bwk@mitre-bedford.ARPA (Barry W. Kort) (02/23/88)
My choice is to classify the Axiom of Choice as synthetic and a posteriori. --Barry Kort
weemba@garnet.berkeley.edu (Obnoxious Math Grad Student) (02/23/88)
In article <25011@linus.UUCP>, bwk@mitre-bedford (Barry W. Kort) writes: >My choice is to classify the Axiom of Choice as synthetic and >a posteriori. I favor analytic a posteriori, myself. "analytic", since all mathematical truths are, and "a posteriori", since AC is based on our derived perceptions of sets. [Oh Greg, I originally missed your article, thanks to the joys of `K'.] ucbvax!garnet!weemba Matthew P Wiener/Brahms Gang/Berkeley CA 94720
g-rh@cca.CCA.COM (Richard Harter) (02/23/88)
In article <7123@agate.BERKELEY.EDU> weemba@garnet.berkeley.edu (Obnoxious Math Grad Student) writes: >In article <25011@linus.UUCP>, bwk@mitre-bedford (Barry W. Kort) writes: >>My choice is to classify the Axiom of Choice as synthetic and >>a posteriori. > >I favor analytic a posteriori, myself. "analytic", since all mathematical >truths are, and "a posteriori", since AC is based on our derived perceptions >of sets. Do I understand this correctly then that the Axiom of Choice is acceptable to Christians, who are not bound by the Law, but not to Jews, since a posteriori knowledge is proscribed in Leviticus? Sorry about that :-) . . . -- In the fields of Hell where the grass grows high Are the graves of dreams allowed to die. Richard Harter, SMDS Inc.
greg@mind.UUCP (greg Nowak) (02/23/88)
In article <7123@agate.BERKELEY.EDU> weemba@garnet.berkeley.edu (Obnoxious Math Grad Student) writes: }In article <25011@linus.UUCP>, bwk@mitre-bedford (Barry W. Kort) writes: }>My choice is to classify the Axiom of Choice as synthetic and }>a posteriori. } }I favor analytic a posteriori, myself. "analytic", since all mathematical }truths are, and "a posteriori", since AC is based on our derived perceptions }of sets. I'm about to violate my own proscription against discussing the Kantian philosophy behind the question. Most people agree that the "analytic a posteriori" class of propositions is empty. Do you have grounds independent of your consideration of AC for considering the class non-empty? Your argument is nonspecific to AC and can easily be extended to the rest of mathematics. To the extent that Kant himself considered geometry to be synthetic a priori, you may be using the terms differently than Kant did. Are you? -- greg
rapaport@sunybcs.uucp (William J. Rapaport) (02/23/88)
In article <7123@agate.BERKELEY.EDU> weemba@garnet.berkeley.edu (Obnoxious Math Grad Student) writes: > >I favor analytic a posteriori, myself. "analytic", since all mathematical >truths are, and "a posteriori", since AC is based on our derived perceptions >of sets. I guess you've not read Kant. All mathematical propositions are synthetic apriori for him. By that, I'd take the Axiom of Choice as synthetic apriori, too. William J. Rapaport Assistant Professor Dept. of Computer Science||internet: rapaport@cs.buffalo.edu SUNY Buffalo ||bitnet: rapaport@sunybcs.bitnet Buffalo, NY 14260 ||uucp: {ames,boulder,decvax,rutgers}!sunybcs!rapaport (716) 636-3193, 3180 ||
bwk@mitre-bedford.ARPA (Barry W. Kort) (02/24/88)
In article <24871@cca.CCA.COM> g-rh@CCA.CCA.COM.UUCP (Richard Harter) writes: >Do I understand this correctly then that the Axiom of Choice is acceptable >to Christians, who are not bound by the Law, but not to Jews, since a >posteriori knowledge is proscribed in Leviticus? Perhaps Richard believes that the Chosen People cannot also be the Choosing People. I choose to believe otherwise. :-) --Barry Kort
greg@mind.UUCP (greg Nowak) (02/24/88)
In article <8768@sunybcs.UUCP> rapaport@gort.UUCP (William J. Rapaport) writes: }In article <7123@agate.BERKELEY.EDU> weemba@garnet.berkeley.edu (Obnoxious Math Grad Student) writes: }> }>I favor analytic a posteriori, myself. "analytic", since all mathematical }>truths are, and "a posteriori", since AC is based on our derived perceptions }>of sets. } }I guess you've not read Kant. All mathematical propositions are }synthetic apriori for him. By that, I'd take the Axiom of Choice as }synthetic apriori, too. This is taking the question perhaps too literally. After Kant, developments in non-Euclidean geometry made it clear that geometry was not synthetic a priori in the way that Kant thought it was. It's pretty clear, pace Matthew's support of analytic a posteriori, that mathematical propositions are either synthetic or analytic a priori. Now that further research has moved geometry up to the level of analytic propositions, the search for synthetic a priori propositions moves on. If anyone is really interested, there's an article by Michael Friedman in the 1985 volume of Philosophical Review on "Kant and Geometry" that discusses the synthetic/analytic distinction quite well. -- greg
g-rh@cca.CCA.COM (Richard Harter) (02/24/88)
In article <25121@linus.UUCP> bwk@mbunix (Barry Kort) writes: >In article <24871@cca.CCA.COM> g-rh@CCA.CCA.COM.UUCP (Richard Harter) writes: >>Do I understand this correctly then that the Axiom of Choice is acceptable >>to Christians, who are not bound by the Law, but not to Jews, since a >>posteriori knowledge is proscribed in Leviticus? > >Perhaps Richard believes that the Chosen People cannot also be the >Choosing People. I choose to believe otherwise. :-) Nay, I was referring to the two forms of carnal knowledge, a priori and a posteriori. Upon reflection, it occurs to me that Jewish physicists may not accept the axiom of choice because of the exclusion principle. However this would depend, I suppose, on whether they were bosons or fermions. Strangely enough, this point was neglected in my education -- perhaps someone can enlighten us on this point. :-) -- In the fields of Hell where the grass grows high Are the graves of dreams allowed to die. Richard Harter, SMDS Inc.
hbe@math.ucla.edu (02/25/88)
In article <8768@sunybcs.UUCP> rapaport@gort.UUCP (William J. Rapaport) writes: >I guess you [weemba@garnet.berkeley.edu} have not read Kant. All mathematical >propositions are synthetic apriori for him. By that, I'd take the Axiom of >Choice as synthetic apriori, too. Kant's classification of mathematical truths as synthetic a priori seems in light of Godel to be more wrong than right. The a priori statements form merely a recursively enumerable set, and hence cannot include even all the true statements of arithmetic. I think that the analytic a posteriori is an unjustly neglected category in the philosophy of mathematics. I am not sure, however, that I want to put the axiom of choice into it. --Herb Enderton, hbe@math.ucla.edu
rar@DUCHAMPS (Bob Riemenschneider) (02/26/88)
=> The a priori statements form merely a recursively enumerable set, ... => => --Herb Enderton, hbe@math.ucla.edu I don't see why this must be the case. Mightn't (some) people come equipped with knowledge of a more complicated set of truths, or with knowledge of a non-effective rule of inference? -- rar
jserembu@sjuvax.UUCP (J. Serembus) (02/26/88)
Analytic a posteriori statements have received serious consideration in philosophical circles. Two philosophers of interest here would be Saul Kripke and Hilary Putnam. Check out Kripke's "Naming and Necessity". -- --------------------------------------------------------------------------------JOHN SEREMBUS PHILOSOPHY DEP'T ST. JOSEPH'S UNIVERSITY --------------------------------------------------------------------------------
greg@mind.UUCP (greg Nowak) (02/26/88)
In article <9734@shemp.CS.UCLA.EDU> hbe@math.ucla.edu (H. Enderton) writes: >In article <8768@sunybcs.UUCP> rapaport@gort.UUCP (William J. Rapaport) writes: >>I guess you [weemba@garnet.berkeley.edu have not read Kant. All mathematical >>propositions are synthetic apriori for him. By that, I'd take the Axiom of >>Choice as synthetic apriori, too. > >Kant's classification of mathematical truths as synthetic a priori seems >in light of Godel to be more wrong than right. The a priori statements >form merely a recursively enumerable set, and hence cannot include even >all the true statements of arithmetic. I think that the analytic >a posteriori is an unjustly neglected category in the philosophy of >mathematics. I am not sure, however, that I want to put the axiom of >choice into it. The "analytic a posteriori" category is neglected because it is EMPTY. Anything which is analytic, i.e. true for semantic reasons, is a priori (independent of experience) -- even if the terms refer to objects we usually encounter with experience. Thus "all bachelors are male" is an analytic a priori statement, because it is true by the meanings of the words -- even though we DO have real-world experience of males and bachelors, we NEED not to judge the truth of the statement. "The sky is blue" is a synthetic a posteriori proposition, because nothing in the definition of what a sky is requires it to be blue. The suggestion about enumerating the set of propositions would seem to be more applicable to the *analytic* propositions than the a priori ones, thus strengthening the claim that there are synthetic a priori propositions. -- greg
hbe@math.ucla.edu (02/26/88)
In article <8802251655.AA06216@duchamps.ads.arpa> rar@DUCHAMPS (Bob Riemenschneider) writes: >=> The a priori statements form merely a recursively enumerable set, ... >I don't see why this must be the case. Mightn't (some) people come >equipped with knowledge of a more complicated set of truths, or with >knowledge of a non-effective rule of inference? For us to *know* a truth of arithmetic, we'll need to see some effectively verifiable supporting argument--a proof. If the set of acceptable proofs is decidable (or even r.e.), then the set of provable statements is r.e. Here we are considering not simply proofs according to some fixed rules, but arguments that could *ever* be accepted (in theory) by human mathematicians. Even so, this still seems to lead to an r.e. set. To "know" that Fermat's last theorem is true, it's not enough to believe it and state it. You need a supporting reason I can check. --H. Enderton, hbe@ucla.math.edu
markh@csd4.milw.wisc.edu (Mark William Hopkins) (02/26/88)
In article <8802251655.AA06216@duchamps.ads.arpa> rar@DUCHAMPS (Bob Riemenschneider) writes: >=> The a priori statements form merely a recursively enumerable set, ... >=> >=> --Herb Enderton, hbe@math.ucla.edu > >I don't see why this must be the case. Mightn't (some) people come >equipped with knowledge of a more complicated set of truths, or with >knowledge of a non-effective rule of inference? Yes, me. ~~~~~~~~~~~~~~~~~~ What difference does it make what we call the axiom of choice? All sets are finite anyway.
jserembu@sjuvax.UUCP (J. Serembus) (02/29/88)
Greg, How would you classify the following the following proposition: "Water is H2O."? It certainly seems to be one gotten from experience and thus a posteriori; yet there also seems to be something necessary about it, hence analytic. -- --------------------------------------------------------------------------------JOHN SEREMBUS PHILOSOPHY DEP'T ST. JOSEPH'S UNIVERSITY --------------------------------------------------------------------------------
rar@DUCHAMPS (Bob Riemenschneider) (03/01/88)
=> For us to *know* a truth of arithmetic, we'll need to see some => effectively verifiable supporting argument--a proof. If the set => of acceptable proofs is decidable (or even r.e.), then the set => of provable statements is r.e. Here we are considering not simply => proofs according to some fixed rules, but arguments that could => *ever* be accepted (in theory) by human mathematicians. Even so, => this still seems to lead to an r.e. set. => => To "know" that Fermat's last theorem is true, it's not enough to => believe it and state it. You need a supporting reason I can check. => => --H. Enderton, hbe@ucla.math.edu Although this sort of foundationalist view of knowledge has been pretty thoroughly discredited, I can see how it might remain attractive when restricted to mathematics. I can also see that the "socially knowable" arithemetic truths are r.e., given some unstated assumptions having to do with the possible future evolution of mathematics and mathematicians (and, of course, Church's thesis). Perhaps my memory's faulty, but I thought you had claimed that the set of all a priori truths was r.e. Do you belive that your argument generalizes, or that all a priori truths are mathematical? Finally, I found the use of `us' and `we' in the reply interesting. Was this meant to exclude the possibility of private mathematical knowledge? E.g., suppose Godel walked into your office--it turns out that he didn't die, he just wanted people to leave him alone--and claimed that, on close inspection of the natural numbers, he perceived that Fermat's last theorem is true. On questioning him, it turns out that he can supply no supporting reason that you can check--he says that he knows it on the basis of observation. (Mystical Platonists of Godel's ilk--for some reason, a lot of the set theorists of my acquaintance seem to fall into this category--draw a clear distinction between "I know" and "I can prove".) Do you really want to claim that it cannot be the case that he knows it because he cannot prove it? -- rar
markh@csd4.milw.wisc.edu (Mark William Hopkins) (03/02/88)
In article <1186@sjuvax.UUCP> jserembu@sjuvax.UUCP (J. Serembus) writes: > > Greg, > > How would you classify the following the following proposition: > > "Water is H2O."? As a mere consequence of a convention concerning the naming of chemical compounds. It's all conventional.
greg@phoenix.Princeton.EDU (Gregory Nowak) (03/03/88)
In article <1186@sjuvax.UUCP> jserembu@sjuvax.UUCP (J. Serembus) writes: > Greg, > How would you classify the following the following proposition: > "Water is H2O."? It certainly seems to be one gotten from experience and thus >a posteriori; yet there also seems to be something necessary about it, hence >analytic. H2O is a chemical compound of three atoms; water is a substance I'm familiar with from everyday experience. If I introduce a spark into a combination of hydrogen and oxygen, a substance will be produced, but I have no way of knowing that it's "water" unless I check. Conversely, none of my experiences of water lead me to conclude that it HAS to be formed of molecules which have two hydrogen atoms and one oxygen atom each. I would consider this statement to be synthetic a posteriori. I suppose I question the assertion that necessary truths are necessarily analytic, and there are some people who have the same doubts about Kripke. (He's in Princeton; maybe I'll go ask him.) I would take as an example of a necessary analytic truth the fact that I am not a nonexistent colorless green dream; as an example of a necessary synthetic truth the fact that the speed of light is 298XXX (I forget the last three digits) meters per second. (We've defined the meter so this is true.)
torkel@sics.se (Torkel Franzen) (03/04/88)
In article <9759@shemp.CS.UCLA.EDU> hbe@math.ucla.edu (H. Enderton) writes: > >For us to *know* a truth of arithmetic, we'll need to see some >effectively verifiable supporting argument--a proof. If the set >of acceptable proofs is decidable (or even r.e.), then the set >of provable statements is r.e. Here we are considering not simply >proofs according to some fixed rules, but arguments that could >*ever* be accepted (in theory) by human mathematicians. Even so, >this still seems to lead to an r.e. set. It makes little sense to say or suppose that "the set of acceptable proofs" is or is not decidable or r.e., for there is no such set. The same is true of "the set of provable statements". "Acceptable proof" is not a concept with a determinate extension, any more than "definable set". (As opposed to "proof in the formal system S", "definable in the formal language L".) The idea expressed in the quoted passage crops up here and there in the philosophical literature. Thus e.g. Putnam: ...the statements that can be proved from axioms which are evident to us can only be a recursively enumerable set (unless an infinite number of irreducibly different principles are at least potentially evident to the human mind, a supposition I find quite incredible). Why are such arguments inconclusive? Well, first, if a principle is not formal, its consequences do not form a recursively enumerable set. Informal principles have no definite set of consequences at all, but only applications (formal principles among them) which are more or less direct, far-fetched, imaginative, convincing, etc. Mach's principle and the set-theoretic reflection principle ("anything true in the universe is true in some set") are examples of such informal principles. In the same vein one may object that there is no definite number at all of principles potentially evident to the human mind, any more than there is a definite number of potential scientific theories or works of art. Finally, it seems only too likely that any principle at all is potentially a c c e p t a b l e to the human mind. The distinction between what is evident and what is merely accepted is dubious when applied to hypothetical principles. It is of course possible to take the view that there nevertheless i s a determinate concept of "proof", "valid argument", "conclusive reasoning" in terms of which it makes good sense to suppose that "the set of provable statements" is or is not recursively enumerable. This very problematic view is held by some in the philosophy of mathematics.
george@scirtp.UUCP (Geo. R. Greene, Jr.) (03/19/88)
> > Greg, > > How would you classify the following the following proposition: > > "Water is H2O."? > > > It certainly seems to be one gotten from experience and thus > a posteriori; yet there also seems to be something necessary about it, hence > analytic. I would classify it as false. One pair of counterexamples leaps immediately to mind. Ice is not normally called water; neither is steam. "Water" is a colloquial term, H2O a technical one. To those who would object that since, even colloquially, ice is frozen water, ice must be water, I would answer simply that in any situation where both frozen and non-frozen water are alternatives, nobody except MAYBE your neighborhood hyper-pedantic philosopher is going to ask you to choose between frozen and non-frozen water. You are going to be asked to choose between ice and water. As though never the twain had met. Water is non-frozen and non-heat-vaporized BY DEFAULT. "Is" is one of the most overloaded words in the language. Sometimes it means equality, sometimes it means subsetting (Bats are mammals), sometimes it means set membership (My pet is A Bat); sometimes it's about holes & pegs (My house is at 5 Fifth St.); sometimes it's just an auxiliary verb (The temperature is rising). The prototypical analytic statement is (A & B) ==> A , i.e., its truth is deducible just from analyzing its parts and noticing that what is concluded is "part of" what has "already" been hypothesized. An interesting consequence of this is that all statements regarding explicitly exhibited substrings are analytically true or false (i.e., by inspection). There is a second class of analytic statements involving stipulated definitions and abbreviations (Bachelors are unmarried men). "'Water' means liquid H2O" is clearly one of those. An interesting consequence of the analyticity of statements like "The first letter of the string 'AB' is 'A'" is that they invite us into the analytic a posteriori; I mean, it is kind of ridiculous to say that we knew the first letter of 'AB' would be 'A' prior to observing or experiencing 'AB', or that we understood strings as a concept prior to experience or theorizing. There is an excellent case to be made for the position that while some things have to be stipulated prior to other things' being discussed (e.g., an alphabet in which to conduct the discussion, a number of truth- values that sentences can have), there is NOTHING that has to be a priori. -- Benchley's Distinction: There are two types of people in the world-- those who think there are two types of people in the world, and those who don't.
rmpinchback@dahlia.waterloo.edu (Reid M. Pinchback) (04/05/88)
In article <1190@scirtp.UUCP> george@scirtp.UUCP (Geo. R. Greene, Jr.) writes: >> >> Greg, >> >> How would you classify the following the following proposition: >> >> "Water is H2O."? >> >> >> It certainly seems to be one gotten from experience and thus >> a posteriori; yet there also seems to be something necessary about it, hence >> analytic. > >I would classify it as false. One pair of counterexamples leaps >immediately to mind. Ice is not normally called water; neither is steam. >"Water" is a colloquial term, H2O a technical one. Is this, perhaps, a confusion between the sense and reference of a term? "Water" and "H2O" may not connote the same thing, but they do denote the same thing. Aside from that, if you want some really disgusting confusion of the a priori/a posteriori distinction (which has been chucked out for quite some time as a false dichotomy), try Kant's approach. He proposed a system where by there existed not only purely analytic and synthetic knowledge, but analytic/synthetic and synthetic/analytic. (Maybe this was partly why philosophers are inclined to discount this dichotomy?). Reid M. Pinchback -----------------