On Gödel's Philosophy of Mathematics
This dissertation comes from PhD Dr. Harold Ravitch at Los Angeles Valley College, in 1968.
Harold, in his abstract tell us his objective in the dissertation is the following one:
You can read it here: ABSTRACT OF THE THESIS On Gödel's Philosophy of Mathematics
One of the interesting parts of his thesis is the so called the vicious circle principle. This theory is developed in a context of metaphysics. Each of the paradoxes trades on a vicious circle in defining an entity which ultimately creates a paradox. Questions of circularity are as old as philosophy., but it was never realized how deeply they could permeate logic and mathematics.
For this reason, according to Gödel, there are four forms of vicious circle inside mathematics:
That also applies to metaphysics.
Harold Ravitch reviews this principle and makes a critique saying: "Gödel has offered a rather complex analysis of the vicious circle principle and its devastating effects on classical mathematics culminating in the conclusion that because it "destroys the derivation of mathematics from logic, he [Gödel] would "consider this rather as a proof that the vicious circle principle is false than that classical mathematics is false."
Thoughts?
Harold, in his abstract tell us his objective in the dissertation is the following one:
An attempt is made to explicate and analyze Kurt Gödel's philosophy of mathematics with emphasis on his defense of classical mathematics, and his rejection of intuitionism, and the vicious circle principle. Gödel's belief in the real existence of mathematical objects is examined. It is argued that one need not accept Gödel's pronounced realism in order to assent to his methodology of mathematics.
You can read it here: ABSTRACT OF THE THESIS On Gödel's Philosophy of Mathematics
One of the interesting parts of his thesis is the so called the vicious circle principle. This theory is developed in a context of metaphysics. Each of the paradoxes trades on a vicious circle in defining an entity which ultimately creates a paradox. Questions of circularity are as old as philosophy., but it was never realized how deeply they could permeate logic and mathematics.
For this reason, according to Gödel, there are four forms of vicious circle inside mathematics:
That also applies to metaphysics.
(1) No totality can contain members definable only in terms of this totality.
(2) No totality can contain members involving this totality.
(3) No totality can contain members presupposing this totality.
(4) Nothing defined in terms of a propositional function can be a possible argument of this function.
Harold Ravitch reviews this principle and makes a critique saying: "Gödel has offered a rather complex analysis of the vicious circle principle and its devastating effects on classical mathematics culminating in the conclusion that because it "destroys the derivation of mathematics from logic, he [Gödel] would "consider this rather as a proof that the vicious circle principle is false than that classical mathematics is false."
Thoughts?
Comments (73)
The problem arises, and is most vexing (vicious), when seeking to justify categorical, self-subsuming, statements or ideas (e.g. criterion problem, global skepticism, radical relativism, value/meaning nihilism, etc) as well as apodictic formalisms like mathematics and logic. Gödel monumentally brings this 'vicious circularity' to the fore in the latter case as, I think, Sextus Emipricus had done in the former. Kant's antinomies perhaps are a bridge. Anyway, very interestion OP, javi; I"m looking forward to see what others – our resident 'mathematics & logic nerds' – make of it.
:up: :100:
To be honest with you I only picked up not only the interesting facts I read but the one I understood. In such difficult dissertation of “maths” and “philosophy” together I have to admit I need to keep reading some books to increase this area.
Quoting 180 Proof
I going to check this out more deeply. It sounds so interesting because I like Roman philosophy or thinkers a lot :up:
So you proffer a critique of Gödel's theory of maths, which raises the question of what that theory actually was. Now I found A Philosophical Argument About the Content of Mathematics; the issue, so far as I can see, is how it could be that, if mathematics is just a syntactic system, devoid of content, which we make up, how could it have application to real world situations - the example used being how the primitive laws of elastic theory could be applicable to building bridges if they are no more than rules for manipulating symbols.
The argument seems to be that a system's being true (about bridges) is on a par with its being consistent, in that we can only show either by bringing in something from outside - this being a consequence of the second incompleteness theorem.
I'd be interested to hear if others think this an accurate account of Gödel's thinking.
If so, two philosophical notions occur to me. The first is that the recent discussion of paraconsistent mathematics might offer a way to see consistency as a special case.
The second is that Wittgenstein's notion in §201 of Philosophical Investigations might be applied here. There would then be a way of applying, say, the fundamental principles of elasticity, that is not just more rules, but found in using them to build bridges.
Anyway, a promising thread, but one in which the detail will be crucial.
The article describes Gödel's account of the "syntactic view," as it clearly states. From the article you linked: "The argument uses the Second Incompleteness Theorem[1] to refute the view that mathematics is devoid of content." (My emphasis).
In other words Gödel was describing a particular point of view in order to refute it.
Gödel himself was a Platonist. He believed that every mathematical proposition has an objective truth value, whether or not that truth value can be determined from a particular axiom system or not.
See this article
"In his philosophical work Gödel formulated and defended mathematical Platonism, the view that mathematics is a descriptive science, or alternatively the view that the concept of mathematical truth is objective."
Quoting Banno
It's an accurate account of his description of a philosophical viewpoint that he did not personally hold, but was describing in order to refute it. Just as if I, a committed globist, described my understanding of flat earth theory.
Perhaps I misunderstood. You said:
Quoting Banno
But in fact that's an accurate account of a position Gödel is refuting, not thinking. Are we more or less in agreement on that?
In any event, on reading the article you linked, it's sufficiently detailed and somewhat confusing to the point that it's not productive to quote-mine it IMO. It's hard to know who's holding what opinion, Carnap or Gödel. Nor do I see how the second incompleteness theorem refutes the syntactic view. to the extent that I follow the quote mining at all. I think the article would benefit from a more clearly description of who is saying what and who agrees or disagrees with who.
What I do know is that Gödel was a Platonist, and believed that (for example) the continuum hypothesis has a definite truth value. Which would not be consistent with a syntactic view of mathematical truth.
Interesting! Because according to Ravitch, who made this dissertation, he referred and applied Russell’s and Poincaré’s solutions of circle paradoxes, writing:
When I check the bibliography used in his dissertation he never mentioned Wittgenstein: https://www.friesian.com/goedel/biblio.htm
Nevertheless, of course I think we should consider Wittgenstein here because supposedly this dissertation was about metaphysics not philosophy of science.
:up: Reading deeply the dissertation, Ravitch wrote that the main goal of Gödel was “believing in the existence” of realism (thus, mathematics). But, he expressed this context as a criticism on Gödel’s philosophy of maths. For this reason quotes David Hume about the meaning of “existence” and “realism” and then wrote the following statement:
I did start reading a bit, and found this nugget:
I shall definitely remember this the next time I cross swords with yet another neo-intuitiionist or constructivist. Hi @sime!
Thanks for linking the article. Interesting that he wrote it in 1968 and was a student of Church himself, as was Alan Turing.
ps -- I noted that the passage you quoted about vicious circles mentions the problems with Dedekind cuts. I'm not entirely sure of exactly what they mean, but I believe that the general idea is that the real numbers are characterized by the least upper bound property: every nonempty set of reals that's bounded above has a least upper bound. The problem is that we're characterizing the real numbers by talking about sets of real numbers. For that reason these kinds of ideas are called impredicative. I don't know much about this subject. I gather that's what this part of the paper is all about. Interesting paper.
Thank you and welcome :up:
Quoting fishfry
Despite the fact Ravitch quotes a lot of interesting teachers or PhD’s, the language and technical paragraphs are so complex. I don’t understand some parts of his dissertation neither so is up to us trying to give our meaning as we are doing here, debating. I am agree with you, this is an interesting academic paper and dissertation. Imagine trying to debate with Ravitch himself about he was thinking back then, but probably he is already dead... after 53 years of his thesis approval.
Update: he retired from teaching in 2019. Look: https://www.coursicle.com/lavc/professors/Harold+Ravitch/
So, for more than 50 years he was a teacher. Probably he has a lot of papers related to this
Even for someone like me, with basic training in math and logic, this makes sense.
Quoting javi2541997
This is intriguing to say the least. Reminds me of this:
[quote= Kurt Patrick Wise]As I shared with my professors years ago when I was in college, if all the evidence in the universe turns against creationism, I would be the first to admit it, but I would still be a creationist because that is what the Word of God seems to indicate.[/quote]
Reject (a principle of) logic (vicious circularity) just so that mathematicians don't lose their sleep over the weak foundations the world of numbers has been built on.
Here's something to ponder upon:
I remember reading a good book on logic about 9, 10 years ago (sorry forgot the title) and it discusses vicious circles otherwise known as circulus in probando - a premise is the conclusion.
It needs to be borne in mind that mathematics is ultimately a system, game-like in nature, where we have complete freedom to choose our starting premises aka axioms. Ergo, since math is by its own admission an axiomatic system, the problem of vicious circularity is N/A, at least not foundationally. After all, mathematicians are openly declaring the fundamental nature of math - it assumes certain propositions (axioms) to be true and builds an edifice of true (mathematical) propositions on them. Math is immune from the charge of vicious circularity.
I thought exactly the same :clap: :lol:
Quoting TheMadFool
Interesting argument and point. What we are talking about here reminds me a lot from Aristotle's syllogisms, but I do not want put it on the table because it looks like nobody likes in our modern Era the syllogisms at all (they are pretty criticised by most of the authors), because this is all about (in my point of view) of how realistic or at least how solid the mathematics can be in our argumentation.
Quoting TheMadFool
Exactly in this part, I guess Gödel and Ravitch concided. Those axioms or propositions are so related with the "realism" itself.
This is exactly the view that Gödel opposed. He believed that mathematics is objective; that mathematical truth is something that we study, not something we make up. If the axioms don't settle a given mathematical question, that's the fault of the axioms. There is truth "out there" waiting to be discovered.
Quoting javi2541997
The exact opposite. The axioms do not determine mathematical truth, in Gödel's view. The classic example being the continuum hypothesis. Gödel said that it has a definite truth value; whether or not it can be proven from the standard axioms.
This is a vast topic. The more I look in to it the more it grows. And it is difficult.
But we might progress a little along the path by being clear who is claiming what.
SO we have Gödel standing up for a more or less classical notion of mathematics as discovering or uncovering things that are in some ill-defined sense independent. This is to be contrasted with intuitionist and constructivist views, roughly that mathematics is stuff we make up as we go along.
I'll come out straight up and say that my own prejudice is that maths is made up as we go along, but I will admit that I do not have sufficient background to argue saliently for this position. I doubt anyone in this forum does.
In the SEP article I cited above, an account is given of an argument from Gödel against intuitionist and constructivist views, which Gödel grouped as the "syntactic view", such that the truth of mathematical propositions is determined by their relation to each other, by their syntax, and not by their dependence on facts.
Gödel is said int he article cited to be arguing against this view; he does this by arguing that any such system must show itself dependent on a mathematical fact: that it is consistent.
SO Gödel is clarifying the syntactic view while rejecting it.
Gödel apparently characterises the syntactic view as consisting of three requirements:
1. Mathematical intuition can be replaced by conventions about the use of symbols and their application
2. There do not exist any mathematical objects or facts, and hence mathematical propositions are void of content
3. The syntactical conception defined by these two assertions is compatible with strict empiricism
His argument appears to be that these three views cannot be consistently held. The argument is unclear, but seems to be that it is inexplicable how a mathematics born from mere "arbitrary" syntax could have empirical import; that (1) and (2) are incompatible with (3). There is something more here, that remains ambiguous; see:
It seems that for Gödel, being consistent and being true are inseparable; that mathematical truth is displayed in mathematic's empirical applicability; and that hence the consistency of mathematics shows that it is empirically applicable and hence incompatible with the syntactic view.
In lay terms, maths can be used to accurately describe real world items like bridges, and so it cannot be just made up.
The Ravitch thesis ( ) looks to vicious circles rather than empirical applicability. This "destroys the derivation of mathematics from logic", and places Gödel in the position of having to choose between vicious circles and "classical" mathematics. Gödel rejects vicious circles.
So back to my own musings. I'd be interested to see if paraconsistent mathematics might present a distinction between consistency and truth that would mitigate against Gödel's almost equating the two. This would count against his argument in the SEP article.
I'd also be interested in work that sets out how mathematical rules might be shown (a philosophically loaded term after Wittgenstein) in empirical situations, rather than expressed in mathematical terms. See The Epistemology of Visual Thinking in Mathematics
(What happened to @Nagase? I'd have liked his opinion here).
All admittedly very speculative, but there it is, for what it's worth.
Yes that's my interpretation of all this. I'm basing it on my prior knowledge that Gödel was a Platonist and held that the continuum hypothesis has a definite truth value, even if it's out of reach of our present axioms. Once you know Gödel was a Platonist, it's easier to understand the complicated arguments in the Ravitch paper and the SEP article you linked.
Quoting Banno
Most definitely. Very deep waters. I know more about the math side and much less, basically nothing, of Gödel's philosophical thought.
Quoting Banno
I hold that view myself from time to time. Then again, is the proposition "5 is prime" made up as we go? I oscillate among Platonism, fictionalism, formalism, and "what difference does it make?" as it suits me in any particular argument. My only strongly felt philosophical stance is that the constructivists are missing something essential; and the Ravitch article showed me that Gödel is on my side.
Quoting Banno
I sort of agree but I'm unclear on one point. It is not my understanding that constructivists argue that mathematical truth is based on the axioms. Rather, constructivists merely say that in order to exist, a mathematical object must be constructed as from an algorithm or some kind of describable procedure. But I'm not sure whether that goes along with saying that all mathematical truth is derived from axioms.
Quoting Banno
Might be but I'm unclear on that too. Gödel believed in mathematical truth, even though he knew no set of axioms can prove itself consistent (second incompleteness theorem). More murkiness.
Quoting Banno
I very much agree with that. The Ravitch article and SEP link are very complicated.
Quoting Banno
Right. A lot of the basic math I was taught turns out to be "impredicative," defining things in terms of themselves. In math nobody cares or talks about it, but it's a bigger deal in philosophy. Gödel seems to be coming down on the side of accepting impredicative definitions and allowing math to just be math.
The classic example of an impredicative definition in math that's so common that math majors never even notice the circularity is the greatest upper bound of a set of real numbers, as I mentioned earlier and as described in the Wiki link.
Quoting Banno
Don't know anything about that which is why I avoided that thread. Interesting to see how this field develops.
Quoting Banno
Wittgy is another subject I don't know enough about to converse on. As he said, Whereof I cannot speak, thereof I should put a sock in it.
I intentionally left axioms out of my post. I've always been struck by the fact that what we select as our axioms is more or less conventional; we might have chosen different axioms. So in that way if one supposes that "all mathematical truth is derived from axioms" all one is doing is insisting on coherence and completeness.
Quoting fishfry
Yep. So both consistency and truth are meta-level aspects of a language. The extent to which they are related is another thing to ponder.
Yes but no nontrivial collection of axioms (strong enough to found the usual arithmetic of the natural numbers) can be complete. The claim that all mathematical truth is derived from axioms was decisively falsified by Gödel. I'm not sure what you are saying here. You can't insist on completeness, it's been proven impossible to do that with any nontrivial system of axioms.
Of course. I'm saying don't bother with axioms.
Edit: so we use natural deduction rather than axiomatic definitions of completes and coherence. That is, rather than showing that no inconsistency is derivable from the axioms, one shows that not inconsistency is derivable using the rules of derivation; and rather than showing that every theorem is derivable from the axioms, one shows that every theorem is derivable with rules of derivation.
Or rather, one shows after Gödel that this is not the case for anything complex enough to do addition.
Now you really have me confused. What does that mean? Can you give me an example? You honestly have me at a loss. The entire conversation is about axiomatic systems and their relation to mathematical truth.
I started logic in a course using Lemmon, but changed Universities and wound up using an axiomatic approach (Hughes and Londey). I fond the difference quite striking, especially since that was the reverse of the historical progression.
I thought we were talking about math, not logic. At least the OP was about math FWIW. Is ND offering an alternative foundation of math? Is it somehow not subject to incompleteness? I'm afraid you've lost me again but I don't know anything about ND.
Logic is just another part of maths.
The point being that nether logic nor maths can or need be derived from axioms. Indeed, in cases that include counting, they can't be, since they cannot be both complete and consistent.
Natural derivation seems more in tune with this, since it does not use axioms; or perhaps because every theorem is an axiom. My understanding is that this was the pedagogy behind it's introduction into undergrad logic courses - it set students up for a better understanding of the issues. But perhaps that is an anachronism on my part? I studied logic forty years ago.
Did you use an axiomatic system?
Math uses axiomatic systems, period. I didn't study much formal logic.
Quoting Banno
Yes but the subject of the OP is axiomatic systems in math. Not that threads don't wander, but I'm no longer understanding your point. Is ND somehow not subject to incompleteness? I tend to doubt that. Underlying set theory is first-order predicate logic. If there's some other basis for doing set theory I am unfamiliar with it. I've heard of ND used in logic, but I can't imagine it gives you different math.
Logic started with axiomatic systems, following on from the Greeks and geometry and so on. Then Gödel showed that no axiomatic system sophisticated enough to include counting could not be shown o be both consistent and complete.
There are two ways to look at this. If one treats of axioms, then we have discreet axiomatic systems, and to each one we might add extra axioms, deriving the whole of mathematics as the asymptotic totality of all the axiomatic systems
Or we might treat it as a series of rules for derivation, never complete but always consistent, to which we add new rules of derivation in a similarly asymptote fashion.
The possibility of paraconsistent mathematics opens to possibility of derivations of complete, inconsistent systems.
Too fast. Why?
Is there a reason to think that natural deduction is not as powerful as axiomatisation?
That is, while mathematicians do use axiomatic systems, is there a reason to think their work could not also be done using natural deduction?
I no longer have any idea where you are going with this. I am sure the fault is all on my side. I can't respond intelligibly because I just don't know what you are saying relative to what the thread is about.
Quoting Banno
Are you saying you can do modern math like that? I'm sure somewhere somebody's making the effort. I can't relate to what you are saying. My apologies.
I don't know, I'm asking you. I don't know anything about it other than that it comes up when students are discussing logic. I glanced at the Wiki article on ND and perhaps you are making a valid point that I'm too ignorant to understand. I never studied formal logic, just picked up the basics from math classes.
So again another field explodes before us - Proof theory.
I'll do some more reading. See https://plato.stanford.edu/entries/proof-theory-development/
But I can see how we might have different pictures of mathematics, which are despite that functionally equivalent: one of maths as a series of axiomatic systems, the other as a language that becomes increasingly complex as new formation rules are added.
Ok. I Googled "natural deduction as basis for math" and the following popped up right at the top:
Now this is just what I call perfectly normal mathematical deduction. That's how mathematicians do proofs. To prove P implies Q we assume P and derive Q. If that's natural deduction, I've been using it all my life! Like the literary character who discovered he'd always been speaking prose.
So if that's all ND is, then it appears to be perfectly standard everyday mathematical reasoning.
I still don't understand the technical distinction being made by the Wiki article on ND, but based on this example, ND is just how everyone does math.
Quoting Banno
I don't think this is what's being said by ND. As the Google example shows, ND is basically normal everyday mathematical reasoning. To show P implies Q we assume P and derive Q.
Ok I waited a bit. I did not understand ND from its Wiki page. But I did completely understand it from the Google description: To prove P implies Q, I assume P and derive Q. This is basic, everyday mathematical practice.
I should point something out. Mathematicians don't use formal logic. They use this kind of casual, everyday reasoning. In formal logic, the things people talk about are foreign to working mathematicians. By the standards of formal logic, no working mathematician has ever seen a proof, if you look at it that way.
So whatever ND is, if it's just "To prove P implies Q assume P and derive Q," then that's what I've been doing all my life and that's what everyone else does. Any distinction between that and some other kind of formal logic is "inside baseball" for logicians and apparently of little interest to mathematicians.
Bottom line, I don't see how ND can add anything to mathematical practice, nor relieve us of the need to start somewhere by writing down our axioms. If you add axioms one at a time making sure they are complete and consistent, you will never get to the arithmetic of the natural numbers, which is known to be incomplete.
Don't know if this was helpful to you, but it sure was to me. "Today I learned" that natural deduction is just the normal type of reasoning done by mathematicians. So in the end, thanks for mentioning it!
That was the point of the explorations into the foundations of maths that led to incompleteness and so on - to give maths the rigidity of logic, conceived of as axiomatic. But what happened is that in including counting in logic, the axiomatic enterprise fell apart; maths is much larger than any given set of axioms.
SO the lesson might be that when the love of axioms tried to tighten up mathematics, it ended up toppling the axioms.
Quoting fishfry
I suspect that in mathematics any true formula can serve as an axiom from which to develop more cool mathematics. That's quite a different thing to an axiom in logic.
Indeed. It was Hilbert who said, "Wir müssen wissen – wir werden wissen ("We must know — we will know."). And as I recall. it was either a few days before or after that, that Gödel announced his incompleteness theorems. The search for mathematical certainty ended in the proof of uncertainty. Be careful what you wish for, or something like that.
Quoting Banno
We don't know what the true formulas are. But it's true that math and logic are very different.
Can you tell me, from your knowledge of ND, is my Google-ish description that "To prove P implies Q we assume P and derive Q," is a fair summary of what it's about? I've seen natural deduction mentioned many times but never knew what it is.
No axioms. That's pivotal. Instead there is a rule of assumption: one can introduce any proposition on chooses at any time, and rules for deriving more theorems from those assumptions
The derive bit is perhaps misleading, since if you assume A and also assume, but not derive, B, then it follows that A?B.
'You can't know'.
Yes but there are no axioms in any deduction system. First you have the rules of deduction, then you add in some rules of set theory, say, and you crank out your theorems. It's like saying gasoline doesn't have a steering wheel. Gasoline is the stuff you put in your car, and the gas makes the car go. In order to apply the deduction system to something you have to write down some axioms. The axioms aren't part of the deduction system. So it's not a pivotal aspect of ND that there are no axioms. To do math, you write down some axioms and then use the rules of deduction to derive theorems. If you have no axioms you have rules of deduction but you can't prove anything.
Unless you mean that we can introduce our axioms in an ad hoc manner, for example, saying "if we assume the axiom of unions and we have sets X and Y then we can conclude that there a set X union Y. Is that what you mean by not having axioms?
I found an answer on math.SE which you may or may not find satisfactory. See the checked answer by Asaf Karagila. I'm repeating it verbatim here:
https://math.stackexchange.com/questions/1962462/g%C3%B6dels-incompleteness-theorem-question-about-self-reference
FWIW we don't "throw out self reference" to fix Russell's paradox. Rather, we outlaw unrestricted comprehension and require restricted comprehension.
That is, if [math]P[/math] is a predicate, we outlaw set specifications of the form [math]\{x : P(x)\}[/math], which says we can form a set out of all the things that satisfy the predicate.
Instead we require that the predicate is used to cut down an existing set. So we have some set [math]X[/math] that already exists, we say that we can form a new set [math]\{x \in X: P(x)\}[/math]. That makes all the difference.
For example if we form the set of everything that's not a member of itself, as in [math]R = \{x : x \notin x\}[/math], we get Russell's paradox.
But if we start with, say, the natural numbers, we may legally form the set [math]S = \{x \in \mathbb N: x \notin x\}[/math], we do NOT get any contradiction.
Let's walk through it Is 0 an element of itself? No, so 0 is in [math]S[/math]. Is 1 an element of itself? No, so 1 is in [math]S[/math]. Continuing like this, we see that [math]S[/math] is just the set of natural numbers. The paradox goes away.
I so mean.
But we call them assumptions.
The question then is, is math discovered or invented?
One of the reasons why some, like Gödel I suppose, believe math is discovered is how math seems to,
1. Describe nature (math models e.g. Minkowski spacetime)
2. Describe nature accurately (we can make very precise predictions to, say, the 15th decimal place)
There's a book titled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences." authored by Eugene Wigner. It's a book that captures this sentiment.
However, we can't/don't know beforehand which mathematical object will prove itself useful in the description of nature. Given this, we need to experiment with math which basically involves tinkering around with the axioms as arbitrarily or as whimsically as possible. The objective? Generate as many mathematical objects as possible so that we can find the one that's the best fit in re nature.
In summary, possibly true that math is a discovery (it's fundamental to nature, the universe) but to find the best possible mathematical object to capture nature's essence as it were, we need to treat math as an invention and play with the axioms. :chin:
Ok, I think we're all clear and in agreement then.
We can start with the inference rules of ND, and, one-by-one, introduce as assumptions the standard axioms of ZF set theory. Of course there are infinitely many axioms because Specification and Replacement are actually axiom schemas, meaning that they each represent one axiom for each of infinitely many predicates. But presumably we can do that. [Specification can be derived from Replacement so technically we only need Replacement].
So far so good. Now as soon as you add the axiom of Infinity, you will have a model of the Peano axioms, and you'll introduce incompleteness. So in the end this is all exactly the same as standard set theory.
I did look this up in the Wiki article on first-order logic, which is the logic used for set theory. It says, in the section, "Hilbert-style systems and natural deduction,"
So as far as I can understand, you get the same set theory either way.
My sense is that these mundane physical considerations were not on Gödel's mind. He believed in the Platonic existence of abstract sets including large cardinals, sets far too large to be of any conceivable interest to the real world. See [url=https://plato.stanford.edu/entries/goedel/#GodVieAxiCon]
2.4.4 Gödel’s view of the Axiom of Constructibility
[/url].
I really can't say what Gödel thought about or believed, since apparently he initially thought the axiom of constructability (the claim that the constructible universe includes all sets) was true, then came to doubt it. But my sense is that he was thinking of the Platonic reality of a very large universe of sets, and was not thinking about the utility of set theory in physics. On the other hand he did do some work in relativity, so who knows.
Harold Ravitch, in his own opinion about Gödel, after finishing the dissertation, wrote this:
Quoting Harold Ravitch
I guess (just my simple opinion) that Gödel was transforming through the years as metamorphosis. I am agree with you that so clearly he was a platonist but it is true that he reflects other points of view in his writings. To be honest I don’t defend we should exclude Gödel from these areas and see him as and “unique”philosopher as Ravitch wrote. Probably there are different versions of him to be consider of: Platonism, realism, relativism, etc...
Way above my paygrade! Thanks though. I hope to advance my knowledge in math ASAP.
I can find only two papers, neither related to this topic. If he taught at a community college it might have been difficult to do research and publish.
Quoting TheMadFool
Most math people are not concerned with this question. I believe as new math appears on the scene it may have elements of both.
Quoting Banno
Some of it is. For example I recently defined an "attractor form" for a certain class of functions. It's trivial stuff, but may not have been around before. Once defined, then its characteristics are explored. It's hard to tell. My advisor fifty years ago went so far as to state, "There is nothing new in mathematics." Who knows?
Yes you are right. There are not articles from Ravitch connected to this thread. There is only one article published here: Springer link.
I found related to this topic the following paper or "scheme":
Principles of Predicate Calculus
Above mine too actually. My point was that Gödel apparently believed in an expansive view of the set-theoretic universe, and that his Platonism was probably motived by that and not by practical considerations such as its use in physics.
FWIW Gödel cooked up a model of set theory called the constructible universe, in which the axiom of choice and the continuum hypothesis are true. That shows that they're consistent with ZF.
So why not just adopt the axiom that the constructible universe is the true universe of sets? If you did that, AC and CH would be theorems and we'd be done. The reason this assumption is not made is that most set theorists believe that the true universe of sets (if there even is such a thing) has way more sets in it than just the constructible ones. Gödel apparently first believed that the constructible universe was the true universe, and later came to not believe that.
Quoting jgill
Or it has been waiting around since the big bang for you to come along and discover it.
Quoting javi2541997
Agreed, his thoughts were probably a lot more complex than the articles about him can capture.
From what I could glean from Wikipedia, a constructible set is one which can be, well, constructed via set theoretic operations (intersection, complement, mainly union I suppose). What would it mean to say some sets aren't constructible? Every set can be disassembled/broken down into its subsets and like a chemical compound, reconstituted back to give the original set.
The empty set? Is it constructible? { } U { } = { } but then that's circular. Aah. {1, 4, a} intersection {p, w} = { } but then how is { } made up of {1, 4, a} and {p, w}?
It's too complicated for me. I give up!
Why the zero appears in your formula? Is it related to “transfinite” concept ??
I think I didn’t get your argument because I am guessing that transfinite recursion is an infinite constructible loop at the base clause. (?)
I am confused but this enigma is making me think a lot.
Wrong article. The Wiki page on "constructible sets" has something to do with topology. Different usage.
This is the relevant page.
https://en.wikipedia.org/wiki/Constructible_universe
The constructible sets are built out of first-order formulas in stages. I don't know enough about this to give a simplified explanation. Basically each stage is built from first-order statements with parameters and quantifiers that range only over the previous stage. It's very logic-y. Wish I could say more but I don't know too much about it. Only that Gödel cooked it up to prove the consistency of AC and CH.
Already read it. Thanks for the link, I understand it better now :up:
@TheMadFool This is also a good link if you're interested.
Contra that, in the item I cited, there are mentions of empiricism and the example of the applicability of the mathematics of elasticity to engineering a bridge; see (3) here.
so again:
Quoting Banno
So if we trust this secondary source - I have no reason not to - Gödel held something like that we can speak of mathematical propositions being true because they are empirically applicable; that truth is at a meta-level to the mathematics itself; and that together these show that maths is not just stuff we make up.
The exact argument remains opaque, but that is the implication.
Yep. Some folk feel a sort of existential angst - why should the mathematics of limits have a use in describing the movement of a cannon ball? Why should the stresses and strains on a bridge be amenable to calculation?
But if it wasn't Young's modulus, wouldn't we just use some other calculation? Don't we choose the mathematics to fit the situation? So why should we be surprised that the mats fits? As if one were surprised to find the end of a screwdriver is just the right shape to drive a screw.
Yes but your (3) is under the heading that says: "Gödel apparently characterises the syntactic view as consisting of three requirements:"
So he is characterizing the syntactic view. But we've already agreed that he's describing the syntactic view in order to disagree with it.
I would be extremely surprised to find that Gödel advocated Platonism because of the use of math in building bridges or whatever. On the contrary, Gödel's Platonism argued for the existence of large cardinals and other highly abstract and decidedly un-physical entities of set theory.
Again, your (3) is under the heading of the view that Gödel is trying to refute, not advocate for. Am I missing something?
Quoting Banno
Two things may be true here:
1) It's perfectly possible that Gödel said that, or believed that; or at the very least, that some third-party interpreted his beliefs that way; and
2) I totally do not believe Gödel himself justified his Platonism on day-to-day physical grounds. Gödel believed in large cardinals, and that's one reason he did not believe we should adopt the axiom of constructability.
In other words: You may be right but I still don't believe it. Both those can be true. I believe many false things.
Quoting Banno
I just can not believe that Gödel paid much attention to the construction of bridges as a justification of large cardinal axioms. Like I say: even if I'm wrong, I'm sticking to my sense of the matter.
Quoting Banno
But wasn't the use of the word empiricism listed under Gödel's description of the syntactic view? The very view that he's describing in order to refute?
I am feeling uneasy expressing strong opinions on things I know nothing about. I don't know what Gödel thought. I do know that he (later in life, at least) came to reject V = L (the axiom that says that the constructible universe is the true universe of sets) because L doesn't have enough sets. And the sets that it's missing are large cardinals, transfinite quantities so large they can't be proven to exist in ZFC and that are as far from empirical concerns as can possibly be.
That's really all I know about it; and I sincerely agree that I could well be completely wrong.
hang on...
here. https://thephilosophyforum.com/discussion/11547/bannos-game/p1
It'll be moved to the lounge as soon as the mods see it, and disappear.
But it seems to me to be a good metaphor for the development of both language and mathematics. More a simplified case than a metaphor.
Folk can add rules - and quickly they will construct paradoxes, but after a dozen or so posts the rules will branch so that some folk are following some rules but not others, sometimes giving justifications for the path they choose, sometimes not.
That's what maths is - sets of rules we have made up, with folk building on the ones they find interesting.
Interesting argument indeed. It is true that the queen only can move according how the rules are established but this is just a basic axiom to pursue equity of opportunities of win between both players. You and me (if we want to) can change the rules in a private play without competition.
Back to mathematics, when you said "5 + 3 equals 8", it reminds me an interesting video that probably you would like about how free of interpretation are our sum orders.
In this context, infinite summation is defined only for converging sequences. If the rules of definition are violated, then, of course, contradictions may be derived. There is no mystery or even problem about that.
The person at the blackboard says, "The problem with infinity is all sorts of weird things happen when you're dealing with infinity". First, that doesn't even mean anything. Second, instead of explaining that the fallacy is in using an undefined notion (infinite summation on a sequence that does not converge), the person at the blackboard doesn't even suggest how we may investigate further to see that there is not an actual conundrum.
The video is yet another example of Internet ignorance and disinformation. That person seems to be teaching a classroom. He should be told by the school administrators to clean up his act: If he wants to present mathematical challenges, then he should provide his students with the benefit of techniques and information for solving the challenges rather than obfuscate with "weird things happen".
:up: :up: :up: :up: :up:
Sorry. Mi fault for sharing it, I thought it could be interesting.
Very nice comeback - thanks! ( I assume it was directed to me? I'll read it that way.)
You can't play chess unless you stipulate a distinction between pawns and queens. That's more on a par with stipulating 3 + 5 does not equal 8.
But you are right that one ought not over stretch the metaphor. I had in mind more the almost organic way the game grows, especially how different branches develop by contradicting some of the rules.
(converging sequences of partial sums: S(n)= a(1)+a(2)+a(3)+...+a(n) -> S, or a(n)->0 fast enough)
It's nit-picking, but there are several "summability" theories of divergent series. They assign "sums" to certain divergent series and must give the proper sum to convergent series.
Summability of Series
And Riemann showed that a conditionally convergent series can be rearranged to "sum" to any value.