Questions of circularity are as old as philosophy., but it was never realized how deeply they could permeate logic and mathematics. — javi2541997

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.

very interestion OP, javi; — 180 Proof

: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.

Sextus Emipricus had done in the former. — 180 Proof

I going to check this out more deeply. It sounds so interesting because I like Roman philosophy or thinkers a lot :up:

Love an OP that both sends one off in a new direction and ties things together.

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 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 form outside - this being a consequence of the second incompleteness theorem. — Banno

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."

I'd be interested to hear if others think this an accurate account of Gödel's thinking. — 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.

Yes; that's what I said.

Yes; that's what I said. — Banno

Perhaps I misunderstood. You said:

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 form 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. — 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.

The second is that Wittgenstein's notion in §201 of Philosophical Investigations might be applied here. — Banno

Interesting! Because according to Ravitch, who made this dissertation, he referred and applied Russell’s and Poincaré’s solutions of circle paradoxes, writing:

The search for a once-and-for-all solution to the paradoxes led Russell, Poincaré, and others to the observation that each of the paradoxes trades on a vicious circle in defining an entity which ultimately creates the paradox. — Harold Ravitch, Ph.D.

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.

But in fact that's an accurate account of a position Gödel is refuting, not thinking — fishfry

: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:

Gödel is primarily concerned with the clarity and force of our conception (ideas) of mathematical objects. One can follow Hume and simply refuse to consider questions involving 'existence' in this context — Professor Carnap and Ravitch

Reading deeply the dissertation ..., — javi2541997

I did start reading a bit, and found this nugget:

Gödel has defended classical mathematics against each of the major programs of restricted methods. He has rejected intuitionism, semi-intuitionism, the vicious circle principle, and constructive or finitist programs in general. has defended classical mathematics against each of the major programs of restricted methods. He has rejected intuitionism, semi-intuitionism, the vicious circle principle, and constructive or finitist programs in general.

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.

Thanks for linking the article. — fishfry

Thank you and welcome :up:

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, — 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.

he is already dead... after 53 years of his thesis approval. — javi2541997

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

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. — javi2541997

Even for someone like me, with basic training in math and logic, this makes sense.

he [Gödel] would "consider this rather as a proof that the vicious circle principle is false than that classical mathematics is false." — javi2541997

This is intriguing to say the least. Reminds me of this:

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. — Kurt Patrick Wise

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.

Even for someone like me, with basic training in math and logic, this makes sense. — TheMadFool

I thought exactly the same :clap: :lol:

circulus in probando - a premise is the conclusion. — 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.

it assumes certain propositions (axioms) to be true and builds an edifice of true (mathematical) propositions on them — TheMadFool

Exactly in this part, I guess Gödel and Ravitch concided. Those axioms or propositions are so related with the "realism" itself.

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. — TheMadFool

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.

Exactly in this part, I guess Gödel and Ravitch concided. Those axioms or propositions are so related with the "realism" itself. — 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:

in whatever manner syntactical rules are formulated, the power and usefulness of the mathematics resulting is proportional to the power of the mathematical intuition necessary for their proof of admissibility. This phenomenon might be called “the non-eliminability of the content of mathematics by the syntactical interpretation.”

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.

SO Gödel is clarifying the syntactic view while rejecting it. — Banno

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.

This is a vast topic. The more I look in to it the more it grows. And it is difficult. — Banno

Most definitely. Very deep waters. I know more about the math side and much less, basically nothing, of Gödel's philosophical thought.

I'll come out straight up and say that my own prejudice is that maths is made up as we go along, — 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.

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. — 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.

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. — 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.

The argument is unclear — Banno

I very much agree with that. The Ravitch article and SEP link are very complicated.

places Gödel in the position of having to choose between vicious circles and "classical" mathematics. Gödel rejects vicious circles. — 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.

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. — Banno

Don't know anything about that which is why I avoided that thread. Interesting to see how this field develops.

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 — 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.

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. — fishfry

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.

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. — 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.

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. — Banno

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.

The claim that all mathematical truth is derived from axioms was decisively falsified by Gödel. I'm not sure what you are saying here. — fishfry

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.

Of course. I'm saying don't bother with axioms. — Banno

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.

Oh - see edit. Lemmon vs. Copi.

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.

https://iep.utm.edu/nat-ded/

https://iep.utm.edu/nat-ded/ — Banno

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.

Oh, sure.

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?

Did you use an axiomatic system? — Banno

Math uses axiomatic systems, period. I didn't study much formal logic.

I studied logic forty years ago. — 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.

Hmm. Your puzzlement has me puzzled, so I will try to articulate some presumptions I had made. Big picture stuff, so this will be lacking in detail...

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.

Math uses axiomatic systems, period. — fishfry

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?

Hmm. Your puzzlement has me puzzled, — Banno

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.

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. — 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.

Is there a reason to think that natural deduction is not as powerful as axiomatisation? — Banno

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.

Ah.

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.