Neo-logicism -- is it really a problem?

[nomadreid]

I am not clear on the status of neo-logicism. Here is what I understand:

The old logicism in the form of a finite axiomatization of mathematics is of course dead, buried by Gödel et.al.

A naïve form of neo-logicism, which would try to reduce mathematics to a single logic, bumps against those holding or not to intuitionism, the limitations of first-order logic, and similar. Mathematics is not monolithic.

However, if we refer to logic (perhaps with uppercase: Logic) as a methodology to distinguish it from any specific logical system, where is the objection? That is, the idea of shifting around is enshrined in such ideas as Kripke semantics, many-worlds semantics, Belief Revision Semantics, etc. Is the only problem that one has not yet formalized all logical foundations of mathematics, or perhaps is the problem a gut feeling of strong Platonism bordering on mysticism? In other reasons, with a broad enough definition of Logic, is there any formal reason to reject neo-logicism in the same way that the Incompleteness Theorems gave a formal reason to reject classical logicism?

 

[jedishrfu]

Perhaps @Demystifier and @Auto-Didact would be interested in your question.

 

[Rive]

I am not clear on the status of neo-logicism.

It is just some things are not useful enough at some point of time to have any impact (even just by provoking objection).
Sometimes it remains so indefinitely. Especially if the topic is flirting with philosophy.

 

[jedishrfu]

In the end, these kinds of notions may limit the growth of a field. You can see that in the attempts in Physics to unify everything under String Theory and how so many resources have gone to that end.

However, its equally true that we do want to unify our knowledge as it can bring a deeper understanding of disparate fields within.

So the jury is out and we just have to wait until someone can figure out a new paradigm to get around Godel's theorem limitations.

CAVEAT: I am not a philosopher, nor professional mathematician. I am a robot and interested in how you humans will solve this riddle.

 

[nomadreid]

It is just some things are not useful enough at some point of time to have any impact (even just by provoking objection).
Sometimes it remains so indefinitely. Especially if the topic is flirting with philosophy.

To rephrase: a lot of philosophy is useless. (That is, this topic doesn't just flirt with philosophy, it is part of philosophy. It is not mathematics, but rather meta-mathematics, or perhaps meta-metamathematics.) Since much metamathematics has turned out useful in turning mathematicians' interests to new mathematics, the "a lot of" is not "all". But perhaps you are right; given the various semantics which were created in response to the earlier form of logicism, perhaps cleaning up the idea of neo-logicism would not lead to any new semantics. The question, then, remains, not as one that will have a direct impact on logic, but it could at least lead to discouraging further annoying attempts at dressing up the Lucas-Penrose fallacy.

we just have to wait until someone can figure out a new paradigm to get around Godel's theorem limitations.

Some would say that the semantics which I mentioned in my original post already get around these limitations by adapting to more flexible and fluid logical systems and semantics.

I am a robot and interested in how you humans will solve this riddle.

It may be robots (computers, androids, etc.) who might benefit the most in our decision which one of the various flexible semantics out there will be adopted. One prejudice that the AI community has apparently already discarded, but which remains with much of the population, are variations of John Searle's "Chinese Room". That is, AI is not to the point where one can convincingly view data processing as having the ability to transcend the stage of imitation or powerful memories combined with basic combinatorics, so it will be a question whether humans general viewpoint of AI will be in advance or behind the development of independent intelligence.

 

[Rive]

That is, this topic doesn't just flirt with philosophy...

I meant that 'topic' as matter of discussion, not as this corner of Physics Forum, but anyway...

The old logicism in the form of a finite axiomatization of mathematics is of course dead, buried by Gödel et.al.

I wonder why would you say that... Isn't this 'neo-logicism' is still about making logic rule all?

 

[nomadreid]

"The old logicism in the form of a finite axiomatization of mathematics is of course dead, buried by Gödel et.al. "

I wonder why would you say that... Isn't this 'neo-logicism' is still about making logic rule all?

Yes, neo-logicism is about logic being the basis of all mathematics, but not via a finite axiomatization, which was Hilbert's dream; Hilbert's logicism is dead, long live neo-logicism à la Kripke.

 

[Rive]

New gun for the old banner: it is the old general who might be dead, but the empire lives on.

 

[nomadreid]

But that is my question -- it seems that the old empire, even under a new general, is beleaguered, as neo-Platonism seems to be the new "in", and neo-logicism is only mentioned in whispers. But why?

 

[Jarvis323]

I'm not an expert in the subject, but I have some thoughts about it. In my opinion, the primary foundations of mathematics are not without considerable doubts, and controversy. It is true that in mathematics, we start from a small set of axioms and then we see what follows. But we have no surety that the axioms we start with will be powerful enough to answer the questions we seek (for example p=np), nor whether they are a good natural basis for reality. In particular, most of modern mathematics, as pointed out here [https://mally.stanford.edu/Papers/neologicism2.pdf] can be considered applied set theory. Intuitively, this seems fine to me since set theory fits naturally with my style of thinking, but there is no strong reason to believe that a mathematical universe that well fits reality should be intuitive to humans.

But can we do better? I don't know, but my gut feeling says that we should be at least diversifying how we approach mathematics a little bit. Not that people don't do this already, but it seems like something that should be done starting at the lowest levels. I think we should have more money going into this type effort, despite its apparent lack of practicality on the surface.

In the long run, some particularly beneficial of ways of thinking about and understanding reality might stem from this, just as has happened following rigorous investigations into the current systems. It might be interesting to see, especially as we continue to learn more about, and grapple with, the strange nature of reality from observation (physics), if we can come up with any appealing new mathematical systems.

I don't know much of anything about neo-logicism, but... Anyways, I could be just talking non-sense.

 

[Rive]

But why?

What big success could this neo-logicism thing provide?

Science develops by exploring the borders. The secondary meaning of this is, that without borders there is no progress, no direction for progress: addition is, that without borders, there is no science. In this regard, Gödel & co. did not kill anything: they finally could give math hard borders - they could finally prove that math is real science, not just playing around with numbers and symbols.

With having this proof, the intent to find a way around Gödel is now very tricky. Anybody trying with this better prove that his/her way still have borders (therefore, it is still science) and also: it can encompass even Gödel (and everything before), with sufficiently useful additions.

Did this neo-watever could do it? Or is it just some meta-stuff, which needs no border (and, consequently: it is no science, but philosophy - at most...)

 

[nomadreid]

Thanks, Jarvis323. No, I would not say you are speaking nonsense (or, in analogy to the equiconsistency of axiom systems, then your ideas are equi-sensible to mine, although I cannot prove that I am talking sense). The link was relevant and very interesting, with a likeable thesis.

I guess it should not be surprising to find out that one of the stumbling blocks as to deciding whether mathematics is derivable from logic depends very much on what one counts as logic... first-order logic alone doesn't cut it, but I don't see the problem with in including, as the linguistic parsing of the phrase would allow, higher order logics or modal logics to be considered as part of logic.

And yes, we will always be doing a kind of leap-frogging, as we look for "reasonable" (which is not the same as "intuitive") axiom systems to solve problems of interest, and if and when we find them, new problems outside the reach of that axiom system would pop-up ( just as in a sort of more meaningful Gödel sentence than the one used in Gödel's proof), and so forth... which is why a flexible system, such as a Kripke frame, can lead to jumping from one axiom system to another ... and why can't this be included as part of logic (obviously not first-order)?

Some of the differences mentioned in that link which people claim to belong to mathematics and not to logic seem unconvincing... for example, the assertion that one cannot prove the existence of any real-world object with logical axioms alone seems very weak... I am not a solipsist, but even if one disagrees with Philosophical Idealism, one has to admit that it is defensible, and hence the Platonic position evidenced by that assertion, while also defensible, is not provable, unless one adopts a definition of existence that contains the conclusion one wants to prove.

Now to Rive:

What big success could this neo-logicism thing provide?

As any unifying principle in any discipline.

With having this proof, the intent to find a way around Gödel is now very tricky. Anybody trying with this better prove that his/her way still have borders (therefore, it is still science) and also: it can encompass even Gödel (and everything before), with sufficiently useful additions.

Did this neo-watever could do it?

Sure: look at any exposition of Kripke frames, for example, and the borders are clearly visible.

Or is it just some meta-stuff, which needs no border

If the "meta-stuff" can be formalized, it becomes "stuff" for a higher-order system, and hence with borders. I agree that fluff that is not formalizable (for whatever reason) is neither science nor even good philosophy, but I am talking about logic, which is by its definition formalizable, and excludes the fluff.

 

[Rive]

Sure: look at any exposition of Kripke frames, for example, and the borders are clearly visible.

I cannot decide on that, but in that sentence was: borders, encompassing nature, usefulness - together.

Just an example from physics: (special) relativity based physics encompassing classical physics: it has limits in being unable to explain both microscopic events and gravity, so these limits are drawing the clear area where special relativity is valid: it is definitely useful in handling high speed objects.

With my limited interest in the topic I can only suggest you to compare the logicism and neo-logicism this way - somehow I doubt that difference would not emerge...

 

[Jarvis323]

I cannot decide on that, but in that sentence was: borders, encompassing nature, usefulness - together.

Just an example from physics: (special) relativity based physics encompassing classical physics: it has limits in being unable to explain both microscopic events and gravity, so these limits are drawing the clear area where special relativity is valid: it is definitely useful in handling high speed objects.

With my limited interest in the topic I can only suggest you to compare the logicism and neo-logicism this way - somehow I doubt that difference would not emerge...

My understanding is that what we have discovered this century is that there is no one finite size axiomatic system that covers everything. And not only this, but each axiomatic system we choose may lead to mathematical unverses with major disagreements, for example, one with the continuum hypothesis vs one without. So we are left with the situation where we have to choose our mathematical universe, despite not knowing very objectively why we should choose one or the other, and there are many alternatives we are likely ignoring which contradict the ones we are living in. How we can live with this, I guess, is by the idea that these different universes with different truths differ only in ways that are far removed from practicality.

And we have no good handle on what is (actually, we are to believe there isn't) a stand alone truth. We can have it this way, or that way depending on the rules we choose. In fact, we could go on endlessly exploring the consequences of many arbitrary, perhaps ridiculous systems. It would be nice to find a way to do away with having only few example sets of sort of limited semi-arbitrarily chosen systems with unknown overlapping borders and incompatibilities. What would be nice if rather than having a few axiomatic systems that we deliberated about, is if we had a sort of system for building axiomatic systems, ideally one that is fundamental. That is, rather than mathematical structures coming just up from axiomatic systems once they are formulated, the axiomatic mathematical systems themselves are structures arising from a lower level system (a sort of space-time of mathematical systems if you will).

If it's even possible, maybe understanding something like this, would be a game changer in the long term for continued mathematical progress? Something like non-finite axiomatic systems, so forth all being objects of study under this field (analogous to how we study infinite sets, groups, etc, in ZFC). Again, I am just rambling, with no certainty I am speaking good sense.

 

[Rive]

Not nonsense, but this kind of existential uncertainty is difficult to make into useful math/science. I have a hunch that at some point this will also burn down to the good old 'condemned to be free' thing as it used to: without definite answers, or with answers promised in the next turn only.

I'm okay having math only and keeping all the rest private. I had my share of this way back, and the only result it brought is that for me promises rather works as warning signs.

 

[nomadreid]

Jarvis323: Indeed, it was Hawking who proposed that we handle physics with whichever mathematical model was convenient, and give up on trying to find a one-size-fit-all model.

Rive: Borders are naturally given in any good formalization in terms of domains etc., and this is very useful in logic as it is in any mathematical theory, just as your example with the present borders between QFT and GR. Pushing this analogy further, just as one is attempting to find a theory of quantum gravity, so too would a more unifying foundation for mathematics be useful -- after all, the earlier work on foundations led to many useful tools in information theory, so one may hope that further work on foundations will also bear fruit in practical realms. As in logic, as in mathematics: not knowing the applications in advance of a new theory (along proper lines, excluding pseudoscience and pseudomathematics) is no reason not to pursue it. (Lack of funding, of course, is a more concrete reason :-) ).

 

[nuuskur]

As a to-be-mathematician, my concern would be that of consistency. Gödel showed, however, that a sufficiently strong axiomatic system is incapable of demonstrating its own consistency. We already have a solid foundation for most of the mathematics. We leave the problem of "expanding $ZF(C)$" to set theorists, because they're good at using the 'forcing method' and dealing with model theoretic problems in general.

is there any formal reason to reject neo-logicism in the same way that the Incompleteness Theorems gave a formal reason to reject classical logicism?

This is a philosophical question. The incompleteness theorems showed, what Hilbert was aiming for, is not possible: axiomatising all of mathematics (whatever that may have meant to him). Do we have any such goal stated for our precious first (or higher) order logic?

As far consistency goes, first order logic is consistent. Any first order pure theory is for that matter, but that's a lot of technical model theoretic mumbo jumbo. So, why would we reject it?

 

[sysprog]

@Jarvis323, and @nomadreid, and @the_others here on this great PF place, I, along with many other computer nerds, think that P~=NP and oh by the way, if the 'smarter than' relation has any validity (I think that it has more than zero validity), I think that Kripke was smarter than I ever in this lifetime will be -- yeah, he, unlike me, got to hang around with the Vienna Circle in the early part of the 20th Century, and I wasn't born until after the start of the second half of that century, but heck, that exceptionally bright young man wrote [had written] a major work on a semantic subset of semiotics when he was/had only 19 and not yet 20 years of age.

 

[nomadreid]

my concern would be that of consistency

your first concern may be consistency in some systems, but since this doesn't get you anywhere in many of the more interesting systems, you have to settle either for equi-consistency, or consistency of system A using the system B which is as strong as or stronger than A.

As far consistency goes, first order logic is consistent.

This is not correct. Nor can one say it is inconsistent. The term doesn't apply. Theories are consistent or inconsistent; first-order logic is a system in which one can formulate theories. ZFC is a theory (which is not provably consistent), but there are other first order theories that are provably consistent.

 

[nuuskur]

This is not correct. Nor can one say it is inconsistent. The term doesn't apply. Theories are consistent or inconsistent; first-order logic is a system in which one can formulate theories. ZFC is a theory (which is not provably consistent), but there are other first order theories that are provably consistent.

Agreed. Seems I have fallen into a common pitfall: assuming first order theory is a subset of first order logic.

 

[nuuskur]

One thing that sticks out to me in the OP:

one has not yet formalized all logical foundations of mathematics

What is a logical foundation of mathematics? Provided, such a thing can be described, is it even safe to assume the "class" of logical foundations can be exhausted?Overall I'm conflicted with the OP. Which formalism do you assume to make your statements? (Likely FOL)

That is, the idea of shifting around is enshrined in such ideas as Kripke semantics, many-worlds semantics, Belief Revision Semantics, etc.

Fair enough, but how does it pertain to the problem of

is there any formal reason to reject neo-logicism in the same way that the Incompleteness Theorems gave a formal reason to reject classical logicism?

Do philosophy majors have a special class that gives instructions on how to make one's arguments as obscure as possible?

However, if we refer to logic (perhaps with uppercase: Logic) as a methodology to distinguish it from any specific logical system, where is the objection?

Objection to What? Why does there have to be an objection in the first place?

I feel like pulling my hair out. I just don't understand. Infuriating.

 

[nomadreid]

What is a logical foundation of mathematics? Provided, such a thing can be described, is it even safe to assume the "class" of logical foundations can be exhausted?

First sentence there: as Hamlet would have it, “ay, there’s the rub.” There is a dispute whether logic can serve as a foundation of mathematics. A Platonist would say that it cannot. So, to answer this question, one would have to figure out what constitutes a foundation of mathematics. Obviously it would be metamathematical, but one could start by taking segments of mathematics. For example, metamathematical statements such as “axiomatic set theory is a foundation of mathematics”, or “category theory is a foundation of mathematics”, are common.

To the second sentence in the above:
First, “class” is a mathematical notion. Foundations of all mathematics would be metamathematical, but foundations of certain mathematical theories can be mathematical. Obviously, these sections cannot be exhausted, which means that whatever foundation one chooses, it would be a flexible one – hence the reference to the attempts of Kripke frames.

Overall I'm conflicted with the OP. Which formalism do you assume to make your statements? (Likely FOL)

To the first sentence: Fine; I put this here to engender discussion, and to learn from constructive criticism.

To the second sentence: No. Metamathematics over a broad enough section of mathematics is not formalizable in FOL. Of course, the model relationship is formalizable in FOL, but to even start formalizing a more restricted version of my question, my guess is that I would need some higher-order modal logic.

Fair enough, but how does it pertain to the problem of "is there any formal reason to reject neo-logicism in the same way that the Incompleteness Theorems gave a formal reason to reject classical logicism?"

The semantics named were largely in response to the wish to deal with the notion of truth, showing that the Tarski-Church theorem on the indefinability of truth was not a death blow to the notion of truth. This theorem has the same basis as the theorems that brought Hilbert’s logicism to a close, but with the trend towards more flexible semantics, the spirit, if not the letter, of Hilbert's idea might be salvaged. It may seem to be a contradiction to talk about a formal (hence, perhaps, logical) reason against neo-logicism when I just said that we were talking about logic, but after all, the formal results of Gödel, Tarski, Church, Robinson, etc. had decisive influences on metamathematics.

Do philosophy majors have a special class that gives instructions on how to make one's arguments as obscure as possible?

I am not a philosophy major. The question here is philosophical, or meta-mathematical, and not directly mathematical, therefore this does not appear in the Mathematics section. (However, non-philosophers may engage in philosophy -- Bertrand Russell, Quine, etc. being prime examples.) However, if someone formalized the issue, thereby clarifying it, I would be all for it. You may notice that I began my first post by stating that I am not clear on the status of neo-logicism, and hence am asking for clarification.

Objection to What? Why does there have to be an objection in the first place?

For example, from https://www.iep.utm.edu/mathplat/, “Mathematical platonism enjoys widespread support and is frequently considered the default metaphysical position with respect to mathematics.” Platonism is, as is explained there, opposed to logicist standpoints. This is recognized in the many attempts to make neo-logicism more "respectable". A small sample from a quick Internet search:
https://pdfs.semanticscholar.org/7630/c73ec0aea2df8c7aabfa63ded8011a188336.pdfhttps://philpapers.org/archive/RAANAI.pdfhttps://mally.stanford.edu/Papers/neologicism2.pdf

I feel like pulling my hair out. I just don't understand. Infuriating.

Well, it can save you the price of a haircut, and at least the experience might warn you against entering the teaching profession. :-)

 

[nuuskur]

text

I appreciate you being patient with me. I also apologise for my pettiness.

the experience might warn you against entering the teaching profession. :-)

Teaching calculus or algebra is not so daunting at all. Maybe because of the simplicity or there's something about mathematics that makes teaching it significantly easier.

 

[nomadreid]

I appreciate you being patient with me. I also apologise for my pettiness.

I appreciate your comment. No problem.

Teaching calculus or algebra is not so daunting at all. Maybe because of the simplicity or there's something about mathematics that makes teaching it significantly easier.

This depends on your audience and situation. I was referring to school teaching below university level, including classes of students who are not of your opinion that mathematics is easy, and feel that they are in school, or in mathematics class specifically, against their will. You are correct that the mathematics itself is much more straightforward than most other subjects, and so if teaching consisted only of lecturing to classes of gifted, interested and hard-working students, then this would be wonderful... but teaching mathematics at most schools below university level involves the actual material only as a small portion of the teacher's tasks, and the rest of it, from discipline to grading (evaluation) to entertainment to psychology to adolescent hormone jumps to dealing with administration and parents to malfunctioning technology to erratic scheduling to competition with the students' other interests to emotional problems to simple laziness to bullying to ... takes up the majority of it, something which requires immense patience to be good at it.