that you accept a proof of a statement A in analysis as proving "if analysis is consistent then A." Now in the case of FLT, it is indeed correct that "if analysis is consistent then FLT" follows from the existence of a proof in analysis of FLT. However, since we have only very poor statistical evidence for the consistency of analysis, on your view, we then have only very poor statistical evidence for any theorem of analysis, as you present the matter. This is a very radical doctrine and goes far beyond ordinary misgivings about consistency. For example, would you say that we have only very poor statistical evidence for our belief that every natural number is the sum of four squares? Or for the validity of Sturm's algorithm for finding the number of real zeros of a polynomial? Furthermore, consistent theories may well have false consequences, viz. consequences not of pi-zero-one form. For example, it is perfectly compatible with FLT being true that it is provable in elementary analysis that there is a counterexample to FLT, even if elementary analysis is consistent. Or, to take a common class of statements: theorems of the form "the algorithm R terminates for every input". The fact that such a theorem is provable in a consistent theory does not imply that it is true. And, in case you wonder about this point, to say that the theorem is true is to say that the algorithm R terminates for every input. >Quite so. Consistency is no more worth remarking on than that the earth >turns. One does not usually bother to preface a mathematical proof with >"provided arithmetic (or analysis, ZF, etc.) is consistent". I'm afraid these remarks of yours further illustrate the ritual character of much talk of consistency, since you apparently silently invoke consistency as a supposed justification of mathematical conclusions, without having taken the trouble to notice that it is insufficient to justify them, except in the case of pi-zero-one statements. >It begins >to matter, when people start claiming (as in the discussion that this >subthread arose from) that arithmetic "really" is true, rather than >merely exhibiting a certain syntactic object (viz. a proof of consistency >of arithmetic in elementary analysis). There is no problem with the >latter. The former is - to me - unclear and unnecessary. For some reason you are intent on trying to make something mysterious out of a perfectly ordinary mathematical use of the word "true". There's no need, you know, to put a "really" before it. If I say that the induction principle is true, what I am saying is simply this, that any set of natural numbers which contains 0 and is closed under the successor operation contains every natural number. If you then ask me, in portentous tones, whether I am really claiming that the induction principle is really true, I don't know what you're talking about. On the other hand I'd be interested to hear if you have any real justification for the idea that conclusions obtained by means of the induction principle have merely very poor, statistical support.
jrk@information-systems.east-anglia.ac.uk (Richard Kennaway CMP RA) (02/05/91)
In message <1991Feb3.205621.10828@sics.se> torkel@sics.se (Torkel Franzen) writes: >Previously you explicitly agreed that the >"statistical" argument for the consistency of elementary arithmetic is >a very poor one, and claimed we have no other. Are you now saying >that it is an excellent argument? Yes. I was earlier swayed by the force of your rhetoric into an unnecessary retreat. I claim (now) that the empirical argument is an excellent one, and that there is no other. (Of course, I might change my mind again in future. What I am claiming, or as you put it - portentously? - "in fact" claiming, is a matter of empirical judgement (for me as well as for you) of which the statements I just made are but the latest evidence available to you. Such is life. If you feel that the changeableness of my views makes them impossible to discuss, that's up to you.) In regard to the first part of my claim, you believe that the empirical evidence (please stop calling it "statistical") is poor. In regard to the second part, that whatever one thinks of the empirical evidence, there is no other, you never quite divulge your attitude, let alone justify it, as I remarked in my last message. You rightly objected to the changeableness of my own views. I am getting rather frustrated at the absence of yours on this matter. You have claimed that my views ascribe a "radical" uncertainty to mathematics which presumably you do not regard it as having. This claim is also in two parts. The second - that mathematics is not so uncertain as you argue that I am arguing it to be - you have only hinted at. So I ask you once more: are you claiming that there is better evidence than the empirical for the consistency of arithmetic? And if so, what is it? If it is "intuition", why do you regard it as reliable? If something else, what? You have asked me why I think there is no other evidence; I can only answer that I see none. Your turn. Lighten my darkness. > But what does a proof in a consistent system prove? I'm afraid this is >entirely unclear from your presentation. Are you saying that a proof in >a consistent theory T of a statement A proves "if T is consistent, then A"? >It doesn't, you know, unless A is a pi-zero-one statement. Can you clarify your question? If A is a theorem of T, then A is satisfied by every model of T. This is so even (trivially) for inconsistent T. If you can be clearer about what you mean by "if T is consistent, then A", then perhaps I can answer your question more clearly. -- Richard Kennaway SYS, University of East Anglia, Norwich, U.K. Internet: jrk@sys.uea.ac.uk uucp: ...mcsun!ukc!uea-sys!jrk
torkel@sics.se (Torkel Franzen) (02/06/91)
In article <4361.9102051326@s4.sys.uea.ac.uk> jrk@information-systems.east- anglia.ac.uk (Richard Kennaway CMP RA) writes: >If you feel that the changeableness of my views makes them impossible >to discuss, that's up to you. By no means, but it helps to know what views one is discussing at a given point. In these exchanges, it is customary to assume that what is claimed in message X+1 is a continuation of what is claimed in message X, unless there is an explicit explanation to the contrary in message X+1. If I could make no assumption whatever that your earlier messages can be taken in conjunction with the latest one, I would have to preface each of my contributions to this enlightening debate with a long statement of what I assume concerning the position I am arguing against. >So I ask you once more: are you claiming that there is better evidence than >the empirical for the consistency of arithmetic? Certainly; of the same kind as the evidence for theorems of ordinary mathematics in general - whatever may be the nature of that evidence. Allow me to recapitulate a bit. This discussion started with your remarking, not without a certain amount of finger-wagging, that you saw nothing in the notion of "intuiting" the consistency of arithmetic. "Dreams require interpretation; fiction requires facts; and mathematical intuition requires proofs." To this I responded that the consistency of arithmetic is as much or as little a matter for "intuition" as the validity of mathematical theorems in general. For, after all, that elementary arithmetic is consistent is a theorem of elementary analysis. In saying that it is *obviously* consistent I meant that there is an *obvious* consistency proof for arithmetic. That is, we don't need to drag in any fancy proof-theoretical stuff like Gentzen's proof. The trivial proof is mathematically less informative, but that doesn't render it any less a proof. And, you will recall, this was my main point in my first articles on the topic: why make a hullabaloo about consistency in particular? If we are unsatisfied with the mathematical evidence for the consistency of arithmetic, this can only serve as a starting point for wondering about what mathematical proofs in ordinary mathematics prove in general. Now my objections to your present standpoint concern two aspects. First, the idea that there is good empirical evidence for the consistency of arithmetic (and perhaps even for that of ZFC?), and second the idea that the consistency of (say) elementary analysis is sufficient to justify it, from the point of ordinary mathematics. As to empirical evidence, this is a rather delicate matter, since there can't be any clearcut refutation of claims of this kind. However, I'll explain why I reject the claim that there is good empirical evidence for the consistency of arithmetic. First note that "arithmetic is inconsistent" is a purely existential assertion. What it means is that it is *possible* to derive a contradiction using the formal rules of first order arithmetic. Here "possible" means "theoretically possible": it has nothing to do with what can in fact be achieved. If, say, the shortest derivation in elementary arithmetic of a contradiction is, when written out in primitive notation (using a standard X font, but a very large VDU), a billion light years long, there is no question of our producing even an extremely compressed version of this derivation. Rather, if we are to be able to realize that arithmetic is inconsistent under such circumstances, we must do so on the basis of general mathematical considerations, presumably of a considerable degree of abstraction. And the claim that we must be able to prove by such means that an inconsistent theory is inconsistent is not one whit more justifiable than the claim that we must be able to prove the consistency of any consistent theory. Now the only empirical evidence you have pointed to consists in the fact that people have done a lot of work in elementary arithmetic (or done work representable in elementary arithmetic) and have never come across any contradiction. By what ordinary empirical criteria is this good evidence? I am not aware of any such criterion. We have checked a pretty large (or, if you like, extremely small) finite subset of a potentially infinite set of derivations without finding a contradiction: that seems to be the sum of the supposed empirical evidence. For some reason you call this "the same sort of evidence we have that the sun will rise tomorrow." Since you make no attempt to suggest the existence of anything corresponding to physical theory in the case of the consistency statement, the only obvious point of your comparison seems to be an appeal to a general principle of the form "what has been, will be". In mathematics at least, I submit, this is a rotten principle. We might of course attempt to argue that it may reasonably (and perhaps "empirically"?) be assumed that a contradiction, if it exists, should be accessible in length rather than astronomical, and should have been discovered by now, since (for example) the axioms of arithmetic after all encompass only a few basic principles: the definitions of addition and multiplication, equality, the induction principle. This is where it is very easy to delude oneself into regarding evidence that is in fact based on our abstract understanding of the axioms as somehow empirical, in the sense that one need not appeal to any such abstract understanding. For the induction schema in elementary arithmetic is indeed very easily understood in its abstract formulation - "every set of (or property of) natural numbers which contains..." etc. This abstract understanding of the principle is quite sufficient to convince me that the theory is consistent. Given that the natural numbers are 0, s(0), s(s(0)), and so on, the induction principle is true of the natural numbers, whatever the nature of those numbers - Platonic objects or pure figments of our imagination. But suppose we reject such appeals to our "abstract understanding" and stay strictly within the realm of the supposedly objective and empirical. Then we must note the following: the induction axioms of elementary arithmetic have an extraordinary degree of complexity. We know that the proof-theoretical strength of the schema A(0) & (x)(A(x)->A(x+1)) -> (x)A(x) depends very sensitively on the allowable logical complexity of the formula A(x). We also know that the complexity of those A(x) that are actually used in ordinary arithmetical proofs is very low. In particular, most or all classical theorems of the form (x)p(x) with p(x) primitive recursive are provable in primitive recursive arithmetic, in which A(x) is constrained to be a primitive recursive predicate. And the consistency of *that* schema is easily provable in elementary arithmetic using a slightly more complicated condition A(x). In fact the empirical evidence for the consistency of the induction schema (in combination with the remaining axioms) only supports, at best, the consistency of a highly restricted version of the schema. When A(x) contains, say, 100 quantifier alterations followed by a primitive recursive relation, there simply isn't anything in ordinary mathematics that can be represented as testing the validity of the corresponding schema. So, to sum up: I claim that those who regard the consistency of arithmetic as well supported on empirical grounds haven't taken their own idea seriously. Now it is of course perfectly legitimate to ask: if the evidence for the consistency of arithmetic or analysis is not empirical, what is it? And does it exist? These are classical questions in the philosophy of mathematics. I don't believe any very good answers have been arrived at; I do believe that references to "empirical" evidence of the kind considered above are worthless. It would take us too far, and involve too much work, to go more deeply into this question, and I'll just assert here that we will get nowhere unless we consider the *actual* use, meaning, teaching, and understanding of mathematical statements and structures in and outside mathematics, rather than our preconceived (and almost inevitably simpleminded) ideas about what mathematical evidence must be like. >Can you clarify your question? If A is a theorem of T, then A is >satisfied by every model of T. This is so even (trivially) for >inconsistent T. If you can be clearer about what you mean by "if T is >consistent, then A", then perhaps I can answer your question more clearly. My question is simple enough. Suppose A is an ordinary mathematical statement. If (a formalized version of) A is proved in a formalized mathematical theory T, what does this tell us? What do we know when we know that A has been proved in T? The answer "we know that A is true in every model of T" is the model-theoretical version of the answer "we know that there is a formal derivation in T of A." If this is all there is to it, there is no other knowledge to be obtained from (formalized) mathematics than this purely formal information, which doesn't even depend on our associating any meaning with the statement A. However, obviously this can't be your view, if only because you said earlier that a proof in an inconsistent system proves nothing - now what do you mean by this? After all, as you yourself point out, a proof of A in a theory T proves that A is true in every model of T, whether or not T is consistent.
jrk@information-systems.east-anglia.ac.uk (Richard Kennaway CMP RA) (02/20/91)
This is a second attempt to post this. Apologies if it did get out the first time. I (Richard Kennaway) asked: >So I ask you once more: are you claiming that there is better evidence than >the empirical for the consistency of arithmetic? And you (Torkel Franzen) cagily replied: > Certainly; of the same kind as the evidence for theorems of >ordinary mathematics in general - whatever may be the nature of that >evidence. I also asked what such evidence might be, concerning which you said: > Now it is of course perfectly legitimate to ask: if the evidence for >the consistency of arithmetic or analysis is not empirical, what is >it? And does it exist? These are classical questions in the philosophy >of mathematics. I don't believe any very good answers have been >arrived at So you believe that (1) There is good evidence but (2) You don't know what it is. So much for the intuitive evidence of consistency. I turn to the empirical evidence, i.e. the fact that no inconsistency has yet turned up. Your rebuttal of the empirical argument, by means of the hierarchy of induction axioms, appears to be this: (1) The induction axiom can be split into a strict hierarchy of stronger and stronger induction axioms. (2) All existing mathematics that can be carried out in arithmetic, can be carried out using only the first few members of this hierarchy (plus the other axioms). (3) Therefore, the absence of contradictions in such mathematics is at best only evidence in favour of the consistency of those first few induction axioms. It is quite possible (leaving aside the mysterious other evidence, such as it might be) that those first few could be consistent, yet later ones inconsistent. There is a problem with this argument, a well-known and fundamental one: what is empirical evidence evidence of? Suppose that, by definition, all A's are B's, and I hypothesise that all B's are C's. If I test this hypothesis by observing some B's, and find that they are all C's, what grounds do I have for believing my hypothesis? If it happens that all the B's I looked at were A's, should I only consider the hypothesis that all A's are C's to be supported, the stronger one being in as much doubt as before? Suppose I only define A after having made the observations, and define it to be exactly the set of observed instances of B? Were one to argue so, one would be denying the possibility of ever supporting a generalisation by particular evidence. Yet that is the situation in your argument. A = the first few induction axioms in the hierarchy, B = the whole lot, C = does not lead to a contradiction. (Or alternatively: A = mathematical arguments which have been performed in arithmetic, B = all possible mathematical arguments which could be performed in arithmetic, C as before.) You see the observations as supporting at most the consistency of A; I take them as supporting the consistency of B. I can't give any satisfactory justification for my choice, any more than you can for the non-empirical evidence. I don't believe anyone else can. This is just the problem of induction (the comp.ai.philosophy relevance), and we are as unlikely to get an answer to it in this discussion as we are to get an answer concerning what other evidence there is for the consistency of arithmetic. [Footnote: not that anyone reading this should need to be told, but just in case: the "induction" referred to in "the problem of induction" has only coincidentally the same name as the "induction axioms" under discussion.] >Since you make no attempt to >suggest the existence of anything corresponding to physical theory in >the case of the consistency statement... Let me rectify this omission. What corresponds in mathematics, and specifically the hierarchy of induction axioms, to that background of physical theory that makes prediction of the sunrise more than just "what has been, will be"? The fact that those induction axioms are not an arbitrary collection, but seem to form a natural whole. Now, I am being very vague (because I cannot be more precise), and as a result this may seem an unutterably feeble argument. But no more feeble than the corresponding claim in the physical sciences. Successive sunrises are as logically independent of each other as are the higher induction axioms from the lower. The connection between sunrises is not in the world but in our theory. What grounds do we have for believing any of the predictions of that theory at all? Those grounds, whatever they might be (and I agree that a mere "what has been, will be" is no good), do we have for believing in the consistency of arithmetic. >...it is very easy to delude oneself into regarding >evidence that is in fact based on our abstract understanding of the >axioms as somehow empirical... If you like, you can call the reason for choosing the conjecture one takes the empirical evidence to support, "intuition" or "abstract understanding", but this doesn't solve the problem, it just gives it a name. (BTW, I suspect delusions in the reverse direction are just as easy.) We seem to have reached philosophical brick walls in each direction. Or perhaps we are on different sides of the same one. I believe (in the sense of today's .signature quote) that the empirical evidence supports the consistency of arithmetic, but cannot say why I conjecture that consistency, rather than that of some weaker theory. You believe there is non-empirical evidence, but cannot say what it is. Can we leave it there? -- Richard Kennaway SYS, University of East Anglia, Norwich, U.K. Internet: jrk@sys.uea.ac.uk uucp: ...mcsun!ukc!uea-sys!jrk "Believe", vb.: Doubt, as in "I believe it's going to rain".
torkel@sics.se (Torkel Franzen) (02/21/91)
In article <6270.9102172336@s4.sys.uea.ac.uk> jrk@information-systems. east-anglia.ac.uk (Richard Kennaway CMP RA) writes: >I can't give any satisfactory justification for my choice, any more than >you can for the non-empirical evidence. Apparently you're saying that the fact that no inconsistency has yet been derived from the axioms actually used so far in mathematics is good evidence that no inconsistency can be derived from the full set of axioms. Since you give no explanation of how this differs from the step from "hypothesis P has been verified for all natural numbers checked so far" to "hypothesis P is true of all numbers" - an argument which we have agreed is even "empirically" poor - and, moreover, say that you have no justification for your claim, there's little I can say about this. You do make some general remarks about generalizations and support, but they're completely pointless in view of the fact that they apply equally to any formal theories. Thus, there is nothing in your remarks on this point that would not equally support the idea that any arbitrary extension T of ZFC is consistent, given that our mathematical theorems are theorems of T. "You see the observations as supporting at most the consistency of A: I take them as supporting the consistency of T." You do also say that >What corresponds in mathematics, and >specifically the hierarchy of induction axioms, to that background of >physical theory that makes prediction of the sunrise more than just "what >has been, will be"? The fact that those induction axioms are not an >arbitrary collection, but seem to form a natural whole. but of course make no attempt to make this out to have anything to do with "empirical evidence". Anybody can claim that the axioms of any theory T seem to form a natural whole. Nothing in what you say is any less arbitrary than the most romantic blathering about intuition, to which you so sternly objected in your first messages on this topic. Your observation that successive sunrises are logically independent of each other again simply ignores the fact that we have a theoretical explanation of sunrises, while you propose no theoretical explanation of the fact that no contradiction has appeared in arithmetic. In short, the vacuous character of your present remarks makes it impossible for me to criticize them in any worthwhile way, and I don't think it's a very meaningful exercise to argue in detail that your remarks are vacuous. Essentially, you rely on the idea that any alternative to your views is equally arbitrary. But this is mere obscurantism. You have preferred to ignore my questions about what, if anything, is proved in mathematical proofs. If we reject the notion of mathematical evidence in favor of your supposed "empirical" evidence, we are left with the idea that the truth of ordinary theorems of mathematics (in so far as they are at all meaningful?) is a matter of their being generalizations that have stood the test of time - a particularly feeble form of mathematical "empiricism". To be sure, J.S.Mill resolutely and bravely argued that the truth of "11+6=17" etc is a matter of empirical experience; and perhaps you wish to follow in his footsteps. If so, there is a considerable body of thinking on this topic that you should take into account.
torkel@sics.se (Torkel Franzen) (02/22/91)
In article <25220.9102201346@s4.sys.uea.ac.uk> jrk@information-systems.east- anglia.ac.uk (Richard Kennaway CMP RA) writes: >This is a second attempt to post this. Apologies if it did get out >the first time. This is getting ridiculous! I only saw this article the first time it appeared thanks to the generous nntp policy of the University of Minnesota, and I've failed since in trying to post both an article and a cancel message. There's been a lot of difficulties lately at my end with getting/sending articles. We're having serious trouble keeping this war up, and maybe it's a sign from heaven that it's time to give this a rest. So I'll just make one more attempt to get out with my second and more temperately worded response to your message. You write: >I can't give any satisfactory justification for my choice, any more than >you can for the non-empirical evidence. I don't believe anyone else can. You seem to be saying that the fact that no inconsistency has yet been found in the axioms actually used by mathematicians is good evidence that no inconsistency exists in the full set of axioms. You don't claim to have any justification for this. You do have a couple of further comments. First, your remarks on generalizations and support. These are not helpful, since they make no distinction at all between theories. Take any theory T extending ZFC. Adapting your own words, we may argue that there is good empirical evidence that T is consistent, given that our mathematical theorems are provable in T. "You see the observations as supporting at most the consistency of PA; I take them as supporting the consistency of T." Your second observation consists in emphasizing that the induction axioms are not an arbitrary collection, but seem to form a natural whole. Here we must ask in what sense this impression is supposed to yield any "empirical evidence" for the consistency of the axioms? Also, for any set theory T there are people who regard its axioms as forming a natural whole. So, as far as the evidence for consistency is concerned, my conclusion is that your grounds for claiming that arithmetic is consistent are no less arbitrary, peculiar, or subjective, and no more empirical in any obvious sense, than just those appeals to intuition which you so sternly rejected in your original postings. You make much of my not presenting any theory of mathematical evidence, but this is surely a bit disingenuous, since you ignore the question of what, if anything, mathematical proofs achieve. As I have emphasized, there is nothing special about consistency theorems from the point of view of mathematical evidence. That 7+6=13, that addition is commutative, that every number is the sum of four squares, that a problem is NP-complete, that a problem is recursively undecidable, that a differential equation has a certain asymptotic behavior - in considering mathematical evidence we must consider what the evidence for such theorems of ordinary mathematics amounts to. All that emerges from your remarks is a determination to say that there is no evidence other than the (in some sense) empirical. There exists a considerable body of thought, not necessarily found in Usenet articles, that does a great deal better.
jls@yoda.Rational.COM (Jim Showalter) (02/23/91)
Of COURSE arithmetic is not consistent. All you need for proof of this
is to look at how the federal "budget" is calculated.
--
***** DISCLAIMER: The opinions expressed herein are my own. Duh. Like you'd
ever be able to find a company (or, for that matter, very many people) with
opinions like mine.
-- "When I want your opinion, I'll beat it out of you."jrk@information-systems.east-anglia.ac.uk (Richard Kennaway CMP RA) (02/24/91)
News glitches have tangled the thread somewhat. Recent messages are: 1. I sent a message dated 17 Feb. 2. I reposted (1) on 20 Feb, having wrongly suspected it didnt get out the first time. 3. In message <1991Feb20.215853.22149@sics.se>, Torkel Franzen replied to (1). 4. In a message of 21 Feb I replied to (3). 5. Finally, in message <1991Feb21.202449.22562@sics.se>, Torkel Franzen replied a second time to (1), suspecting that message (3) didnt get out. Notwithstanding your preference for a reply to (5) rather than (3), I must first comment on one more point from (3): "obscurantism". This is the pot calling the kettle black with a vengeance, coming from someone who believes that there is excellent evidence for the consistency of arithmetic, and not only cannot say what it is, but believes no-one else can either (to quote you more precisely: "I don't believe any very good answers have been arrived at"). I am reminded of a notorious non-solution to the theological problem of evil, which consists in saying that indeed God is good, but good in some mysterious sense that no-one understands. Now going back the question of the coherence of physical theory vs. that of mathematics. I am dissatisfied with my previous attempts to address this. As usual, if the following conflicts with my previously expressed views, it supersedes them. One has a reason for expecting the sun to rise again, other than that it has done so previously: its rise is also predicted on the basis of a great deal of other background beliefs about the world. However, if those background beliefs are brought into the foreground, and one asks why the entire body of belief relevant to sunrises should continue to apply, then one is in the same position as one is with respect to the consistency of the entire theory of arithmetic. There is no longer any separate background to provide extra support for a particular prediction. So why do I believe in the predictions of physical theory? I don't know. Why do I believe in the consistency of arithmetic? I don't know that either. Which brings me to the gist of message (5): >So, as far as the evidence for consistency is concerned, my conclusion >is that your grounds for claiming that arithmetic is consistent are no >less arbitrary, peculiar, or subjective, and no more empirical in any >obvious sense, than just those appeals to intuition which you so sternly >rejected in your original postings. As in my message (4), I agree, as far as that goes; however, I would claim at the same time that your grounds for expecting the next sunrise have the same character. If empirical evidence is, as you claim, worthless in mathematics, it is worthless in the physical sciences also. You can believe the axioms of arithmetic, and therefore believe in their consistency and the truth of arithmetic theorems; or you can apply various degrees of doubt to the various subtheories, which will carry over to their theorems. Likewise, you can believe in physical theory, and believe in its various predictions, such as that the sun will rise again; or be variously doubtful about its different parts and correspondingly doubtful about their predictions. Now when the theorems are found to be true in particular instances, and their study to have unearthed no inconsistency, what does this tell us about the mathematical theory or its subtheories? When the predictions are found to be satisfied in particular instances, what does this tell us about the physical theory or its subtheories? The two problems are, it seems to me, the same problem, the problem of what evidence is evidence of, and no good answer has been given to it. In practice, one must make decisions about how much of a physical or mathematical theory to believe (or at least to adopt as a working hypothesis and stop worrying about pro tem), without having any solution to the problem of how such decisions can be made. In such matters, people ascribe their choices to something they call "intuition", which only gives the problem a name, but does not solve it. And for me at least, such intuitive understanding of and confidence in a mathematical theory comes after and arises from working with it and seeing it work, not from contemplating its axioms and seeing that they are true. The latter provides me at best with prima facie evidence that the axioms may have succeeded in describing the perhaps vague concepts that they were intended to, and may be worth working with further - no more. >There exists a considerable body of thought, >not necessarily found in Usenet articles No kidding! :-) -- Richard Kennaway SYS, University of East Anglia, Norwich, U.K. Internet: jrk@sys.uea.ac.uk uucp: ...mcsun!ukc!uea-sys!jrk "But...surely you believe that tables and chairs exist? What's this, if it isn't a table?" he said, rapping on the table. "A working hypothesis," I replied. Then Vimalakirti's doorkeeper disappeared again.
torkel@sics.se (Torkel Franzen) (02/26/91)
In article <1355.9102231618@s4.sys.uea.ac.uk> jrk@information-systems. east-anglia.ac.uk (Richard Kennaway CMP RA) writes: >Notwithstanding your preference for a reply to (5) rather than (3), I >must first comment on one more point from (3): "obscurantism". >This is the pot calling the kettle black with a vengeance, coming from >someone who believes that there is excellent evidence for the consistency >of arithmetic, and not only cannot say what it is, but believes no-one >else can either. What I was talking about in referring to the lack of "good answers" was of course a philosophical or theoretical analysis of mathematical evidence. As for saying what the evidence for the consistency of arithmetic is, I thought I had done so: it's no different from the evidence for mathematical theorems in general: we have proved it, using valid axioms and principles of reasoning. So how do we know that the axioms are true and the principles of reasoning valid? This is where it is difficult to say anything very interesting - which is why these are classical questions in the philosophy of mathematics. As I put it, concerning the induction principle: given that the natural numbers are s(0), s(s(0)), etc, it follows that anything that is true of 0 and is true of s(x) whenever it is true of x is true of all natural numbers. This is not mysterious - it's just that there is nothing to say to make this more convincing than it already is. What should we call the evidence for this assertion? Reflection on the meaning of our concepts? Seeing into mathematical reality? Understanding the rules of language? Visualizing hypothetical entities? These alternatives, and others, have been pursued in the philosophy of mathematics. The induction principle itself is, however, far better understood and less problematic than any of the answers to the philosophical questions. Other basic mathematical principles, such as the axiom of choice or the existence of the power set of N are of course far less transparent, and in many cases the question of what we do or do not have evidence for is to a considerable extent a matter of subjective judgment. It sometimes appears that you assume that mathematical evidence, when invoked, must be thought apodictic, incontrovertible, unshakeable. This has been no part of my argument. I have claimed that your proposed empirical evidence for the truth of a particular mathematical theorem ("arithmetic is consistent") is worthless, and that there is such a thing as mathematical evidence. Constructivists and intuitionistists of various schools have a different idea of what is good mathematical evidence than I have, and as I pointed out in an earlier article, I have no quarrel with their views in this particular context. >If empirical evidence is, as you claim, worthless in mathematics, it is >worthless in the physical sciences also. What empirical evidence means in physics and the other sciences is a large question which I won't try to enter into. Of course the fact that the sun has risen so far is not good empirical evidence that it will rise forever - on the contrary, we believe that it will eventually rise no more. What I claim is that the empirical fact that no inconsistency has been found in arithmetic so far is worthless as evidence for the assertion that there exists no inconsistency in arithmetic. Since you presented no other empirical evidence for the consistency of arithmetic, I make no claims regarding empirical evidence in general - nothing having been said as to what would in general be meant by empirical evidence for the truth of mathematical theorems. Your remaining remarks seem to me to dilute the issues to a point where I see little to argue against. You admit to an "intuitive understanding of and confidence in a mathematical theory", but seem reluctant to say that this yields any kind of evidence for the truth of mathematical theorems. I find it hard to take this seriously. There is no need to suppose that this intuitive understanding and confidence results from some kind of pure contemplation, nor that it is infallible or unchangeable. But to declare (to return to the very special case of the consistency of arithmetic) that the ordinary mathematical evidence for the truth of a theorem - i.e. its proof - is no evidence at all, that is what I call obscurantism. So, to sum up. I am not concerned with arguing the philosophy or theory of mathematical evidence here. I do claim that the empirical evidence for the consistency of arithmetic (consisting in the empirical fact that no inconsistency has been deduced) is very poor, and with this you seem, this time around, to agree. (Or so it would appear, in view of your low opinion of appeals to "seeing", which you concede to be no worse than your appeal to the empirical.) I also claim that there is nothing special about consistency theorems when one considers questions of mathematical evidence. (This was earlier formulated in polemical terms as a rejection of the "rituals" surrounding talk of consistency.) This you don't seem to deny. And finally, I claim that it is obscurantism to pretend that there is no other evidence for the truth of mathematical theorems than the "empirical". Here you apparently disagree, but I don't see how to take the matter further with the material now at hand.
jrk@information-systems.east-anglia.ac.uk (Richard Kennaway CMP RA) (02/28/91)
In a message which will eventually appear, torkel@sics.se (Torkel Franzen) will write: >But to declare (to return to the very >special case of the consistency of arithmetic) that the ordinary mathematical >evidence for the truth of a theorem - i.e. its proof - is no evidence >at all, that is what I call obscurantism. I'm not saying it's no evidence at all, but that it is only evidence relative to the consistency of the system in which the proof is conducted. If one accepts that consistency, the proof is perfectly good evidence; to the extent that one doubts it, so is that evidence weakened. What is special about consistency theorems is that they are consistency theorems, and that they generally cannot be proved in systems weaker than those whereof they speak. As a result, a proof of the consistency of a system X conducted in a system Y containing X does not give one any reason to believe in that consistency that one did not have already. If one is more sceptical of Y than of X, it is worth even less than one's preexisting convictions, and is more interesting as a fact about Y than about X. The proof is worthless as evidence to support one's belief in the consistency of X. In contrast, if one does not know that, say, the four-squares theorem is a theorem, it will be unlikely to appear obvious, and its truth will be entirely uncertain; on reading a proof, one becomes as certain of its truth as one is of the truth of the system in which the proof is made. The fact that consistency theorems - some of them at least - can be seen as "just" statements of number theory does not alter this. Once one has read them as asserting consistency of system X, that reading vitiates the usefulness of proving them in system Y. >You admit to an "intuitive >understanding of and confidence in a mathematical theory", but seem >reluctant to say that this yields any kind of evidence for the truth >of mathematical theorems. I find it hard to take this seriously. I am indeed reluctant. But I suspect it is largely a matter of personal taste, with, as you say, little to argue over. I am far more impressed by the fact that arithmetic is still standing after all these years than by seeing the truth of the axioms. In message <1991Feb24.095043.18175@sics.se>, torkel@sics.se (Torkel Franzen) writes: [concerning Wette] > In its details, his proof is clearly something out of the ordinary. That >is, it is not the kind of thing that is done in ordinary mathematics. If >only for this reason, truly empirically-minded scientists should give it >close attention. I don't think I'll bite. From this and other things I've heard about his proofs, it sounds like something I would be fascinated to learn more about - but only if I don't have to lift a finger to find out. I'd rate his chances at about the same level as the people who claim to make antigravity machines with gyroscopes. -- Richard Kennaway SYS, University of East Anglia, Norwich, U.K. Internet: jrk@sys.uea.ac.uk uucp: ...mcsun!ukc!uea-sys!jrk "Gods these days, they want FTL travel, causality, and Lorentz invariance. I tell 'em to pick two out of three and call me back."
ag@sics.se (Anders G|ransson) (03/01/91)
Excuse me for butting in! To simplify this it seems to me that Richard essentially holds the position that trust in first order peano-arithmetic can be based only on the (empirical) observation that no contradiction has yet been observed. My point is simply that nobody uses first order peano-arithmetic for proving theorems of arithmetic. What _is_ used must be just about the same methods as Gauss used (whatever they were they were not first order Peano-arithmetic). A similar case would be to trust an engine, regardless of how it is constructed (how the blueprints look), because it has up till now worked. Torkel holds that something after all can be gotten out of the blueprint (with the aid of something like general mechanics). What I am suspcious of in Richards argument is that so to speak the engine he trusts in because it works has never been assigned to any real mission. Only the blueprint is in reality used. -- name(!): Anders G|ransson