ZFC etc. (was Re: Report on Philosophies of Physicists)
cj...@minster.york.ac.uk
but it does seem that some of the particpants might be able to help me
a couple of things straight in my own mind. Maybe the questions have
already been answered, and I just didn't see it...
1) We all know that ZFC is consistent (or maybe just those of us who
are sufficiently naive): I want to know why we have this confidence.
It just the case that lots of very intelligent people have failed
to find an inconsistency? Or are there informal arguments (as there
are for Church's thesis) to suggest consistency.
2) We know that the continuum hypothesis is independent of ZFC. Godel
provided the model (constructible sets) in which the CH is true, and
Cohen provided the model (via forcing) in which the CH is false.
Are the proofs by Cohen and Godel are formal proofs? (I would think
so.) Can we identify the formal system in which the proofs were
performed?
Thanks for any informed comment.
Felicitations -- Chris Ho-Stuart
(PS this is fall-out from the alt.atheism thread, which has evolved to
the point where sci.math is the better location)
Torkel Franzen
>We all know that ZFC is consistent (or maybe just those of us who
>are sufficiently naive): I want to know why we have this confidence.
>It just the case that lots of very intelligent people have failed
>to find an inconsistency? Or are there informal arguments (as there
>are for Church's thesis) to suggest consistency.
Some people suggest that that there are "statistical" reasons for taking
ZFC to be consistent. That is, no inconsistency has been found in spite
of the theory having been used a lot, so it is reasonable to think that
no inconsistency exists. In my opinion this argument is worthless.
Others hold that we have good grounds for believing the theory to be
consistent, namely that we know or "can imagine" that all the axioms are
true. I belong, more or less, to this category, but I would also want
to emphasize that this argument implies a lot more. Consistency, after all,
isn't all that interesting in itself. For example, the consistency of ZFC
implies that every theorem of ZFC of the form "the Diophantine equation ...
has no solution" is true, but it does not imply that every theorem of
ZFC of the form "the Diophantine equation ... has a solution" is true.
Taking the axioms of ZFC to be true or "possibly true" in the sense here
at issue implies, for one thing, taking every arithmetical consequence of
ZFC to be true.
Yet others are frankly dubious about even the consistency of ZFC. I don't
think there is any compelling argument to convince any of these people that
ZFC is consistent.
>Are the proofs by Cohen and Godel are formal proofs? (I would think
>so.) Can we identify the formal system in which the proofs were
>performed?
The proofs you are talking about are as formal as anything in mathematics.
On logical grounds, it is known that the statement "if ZFC is consistent,
CH is undecidable in ZFC" is provable by an elementary combinatorial proof.
John C. Baez
>Some people suggest that that there are "statistical" reasons for taking
>ZFC to be consistent. That is, no inconsistency has been found in spite
>of the theory having been used a lot, so it is reasonable to think that
In my opinion this argument is fine and is the only way we know anything
at all. (Ugh, now I'll have a pack of epistemologists, led by Torkel,
on my tail!) Just as the laws of physics seem to be true (but might not
be) since so far they seem always to have held, so too the consistency
of mathematics is an empirical fact. I claim that this holds, not only
for systems such as ZFC, but even systems such as the propositional
calculus for which there is a formal "proof" of consistency. The
propositional calculus might be inconsistent and we have somehow failed
to notice this so far; whatever subtle flaw it had would also afflict
the reasoning of the consistency proof, and when someone finally noticed
a certain gigantic proposition that could be proved both true and false,
we would see how one could construct a formal proof of the inconsistency
of the propositional calculus.
Torkel Franzen
(John C. Baez) writes:
>The propositional calculus might be inconsistent and we have somehow failed
>to notice this so far;
This is an observation of the same kind as "perhaps the fundamental
theorem of arithmetic is not really true", or "perhaps Euclid's
algorithm sometimes gives the wrong answer", and so on. Not very
exciting stuff.
Randall Holmes
>but it does seem that some of the particpants might be able to help me
>a couple of things straight in my own mind. Maybe the questions have
>already been answered, and I just didn't see it...
>
>1) We all know that ZFC is consistent (or maybe just those of us who
>are sufficiently naive): I want to know why we have this confidence.
There is an intuitive picture of how the universe of ZFC is
"constructed" by taking iterated power sets of a base set, which seems
reasonably convincing.
>
>It just the case that lots of very intelligent people have failed
>to find an inconsistency? Or are there informal arguments (as there
>are for Church's thesis) to suggest consistency.
See above. But the "intuition" remains informal. There are possible
objections, on the grounds of impredicativity, for instance.
>
>2) We know that the continuum hypothesis is independent of ZFC. Godel
>provided the model (constructible sets) in which the CH is true, and
>Cohen provided the model (via forcing) in which the CH is false.
>
>Are the proofs by Cohen and Godel are formal proofs? (I would think
>so.) Can we identify the formal system in which the proofs were
>performed?
They are rigorous arguments. No one ever does proofs in formal
systems, in practice. The Godel proof is a construction in ZF (normal
set theory without choice). The Cohen proof is a construction in ZFC
(Choice is used, I believe).
>
>Thanks for any informed comment.
>
>Felicitations -- Chris Ho-Stuart
>
>(PS this is fall-out from the alt.atheism thread, which has evolved to
>the point where sci.math is the better location)
--
The opinions expressed | --Sincerely,
above are not the "official" | M. Randall Holmes
opinions of any person | Math. Dept., Boise State Univ.
or institution. | hol...@opal.idbsu.edu
Michael Weiss
hol...@opal.idbsu.edu (Randall Holmes) answers the question:
>
>Are the proofs by Cohen and Godel are formal proofs? (I would think
>so.) Can we identify the formal system in which the proofs were
>performed?
thus:
They are rigorous arguments. No one ever does proofs in formal
systems, in practice. The Godel proof is a construction in ZF (normal
set theory without choice). The Cohen proof is a construction in ZFC
(Choice is used, I believe).
The relative consistency results of Godel and Cohen can be regarded as
purely combinatorial statements about pushing symbols around,
and as such can be expressed in the language of Peano arithmetic.
In principle, the proofs of these relative consistency statements could be
carried out in Peano arithmetic. Cohen discusses this briefly in the last
chapter of his book "Set Theory and the Continuum Hypothesis".
I believe the treatment in Shoenfield's book "Mathematical Logic" makes the
same point.
In practice, the proofs become far more intuitive if we adopt a less
puritan attitude, and talk about models of ZF. Cohen makes free use of
axiom SM (="there exists a model of ZF whose universe is a set and whose
element-of relation is the standard element-of relation".) Cohen proves
with ZF+V=L+SM that there is (for example) a model of ZFC + not CH.
Naturally this implies the relative consistency result, Con(ZF) -->
Con(ZFC+not CH). But if all you want is the relative consistency result,
then a much weaker set of axioms (such as Peano arithmetic) will do.
John C. Baez
I'm sorry not to have excited Torkel, but as the original questioner
was wondering whether the evidence for the consistency of ZFC was
only empirical, I thought this was worth noting, since some people
seem to think it is *impossible* that the propositional calculus
is inconsistent.
Randall Holmes
>In article <1992Sep15.1...@guinness.idbsu.edu>
>hol...@opal.idbsu.edu (Randall Holmes) answers the question:
> >
> >Are the proofs by Cohen and Godel are formal proofs? (I would think
> >so.) Can we identify the formal system in which the proofs were
> >performed?
>
>thus:
>
> They are rigorous arguments. No one ever does proofs in formal
> systems, in practice. The Godel proof is a construction in ZF (normal
> set theory without choice). The Cohen proof is a construction in ZFC
> (Choice is used, I believe).
>
>The relative consistency results of Godel and Cohen can be regarded as
>purely combinatorial statements about pushing symbols around,
>and as such can be expressed in the language of Peano arithmetic.
>
>In principle, the proofs of these relative consistency statements could be
>carried out in Peano arithmetic. Cohen discusses this briefly in the last
>chapter of his book "Set Theory and the Continuum Hypothesis".
>I believe the treatment in Shoenfield's book "Mathematical Logic" makes the
>same point.
I know that, and I thought of pointing it out, but I think it would be
deceptive to say that Godel and Cohen carried out their arguments in PA.
>
>In practice, the proofs become far more intuitive if we adopt a less
>puritan attitude, and talk about models of ZF. Cohen makes free use of
>axiom SM (="there exists a model of ZF whose universe is a set and whose
>element-of relation is the standard element-of relation".) Cohen proves
>with ZF+V=L+SM that there is (for example) a model of ZFC + not CH.
>Naturally this implies the relative consistency result, Con(ZF) -->
>Con(ZFC+not CH). But if all you want is the relative consistency result,
>then a much weaker set of axioms (such as Peano arithmetic) will do.
Thomas Clarke
Mathematical Truth which mathematicians seek to discover.
Mathematical Truth, however, like physics, is stranger than
we can ever know. Thus the varous mathematical theories that
we INVENT will always be mere shadows of Mathematical Truth
cast on the cave wall.
Thus, I don't see that the various discoveries/inventions
concerning the independence of ZF, C, CH etc eliminate
mathematical Platonism.
--
Thomas Clarke
Institute for Simulation and Training, University of Central FL
12424 Research Parkway, Suite 300, Orlando, FL 32826
(407)658-5030, FAX: (407)658-5059, cla...@acme.ucf.edu
Gerald Edgar
What is Platonism?
This book is a modern view (and survey of older views):
Realism in mathematics / Penelope Maddy. Oxford : Clarendon Press ; New
York : Oxford University Press, 1990. ix, 204 p. ; 22 cm.
Includes bibliographical references (p. [182]-197). Includes index.
SUB: 1. Mathematics--Philosophy
LC CARD #: 89-49346 TITLE #: 4704507 OCLC #: 20797041 &wq901211
--
Gerald A. Edgar Internet: ed...@mps.ohio-state.edu
Department of Mathematics Bitnet: EDGAR@OHSTPY
The Ohio State University telephone: 614-292-0395 (Office)
Columbus, OH 43210 -292-4975 (Math. Dept.) -292-1479 (Dept. Fax)
Randall Holmes
Like me. I can imagine ~Con(PA), though (although I certainly don't
believe it!).
Torkel Franzen
(John C. Baez) writes:
It's no more and no less impossible than any other elementary
mathematical theorem - for example, "addition of natural numbers
is commutative" - being false. When somebody advances the
reflection that maybe these theorems are false, I don't expect
anything very interesting to follow, since I can only imagine that
some kind of general philosophical skepticism lies behind it.
John C. Baez
Agreed. As you may have noted, NOTHING followed this remark of mine: I
just made it for the sake of those who hadn't thought of it, not
sophisticates such as yourself.
Torkel Franzen
(John C. Baez) writes:
>Agreed. As you may have noted, NOTHING followed this remark of mine: I
>just made it for the sake of those who hadn't thought of it, not
>sophisticates such as yourself.
But surely making a remark like this without anything following is
just a piece of arbitrary mystification? After all, those who think
that it is impossible that propositional logic should be inconsistent
are simply right in any ordinary terms. It is mathematically
impossible, which is about as impossible as things get. Saying that
propositional logic may after all be inconsistent, with a wise
owl-like look perhaps, to people who are not "sophisticates" and
then saying nothing, is to my mind slightly fraudulent, since some of
these people might get the impression that you have said something
more profound than "maybe there is a very large natural number that is
an even power of three".
John C. Baez
>impossible [for the propositional calculus to be inconsistent], which
>is about as impossible as things get.
Do you think it is mathematically impossible for ZFC to be
inconsistent? If not, please explain the difference in your
attitude towards prop. calc. and ZFC. It seems to me that there
is at best a difference of degree. Actually, I forget your views on
this.
>Saying that
>propositional logic may after all be inconsistent, with a wise
>owl-like look perhaps, to people who are not "sophisticates" and
>then saying nothing, is to my mind slightly fraudulent....
Henceforth I should include a GIF to make clear that my typical
expression while posting to usenet is not a wise owl-like look, but a
half-suppressed grin.
Torkel Franzen
(John C. Baez) writes:
>Do you think it is mathematically impossible for ZFC to be
>inconsistent? If not, please explain the difference in your
>attitude towards prop. calc. and ZFC. It seems to me that there
>is at best a difference of degree.
Well, the difference between being hit by a ton of bricks and by a
pigeon feather is a difference of degree. So with "mathematical
impossibility". I have no objection to your pointing out - whether
owl-like or grinning - to "non-sophisticates" that propositional logic
may be inconsistent, as long as you make it clear that this is on a
par with the reflection that there may be a very large even natural
number that is a power of three.
Vaughan R. Pratt
>In article <TORKEL.92S...@isis.sics.se> tor...@sics.se (Torkel Franzen) writes:
>>It is mathematically
>>impossible [for the propositional calculus to be inconsistent], which
>>is about as impossible as things get.
>
>Do you think it is mathematically impossible for ZFC to be
>inconsistent? If not, please explain the difference in your
>attitude towards prop. calc. and ZFC. It seems to me that there
>is at best a difference of degree. Actually, I forget your views on
>this.
====WIN $10,000==== (details at bottom)
Gosh, here I am siding with Torkel. Sounds inconsistent to me.
Some residual room for doubt about ZFC remains despite 70 years of
trouble-free mileage on it (like a Swedish car driven by Olive Oil or
whatever she's called in Sweden). Torkel may have some sort of problem
with such doubters, but I don't think we're all that rare, and many of
us are quite reasonable people.
One of the many characterizations of the propositional calculus is as
the equational theory of finite sets. With this characterization the
meaning of its inconsistency is that there exist no sets, not even the
empty set.
BUT anyone who worries that Z might have some problem is surely deep
into what Torkel sees as the looney fringe of philosophical
navel-contemplation. On the other hand, if it does prove inconsistent,
mathematics is by no means over. We would probably start by trying
less onerous notions of power set---I have some ideas of my own as to
elegant ways to proceed in this event, though I regard that outcome as
extremely unlikely.
But if propositional calculus collapses, Torkel is exactly right:
mathematics disappears. Once 0 = 1 all hope of either arithmetical or
even two-valued thought goes away, along with the ability to tell
whether you are dead or alive.
Let me make this more concrete as follows.
I will bet $100 in favor of ZF being found inconsistent within the next
20 years, with odds of 100:1 against. That is, the parties regard ZF
inconsistent as a rather unlikely outcome, though I'm not ruling it out
at those odds. If you win you get my $100, if you lose you owe me
$10,000.
I will bet $10,000 *against* Z being found inconsistent within the next
20 years, with odds of 1000:1 against. That is, I'm saying that Z
inconsistent is extremely unlikely, and if you think you can make money
at any crazy odds on this bet you're the one who's crazy. If you win
you get my $10,000, if I win I get only your $10.
These amounts are in US dollars for 2012. We agree that our respective
estates will honor the debt.
(I picked the amounts assuming roughly 5-fold inflation over that
period.)
Let me know if you wish to accept either bet. I will keep your name
private if requested, but not the number of people accepting each bet.
These are meaningful bets. It would be meaningless to propose a bet
about the propositional calculus. That can only become inconsistent
when all the lights go out.
--
======================================================| God found the positive
Vaughan Pratt pr...@cs.Stanford.EDU 415-494-2545 | integers, zero was
======================================================| there when He arrived.
Richard Zach
|> But if propositional calculus collapses, Torkel is exactly right:
|> mathematics disappears. Once 0 = 1 all hope of either arithmetical or
|> even two-valued thought goes away, along with the ability to tell
|> whether you are dead or alive.
What if the proof of the inconsistency of propositional
logic would not carry over to, say, intuitionistic
logic? We would then go on doing intuitionistic mathematics,
wouldn't we?
--
Richard Zach [za...@csdec1.tuwien.ac.at]
Technische Universitaet Wien, Institut fuer Computersprachen
Resselgasse 3/185.2, A-1040 Vienna, Austria/Europe
Torkel Franzen
(Richard Zach) writes:
>What if the proof of the inconsistency of propositional
>logic would not carry over to, say, intuitionistic
>logic? We would then go on doing intuitionistic mathematics,
>wouldn't we?
You seem to be assuming a situation in which we have proved classical
logic to be inconsistent, but have not proved intuitionistic logic to be
inconsistent. In that case, you suggest, we can still go on using
intuitionistic logic.
This reflection illustrates the unfortunate effect that talk of
"consistency" so often has on the thought processes of mathematicians.
The situation you describe doesn't make any sense, either classically
or intuitionistically. The consistency of classical propositional (or
for that matter predicate) logic is a very elementary and trivial
result in mathematics, and its proof is intuitionistically valid. So
it makes as much sense (classically or intuitionistically) to
speculate about a proof that classical logic is inconsistent as it
does to speculate about a proof that there are two (large) natural
numbers a and b such that a+b>b+a.
But perhaps I am being hasty. Your remarks make sense if what you are
assuming is that we manage to prove, in classical mathematics, that
classical propositional logic is inconsistent, although this proof is not
valid in intuitionistic mathematics. Then what you are assuming is that
classical mathematics turns out to be inconsistent, and this, to be sure,
is compatible with intuitionistic mathematics being consistent. But of
course in such a case it would still be a theorem of intuitionistic
mathematics that classical logic is consistent.
John C. Baez
>One of the many characterizations of the propositional calculus is as
>the equational theory of finite sets. With this characterization the
>meaning of its inconsistency is that there exist no sets, not even the
>empty set.
That would put lots of out of a job. :-)
>BUT anyone who worries that Z might have some problem is surely deep
>into what Torkel sees as the looney fringe of philosophical
>navel-contemplation.
I do my fair share of looney philosophical navel-contemplation, but
just in case anyone is about to call up the chair of my department
and report me, let me emphasize that I never "worry" that the
propositional calculus or ZFC is inconsistent.
>On the other hand, if it does prove inconsistent,
>mathematics is by no means over. We would probably start by trying
>less onerous notions of power set---I have some ideas of my own as to
>elegant ways to proceed in this event, though I regard that outcome as
>extremely unlikely.
>But if propositional calculus collapses, Torkel is exactly right:
>mathematics disappears. Once 0 = 1 all hope of either arithmetical or
>even two-valued thought goes away, along with the ability to tell
>whether you are dead or alive.
Imagine the following story: I know I'm alive. Then someone shows me an
enormous deduction in the propositional calculus (axiomatized in ones
favorite manner) which has as the final theorem P & not(P). I am
told that this deduction has been checked by computers many times
over, and I get a copy in electronic form and write my own program
and check it and, sure enough, it checks! Do I then somehow all of a
sudden stop knowing I'm alive?
Of course, you may rightly accuse me here of telling a tale based on the
most counterfactual of counterfactuals. Telling any story about "what
if we proved the propositional calculus is inconsistent" is indeed
very, very silly.
>I will bet $100 in favor of ZF being found inconsistent within the next
>20 years, with odds of 100:1 against. That is, the parties regard ZF
>inconsistent as a rather unlikely outcome, though I'm not ruling it out
>at those odds. If you win you get my $100, if you lose you owe me
>$10,000.
>I will bet $10,000 *against* Z being found inconsistent within the next
>20 years, with odds of 1000:1 against. That is, I'm saying that Z
>inconsistent is extremely unlikely, and if you think you can make money
>at any crazy odds on this bet you're the one who's crazy. If you win
>you get my $10,000, if I win I get only your $10.
>Let me know if you wish to accept either bet. I will keep your name
>private if requested, but not the number of people accepting each bet.
I would worry that you may have a lot of people taking you up on your
offer of $100. I won't drive you further into debt by taking you up
on that one.
>These are meaningful bets. It would be meaningless to propose a bet
>about the propositional calculus. That can only become inconsistent
>when all the lights go out.
I will bet anyone a quarter at even odds that *if* the propositional
calculus is shown to be inconsistent, I can find a very large integer
power of 3 that is even. 8-O (<--- wise owl face)
Vaughan R. Pratt
>pr...@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>
>>One of the many characterizations of the propositional calculus is as
>>the equational theory of finite sets. With this characterization the
>>meaning of its inconsistency is that there exist no sets, not even the
>>empty set.
>
Jesus, how did that get back in my message, I thought I deleted it back
when I had doubts about the argument. Luckily I believe the argument
once again (same proof I think, hard to read it in the dark :-).
>That would put lots of out of a job. :-)
Not at all, see the intuitionist's answer and my reply agreeing without
having to resort to logic ("only" category theory, which some may not
regard as math though I do).
>Imagine the following story: I know I'm alive. Then someone shows me an
>enormous deduction in the propositional calculus (axiomatized in ones
>favorite manner) which has as the final theorem P & not(P). I am
>told that this deduction has been checked by computers many times
>over, and I get a copy in electronic form and write my own program
>and check it and, sure enough, it checks! Do I then somehow all of a
>sudden stop knowing I'm alive?
As Torkel says, you should trust PC more than your program. All these
programs have bugs, at least if they're using US or Japanese logic
gates, which are Boolean. But if you use intuitionistic gates and
intuitionistic reasoning and you *still* have the inconsistency, then
you do begin to wonder about PC, hence about the existence of finite
sets (retreat to posets). (My guess is that Torkel can't possibly be
taking me seriously at this point: since he believes in the consistency
of ZFC this entails belief in the existence of much larger cardinals
than we're contemplating here. He would do extremely well in those
drama-class trust exercises where everyone forms a circle around you,
you close your eyes, stiffen your body, keep your feet on one spot on
the floor, and let people push you from person to person.)
>Of course, you may rightly accuse me here of telling a tale based on the
>most counterfactual of counterfactuals. Telling any story about "what
>if we proved the propositional calculus is inconsistent" is indeed
>very, very silly.
My attitude is that any ostensibly mathematical statement that isn't
obviously equivalent to denying the existence of all mathematics
deserves an analysis of its status as a statement *of* mathematics.
Any statement where the "obvious" answer does not have an equally
obvious proof indicates a potential bug in our understanding of
mathematics.
>I would worry that you may have a lot of people taking you up on your
>offer of $100. I won't drive you further into debt by taking you up
>on that one.
If I tell you there are no takers yet will that make you feel better
about taking advantage of me? Or are you really worried there's some
chance you'll lose $10,000? It's extremely unlikely, and anyway think
of inflation, think of it as just $2,000, you'll easily be able to
afford it then. And it will be worth it, you'll be famous as the first
(I hope not only) person in the world to lose real money on account of
ZFC being inconsistent.
>I will bet anyone a quarter at even odds that *if* the propositional
>calculus is shown to be inconsistent, I can find a very large integer
>power of 3 that is even. 8-O (<--- wise owl face)
At those odds you're on. It's hard to find anything with the lights
out. ("o (<--- Robotman face)
--
=================================================== Trouble is, son
Vaughan Pratt pr...@cs.Stanford.EDU 415-494-2545 The farther you run
=================================================== The more you feel undefined
Vaughan R. Pratt
(I canceled an earlier post with dateline Sun, 20 Sep 1992 01:30:03 GMT
on this question. If you run across it anyway please note that I've
thoroughly disowned that drivel. My only excuse is that contemplating
the inconsistency of classical logic tends to encourage sloppy
reasoning :-) I hope the following fares better. -vp)
In article <TORKEL.92S...@lludd.sics.se> tor...@sics.se (Torkel Franzen) writes:
>In article <19f4cj...@nestroy.wu-wien.ac.at> r...@mips.complang.tuwien.ac.at
> (Richard Zach) writes:
>
> >What if the proof of the inconsistency of propositional
> >logic would not carry over to, say, intuitionistic
> >logic? We would then go on doing intuitionistic mathematics,
> >wouldn't we?
>
>
> This reflection illustrates the unfortunate effect that talk of
>"consistency" so often has on the thought processes of mathematicians.
>[more in this vein]
>
> But perhaps I am being hasty.
No, just you.
>Your remarks make sense if what you are
>assuming is that we manage to prove, in classical mathematics, that
>classical propositional logic is inconsistent, although this proof is not
(Note Torkel's use of "classical" (= Boolean). "Propositional" logic
on its own means merely zeroth-order logic, distinguishing it from
first order logic which speaks not only of propositions but of
individuals. A propositional logic may be classical (Boolean),
intuitionistic (Brouwerian, Heyting), modal, etc., one needs to specify
which.
>valid in intuitionistic mathematics. Then what you are assuming is that
>classical mathematics turns out to be inconsistent, and this, to be sure,
>is compatible with intuitionistic mathematics being consistent. But of
>course in such a case it would still be a theorem of intuitionistic
>mathematics that classical logic is consistent.
It is hard to judge any such inference made under the premise that
proposition calculus is inconsistent. What statements or inferences can
still be trusted in the face of such a calamity?
Nevertheless the concern that propositional calculus might be
inconsistent is not an unreasonable one in the light of the fact that
classical logic, unlike intuitionistic logic, is complete in the sense
that adding *any* equation to the theory without also adding new
operations to the language such as modalities, renders the resulting
theory inconsistent. That this is not the case for intuitionistic
logic is illustrated by the example ~~x = x, which when added to the
axioms of intuitionistic logic yields classical logic, normally
understood to be consistent. This complete lack of any protective
buffer for classical logic means that if classical logic turns out to
contain *any* equation we might have somehow hitherto overlooked, then
it must be inconsistent. (Of course in the same spirit we might have
"overlooked" ~~x=x as an intuitionistic theorem, but then classical and
intuitionistic logic would coincide, closing off the above escape
route. And besides, if one oversight is so unlikely, how much more
unlikely are the two needed to make intuitionistic logic inconsistent
in this way?)
With these caveats in mind, let me point out that when each
propositional variable and constant appearing in a proof in classical
propositional logic is replaced by its double negation, one obtains a
proof that is not only classically sound but also intuitionistically
sound. This is because the image of double negation in a Heyting
algebra A (the semantics of intuitionistic logic) is a subalgebra of A
that is a Boolean algebra (the semantics of classical logic). The
proof is intuitionistically sound because no non-Boolean
counterexamples to the reasoning of the proof can be found:
intuitionistic truth values are neutralized to classical ones at the
variables and constants before they are allowed to wander around in the
substance of the proof.
(This principle may be observed at work in natural language, where
double negatives convey to the listener the distinct impression that
the speaker is trying to weaken his statement. I can imagine no
clearer evidence for the thesis that the logic of ordinary speech is
not classical.)
Now translate any classical proof of 0=1 in this way to an
intuitionistic proof of ~~0 = ~~1. But ~0 = 1 and ~1 = 0 are theorems
of intuitionistic logic, whence 0=1 is a theorem of intuitionistic
logic as well. But now we can argue intuitionistically that x = x&1 =
x&0 = 0, i.e. intuitionistic logic is inconsistent. (This can be
reformulated in other frameworks, I just find the equational one most
convenient to work in.)
This argument is perfectly constructive. Nor does it depend on numbers
and induction: the translation is applied locally at each variable and
constant, and soundness of the translation does not require induction
either, being verifiable locally. (etc. etc.)
Hence if classical propositional logic does prove inconsistent,
retreating to intuitionistic logic will afford little relief since even
the minimal amount of mathematics used here will not be available to
you.
John C. Baez
>In article <1992Sep19.1...@galois.mit.edu> jb...@riesz.mit.edu (John C. Baez) writes:
>>I would worry that you may have a lot of people taking you up on your
>>offer of $100. I won't drive you further into debt by taking you up
>>on that one.
>
>If I tell you there are no takers yet will that make you feel better
>about taking advantage of me? Or are you really worried there's some
>chance you'll lose $10,000? It's extremely unlikely, and anyway think
>of inflation, think of it as just $2,000, you'll easily be able to
>afford it then. And it will be worth it, you'll be famous as the first
>(I hope not only) person in the world to lose real money on account of
>ZFC being inconsistent.
Okay, to be frank, when I contemplate this bet, I can't endure the
thought of telling my fiancee: "Honey I just made a bet where if I win
I get $100, but if I lose I lose $10,000." She'd think I'm nuts, even
if I explained to her that nobody has found a contradiction so far, etc.
etc.. Of course I could just not tell her at all but I don't think it'd
be nice to make such a bet and just not tell her in the hope that I
will win.
As you could see from sci.physics, I have no problem with bets in the
<$100 range, which are essentially peanuts. I don't know if some
scaled-down version of the above bet would appeal to you, or whether
you'd regard it as too dull.
You're on for the bet of one quarter, by the way. Not that it matters.
Vaughan R. Pratt
>As you could see from sci.physics, I have no problem with bets in the
><$100 range, which are essentially peanuts.
Fine. Assuming you could commit $100 in 2012 dollars without feeling
obligated to mention it to your fiancee, that makes it $1 to you if
win, $100 to me if I win. ($1 will not be peanuts in 2012, a can of
peanuts will be considerably more.)
Give it some thought. Even if the amount is ok now, the odds might
still not be to your taste.
John C. Baez
>In article <1992Sep20....@galois.mit.edu> jb...@riesz.mit.edu (John C. Baez) writes:
>>As you could see from sci.physics, I have no problem with bets in the
>><$100 range, which are essentially peanuts.
>
>Fine. Assuming you could commit $100 in 2012 dollars without feeling
>obligated to mention it to your fiancee, that makes it $1 to you if
>win, $100 to me if I win. ($1 will not be peanuts in 2012, a can of
>peanuts will be considerably more.)
>
>Give it some thought. Even if the amount is ok now, the odds might
>still not be to your taste.
The amount is fine and the odds are okay too. Consider it done!
(I am starting a file in which I save all these bets I'm making.
In the future I hope to endow a prize using the money I make
this way. :-) It will be called the "Peanut Prize.")
Karl Heuer
>Okay, to be frank, when I contemplate this bet, I can't endure the
>thought of telling my fiancee: "Honey I just made a bet where if I win
I was going to recommend that you tell your fiancee that she could hedge for
you by taking the other bet, in which case the two $10,000 transfers would
cancel (assuming you can prevent the government from taking a large slice of
both), and since the "insurance" would cost only $10, you'd be $90 ahead.
But I just checked back to the original article, and noticed that the two
events aren't complementary: if Z is consistent but ZF is not, you'd still
lose $10,010. Oh well. (Be sure to tell us how many people accept both
bets, anyway.)
Karl Heuer ka...@ima.isc.com uunet!ima!karl
G.J. McCaughan
that ZF will not be proved inconsistent by 2012. (Of course "proved" means
"proved to the satisfaction of the general mathematical community"; cranks
don't count!)
--
Gareth McCaughan Dept. of Pure Mathematics & Mathematical Statistics,
gj...@cus.cam.ac.uk Cambridge University, England. [Research student]
Thomas Clarke
Baez) writes:
> Imagine the following story: I know I'm alive. Then someone shows me an
^awake^
> enormous deduction in the propositional calculus (axiomatized in ones
> favorite manner) which has as the final theorem P & not(P). I am
> told that this deduction has been checked by computers many times
> over, and I get a copy in electronic form and write my own program
> and check it and, sure enough, it checks! Do I then somehow all of a
> sudden stop knowing I'm alive?
^wake up^
I sometimes have dreams like this. This inconsistency is the
tip off that you are dreaming. I sometimes worry that
something inconsistent will turn up while I think I am awake.
John C. Baez
>Oh, well if you're prepared to make smaller bets, then I bet $10 against $1000
>that ZF will not be proved inconsistent by 2012. (Of course "proved" means
>"proved to the satisfaction of the general mathematical community"; cranks
>don't count!)
Is "you" supposed to mean me? I'm sorry, I won't make this bet since
I am very certain that ZF will *not* be proved inconsistent by 2012; I
think the chance is much less than 10%, for example. That's why I was
willing to bet $100 to Pratt's $1 that ZF would not be proved
inconsistent.
Hmm, by the way, looking at my records I see that I don't know which
system I bet on with Pratt - ZF or Z. Vaughan?
I am embarassed to admit that I don't even know what "Z" is.
G.J. McCaughan
No, the above was addressed to Vaughan. Of course I agree with you about
the unlikeliness of ZF being proved inconsistent by 2012. (You probably
mean "1%", by the way.)
It was ZF on which you bet, I think; Z is ZF without the Replacement axiom.
Michael Zeleny
Suppose logic is inconsistent. Then there is a contradiction. #
A similar proof may be used to demonstrate consistency of mathematics.
Thanks to Gabriel Stolzenberg, who will credit it to someone else.
cordially,
mikhail zel...@husc.harvard.edu
"Un de mes plus grands plaisirs est de jurer Dieu quand je bande."
John C. Baez
>[as] V. Pratt
>and R. Holmes pointed out, namely that classical prop.
>logic and intuit. logic are equiconsistent.
Just checking: does the (informal) proof of this rely on
any classical logic, or can it be done using only intuitionistic
logic? I.e., we need a constructive method for transforming
a classical proof of p & ~p into an intutionistic proof of the
same.
One has to be quite careful with this stuff, and I'm utterly out of
my depth, but my point is that if the classical propositional calculus
is inconsistent, all metatheorems *about* logic which use the full
strength of the classical propositional calculus have to be treated
as flawed. One might hope to think about this calamitous possibility
by proving some metatheorems using reasoning that is strictly weaker
than classical logic.
>I think that the reason that we will not be able to
>prove propositional logic inconsistent, is that the
>very question is nonsensical. Without propositional
>logic, what does it mean to be consistent or inconsistent?
While the question is undoubtedly a matter of paranoid
philosophical navel-gazing verging on lunacy, I wouldn't
say that it's nonsensical. Here's what I mean for the classical
propositional calculus to be inconsistent. Write down your favorite axioms
like p -> ~~p etc., and take modus ponens as your rule of deduction.
Now suppose that someone writes down a proof of p & ~p using these
rules. No doubt it will be long since we haven't found it yet.
Of course everyone is utterly skeptical of it. But suppose that
it is discovered by famous logician, not a crank. Then people (or at
first computers) might be persuaded to check it. They check it over
and over again and can't find any flaw.
If one wanted to make a philosophical "federal case" out of this, which I
definitely don't, one might say that the concept of "following a rule"
is more primordial than the rules of Boolean logic, and all one needs
is the former concept to make sense of the idea of "proving the
classical propsitional calculus is inconsistent."
Of course, if you want a little more wiggle room, you can play around
with a story in which the classical predicate calculus is inconsistent, but
not the propositional calculus.
Mark Fulk
>In article <1992Sep22.1...@email.tuwien.ac.at> za...@csdec1.tuwien.ac.at writes:
>>[as] V. Pratt
>>and R. Holmes pointed out, namely that classical prop.
>>logic and intuit. logic are equiconsistent.
>
>Just checking: does the (informal) proof of this rely on
>any classical logic, or can it be done using only intuitionistic
>logic? I.e., we need a constructive method for transforming
>a classical proof of p & ~p into an intutionistic proof of the
>same.
That, in fact, is what we have. The construction, which was posted here
a while ago, is effective; in fact, it is very simple: just replace every
letter $x$ appearing in the proof with $~~x$, in order to prove
$~~p and ~~~p$. From this it is a few (fixed) steps to get to $p and ~p$.
>
>>I think that the reason that we will not be able to
>>prove propositional logic inconsistent, is that the
>>very question is nonsensical. Without propositional
>>logic, what does it mean to be consistent or inconsistent?
>
>While the question is undoubtedly a matter of paranoid
>philosophical navel-gazing verging on lunacy, I wouldn't
>say that it's nonsensical. Here's what I mean for the classical
>propositional calculus to be inconsistent. Write down your favorite axioms
>like p -> ~~p etc., and take modus ponens as your rule of deduction.
>Now suppose that someone writes down a proof of p & ~p using these rules.
This will do; more usually, one asks that every formula be provable.
(E. Wette claims to have proved the inconsistency of propositional logic.
For some reason, the proof needs to be presented in person, using several
colors of chalk. B. Daley claims that the critical step occurs at a point
when the colors run together :-). Wette, when pressed, cites authority
to support his proof; the authority is his daughter.)
>Of course, if you want a little more wiggle room, you can play around
>with a story in which the classical predicate calculus is inconsistent, but
>not the propositional calculus.
This is not possible, as any model of propositional calculus can be used
to construct a model of predicate calculus; the resulting model has
one individual, and the predicates come to stand for the the
sentential variables of the propositional model.
--
Mark A. Fulk University of Rochester
Computer Science Department fu...@cs.rochester.edu
Torkel Franzen
(Mark Fulk) writes:
>That, in fact, is what we have. The construction, which was posted here
>a while ago, is effective; in fact, it is very simple: just replace every
>letter $x$ appearing in the proof with $~~x$, in order to prove
>$~~p and ~~~p$. From this it is a few (fixed) steps to get to $p and ~p$.
It's completely pointless to say that classical and intuitionistic
propositional logic are intuitionistically equiconsistent. It's like
saying, "even intuitionistically, it is provable that if 0=0, then 1=1".
The consistency of classical propositional (or predicate) logic is just
a trivial intuitionistic theorem.
Vaughan R. Pratt
FURTHER DETAILS CONCERNING THE $10,000 BET APPEAR BELOW,
including a more precise specification of the problem
for the benefit of interested programmers and others.
In article <1992Sep21....@galois.mit.edu> jb...@riesz.mit.edu (John C. Baez) writes:
>Hmm, by the way, looking at my records I see that I don't know which
>system I bet on with Pratt - ZF or Z. Vaughan?
ZF.
John's touching concern for his fiancee caught me at a moment of
weakness, future bets will be at the originally stated rate. Thus far
I have one bet at the official rate, plus John's freebie. Remember, if
you really are sure that ZF is consistent then this is an easy $100
that goes to *you*, you don't even have to think about the down side.
When you're cashing your check for $100 John will be cashing his $1
check, worth about 20 cents by then---it may cost him more than that to
cash it.
>I am embarassed to admit that I don't even know what "Z" is.
ZF without replacement (F).
Replacement just says that all expressible images exist. That is, for
every set a the image f(a) of a, namely the set
{ f(x) | x in a, f(x) defined},
exists. Here f is any expressible partial function, defined as a
predicate p(x,y) of ZF holding of at most one y for each x, namely
y=f(x) when such y exists. Formally:
(Aa)[(Au)(Av)(Aw)[p(u,v)&p(u,w) -> v=w]
-> (Eb)(Ay)[ y in b <--> (Ex)[ x in a & p(x,y)]]]
(This is not an axiom but an axiom schema, namely one axiom for every
instantiation of p(-,-) with a suitable ZFC formula. (Au) means "for
all u" and (Eb) means "there exists b.")
One might wonder why a schema, why not just treat p as a set of pairs
and quantify over all such p? Well, for those p that already exist
(the only kind quantification can know about), Replacement is already
true in Z, where you can construct the desired image f(a) as follows
(all terms standard).
restrict(p,a) = (a x range(p)) intersect p
image(p,a) = range(restrict(p,a))
Without Replacement you can construct a model of Z as the union of the
set of elements of a sequence whose first element is the set of natural
numbers (at least I think that's enough) and whose i-th element is the
powerset of its predecessor. Call this model M.
This construction seems innocent enough, which is why I was willing to
bet $10,000 at 1000:1 (my second bet, repeated below) that Z would not
be found to be inconsistent by 2012.
No takers on this bet so far. Just checking everyone's intuition,
looks good.
M is not very large as large cardinals go, Aleph_omega assuming CH.
Some reasonable-sounding theorems of analysis, e.g. " "every symmetric
Borel relation either contains or is disjoint from a Borel function"
require induction up to this cardinal and hence depend on its
existence.
The sequence in the construction of M is an expressible function.
Hence when Replacement becomes a principle the sequence becomes a set,
whence so does its union M, call it the set m. We have no reason to
suppose m is an element of m (indeed another axiom, Foundations, tells
us it cannot be, but we do not need to use this). Hence this
construction does not to our knowledge produce a model of ZF, and hence
does not work as a proof of consistency of ZF the way it did for Z.
Can we fix this by taking the union of some sequence longer than
omega? No, because no matter how complicated a sequence, or more
generally a family, we come up with, it can always be made into a set
by Replacement. We have to come up with some other way of constructing
models than as sequences or families of sets. I don't know much beyond
this point, perhaps someone more familiar with this area can tell us
what sorts of plausible constructions have been considered.
PROBLEM SPECIFICATION
Let me make the problem underlying the bet very concrete. Here are two
easily understandable statements.
A. There exists an NxN 0-1 matrix, N the natural numbers, such that
every axiom of ZF is true when the variables in the axiom are
interpreted as ranging over N and the membership predicate symbol is
interpreted so that i is a member of j iff E[i,j] = 1. (So no need to
think about uncountable models, or models containing "sets". On the
other hand E is too big to write out explicitly all at once, and
quantifiers will take forever to evaluate if done naively!)
B. An ideal computer search for an inconsistency in ZF is guaranteed
to succeed in a finite time. (Ideal = correct code and no time or
space limits. Bugs are up to you. Space: well, how big a proof do you
think you're looking for? Time would seem to be your main obstacle.)
B is recognizable as asserting the inconsistency of ZF. By
completeness of first-order logic, A is asserting the consistency of
ZF, i.e. A = not-B.
Bet 1 is about A vs. B, bet 2 is the same with Z in place of ZF. Z is
ZF less F.
We know by Goedel incompleteness that A is not provable in ZF (ditto
with Z for ZF everywhere). (On the other hand if either instance of B
is provable it is provable in a lot less than Z.) Hence if you plan on
trying to prove A you will need some insight about the raw mechanics of
A not provable in ZF. As you can see those mechanics are very simple,
apart from the big mess of formulas asserting ZF, so you have as good a
chance as any mathematician at finding such an insight, possibly more
if you don't understand what the axioms mean.
You will find the axioms of ZF explained quite readably as well as
formally in Shoenfield's article, pp. 321-341 of Barwise's "Handbook
of Mathematical Logic", North Holland, and *only* formally (!) as
Axioms 1-6 in Takeuti and Zaring, "Introduction to Axiomatic Set
Theory" Springer-Verlag (my formal statement of F is their Axiom 5,
Shoenfield's version of F is oversimplified).
I would of course appreciate hearing about any substantive
discrepancies between these two axiomatizations.
This is pretty much the complete specification of the problem as I
understand it.
================================================
Here is a repeat of the bet as posted on sci.math on Sept. 19, 1992.
--------------
I will bet $100 in favor of ZF being found inconsistent within the next
20 years, with odds of 100:1 against. That is, the parties regard ZF
inconsistent as a rather unlikely outcome, though I'm not ruling it out
at those odds. If you win you get my $100, if you lose you owe me
$10,000.
I will bet $10,000 *against* Z being found inconsistent within the next
20 years, with odds of 1000:1 against. That is, I'm saying that Z
inconsistent is extremely unlikely, and if you think you can make money
at any crazy odds on this bet you're the one who's crazy. If you win
you get my $10,000, if I win I get only your $10.
These amounts are in US dollars for 2012. We agree that our respective
estates will honor the debt.
(I picked the amounts assuming roughly 5-fold inflation over that
period.)
Let me know if you wish to accept either bet. I will keep your name
private if requested, but not the number of people accepting each bet.
These are meaningful bets. It would be meaningless to propose a bet
about the propositional calculus. That can only become inconsistent
when all the lights go out.
--------------
Points of clarification added Sept 22:
By "within the next 20 years" I mean by Sept. 19, 2012. Both bets will
be payable on Sept. 22, 2012. Alternatively the loser may pay a
geometrically smaller amount earlier, namely 1/5 the owed amount in
1992, 8.38% more each year later.
By "found" I mean its ordinary mathematical usage of having a proof
that has convinced a majority of reputable mathematicians. This could
not happen for either of these bets unless either the proof were
derivable in Z or reputations didn't mean as much in 2012.
If there are sufficiently many takers for either bet I will stop taking
bets thereafter. (There is no risk of this happening so far.) I
currently have no plans to scale down the amounts of the bets; in the
unlikely event this changes I will announce it on sci.math. The most
likely change would be $100->$10 and $10,000->$1,000 in bet 1. Pleast
let me know by email (pr...@cs.stanford.edu) if you would be interested
at that rate but not the current rate.
Richard Zach
|> (E. Wette claims to have proved the inconsistency of propositional logic.
|> For some reason, the proof needs to be presented in person, using several
|> colors of chalk. B. Daley claims that the critical step occurs at a point
|> when the colors run together :-). Wette, when pressed, cites authority
|> to support his proof; the authority is his daughter.)
This has been published in the Proceedings to the Symposium honoring
G"odel on his 60th birthday or so. According to what I've heard,
G"odel, when asked about the paper, said that "it would be interesting
to publish this." Austrian sarcasm not always being recognizable
as such, the paper was accepted.
Keith Ramsay
>Suppose logic is inconsistent. Then there is a contradiction. #
>Thanks to Gabriel Stolzenberg, who will credit it to someone else.
He says that Fred Richman told it to him, attributing it to Don Cook.
Keith Ramsay Even if a proof from given axioms can, in
ram...@unixg.ubc.ca principle, be found mechanically, is it not
often found in ways not all that different
from finding a new axiom? -Georg Kreisel
Torkel Franzen
.ac.at (Richard Zach) writes:
>Hence, it seems arguable to me, whether the statement that
>there might be a even large number being a power of three
>is on a par with the consistency question of logic.
>Numbers, induction, etc. are on a higher level
>than propositional logic.
The assertion "propositional logic is consistent" is a mathematical
assertion, proved by a simple induction. From a mathematical point of
view it is no different from any other combinatorial assertion proved
by a simple induction. From a philosophical point of view, the proof
of the theorem is dubious precisely to the extent that any other
elementary proof by induction is dubious.
When mathematicians or other people mutter mysteriously about
the spectre of inconsistency, they are pretty plainly not interpreting
such statements as "propositional logic is inconsistent" as mathematical
statements. Rather, they seem to interpret them as "there is some
flaw in our minds, or in our thinking, whereby all that we think we
have created is but a figment, incoherent in its very structure".
Now it is of course perfectly in order to give vent to such reflections.
However, the consistency statements involved in these reflections
should not be confused with mathematical statements.
Richard Zach
|> In article <1992Sep22.1...@email.tuwien.ac.at> r...@mips.complang.tuwien
|> The assertion "propositional logic is consistent" is a mathematical
|> assertion, proved by a simple induction. From a mathematical point of
|> view it is no different from any other combinatorial assertion proved
|> by a simple induction.
*If* logic and induction are consistent. If logic is
consistent and induction is false, then maybe we find a
large even number being a power of three.
We may not be able to prove the consistency of
propositional calculus, but that's irrelevant.
You are right if you are assuming that the "mathematical point of
view" is consistent. But how can you do this if you are
doubting things that are fundamental to this standpoint?
|> From a philosophical point of view, the proof
|> of the theorem is dubious precisely to the extent that any other
|> elementary proof by induction is dubious.
|> When mathematicians or other people mutter mysteriously about
|> the spectre of inconsistency, they are pretty plainly not interpreting
|> such statements as "propositional logic is inconsistent" as mathematical
|> statements. Rather, they seem to interpret them as "there is some
|> flaw in our minds, or in our thinking, whereby all that we think we
|> have created is but a figment, incoherent in its very structure".
|> Now it is of course perfectly in order to give vent to such reflections.
|> However, the consistency statements involved in these reflections
|> should not be confused with mathematical statements.
"Propositional logic is inconsistent" is a mathematical assertion.
The problem is that propositonal logic also has a very precisely
defined semantics that seems to correspond to a fragment
of human "logical" reasoning. If we had a proof of the inconsistency
of propositional logic, then we would also have an example
of human logical reasoning that yields a contradiction.
Hence, "there is some
flaw in our minds, or in our thinking, whereby all that we think we
have created is but a figment, incoherent in its very structure"
*follows* from a mathematical proof of the mathematical
assertion that "Propositional logic is inconsistent".
On the other hand, if we find a very large even number being a
power of three, then this only means that our understanding
of concepts such as "natural number", "induction"
etc. are somehow flawed. When I said that this is on a different
level than the consistency of propositional logic,
I assumed that "logical thinking" is innate to humans
while reasoning in PA is not.
Torkel Franzen
ac.at (Richard Zach) writes:
>You are right if you are assuming that the "mathematical point of
>view" is consistent. But how can you do this if you are
>doubting things that are fundamental to this standpoint?
I'm not assuming anything in particular. I'm just asking on what
grounds you would regard one particular elementary inductive proof
as more doubtful than another.
>On the other hand, if we find a very large even number being a
>power of three, then this only means that our understanding
>of concepts such as "natural number", "induction"
>etc. are somehow flawed.
This "only" appears to me essentially arbitrary. For my part, I
don't see that the existence of an enormous inconsistency in
propositional logic would indicate a flaw in my thinking any more
"fundamental" than that indicated by the existence of a large even
Randall Holmes
I suspect that reasoning in what amounts to a substantial fragment of
PA _is_ innate to human beings (in our language processing, not our
specifically "mathematical" abilities, if any).
>
>--
>Richard Zach [za...@csdec1.tuwien.ac.at]
>Technische Universitaet Wien, Institut fuer Computersprachen
>Resselgasse 3/185.2, A-1040 Vienna, Austria/Europe
--
The opinions expressed | --Sincerely,
above are not the "official" | M. Randall Holmes
opinions of any person | Math. Dept., Boise State Univ.
or institution. | hol...@opal.idbsu.edu
John C. Baez
> When mathematicians or other people mutter mysteriously about
>the spectre of inconsistency, they are pretty plainly not interpreting
>such statements as "propositional logic is inconsistent" as mathematical
>statements. Rather, they seem to interpret them as "there is some
>flaw in our minds, or in our thinking, whereby all that we think we
>have created is but a figment, incoherent in its very structure".
I don't know if my posts count as "mysterious mutterings" (you do seem
to project so - first it was a "wise owl-like expression") but I agree
that the inconsistency of the propositional calculus is a mathematical
statement, almost certainly false. If it were true, we must have
screwed up colossally. This would, in my opinion, be indicative of some
flaw in our minds.
Vaughan R. Pratt
Franzen) writes:
> When mathematicians or other people mutter mysteriously about
>the spectre of inconsistency,
Why are concerns about inconsistency ``mysterious mutterings?'' Frege
and Quine ran into inconsistencies. Was their reaction to mutter
mysteriously?
Belief of a mathematician in the consistency of ZF is the belief of a
Christian in heaven after death. There is no more basis for the
existence of a model of ZF than for the existence of heaven. You are
making mathematics into a religion.
I'm admittedly biased here, since if ZF is found inconsistent by 2012 I
win $10,000. If Torkel's saying it three times isn't enough to make it
true, maybe a bet of this size is.
Torkel Franzen
(John C. Baez) writes:
>I agree
>that the inconsistency of the propositional calculus is a mathematical
>statement, almost certainly false. If it were true, we must have
>screwed up colossally. This would, in my opinion, be indicative of some
>flaw in our minds.
I see that you have extremely strict standards of certainty, as
befits a devotee of quantum mechanics. What you say makes perfect sense,
but it sidesteps the apparent confusion I had in mind. Let me
explain.
What you say about the consistency of propositional logic can also
be said about, say, the theorem "every formula in propositional logic
has a disjunctive normal form". If this is false, our minds are
seriously flawed. However, it never occurs to anybody to try to read
any assertion that our minds are not flawed into a proof of this
latter theorem.
In sharp contrast, you find, again and again, reflections along the
following lines: how can we trust the proof that propositional logic
is consistent, if what we are putting in question is precisely whether
our thinking is consistent? This makes sense if you are
trying to interpret the statement proved - i.e. "propositional logic
is consistent" - as some sort of assurance that our thinking is ok,
our minds unflawed, etc. But it is nothing of the sort. Such assurances
must be sought outside mathematics. What is proved is just this:
propositional logic is consistent. The statement is an ordinary
mathematical statement, the proof is an ordinary mathematical proof.
There is no more point in agonizing over this particular statement and
proof than there is in agonizing over the theorem that every formula
has a disjunctive normal form.
Again, consider the baffling fact that mathematicians in this kind
of context tend to attach a peculiar significance to the theorem
"if intuitionistic logic is consistent, then classical logic is
consistent". This would make sense if the truth of this implication
were more unproblematic than the truth of its antecedent and
consequent. But it is not. In classical and intuitionistic mathematics
both, the two statements "classical logic is consistent" and
"intuitionistic logic is consistent" are provable outright, by arguments
not one whit more complicated than any argument for the implication.
Hence it is mere "mystic muttering" when the relative consistency
theorem is regarded as an argument against the possibility of
classical but not intuitionistic logic being inconsistent. If you
assume that classical logic is inconsistent, you may as well assume
that the relative consistency statement is false. The one assumption
makes precisely as much sense as the other.
With this I rest my case for the claim that mathematicians tend to
throw their mathematical common sense to the winds when consistency
is at issue.
Torkel Franzen
Stanford.EDU (Vaughan R. Pratt) writes:
>Why are concerns about inconsistency ``mysterious mutterings?'' Frege
>and Quine ran into inconsistencies. Was their reaction to mutter
>mysteriously?
You quite misunderstand my comments, which were only concerned with
the peculiar distinctions made by mathematicians between trivial
theorems of the form "T is consistent" and other trivial theorems.
My reply to John Baez should make my point clear.
John C. Baez
>(John C. Baez) writes:
>
> >I agree
> >that the inconsistency of the propositional calculus is a mathematical
> >statement, almost certainly false. If it were true, we must have
> >screwed up colossally. This would, in my opinion, be indicative of some
> >flaw in our minds.
>
> I see that you have extremely strict standards of certainty, as
>befits a devotee of quantum mechanics.
Indeed, just as a good doctor is always to be found among the unhealthy.
>What you say makes perfect sense,
>but it sidesteps the apparent confusion I had in mind. Let me
>explain.
Sorry to have sidestepped a confusion. It occaisionally happens.
Go right ahead...
> What you say about the consistency of propositional logic can also
>be said about, say, the theorem "every formula in propositional logic
>has a disjunctive normal form". If this is false, our minds are
>seriously flawed. However, it never occurs to anybody to try to read
>any assertion that our minds are not flawed into a proof of this
>latter theorem.
> In sharp contrast, you find, again and again, reflections along the
>following lines: how can we trust the proof that propositional logic
>is consistent, if what we are putting in question is precisely whether
>our thinking is consistent? This makes sense if you are
>trying to interpret the statement proved - i.e. "propositional logic
>is consistent" - as some sort of assurance that our thinking is ok,
>our minds unflawed, etc. But it is nothing of the sort. Such assurances
>must be sought outside mathematics. What is proved is just this:
>propositional logic is consistent. The statement is an ordinary
>mathematical statement, the proof is an ordinary mathematical proof.
>There is no more point in agonizing over this particular statement and
>proof than there is in agonizing over the theorem that every formula
>has a disjunctive normal form.
If anyone is suffering here it seems to be you. It seems to me that,
if one is in the mood to think about "what if something that
everyone agrees is true turned out to be false," picking the consistency
of propositional logic as an example has a certain touch of class.
One could also pick the algorithm for long division, but this doesn't
sound as nice. It's just the same as in the oft-cited but rarely
seen proof by contradiction which ends "1 = 0". People could equally
say one has obtained a contradiction whenever one obtains
324578934948 - 12^101 = 19.3
but for esthetic reasons the former is preferable. Of course you
are right in noting that esthetics of this kind can easily be
mystifying. One can imagine, for example, a student who came to
believe that the statement 1 \= 0 held a privileged place arithmetic as
a kind of bastion of consistency. (This was precisely my
joke about the "Fundamental Theorem of Mathematics," by the way.)
In any event, perhaps if I continue to mystify the readers of sci.math
with my wise, owl-like mutterings, and you continue to attempt to
demystify them, they will be entertained and edified.
> Again, consider the baffling fact that mathematicians in this kind
>of context tend to attach a peculiar significance to the theorem
>"if intuitionistic logic is consistent, then classical logic is
>consistent". This would make sense if the truth of this implication
>were more unproblematic than the truth of its antecedent and
>consequent. But it is not. In classical and intuitionistic mathematics
>both, the two statements "classical logic is consistent" and
>"intuitionistic logic is consistent" are provable outright, by arguments
>not one whit more complicated than any argument for the implication.
Hear, hear! Does anyone, by the way, ever do a careful analysis of
the types of reasoning involved in proving these metatheorems, so one
could make your "not one whit" quite precise? For example, perhaps
in the proof of consistency of classical logic one uses the obviously
correct principles of reasoning A, B, C, and D, while in the the
relative consistency proof one also uses principle E - then one might
say in fact that the relative consistency proof is in fact precisely
one whit *more* complicated. Of course there are lots of choices
for what one might take as such principles, but logicians seems to
enjoy worrying about such things endlelssly so someone may have done so.
I have trouble imagining that you, Torkel, would be interested in such
nitpickery, but someone might be.
Frank Adams
>be said about, say, the theorem "every formula in propositional logic
>has a disjunctive normal form". If this is false, our minds are
>seriously flawed. However, it never occurs to anybody to try to read
>any assertion that our minds are not flawed into a proof of this
>latter theorem.
>
> In sharp contrast, you find, again and again, reflections along the
>following lines: how can we trust the proof that propositional logic
>is consistent, if what we are putting in question is precisely whether
>our thinking is consistent? This makes sense if you are
>trying to interpret the statement proved - i.e. "propositional logic
>is consistent" - as some sort of assurance that our thinking is ok,
>our minds unflawed, etc. But it is nothing of the sort. Such assurances
>must be sought outside mathematics. What is proved is just this:
>propositional logic is consistent. The statement is an ordinary
>mathematical statement, the proof is an ordinary mathematical proof.
>There is no more point in agonizing over this particular statement and
>proof than there is in agonizing over the theorem that every formula
>has a disjunctive normal form.
The point is that in proving the consistency of propositional logic, we
are using reasoning techniques which include all the techniques built
in to propositional logic. Now, the main concern I have seen with this
kind of statement is to wonder what the point is; we don't believe the
proof unless we already believe what we are trying to prove.
As for the other problem, let us step back for a moment to the point
where neither of the theorems in question had been proved. In this
case, their status is very different. If we were to have proved that
some formulae don't have a disjunctive normal form, this would have
been an interesting fact. If we had proved that propositional logic
was inconsistent, it would have been a catastrophe -- there was
something wrong with our reasoning process.
(In fact, just this kind of catastrophe occurred with naive set theory
-- set theory with the axiom "for any property P, there is a set S of
all sets having property P. This did invalidate all, or even very
much, of the reasoning process of mathematics up to that time; but it
was nevertheless probably the most event in the history of
mathematics.)
The argument is not "we've proved this is true; if it turned out to be
false it would be a real problem", but "we've assumed this is true (and
incidently we've proved it); if it turned out to be false it would be a
real problem".
Keith Ramsay
(Torkel Franzen) writes:
>There is no more point in agonizing over this particular statement and
>proof than there is in agonizing over the theorem that every formula
>has a disjunctive normal form.
In article <1992Sep25.1...@galois.mit.edu> jb...@riesz.mit.edu
(John C. Baez) writes:
>If anyone is suffering here it seems to be you.
I have to side with Torkel Franzen in this business (my compulsions
are forcing me to...). :-) When John Baez suggested that predicate
calculus "could be inconsistent", Torkel Franzen, as I recall, replied
by saying that this was so in the same sense as it was possible that a
large power of 2 could turn out to be a multiple of 3 (or perhaps it
was vice-versa). This is a concise, apt way of describing the nature
of the doubt involved. The discussion could have ended there.
It didn't. Someone brought up the comparison with doubts about the
consistency of ZFC, as if this were comparably doubtful somehow! And
it went on from there....
It appears to me that there is indeed a general pattern at work in
discussions of this kind. We all have beliefs which we hold with some
degree of doubt, which varies from belief to belief. Part of the
problem is, however, that people are not very consistent about
comparing degrees of likelihood.
For example, an experiment was once performed in which people were
asked to compare the likelihoods of various possibilities. One of the
more interesting comparisons was between the likelihood of a woman
being a banker, and the likelihood of a woman being a feminist *and* a
banker. A certain number of people rated it as being *more* likely
that the woman was a feminist *and* a banker, than that she was a
banker. Now, anyone who is familiar with probability knows that this
is absurd; being a feminist *and* a banker implies already being a
banker, and hence cannot be any more likely.
What was thought to be happening was that the people who answered this
way were describing, rather, their belief that the conditional
probability of her being a banker, if she is a feminist, is greater
than if she is not a banker. Perhaps these are people who don't
usually expect women to be feminists or bankers, and the possibility
of a woman being a feminist puts them in a frame of mind in which
these assumptions are more easily doubted. The picture as a whole,
that of a feminist banker, "fits" together better, ergo, is more
"likely".
Generally speaking, people adjust the manner in which they examine
doubts, depending on how deep the assumptions are which are being
doubted. In the context of everyday life, one can ask someone how
certain they are that their spouses love them, and they may tell you
that it is certain that their spouses love them. But some of the same
type of people, if you ask them whether they believe in the existence
of other minds, will agree that it is not entirely certain that other
minds exist. "Maybe so", "who can know", "how can anyone know
anything", "it is a matter of faith".
It is as if there is a hierarchy of doubt. When examining an
assumption on one level, it is as if one suspends it, and examines it
relative to the assumptions on lower levels. The "seriousness" with
which the doubt is taken depends partly upon the relation with the
more basic assumptions, rather than being exclusively determined by
the straight "likelihood" of the assumption being incorrect.
Questions of "inconsistency" appear very often to get "special"
treatment. People often seem not to regard them as being ordinary
mathematical assertions, of the same kind as "pi is irrational" and so
on. It is to be emphasized: formal consistency questions *are*
essentially questions about strings of elements of a finite set, with
certain combinatorial properties.
If I posted a claim to the effect "a straightedge and compass
trisection still might be possible" people would likely consider this
idiotic. "We've *proven* that it is impossible, dummy! Get a life!"
This would be discussed in the ordinary mathematical context, in which
all the (extremely safe) assumptions required are not subjected to
continual skeptical doubt.
On the other hand, when a question of "consistency" is raised, it is
automatically treated in the context of doubts about the validity of
the system, which bumps it down to the status of a "deeper" doubt. I
suspect this is the crucial psychological effect at work, although I
can't think of any way to confirm it. Moreover, sometimes people treat
Godel's theorem as though it confirmed that consistency statements are
inherently especially dubious. It is "more likely" that a system is
inconsistent and wrong than that it simply is wrong, it seems. :-)
>It seems to me that,
>if one is in the mood to think about "what if something that
>everyone agrees is true turned out to be false," picking the consistency
>of propositional logic as an example has a certain touch of class.
Well, okay, if you like. I prefer more imaginative doubts, which I am
having a harder time thinking up right now. Try imagining instead that
there is a conspiracy, and that *lots* of "well known" results which
have been proven in the literature, and which we take on trust,
actually are fake, designed to fool us. One day this summer, for
example, I checked to make sure that pi really did start with
3.14159..., without depending upon my calculator being "safe". Sheer
paranoia is so much more entertaining than skeptical doubt. :-)
Keith Ramsay
ram...@unixg.ubc.ca
Torkel Franzen
(Frank Adams) writes:
>The point is that in proving the consistency of propositional logic, we
>are using reasoning techniques which include all the techniques built
>in to propositional logic. Now, the main concern I have seen with this
>kind of statement is to wonder what the point is; we don't believe the
>proof unless we already believe what we are trying to prove.
Let us try to keep the two statements involved apart: the
combinatorial statement "logic is consistent", and a statement
expressing trust in our methods of reasoning. Very many people
"believe in" ordinary logic (and trust that it cannot lead them into
contradiction) without even being able to formulate the combinatorial
statement. Now the combinatorial statement, if we choose a suitable
formalization of logic, will be easily proved by using just those
rules of ordinary logic on which we normally rely and which are
formalized in the system. Your reflection above makes good sense if we
try to read into the proof a justification of our trust in our
ordinary modes of reasoning. It is then relevant to observe that the
proof is nothing of the kind. What it is is a simple proof of a simple
combinatorial result, as trustworthy as any other proof of comparable
complexity and level of abstraction. So what is the point of the
proof? It has as much of a point as any of thousands of routine
verifications of expected results in mathematics. It is only the
"mystique of consistency" that inspires people with the idea that
consistency proofs must be either non-trivial or pointless.
>As for the other problem, let us step back for a moment to the point
>where neither of the theorems in question had been proved. In this
>case, their status is very different. If we were to have proved that
>some formulae don't have a disjunctive normal form, this would have
>been an interesting fact. If we had proved that propositional logic
>was inconsistent, it would have been a catastrophe -- there was
>something wrong with our reasoning process.
This is quite arbitrary, as so often in counterfactual reflections
of this kind. We may as well say that if we had proved that
propositional logic (i.e. some formal system so designated) was
inconsistent, it would have established the interesting fact that
"propositional logic" was not in fact a correct formalization of our
methods of reasoning.
>(In fact, just this kind of catastrophe occurred with naive set theory
>-- set theory with the axiom "for any property P, there is a set S of
>all sets having property P. This did invalidate all, or even very
>much, of the reasoning process of mathematics up to that time; but it
>was nevertheless probably the most event in the history of
>mathematics.)
The only person who stated an unrestricted comprehension axiom was
Frege, who did not do set theory, and whose system had had no impact
whatever on mathematics. Neither Cantor nor any other mathematician
took the view that the set-theoretic paradoxes invalidated even
set-theoretical reasoning. But this, of course, is a different topic.
J. Fritz
[...]
> Without Replacement you can construct a model of Z as the union of the
> set of elements of a sequence whose first element is the set of natural
> numbers (at least I think that's enough) and whose i-th element is the
> powerset of its predecessor. Call this model M.
[...]
> I will bet $10,000 *against* Z being found inconsistent within the next
> 20 years, with odds of 1000:1 against. That is, I'm saying that Z
> inconsistent is extremely unlikely, and if you think you can make money
> at any crazy odds on this bet you're the one who's crazy. If you win
> you get my $10,000, if I win I get only your $10.
[...]
At those odds I can't resist. In defense of my sanity, let me say that
while I tend to be a Platonist with respect to the integers, I have no
such conviction about the existence of uncountable sets. (I don't mind
working with them, mind you--I just don't really believe they exist.)
So I think there's a very slight chance that the second step in the
construction of M can't be done. Furthermore, I seem to recall that Z
and ZC are equiconsistent, just as ZF and ZFC are. If that's so, then
if the reals can exist, but a well-ordering of them cannot, I'd still
win.
P.S. Just for clarification, I do assume that by Z we mean ZF without
Replacement, but with Separation; that is, for every predicate p of
one variable, we have:
(Aa)(Eb)(Ay)[y in b <--> y in a & p(y)]
- Fritz (fr...@ruby.vcu.edu)
Disclaimer: nothing I say represents my employer
David Gudeman
]In article <1992Sep25.2...@Cookie.secapl.com> fr...@Cookie.secapl.com
] (Frank Adams) writes:
] >(In fact, just this kind of catastrophe occurred with naive set theory
] >-- set theory with the axiom "for any property P, there is a set S of
] >all sets having property P. This did invalidate all, or even very
] >much, of the reasoning process of mathematics up to that time; but it
] >was nevertheless probably the most event in the history of
] >mathematics.)
]
] The only person who stated an unrestricted comprehension axiom was
]Frege, who did not do set theory, and whose system had had no impact
]whatever on mathematics. Neither Cantor nor any other mathematician
]took the view that the set-theoretic paradoxes invalidated even
]set-theoretical reasoning. But this, of course, is a different topic.
I must take exception with both of these statements. First Russell's
Paradox was only the last nail in the coffin. There were already
several paradoxes running about in the mathematics and logic
community, and logicians were already working on ways to avoid the
paradoxes (mathematicians tended --quite properly-- to ignore them).
In the second place, it did not invalidate anything except for naive
set theory and naive omega-order logic. In the third place the
paradoxes caused problems for _informal_ theories, not for formal
theories like the predicate calculus. The question of the consistency
of a syntactic formal system is quite independent of the question of
the consistency of the reasoning it is intended to reflect.
As for the unrestricted comprehension axioms, set theory was not
axiomatized until many years after Russell's paradox was first
published, but the previous researchers certainly did assume the
existence of an unrestricted comprehension axiom. They only began
restricting it when the various paradoxes began to arise. In fact
Cantor's proof that the cardinality of the set of subsets of S is
larger than the cardinality of S relies pivotally on the creation of a
set that degenerates to Russell's paradoxical set if the theorem were
not true.
My assertions are documented in
.E Jean van Heijenoort
.J From Frege to Go:del A Source Book in Mathematical Logic, 1879-1931
.I Harvard University Press
.C Cambridge, Massachusetts
.D 1967
A book I highly recommend to anyone who wants to learn about these
things.
--
David Gudeman
gud...@cs.arizona.edu
Torkel Franzen
(David Gudeman) writes:
>but the previous researchers certainly did assume the
>existence of an unrestricted comprehension axiom. They only began
>restricting it when the various paradoxes began to arise.
Which previous researchers, and how does it emerge that they assumed
an unrestricted comprehension axiom?
>In fact
>Cantor's proof that the cardinality of the set of subsets of S is
>larger than the cardinality of S relies pivotally on the creation of a
>set that degenerates to Russell's paradoxical set if the theorem were
>not true.
It's unclear what you mean to say here. Russell did indeed derive his
paradox by applying Cantor's proof to a supposed universal set. What
does this have to do with whether or not anybody assumed an unrestricted
comprehension axiom?
Christopher P Menzel
>] The only person who stated an unrestricted comprehension axiom was
>]Frege, who did not do set theory, and whose system had had no impact
>]whatever on mathematics. Neither Cantor nor any other mathematician
>]took the view that the set-theoretic paradoxes invalidated even
>]set-theoretical reasoning. But this, of course, is a different topic.
>
>axiomatized until many years after Russell's paradox was first
Only if 5 counts as many. Russell first published his paradox in The
Principles of Mathematics in 1903; Zermelo's initial axiomatization of
set theory (which included the restricted comprehension axiom, better
known as Separation) appeared in 1908 in the paper "Untersuchungen
"uber die Grundlagen der Mengenlehre, I." It's translated in van
Heijenoort.
--Chris Menzel
Department of Philosophy
Texas A&M University
Randall Holmes
No, it did not. The effect on existing mathematics of the paradoxes
of set theory was zero. It forced the program of founding mathematics
on arithmetic instead of geometry, and thus ultimately on set theory,
which was not very old at that point, to take a different tack.
but it
>was nevertheless probably the most event in the history of
>mathematics.)
Is the word "important" missing here? It was not the most important
event in the history of math by a long shot. It was _an_ important
event, to be sure. It was the most important event in the history of
the philosophical program for mathematics described above. I am a
mathematical logician; I work with this stuff.
>
>The argument is not "we've proved this is true; if it turned out to be
>false it would be a real problem", but "we've assumed this is true (and
>incidently we've proved it); if it turned out to be false it would be a
>real problem".
Frank Adams
>In article <1992Sep25.2...@Cookie.secapl.com> fr...@Cookie.secapl.com
> (Frank Adams) writes:
>
> >The point is that in proving the consistency of propositional logic, we
> >are using reasoning techniques which include all the techniques built
> >in to propositional logic.
>
> >where neither of the theorems in question had been proved. In this
> >case, their status is very different. If we were to have proved that
> >some formulae don't have a disjunctive normal form, this would have
> >been an interesting fact. If we had proved that propositional logic
> >was inconsistent, it would have been a catastrophe -- there was
> >something wrong with our reasoning process.
>
> This is quite arbitrary, as so often in counterfactual reflections
>of this kind. We may as well say that if we had proved that
>propositional logic (i.e. some formal system so designated) was
>inconsistent, it would have established the interesting fact that
>"propositional logic" was not in fact a correct formalization of our
>methods of reasoning.
I'll go you one better: this is a *better* thing to say. It's still a very
different thing from a simple "that happens to be false", which disproof of
the other conjecture would have elicited. Or to put it another way, our
reasoning process has led us to believe very strongly that propositional
logic *does* correctly formalize some of our methods of reasoning. If it
were inconsistent, our reasoning would have failed in a major way.
> Neither Cantor nor any other mathematician
>took the view that the set-theoretic paradoxes invalidated even
>set-theoretical reasoning. But this, of course, is a different topic.
>
It did cause many mathematicians to *worry* that the paradoxes might
invalidate set-theoretical reasoning, however.
Torkel Franzen
com (Frank Adams) writes:
>I'll go you one better: this is a *better* thing to say. It's still a very
>different thing from a simple "that happens to be false", which disproof of
>the other conjecture would have elicited.
Saying that a disproof of "all formulas have a disjunctive normal
form" would only have elicited a reaction on the lines of "that
happens to be false" is not really by itself an intelligible
description of a possible scenario. Are you stipulating that
"classical propositional logic" means the same in your hypothetical
world as it does here and now? And if so, are you not assuming that it
could still be proved in your hypothetical world that every formula
does have a disjunctive normal form? Then the only difference
between "propositional logic is inconsistent" and "there is a formula
in propositional logic that has no disjunctive normal form" would be
that people tend to react automatically to "propositional logic is
inconsistent" as "catastrophic". The actual inconsistency in our
reasoning would be exactly the same in the two cases. Or are you
assuming that "classical propositional logic" means something different in
your hypothetical world? Then there is not necessarily anything even
mildly disturbing, in that world, about propositional logic being
inconsistent.
Frank Adams
>In article <1992Sep25.2...@Cookie.secapl.com> fr...@Cookie.secapl.com (Frank Adams) writes:
>>was nevertheless probably the most [important] event in the history of
>>mathematics.
>
> It was not the most important
>event in the history of math by a long shot.
All right; what was the most important event in the history of mathematics?
My vote is in; what do the rest of you think?
Terry Tao
Feynman said that in his HO Tartagalia's solving of the cubic. Not because
that this was really important in itself, but because it showed that we could
beat the ancient Greeks.
I don't think there were many single events, though: more of a long, gradual
process. The gradual acceptance of Arabic numerals, for example, or symbolic
algebra. Individuals did do a huge amount of work which then changed mathematics
forever, but not with one single achievement.
David Petry
>All right; what was the most important event in the history of mathematics?
>My vote is in; what do the rest of you think?
Gauss' birthday?
G.J. McCaughan
>All right; what was the most important event in the history of mathematics?
>My vote is in; what do the rest of you think?
I don't have a single event I'd vote for. High contenders:
* Discovery of irrational numbers
* Discovery of coordinate geometry
* [not an event as such] Realisation that issues of convergence etc. are
harder than they look, leading to rigorisation of analysis
* Riemann's paper on prime numbers and the zeta-function
* Discovery of the calculus
* Cantor's proof that 2^x>x
* Discovery of the Lebesgue integral and measure
* Discovery of generating functions
* Birth of the idea of proof from axioms ("Elements" or soon before)
* Birth of Gauss (OK, not the sort of event you meant)
I'm sure there are plenty of others I haven't mentioned. In each case the
influence of the event on future mathematics was very great.
--
Gareth McCaughan Dept. of Pure Mathematics & Mathematical Statistics,
gj...@cus.cam.ac.uk Cambridge University, England. [Research student]
norman danner
of disrespect...]
Hand's down, the first and second incompleteness theorems.
I think the most important result really depends on the area of mathematics.
In particular, most analysists will proabaly disagree with me. But I
think there's a strong argument to be made that the incompleteness theorems
were the most important results when it comes to philosophy of mathematics.
Just my $.02
--
Norman Danner |"There are three kinds of mathematicians:
"The guy with the hair." | Those who can count, and those who can't."
nda...@indiana.edu | --???
-----------------------------------------------------------------------------
Herman Rubin
I would place Diophantus' discovery of the idea of using symbols for quantities
as the most important. Second, I would put the discovery that an axiomatic
approach could have lots of consequences, which leads to the idea that a
simple structure can be very powerful.
--
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hru...@snap.stat.purdue.edu (Internet, bitnet)
{purdue,pur-ee}!snap.stat!hrubin(UUCP)
Michael Zeleny
>In article <1992Sep28....@guinness.idbsu.edu>
>hol...@opal.idbsu.edu (Randall Holmes) writes:
>>In article <1992Sep25.2...@Cookie.secapl.com>
>>fr...@Cookie.secapl.com (Frank Adams) writes:
FA:
>>>[the proof of the inconsistency of naive set theory]
>>>was nevertheless probably the most [important] event in the history of
>>>mathematics.
RH:
>> It was not the most important
>>event in the history of math by a long shot.
Getting pretty authoritative in your old age, aren't you?
FA:
>All right; what was the most important event in the history of mathematics?
>My vote is in; what do the rest of you think?
The discovery of zero by a swarthy individual whose name escapes me at the
moment, closely followed by Galileo's discovery of the paradoxes of the
infinite.
cordially,
mikhail zel...@husc.harvard.edu
"Un de mes plus grands plaisirs est de jurer Dieu quand je bande."
Bill Taylor
Its supreme significance is affirmed by its unique (apocryphal?) status as
the only time a mathematician has been killed for making a shattering mathematical discovery !
Gerald Edgar
The discovery by the Greeks that: (1) simple (obvious) geometric facts
require proof; and (2) it is possible to provide proofs for them.
--
Gerald A. Edgar Internet: ed...@mps.ohio-state.edu
Department of Mathematics Bitnet: EDGAR@OHSTPY
The Ohio State University telephone: 614-292-0395 (Office)
Columbus, OH 43210 -292-4975 (Math. Dept.) -292-1479 (Dept. Fax)
Charles Yeomans
>In article <1992Sep28.1...@Cookie.secapl.com> fr...@Cookie.secapl.com (Frank Adams) writes:
>
>>All right; what was the most important event in the history of mathematics?
>>My vote is in; what do the rest of you think?
>
>I don't have a single event I'd vote for. High contenders:
>
> * Discovery of irrational numbers
> * Discovery of coordinate geometry
> * [not an event as such] Realisation that issues of convergence etc. are
> harder than they look, leading to rigorisation of analysis
> * Riemann's paper on prime numbers and the zeta-function
> * Discovery of the calculus
> * Cantor's proof that 2^x>x
> * Discovery of the Lebesgue integral and measure
> * Discovery of generating functions
> * Birth of the idea of proof from axioms ("Elements" or soon before)
> * Birth of Gauss (OK, not the sort of event you meant)
>
I pick the appearance of Springer-Verlag in mathematics publishing, based on
the preponderance of the color yellow on my bookshelves.
Charles Yeomans
cyeo...@ms.uky.edu
yeo...@austin.onu.edu
Robert J Harley
on his fingers.
__ '. .` __
__/_/\___________________________________________________oOo_U_oOo_____/\_\__
/_/\_\/ B-) :o) :-b I don't vote--- _|___|___|___|_ \/_/\_\
\_\/_/\ H a r l e y it only encourages them. _|___|___|_ /\_\/_/
/_/\_\/ _|___|_ \/_/\_\
\_\/_/\__ro...@vlsi.cs.caltech.edu_____________________________________/\_\/_/
\_\/ \/_/
Randall Holmes
This was my immediate reaction, too, although I'm not certain.
Frank Adams
>com (Frank Adams) writes:
>
> >I'll go you one better: this is a *better* thing to say. It's still a very
> >different thing from a simple "that happens to be false", which disproof of
> >the other conjecture would have elicited.
>
> Saying that a disproof of "all formulas have a disjunctive normal
>form" would only have elicited a reaction on the lines of "that
>happens to be false" is not really by itself an intelligible
>description of a possible scenario. Are you stipulating that
>"classical propositional logic" means the same in your hypothetical
>world as it does here and now? And if so, are you not assuming that it
>could still be proved in your hypothetical world that every formula
>does have a disjunctive normal form?
Try substituting Fermat's Last Theorem for your example. A proof that FLT
was false would be interesting. By contrast, a proof that ZFC was
inconsistent would be a crisis (not a catastrophe; we depend on ZFC much
less than on propositional logic).
Similarly, in my hypothetical world where the disjunctive normal form
theorem was not proved, its disproof would not be a crisis. The subsequent
proof, whenever it was found, would be a very serious crisis. But that is
*because* it is an inconsistency.
Marko Amnell
(Randall Holmes) writes:
>In article <BvC6r...@cantua.canterbury.ac.nz> w...@math.canterbury.ac.nz (Bill Taylor) writes:
>>Got to be the discovery of irrationals by a Pythagorean (whose name escapes me).
>>
>>Its supreme significance is affirmed by its unique (apocryphal?) status as
>>the only time a mathematician has been killed for making a shattering mathematical discovery !
>This was my immediate reaction, too, although I'm not certain.
The discovery of irrational numbers was probably made by some Pythagorean
before 400 B.C. Proclus, a fifth century Neoplatonist, attributes the
discovery of 'the theory or study of irrationals (ten ton alogon
pragmateian)' to Pythagoras himself, but scholars now agree that this is
not credible. We know it must have taken place before 400 B.C. since
Plato writes in the _Theatetus_ that his teacher, Theodorus of Cyrene,
was already proving the irrationality of the square roots of three, five
and seventeen. An anonymous scholiast to Euclid makes the famous claim
that 'the first of the Pythagoreans who made public the investigation of
these matters perished in a shipwreck'. The attribution to the
Pythagoreans is backed up by a commentary in an Arabic translation that
probably originated from Pappus, a mathematician who worked in
Alexandria at the end of the third century A.D. The commentary notes
that the theory of irrationals 'had its origins in the school of
Pythagoras'. The name of the discover is lost to us.
--
Marko Amnell
amn...@klaava.helsinki.fi
Graduate Student in Philosophy
Torkel Franzen
com (Frank Adams) writes:
So what you are saying is that if we had somehow managed to avoid
proving that every formula has a disjunctive normal form, a proof that
some formulas do not have a disjunctive normal form would not have
caused an immediate furor, unlike a proof that propositional logic was
inconsistent. All that this seems to amount to is that we need to
think a little bit to see that every formula has a disjunctive normal
form, whereas the consistency of propositional logic is immediate.
Now, I've all but forgotten what was supposed to be the point of this.
Let's see. I said that a disproof of "every formula has a disjunctive
normal form" would show that our minds are seriously flawed, every bit
as much as a proof that propositional logic is inconsistent. Is there
anything in your present reflections that tells against this?
Hans de Vreught
>Got to be the discovery of irrationals by a Pythagorean (whose name escapes me)
>Its supreme significance is affirmed by its unique (apocryphal?) status as
>the only time a mathematician has been killed for making a shattering
>mathematical discovery !
His name was Hippasos of Metapontum. The legend goes that his disclosure of
the discovery of irrational numbers resulted in the death penalty by drowning.
It certainly was his most important event in the history of mathematics...
--
Hans de Vreught | John von Neumann:
hd...@dutiaa.tudelft.nl | Young man, in mathematics
Delft University of Technology (TWI-ThI) | you don't understand things,
The Netherlands | you just get used to them.
Marko Amnell
(Hans de Vreught) writes:
>w...@math.canterbury.ac.nz (Bill Taylor) writes:
>
>>Got to be the discovery of irrationals by a Pythagorean (whose name escapes me)
>>Its supreme significance is affirmed by its unique (apocryphal?) status as
>>the only time a mathematician has been killed for making a shattering
>>mathematical discovery !
>
>His name was Hippasos of Metapontum. The legend goes that his disclosure of
>the discovery of irrational numbers resulted in the death penalty by drowning.
>It certainly was his most important event in the history of mathematics...
The whole story is pure legend. I challenge you to cite any reputable
scholarly evidence that Hippasus discovered the irrational numbers. As
someone pointed out earlier in another thread in sci.physics, the
discoverer was not executed, but perhaps did perish in a shipwreck.
It is true that the Pythagoreans kept their discovery secret for some
time because of unpleasant consequences in geometry...
The alleged mathematical achievements of Hippasus are addressed in Iamblichus'
_On the Pythagorean Way of Life_. I will cite two relevant passages
About Hippasus they say that he was one of the Pythagoreans but that
because he was the first to publish and construct the sphere of the
twelve pentagons he died at sea as an impious man. He acquired the
reputation of discovering it, although everything belongs to That
Man (that is how they refer to Pythagoras, never calling him by his
name). Iamblichus 88.
Some say that the divinity punished those who made Pythagoreans' views
public. For the man who revealed the construction of the vigintangle
perished at sea an an impious man. (The vigintangle is the
dodecahedron, one of the so-called five solid figures, when it extends
into a shpere.) Others said that it was the man who spoke about
irrationality and incommensurability who suffered this fate.
Iamblichus 247.
Hippasus was an average Presocratic cosmologist, he was certainly not
a great mathematician. The name of the discoverer of irrational numbers
is lost to us. Let us simply call him That Man. :-)
Torkel Franzen
com (Frank Adams) writes:
>A proof of the
>inconsistency of propositional logic is a more direct attack on the
>consistency of our reasoning process, thus considerations of possible
>mental flaws are more reasonable when discussed in the context of that
>theorem.
I can't make any sense of this whatever. You seem to be insisting
that an elementary proof and disproof of one statement "is a more
direct attack on the consistency of our reasoning process" than an
elementary proof and disproof of another statement. Well, whatever
this may mean, I have no inclination to dispute it.
Frank Adams
>Let's see. I said that a disproof of "every formula has a disjunctive
>normal form" would show that our minds are seriously flawed, every bit
>as much as a proof that propositional logic is inconsistent. Is there
>anything in your present reflections that tells against this?
No, but you go on to argue that there is no more point to talking about our
minds being seriously flawed in one case than the other. This is the point
I was arguing against.
For the last time: it is inconsistencies in our reasoning process which
lead us to doubt the perfection of our minds. A disproof of "every formula
has a disjunctive normal form" would be a problem only in that, in
conjunction with its proof, it constitutes an inconsistency. A proof of the
Hans de Vreught
>In article <hdev.717797409@dutiag> hd...@dutiag.tudelft.nl
>(Hans de Vreught) writes:
>>w...@math.canterbury.ac.nz (Bill Taylor) writes:
>>
>>>Got to be the discovery of irrationals by a Pythagorean (whose name escapes me)
>>>Its supreme significance is affirmed by its unique (apocryphal?) status as
>>>the only time a mathematician has been killed for making a shattering
>>>mathematical discovery !
>>
>>His name was Hippasos of Metapontum. The legend goes that his disclosure of
>>the discovery of irrational numbers resulted in the death penalty by drowning.
>>It certainly was his most important event in the history of mathematics...
Please note I said the disclosure not the discovery and furthermore I spoke of
the legend.
>The whole story is pure legend. I challenge you to cite any reputable
>scholarly evidence that Hippasus discovered the irrational numbers. As
>someone pointed out earlier in another thread in sci.physics, the
>discoverer was not executed, but perhaps did perish in a shipwreck.
>It is true that the Pythagoreans kept their discovery secret for some
>time because of unpleasant consequences in geometry...
OK, I found it in a Dutch book of an old professor of mine. He has worked on
the translation of several old papers and has worked in several teams on this
subject. To be short: a respectable scientist. The book gives some highlights
in the history of math using the original texts. The book is called:
A.W. Grootendorst, "Grepen uit de Geschidenis van de Wiskunde",
Delftse Uitgevers Maatschappij, 1988.
On page 49: "De ontdekking daarvan wordt toegeschreven aan Hippasos van
Metapontum, volgeling van Pythagoras en het verhaal gaat dat hij als straf
voor de bekendmaking daarvan de verdrinkingsdood heeft gevonden." Translated in
English: "The discovery of that (irrational numbers) is attributed to Hippasos
of Metapontum, a follower of Pythagoras and the story goes that for the
disclosure he was penalized by being drowned." Although Grootendorst used the
word "story", he meant "legend" (actually in the Dutch sentence it is used in
that context).
So there is my justification. I don't see why it is so much different from the
story of the shipwreck. I can easily imagine how the legend could emerge from
that story. But I did clearly state it was a legend. Maybe it's true but
probably it's not. Either way it is a nice story.
Personally I think you jumped the gun too soon. Well, in any case I end it
here.
Frank Adams
A hypothetical proof of the inconsistency of the propositional calculus is
a problem ALL BY ITSELF, because we believe that the propositional calculus
is consistent even BEFORE we prove it, because it represents a formalization
of part of our reasoning process.
By contrast, a proof of the impossibility of generally finding a conjunctive
normal form is a problem ONLY IN CONJUNCTION WITH THE PROOF THAT SUCH A FORM
ALWAYS EXISTS.
One is a direct assault on our reasoning processes, one is more indirect.
That's all there is to it.
Philip Bond
system. Previous cultures ( Greek, Roman etc. ) failed to develop beyond
geometry mainly due to a lack of this system. With the invention of
decimal representation mankind really took a giant leap forward. Of
course it took time to develop but I think ( my opinion only ) that the
Arab mathematicians should be given credit for bringing the system to
fruition.
The second development was the calculus.
Phil.
My humble opinion only. Usual disclaimers apply.
Andy Bennett
>My vote is split : The first event was the development of the decimal
>system. Previous cultures ( Greek, Roman etc. ) failed to develop beyond
>geometry mainly due to a lack of this system. With the invention of
>decimal representation mankind really took a giant leap forward. Of
>course it took time to develop but I think ( my opinion only ) that the
>Arab mathematicians should be given credit for bringing the system to
>fruition.
One problem with this "event." Positional value systems actually evolved
earlier than the Greeks and were even known and used by greek scientists
of the hellenistic period.
The babylonians used a positional number system on a base of 60. They
inherited the system from the sumerians but they then developed a nice
theory of algebra using it. The system did include both integer and
non-integer values (i.e. 1/2 was written as .(30) representing 30/60)
See "The Exact Sciences in Antiquity" by Neugebauer. Greek scientists
were aware of the system at least after Alexander's conquest of Persia
if not before and used the babylonian cuneiform for computations in their
"scientific" work, for example in Ptolemy's Almageist. The influence on
astronomy survives in our modern division of degrees into 60 minutes and
minutes into 60 seconds, along with the corresponding divisions of time.
The system was not used in "mathematical" work for philosophic reasons,
even though greek "scientists" and "mathematicians" were usually one and
the same people. This displays the two-edged nature of the Greek philosophy
of mathematics depending on proof. While it let the Greeks raise mathematics
from a set of tricks to an understood system of knowledge, it hindered them
from adopting elements outside the tradition that would be useful. You may
draw your own lessons for modern mathematics from this.
--
Andrew G. Bennett ben...@math.ksu.edu If you count too
Dept. of Mathematics Voice: (913) 532-6750 much you turn
Kansas State University Fax: (913) 532-7004 purple. - SARAH
Manhattan, KS 66502 STRICTLY MY OWN OPINIONS
Gregory McColm
>In article <1992Sep28.1...@Cookie.secapl.com>, fr...@Cookie.secapl.com (Frank Adams) writes:
>|> In article <1992Sep28....@guinness.idbsu.edu> hol...@opal.idbsu.edu (Randall Holmes) writes:
>|> >In article <1992Sep25.2...@Cookie.secapl.com> fr...@Cookie.secapl.com (Frank Adams) writes:
>|> >>[the proof of the inconsistency of naive set theory]
>|> >>was nevertheless probably the most [important] event in the history of
>|> >>mathematics.
>|> >
>|> > It was not the most important
>|> >event in the history of math by a long shot.
>|>
>|> All right; what was the most important event in the history of mathematics?
>|> My vote is in; what do the rest of you think?
>
>Feynman said that in his HO Tartagalia's solving of the cubic. Not because
>that this was really important in itself, but because it showed that we could
>beat the ancient Greeks.
>
The discovery of noncommensurables, undermining the
first foundation of mathematics (number theory) and
the discovery of nonEuclidean geometry, undermining
the second foundation of mathematics (geometry) were
both more important than Russell's paradox (which
I gather is what is meant by "the inconsistency of
naive set theory"). Despite the paradox, the third
foundation of mathematics (set theory) still stands
and is still accepted.
-----Greg McColm
Frank Adams
Since most respondents made no effort to choose just one, I am counting
nominations, not votes.
Essentially, only 3 items got more than one nomination apiece. Clearly,
there is no consensus here.
4 The discovery of irrationals
3 The invention/discovery of the calculus.
1 The discovery that an axiomatic approach could have lots of consequences
1 Birth of the idea of proof from axioms ("Elements" or soon before)
1 The discovery by the Greeks that: (1) simple (obvious) geometric facts
require proof; and (2) it is possible to provide proofs for them.
1 The decision that theorems need proofs
1 The publication of The Elements (Stoicheia) of Euclid.
1 Russell's paradox.
1 The Pythagorian theorem.
1 The solution of higher degree equations;
1 Diophantus' discovery of the idea of using symbols for quantities
1 Discovery of coordinate geometry
1 Realisation that issues of convergence etc. are harder than they look,
leading to rigorisation of analysis
1 Riemann's paper on prime numbers and the zeta-function
1 Cantor's proof that 2^x>x
1 Discovery of the Lebesgue integral and measure
1 Discovery of generating functions
1 The first and second incompleteness theorems.
1 The discovery of zero
1 The discovery of the paradoxes of the infinite.
1 Counting on one's fingers.
1 The development of the decimal system.
1 The discovery of nonEuclidean geometry
1 The discovery that ordinal and cardinal finite numbers coincide.
1 The discovery that the numbers go on forever
1 The discovery of idealizable shapes---circles, triangles, etc
1 Number as the first foundation of mathematics.
1 Geometry as the second foundation of mathematics.
1 The use of idealized forms to represent scientific problems
1 The notion that theories merely describe, but do not explain or provide The
Truth.
Ben Fok the lemming
Randall Holmes
It doesn't rate much more than a footnote so far (relative to other
things brought up in this thread).
John C. Baez
Fractals?" Let's hope that this is a joke... since perhaps it *is* the
most important event for the lemmings among us.
Dan Pehoushek
Dan Pehoushek
Bandwidth? What's that?