- Geometry: Use Euclidean Geometry when appropriate, for example if you are designing a bridge, use Riemannian geometry when looking at space time, and use geometries when they are appropriate. So there is no correct geometry, its more of a right tool for the right job thing. So far Set Theory does not seem to have a strong enough connection to the real world for this to make sense. I supposed you use ZFC when dealing with most of mathematics, but I doubt you would ever say something like: When dealing with Quantum Mechanics its best to assume AD. So what can you use to decide what axioms to use? You may decide what axioms to use based on your tastes. For example see this prior blog posting. This is good for an individual but will not really work for the whole community. For example, I happen to like AD since I like a world where the Banach Tarski paradox is false. But that's just me.
- People concerned with these issues in the early 1900's were much more passionate then we are today. They had strong opinions on foundations and on non-constructive proofs. Mathematicians commonly carried firearms. We are far less passionate today on these issues. As an example, there are today people who study constructive proofs and prefer them, but I doubt anyone today would reject a theorem that was proven nonconstructively. Why the change of heart? Possibly Godel's theorem, but also the fact that people in different parts of math can't talk to each other so they can't argue.
- Another axiom of interest: The existence of Inaccessible cardinals. MOTIVATION: Take omega. If |X| < omega then |powerset(X)| < omega. Does any other cardinal have this property? Why should omega be so unique? Kappa is an inaccessible cardinal is such that if |X| < Kappa then |powerset(X)| < Kappa. Do such cardinals exist? The existence of an inaccessible cardinal large than omega cannot be proven in ZFC. An inaccessible cardinal would be a model of ZFC and hence would prove that ZFC is consistent (omega does not prove ZFC consistent since no proper subset of omega is infinite). It is known that ZFC cannot prove its own consistency (I think that's true of any theory but there may be some conditions.)
- Penelope Maddy has two nice articles on why mathematicians believe what they do: believing the axioms I believing the axioms II Also good to read: Shelah's Logical Dreams
Computational Complexity and other fun stuff in math and computer science from Lance Fortnow and Bill Gasarch
Tuesday, January 05, 2010
Axioms: What should we believe?
Some misc thoughts on set theory inspired by yesterdays comments
and other things.
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So what is your view on statement A?
ReplyDelete(A): The continuum hypothesis is true in the cumulative hierarchy.
Do you think it is true, false, or meaningless? Years ago, when I thought about these things, I was convinced it was not meaningless. The cumulative hierarchy, V, is a particular model, neglecting the fact that it is a proper class and not a set, and therefore CH should be true or false for that model, right? Or is the "class-ness" of V important?
If one is any sort of mathematical "realist" (I am one) than one is committed to the idea that CH must be true or false for a particular model, right?
Independence is weakness of a particular mathematical "experimental apparatus", i.e. a given set of axioms. But mathematical objects themselves, which certainly rise above any particular axiomatic description, cannot suffer from "semantic incompleteness", can they?
I haven't thought about these things in years so maybe technical developments have occurred that make my comments irrelevant.
It seems to me that that statement A addresses the very nature of mathematical objects.
I don't know where you're from, but several of the mathematicians where I work do carry firearms, at least off campus.
ReplyDeleteI look at ZF[C] and its extensions (say by large cardinals) as partial axiomatizations of all of mathematics, as describing a context in which all the consistent theories can be given meaning. Not sure how much sense that makes.
ReplyDeleteOne might make certain assumptions about which subsets of the universe can be "physically realized", for example assuming that only measurable sets or Borel sets are physically meaningful. This would disallow Banach-Tarski for such meaningful sets, while allowing for it in the more abstract world of all possible sets.
I'm inclined to agree with antianticamper that CH has a well-defined truth value even though models of ZFC + CH and ZFC + (not CH) can be built. Whether we restrict ourselves to V doesn't matter, though. If there's a bijection between ℵ_1 and P(ω), both of which are objects in V, then that bijection is itself in V.
Minor technical point: What you've defined as inaccessible is in fact called strong limit. Strong limit cardinals do exist, and in fact form a proper class; if you look at the Beth hierarchy, where beth_0 = ℵ_0, beth_{α+1} = |P(beth_α)|, beth_β = sup {beth_α : α<β} for β a limit ordinal, then the strong limit cardinals are exactly the cardinals beth_β for β limit.
ReplyDeleteNow if you add the assumption that κ is regular, i.e. not the union of a smaller number of smaller sets, then you have inaccesibility.
I'm inclined to accept the existence of inaccessibles and their mild generalizations (such as α-inaccessible and Mahlo) simply because they're what you get when you start treating proper classes as legitimate mathematical objects and take unions and powersets and so forth.
From an engineering point of view, please let me say that GASARCH got it exactly write with his point #1, using the right tool for the right job. And this has the important corollary that oftentimes the right tool is the one that's easiest to learn.
ReplyDeleteThese ideas seem safe enough, but definitely they can arouse passions.
For example, this quarter's UW/QSE seminar So You Want to be a Quantum Systems Engineer is (provisionally) taking the point of view that the traditional pedagogic principle "in learning quantum mechanics it's best to adopt a vector-space framework"---which has served admirably at the undergraduate level---should be regarded as outdated 20th century mumpsimus.
The seminar will (provisionally) replace this mumpsimus with the 21st century sumpsimus "in learning quantum mechanics it's best to adopt a symplectic/metric framework."
We're not entirely sure this replacement will work ... so we're going to try it and see.
As with ZFC/AD/AC issues, there is nothing physical at stake. But adademic culture most definitely is at stake ... and for reasons that Doron Zeilberger has set forth (and that GASARCH alludes to in #2) ... there are (fortunately) plenty of folks who have passion for these issues.
This passion is good (IMHO), it being neither necessary nor desirable that everyone think the same way. It is only when students and professors alike cease to care very much, about the frameworks within which we work and teach, that we need begin to worry.
That is why I am very grateful and appreciative of this blog (and many other mathematical blogs too) for fanning these vitalizing flames of passion! :)
Interesting post. Thanks for the pointer to this "Penelopaper".
ReplyDelete1. "...People concerned with these issues in the early 1900's were much more passionate then we are today. They had strong opinions on foundations and on non-constructive proofs. Mathematicians commonly carried firearms. We are far less passionate today on these issues..."
At a theoretical level mathematics is the study of formal or symbolic objects or systems where we create the systems (by stating the axioms and/or definitions)and discover its consequences (theorems). At this level any consistent system of axioms would be acceptable.
At a practical level, in societies with freedom, we use formal systems to convince others that our views are the correct views. At this level only realistic axioms everyone can agree on would be acceptable.
Since several years there has been a mathematical explosion and the accent is marked in innovation, so everyone wants to be a formal systems creator, a system which probably only the creator will work on, so who cares today about convincing others about his axioms?
2."So far Set Theory does not seem to have a strong enough connection to the real world for this to make sense".
Coincidentally i bought the other day the book "Understanding the Infinite" from Shaughan Lavine. I´m not yet in the second part where he developpes his thesis (the first part is historical)but it seems that he does not agree with your comment.
Personaly, as a finitist i´m was not interested in infinite sets until i found by accident one interesting infinite object (a graph not a set). For me infinite objects are more processes that never terminates...
Proaonuiq said: "At a practical level, in societies with freedom, we use formal systems to convince others that our views are the correct views. At this level only realistic axioms everyone can agree on would be acceptable."
ReplyDeleteProaonuiq, that is IMHO wonderfully accurate summary not only of 21st century mathematics, but of 21st century engineering too.
The correspondence is induced by two natural isomorphisms: "formal system ⇔ simulation framework," and "correct views ⇔ feasible enterprises."
This isomorphism explains why global-scale engineering-and-science enterprises (the Genome Project, VLSI design, quantum spin biomicroscopy) increasingly resemble global-scale mathematical enterprises (the Langlands Program, QIT/QIP), especially in the increasing overlap of their mathematical toolsets.
Back in 1947, von Neumann could argue (quite passionately) that this isomorphism had beneficial effects, "rejuvinating return to the source: the reinjection of more or less directly empirical ideas [into mathematics]."
Nowadays this mapping is growing ever more natural and intimate, and AFAICT is becoming an effective counterbalance to the mathematical fragmentation that Doron Zeilberger deplores.
This is IMHO fortunate, because without such mechanisms, we would collectively be stuck in the disastrous position of "Pick any two from: (1) a planetary population of 10^10, (2) the creation of family-supporting jobs for the young mathematicians of this planet, (3) a reasonably unified culture of mathematics."
Or more soberingly: pick one from the list. Or still more soberingly: pick none.
It is tempting to say of these dystopian outcomes (with Luke Skywalker), "No, that can't be true ... that's impossible!" But doesn't the history of mathematics provide scant logical grounds for this optimism? We need only think of David Hilbert's sad passing, in 1943, in Göttingen.
Our planet urgently needs as much (justified) optimism and (feasible) enterprises and (unifying) narratives as we can possibly create; the mathematics of the 21st century, conjoined with the creative young minds that our planet possesses in super-abundance, are (IMHO) the most fertile resources that our planet possesses.
So if anyone wonders why engineers read mathematical blogs ... it's because in this century, at a fundamental level, we are all of us in the same business.
I think that if you're doing mathematics that is supposed to describe the physical world (or the behavior of physical computing machines), but the answer turns out to depend on the truth or falsity of AD or greater-than-countable-AC or the existence of Mahlo cardinals or whatever, then something is wrong with your assumptions.
ReplyDeleteJohn, i ´ve read Zeibelger short essay and i can not resist to quote his conclusion:
ReplyDelete"For the good of future mathematics we need generalists and strategians who can see the big picture. Narrow specialists and tacticians would soon be superseded by computers.
So let's get to work, and try to become mathematicians rather than topological algebraic Lie theorists, algebraic analytic number theorists, pseudo-spectral graph theorists etc."
I´m not sure that this explosion and fragmentation is so bad provided that it leads later to a greater unification. We are now exploring a new wide jungle through tortous trails and explorers can not be but a few; but as you point also highways are being produced (IMO good surveys or clarifying general narratives on a subject could be as great as new formalizations or the discovery of deep theorems within a formal system).
On the other hand i´ve been surprised seeing how close is Zeibelger in his views in this essay to the mathematician i always thought was his opposite in views, the infinitist Dieudonné (the author of the still highly advisable "Mathematics, the music of reason" and Panorama, as bird views of mathematics).
So the famous sentence "for every Dieudonné there is a Zeibelger" is not valid anymore.
Shelah's article is really awesome!
ReplyDelete@proaonuiq: I really like your comment. In fact we really can't avoid "infinite objects". Widely accepted and basic things like regular or context-free languages are often infinite. Also the set of all predicates on a countable-infinite language is uncountable.
The need of a sound theory to model processes that do not terminate also lead to dealing with infinite objects. This is already well-accepted among practitioners like formal method (software engineering) researchers.
My aparently clean, blameless and purely metamathematical comment (althought i must recognise it, full of typos)in this post has been deleted. This is not the only deleted comment on this, one of my favorites, blog. Another clean comment about the availability of taxis in China has also been deleted.
ReplyDeleteThis puts an end to (a promising career ?) as an sporadic blog commentator. My apologies if i have offended to anyone. Keep on with the interesting onversation !