lew@ihlpa.UUCP (Lew Mammel, Jr.) (03/06/85)
I watched the title show last night and found it interesting mainly for the chance to see some famous mathematicians whose names I'd heard. In particular Raymond Smullyan, Paul Erdos, and Rene Thom, as well as a member of Bourbaki, whose name I didn't know. They didn't say anything about catastrophe theory, incidentally. They showed Smullyan talking to a high school class and I was surprised to hear him proclaim that most mathematicians were Platonists and believed that the Continuum hypothesis "either was or wasn't true". He compared it to the question of whether a bridge could carry a certain load. Either it would or it wouldn't, regardless of whether you had the tools to "decide" the case abstractly. I thought that mathematicians accepted that you could take your pick, but Smullyan put these in the minority. Is this a reflection of the "constructionist" schism? Lew Mammel, Jr. ihnp4!ihlpa!lew
cjh@petsd.UUCP (Chris Henrich) (03/07/85)
[] Lew Mammel asks some interesting questions, which I will paraphrase before giving my $.02 worth. 1. Are most mathematicians Platonists, in the sense of believing that mathematics is about some objective reality, not just participation in a game of manipulating symbols? I think so. Existentially, the answer is "yes, surely" - mathematicians do their thing as if it did have an objective meaning. In particular, though parts of mathematics can be formalized, i.e. described in terms of rules for manipulating symbols, the rules have not remained constant over the history of mathematics, and there have been lively debates over what are the "right" rules. These debates must appeal to some standard outside the rules themselves. The experience of doing mathematics certainly feels to me as if it is connected to an outside reality. There are too many puzzles and mysteries and surprises in math to account for on the premise that it is all subjective. When you try to solve a problem, especially on the frontier of research, you are wrestling with something that is very tough and subtle. Formality and formalisms are good techniques for doing this, but they are not the thing with which you are wrestling. Then, there is the relevance of mathematics to the natural sciences, what Wigner called "The Unreasonable Effectiveness" of mathematics. If mathematics were merely a game, like chess, then the first students of subatomic physics would have been no more likely to find mathematical phenomena inside the atom than to find chess pieces there. Finally, I think mathematical logicians are existential Platonists also. They study formalisms, to be sure. But they use mathematical methods (look at Goedel for instance) and regard their conclusions are "really" true. 2. What is the standing of a statement like the continuum Hypothesis, which has been shown to be "undecidable;" is it really either true or false, only we cannot get at the truth about it? The work of Goedel and Cohen showed that you can have a consistent set theory with or without this axiom. The situation is like that in geometry, where you can have a consistent theory with or without the parallel postulate. When mathematicians got used to this discovery, they found that non-euclidean geometries were as interesting and elegant as the Euclidean kind, and actually more useful for some applications. The consistency of set theory without the Continuum Hypothesis was discovered much more recently, and it is my impression that non-Cantorian set theories still seem exotic and "pathological" (i.e. you only see them if you go looking for trouble). But this is a matter of taste. 3. How does the question of the Continuum hypothesis relate to issues such as "constructivism?" I think they are loosely coupled. The "constructivist" position, at its strongest, is that a statement, that something with property X exists, ought not to be made unless a means of "constructing" such a thing can be given. Even if the contrary statement can be shown to result in a contradiction, the affirmative statement is not really true until the "construction" is provided. This is a minority position among mathematicians. It bears on set-theoretic questions, notably the Axiom of Choice, which asserts the existence of lots of things without making the slightest effort to construct them. Regards, Chris -- Full-Name: Christopher J. Henrich UUCP: ..!(cornell | ariel | ukc | houxz)!vax135!petsd!cjh US Mail: MS 313; Perkin-Elmer; 106 Apple St; Tinton Falls, NJ 07724 Phone: (201) 870-5853
mrh@cybvax0.UUCP (Mike Huybensz) (03/08/85)
I was really disapointed by this show. This was one of the worst Nova episodes I've ever seen. (Another two were the ESP and meteorite/extinction episodes.) I enjoyed seeing Smullyan, and really liked the presentation of the paradox of the set of all sets that do not contain themselves. (I still chuckle at the image of the librarian holding a big catalog and scratching his head trying to decide which stack of catalogs to put it in.) But the vast majoprity of the rest of the program made extremely little use of the visual qualities of television, too much use of talking heads, and taught virtually nothing about mathematics. It was more like a gossip show. Nova is essentially the only general-science show on the air, and sometimes is extremely good. But this one episode was disappointing. I wonder why AAAS hasn't started a general science television show yet. They've already wet their feet with Science '85 magazine, which is by far the best of the popular science magazines. There's alot of room for corporate sponsorship of general science programming. -- Mike Huybensz ...decvax!genrad!mit-eddie!cybvax0!mrh
lambert@boring.UUCP (03/10/85)
Lev Mammel expressed surprise at Smullyan's statement that most mathemati-
cians were Platonists and so would believe that the Continuum Hypothesis
(CH) is either true or false, rather than that you could take your pick.
Chris Henrich paraphrases `Platonism' as: `believing that mathematics is
about some objective reality, not just participation in a game of manipu-
lating symbols'.
There is something funny here. The schism in mathematics is one between
`classical mathematics' on the one hand, and intuitionism or constructivism
on the other hand. The latter two represent different schools, but are re-
lated in their criticism of classical mathematics. The protagonist of the
intuitionist school was Luitzen Brouwer, and his main target was David Hil-
bert, whom he attacked for his `Formalist' position, that is ... seeing
mathematics as a game of manipulating symbols. This would make Brouwer,
rather than Hilbert, the more Platonic of the two. But if mathematician A
chides mathematician B for being a Platonist, you can be sure A is an in-
tuitionist or constructivist. (For a proper understanding it is necessary
to take into account that the great Brouwer-Hilbert debate was on the prop-
er *foundations* for mathematics.)
Anyway, my experience is (i) that most mathematicians are indeed Platonists
in the sense that they believe that any proposition is either `objectively'
true or `objectively' false, independent of the existence of effective
methods of verifying or falsifying it, but also (ii) that they are not
really aware of the contents of existing criticism and are not prepared to
parry it. The discussion is complicated by the fact that no-one can define
the notion of `objective' here. However, intuitionists or constructivist
certainly do not believe you can just take your pick (except where it comes
to definitions, but that is universally agreed). This would be ill-advised
too, since the fact that we cannot currently prove or disprove some propo-
sition does not mean that we will not some day succeed in doing so.
The same must apply to classical mathematicians. Even though CH is in-
dependent of the axioms of ZF Set Theory, it is conceivable (although high-
ly implausible) that someone will some day come up with new methods of
mathematical reasoning that are *obviously* valid, using which CH can be
decided. The situation is different from the one concerning Euclid's Fifth
Postulate. None of Euclid's axioms is `true', obvious or not. There are
`geometries' in which there can be several lines through two given points
(e.g., great circles on a sphere). However, a `mathematics' in which both
a proposition and its negation can be true is unacceptable to classicists,
intuitionists and constructivists alike.
A good question to discuss Platonism in mathematics is the question whether
the cardinality of the continuum (c) is greater than that of the natural
numbers (Aleph0). This has some bearing on CH. The usual definition of
two sets having the same cardinality is the existence of a bijection. I
give a slightly different definition, that is equivalent to the original
one for classical mathematicians (who accept Zorn's Lemma or the Axiom of
Choice) but makes a difference to constructivists:
Define A <= B, for two sets A and B, to mean: there is an injective mapping
from A to B. This relation is reflexive and transitive. Define A ~ B to
mean A <= B & B <= A. This relation is an equivalence relation.
As an example, take A to be the set of Goedel numbers of never-halting Tur-
ing Machines (TM's) and B to be the set of natural numbers. Obviously, A
<= B. But we can, for each natural number n, construct a TM that outputs
(in some coding) n.000000... (i.e., a decimal representation of n followed
by an infinite sequence of zeros). So also B <= A, and therefore A ~ B.
However, we cannot effectively construct a bijection between A and B. For
B would then be recursively enumerable. Since the complement of B is also
recursively enumerable, this would then imply that the Halting Problem is
solvable.
Now consider the following `axiom': All real numbers are computable (mean-
ing: there exists a never-halting TM that outputs a decimal expansion).
Is this axiom false? There is one thing I am sure of: you cannot come up
with a counterexample that will satisfy a constructivist. For a construc-
tive definition of a purported counterexample real number can be turned
into the construction of a TM computing that number (or we would have
disproven Church's Thesis).
If one assumes the axiom to be true, then we have R~N, where R denotes the
set of real numbers, and N the set of integers. So c = Aleph0. The usual
diagonalization argument to show that c > Aleph0 does not work, because it
produces an uncomputable number. It shows, however, that the Halting Prob-
lem is unsolvable. (Assume it to be solvable. Enumerate all never-halting
TM's, let d[i] be the i-th digit output by the i-th machine, and construct
a TM whose i-th digit is d[i] mod 5. Let k be the number of this machine
in the enumeration. Then d[k] = d[k] mod 5. Contradiction.) So the con-
nection between the two arguments is deeper than the surface.
A classical mathematician will now say: Although the diagonal number is not
computable, to me it is an acceptable definition. But how can we interpret
this unless we assume the belief that the diagonal real number `exists' in
some sense. But what is this sense?
What the constructivist and classicist have in common is that they see that
to assume that one can enumerate all `acceptable' real numbers in an `ac-
ceptable' way leads to a contradiction.
Now it is not uncommon that some assumption leads to a contradiction.
Famous examples are the assumption of the existence of the set of all sets
(what is its cardinality?), or of the set of all sets that do not shave
themselves. (Interestingly enough, these contradictions are shown by diag-
onal arguments of again the same form.) The usual way taken out is to ex-
plain these counterexamples away, by saying, e.g., "the set of all sets
does not exist". It is not a priori clear why one cannot admit the set of
all sets. However, it leads to a contradiction, so let us put some fences
around the roads that lead to this pit. So there are fewer `acceptable'
sets than we would, naively, have thought. It would not have been strange
if, in history, mathematicians would also have put fences around the defin-
ition of enumeration, leading to fewer `acceptable' real numbers. (This
does, by the way, not automatically lead to the constructivist position.)
In particular, they could have taken the position: since the argument
showing that 2^n > n for each cardinal n leads to a contradiction if we
take n = Sjin, the cardinal of the set of all sets, there is something
wrong with the proof, and we must not admit unrestricted powersets.
This is not the way it happened. Instead, it was decided to be liberal
here and assume higher cardinals than Aleph0.
Before I can talk about the limit of a given sequence, or the smallest
number with a certain property, I must show that that limit or that number
exists. Not to require this leads to contradictions. For example: Let n
be the smallest number that is both equal to 0 and to 1. So n = 0 and n =
1. Therefore, 0 = 1. In general, mathematicians must show that the ob-
jects they define exist. So before we define Aleph1 as the smallest cardi-
nal exceeding Aleph0, we must at the very least show the existence of car-
dinals exceeding Aleph0. This requires showing the existence of the unres-
tricted powerset of at least one infinite set. But how is this done? Why
is just assuming its existence any more acceptable than just assuming the
existence of the set of all sets? Because one does not lead to a contrad-
iction, whereas the other one does? That is a rather weak justification,
even if one could rigorously prove that the assumption would not lead to a
contradiction. However, this has never been shown anyway, nor can it be
shown. One can certainly play an interesting formal game with the higher
cardinals. But is it more than a game? For example, can we take the
statement (by H.W. Lenstra) seriously that pure mathematics = not-yet-applied
mathematics? Is it conceivable that one day we might find an application
of Aleph1? That day I too will believe it exists:-)
--
Lambert Meertens
...!{seismo,philabs,decvax}!lambert@mcvax.UUCP
CWI (Centre for Mathematics and Computer Science), Amsterdamgjk@talcott.UUCP (Greg Kuperberg) (03/11/85)
> The same must apply to classical mathematicians. Even though CH is in- > dependent of the axioms of ZF Set Theory, it is conceivable (although high- > ly implausible) that someone will some day come up with new methods of > mathematical reasoning that are *obviously* valid, using which CH can be > decided. The situation is different from the one concerning Euclid's Fifth > Postulate. None of Euclid's axioms is `true', obvious or not. There are > `geometries' in which there can be several lines through two given points > (e.g., great circles on a sphere). However, a `mathematics' in which both > a proposition and its negation can be true is unacceptable to classicists, > intuitionists and constructivists alike. ... > Lambert Meertens Really? I had always thought that there were two "classes" of sets: in one class, the CH is true, and in the other it is false. That is, just like there are other geometries in which Euclid's fifth is false, there are "universes" in which CH is true and other "universes" in which it is false. If this duality is valid, how can one possibly come up with new methods by which CH can be decided? Or is this duality valid in the first place? --- Greg Kuperberg harvard!talcott!gjk "2*x^5-10*x+5=0 is not solvable by radicals." -Evariste Galois.
play@mcvax.UUCP (Andries Brouwer) (03/12/85)
In article <350@talcott.UUCP> gjk@talcott.UUCP (Greg Kuperberg) writes: >> The same must apply to classical mathematicians. Even though CH is in- >> dependent of the axioms of ZF Set Theory, it is conceivable (although high- >> ly implausible) that someone will some day come up with new methods of >> mathematical reasoning that are *obviously* valid, using which CH can be >> decided. The situation is different from the one concerning Euclid's Fifth >> Postulate. None of Euclid's axioms is `true', obvious or not. There are >> `geometries' in which there can be several lines through two given points >> (e.g., great circles on a sphere). However, a `mathematics' in which both >> a proposition and its negation can be true is unacceptable to classicists, >> intuitionists and constructivists alike. >... >> Lambert Meertens > >Really? I had always thought that there were two kinds of universe: >in one class, the CH is true, and in the other it is false. That is, just >like there are other geometries in which Euclid's fifth is false, there are >"universes" in which CH is true and other "universes" in which it is false. >If this duality is valid, how can one possibly come up with new methods by >which CH can be decided? Or is this duality valid in the first place? >--- > Greg Kuperberg In a mathematical theory you use axioms and inference rules to arrive at theorems. The axioms are usually made explicit; they are not considered obvious, but are assumed as a starting point. The rules of inference are seldom made explicit - it is assumed that mathematicians can recognise valid reasoning. What Lambert says (I think) is that it is conceivable (but unlikely) that somebody comes up with an obviously valid inference rule that would enable us to decide CH from the ZFC axioms. I do not agree, since the current inference rules already allow you to construct models in which CH holds and models in which it doesnt hold. Stronger inference rules deciding CH would thus lead to a contradiction. Something that is conceivable to me is that one might come up with new axioms for Set Theory that are *obviously* valid and would decide CH.
ndiamond@watdaisy.UUCP (Norman Diamond) (03/12/85)
> > ... it is conceivable (although highly implausible) that someone will some > > day come up with new methods of mathematical reasoning that are *obviously* > > valid, using which CH can be decided. > > -- Lambert Meertens > > Really? I had always thought that there were two "classes" of sets: in > one class, the CH is true, and in the other it is false. > ... Or is this duality valid in the first place? > -- Greg Kuperberg That duality is valid in ZF set theory. It is conceivable (although highly implausible) that someone will some day come up with a new set theory, that will be *obviously* valid, that will not be ZF set theory. It would have different axioms and a different set of provable theorems. Obviously 8-)----- most of the theorems will be the same as today's, but.... ^ | (that's sticking my neck out) -- Norman Diamond UUCP: {decvax|utzoo|ihnp4|allegra}!watmath!watdaisy!ndiamond CSNET: ndiamond%watdaisy@waterloo.csnet ARPA: ndiamond%watdaisy%waterloo.csnet@csnet-relay.arpa "Opinions are those of the keyboard, and do not reflect on me or higher-ups."
lambert@boring.UUCP (03/14/85)
>>> The same must apply to classical mathematicians. Even though CH is in- >>> dependent of the axioms of ZF Set Theory, it is conceivable (although high- >>> ly implausible) that someone will some day come up with new methods of >>> mathematical reasoning that are *obviously* valid, using which CH can be >>> decided. [...] >>> Lambert Meertens >> Really? I had always thought that there were two kinds of universe: >> in one class, the CH is true, and in the other it is false. [...] >> If this duality is valid, how can one possibly come up with new methods by >> which CH can be decided? Or is this duality valid in the first place? >> Greg Kuperberg > In a mathematical theory you use axioms and inference rules to arrive > at theorems. The axioms are usually made explicit; they are not considered > obvious, but are assumed as a starting point. The rules of inference are > seldom made explicit - it is assumed that mathematicians can recognise > valid reasoning. What Lambert says (I think) is that it is conceivable > (but unlikely) that somebody comes up with an obviously valid inference > rule that would enable us to decide CH from the ZFC axioms. > I do not agree, since the current inference rules already allow you to > construct models in which CH holds and models in which it doesnt hold. > Stronger inference rules deciding CH would thus lead to a contradiction. > Something that is conceivable to me is that one might come up with new > axioms for Set Theory that are *obviously* valid and would decide CH. > Andries Brouwer To me, there is an essential difference between (i) mathematical *reasoning*, in which propositions, natural numbers, sets, etc. are used as *tools*, and (ii) mathematical *theories*, in which abstract objects with certain properties are studied (using mathematical reasoning), such as algebra or set theory, or (mathematical formal) logic. In the latter, for instance, a mathematician uses mathematical reasoning, including propositions and inferences, to study properties of gottloboids (which are abstract objects intended to formalize mathematical theories and mathematical reasoning, and which are often called--somewhat confusing-- `theories'). In set theory, zermeloids are studied. These are supposed to formalize sets. But they cannot form the *foundation* for the use of sets in mathematical reasoning, if only since it is impossible to discuss the meaning of a theory without using the notion of `set'. To make the issue more complicated, set theory is formalized in one particular gottloboid, such as ZFC. For the same reason, it is impossible to found mathematics using gottloboids as a basis. If the axioms and inference rules of a gottloboid are correct, in the sense that they are valid for the part of mathematics the gottloboid is supposed to formalize, then we may translate the string of symbols obtained by applying the rules of the formal game into a mathematical proposition that we know to be valid: a theorem. This is such a common operation that we are usually not aware of the fact that there are two levels here, the `normal' mathematical level and a metamathematical level. In fact, the excursion to the metalevel is never necessary to reach a conclusion on the normal level, only possibly convenient. It is, of course, essential for metatheorems (such as Goedel's and Cohen's results). So I reject an approach in which sets are defined as `whatever objects satisfy the axioms of ZF Set Theory' as unsound and viciously circular. (Stronger, I have no reason to assume or believe that these zermeloids faithfully formalize sets, since I cannot think of a reasoning that makes the Axiom of Choice plausible, let alone of a justification for it--but that is beside the issue; assume for simplicity's sake that I believe in it.) Given any set of axioms, there may be other systems satisfying the axioms (called `models' in mathematical logic) than the particular one it was supposed to formalize. This means that some propositions may exist whose formal expression is formally undecidable from the set of formal axioms. But in no way does this imply that they are not possibly plain right or wrong. For example, for any acceptable gottloboid formalizing natural numbers, say FOL+PA, Goedel's Incompleteness Theorem can be translated into a formal proposition GIT that is formally undecidable, meaning that there exists a model for FOL+PA in which GIT is `false'. But the mathematical proposition formalized by GIT is true, valid, right, correct. (It is a red herring to call these models `universes' as though they all have equal status: there is the original system that we tried to formalize, and there are bogus, non-standard models that are an artefact of the inherent limitations of formalization.) Similarly, the formal undecidability of CH does not imply anything about the status of CH; it still might be plain right, or plain wrong. (In case you are interested in my position, I actually think that CH is devoid of meaning.) I do not see an essential difference between axioms and inference rules; an axiom is simply an inference rule with an empty set of antecedents. I think Andries is wrong when he says that adding inference rules to ZFC that make CH decidable would lead to a contradiction. In particular, if adding CH itself as an axiom leads to a contradiction, then one can formally infer `not CH'. (In case ZFC+CH is only omega-inconsistent, standard metamathematical reasoning would still lead to the conclusion that CH is false.) -- Lambert Meertens ...!{seismo,philabs,decvax}!lambert@mcvax.UUCP CWI (Centre for Mathematics and Computer Science), Amsterdam
play@mcvax.UUCP (Andries Brouwer) (03/14/85)
>I do not see an essential difference between axioms and inference rules; an >axiom is simply an inference rule with an empty set of antecedents. I >think Andries is wrong when he says that adding inference rules to ZFC that >make CH decidable would lead to a contradiction. In particular, if adding >CH itself as an axiom leads to a contradiction, then one can formally infer >`not CH'. (In case ZFC+CH is only omega-inconsistent, standard >metamathematical reasoning would still lead to the conclusion that CH >is false.) > > Lambert Meertens But I *do* see a difference between an axiom and a rule of inference. The former is a requirement on one particular situation; the latter (in the context of our discussion) formalizes a way of reasoning that is supposed to be universally valid.
gjk@talcott.UUCP (Greg Kuperberg) (03/16/85)
>> = Lambert Meertens > = Andries Brouwer >>I do not see an essential difference between axioms and inference rules; an >>axiom is simply an inference rule with an empty set of antecedents. > >But I *do* see a difference between an axiom and a rule of inference. >The former is a requirement on one particular situation; the latter >(in the context of our discussion) formalizes a way of reasoning >that is supposed to be universally valid. In the particular case of CH, I think that Lambert is right in saying that one could conceivably add new rules of inference that would make CH decidable without causing a contradiction. But about axioms versus inference rules in general I'm not so sure. My only objection is that inference rules are much more fundamental than axioms; if you changed De Morgan's law then the new rules might well be so radical as to not be useful. Certainly in the case of an unproved result, such as the Riemann Conjecture, the mathematical community can, and does, treat is as an axiom of sorts. (There are papers which start off with "Assuming the Riemann Conjecture..." just as there are papers which start off with "Assuming the Contiuum Hypothesis....") The reason is that if RC turns out to be true, then all these theorems will be very useful (of course, if RC is false, then they will be garbage, but that's another story). On the other hand, one does not have the privilege at the current time to change the rules of inference so that RC is decidable in an easy way. And if one did this, all of the "theorems" that would result would not necessarily be useful if RC were to be decided by conventional means. In short, at the current time most open problems turn out to be true or false, rather than unsolvable, so I don't see the utility in changing the rules of inference just yet. Of course, in the year 2150 say, there might be a "Goedelian crises", whereby there will be lots of new axioms and few new theorems. At that point we might see the light and switch to a different set of inference rules. --- Greg Kuperberg harvard!talcott!gjk "No Marxist can deny that the interests of socialism are higher than the interests of the right of nations to self-determination." -Lenin, 1918
sullivan@harvard.ARPA (John Sullivan) (03/16/85)
> Similarly, the formal > undecidability of CH does not imply anything about the status of CH; it > still might be plain right, or plain wrong. (In case you are interested in > my position, I actually think that CH is devoid of meaning.) > > Lambert Meertens CH might be plain right or plain wrong only if you think there are real objects out there which are uncountably infinite sets. The undecidability of CH says there are models for ZF+CH and ZF+~CH. It may turn out that as we increase our understanding we can come up with a new set of axioms (more "obvious" than CH) which will allow us to deduce either CH or ~CH (and, we hope, not both!). Then it would be plain right or plain wrong. But I think it is likely that this will never happen, since we don't get much real-world experience with infinite sets. Compare the situation with non-Euclidean geometry. Is the parallel postulate plain right or plain wrong? I don't think so. Even though we have much better intuition about geometry than about uncountable sets, I don't think that helps too much. Both systems have useful applications to the real world. John M. Sullivan sullivan@harvad
gwyn@brl-tgr.ARPA (Doug Gwyn <gwyn>) (03/17/85)
> Something that is conceivable to me is that one might come up with new > axioms for Set Theory that are *obviously* valid and would decide CH. So, what does category theory have to say about CH?
gjerawlins@watdaisy.UUCP (Gregory J.E. Rawlins) (03/18/85)
In article <6355@boring.UUCP> lambert@boring.UUCP (Lambert Meertens) writes: > [...much of the article deleted, hopefully I captured the main points..] >In set theory, zermeloids are studied. These are supposed to formalize sets. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ >But they cannot form the *foundation* for the use of sets >in mathematical reasoning, if only since it is impossible to discuss the >meaning of a theory without using the notion of `set'. > [.....] >So I reject an approach in which sets are defined as `whatever objects >satisfy the axioms of ZF Set Theory' as unsound and viciously circular. > [.....] >(It is a red herring to call these models `universes' as though they all >have equal status: there is the original system that we tried to formalize ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ >and there are bogus, non-standard models that are an artefact >of the inherent limitations of formalization.) Similarly, the formal >undecidability of CH does not imply anything about the status of CH; it >still might be plain right, or plain wrong. (In case you are interested in >my position, I actually think that CH is devoid of meaning.) With all due respect, the above argument just tells me that you are a Platonist (in the sense that you assume the a priori existence of objects whose essence we then try to *model* by formal axioms and inference rules). Some mathematicians are not; they are quite happy making up (formal) axioms and inference rules and churning out theorems. Personally I agree with you in that when I do mathematics *it is as if* the things (numbers, sets, circles, whatever) exist, but I can see no logical reason why they must exist. Your argument does not convince because you implicitly assume that such things as numbers do in fact exist, the fact that I happen to share that belief (as do many mathematicians) does not imply that it is a true statement. Logically I see no fallacy in having undecidable hypotheses, and I don't see that it need interfer with the process of doing mathematics. So I recommend that we get back to "discovering" things about the "real world", since it is impossible to refute an argument whose basis axioms are not purported to correspond to anything at all. Cheers. -- Gregory Rawlins CS Dept.,U.Waterloo,Waterloo,Ont.N2L3G1 (519)884-3852 gjerawlins%watdaisy@waterloo.csnet CSNET gjerawlins%watdaisy%waterloo.csnet@csnet-relay.arpa ARPA {allegra|clyde|linus|inhp4|decvax}!watmath!watdaisy!gjerawlins UUCP
palmq@sdcrdcf.UUCP (Paul.H. Palmquist) (03/19/85)
I also liked the 'Mathematical Mystery Tour'. The name of the member of Bourbaki is Jean A. Dieudonne who is retired and living in Nice, France. His works include Algebraic Topology and Functional Analysis. He was one of the hosts at the International Congress of Mathematicians held in France in 1970. I doubt that Smullyan is a 'Constructionist', as they are most noted for their doubts about analysis, the infinite processes. Smullyan probably reflects the majority of mathematicians, i.e., 'working mathematicians' who *use* mathematics without worrying about *foundations*. The 'constructionists' would dismiss most of the unsolved problems mentioned in the program as meaningless. In case you missed it: 'For a transcript of the program send $4.00 to NOVA Mathematical Mystery Tour Box 322 Boston, MA 02134 and be sure to mention the title.' Paul H. Palmquist (sdcrdcf!palmq)