The headline question is mainly a rhetorical question.

Paul Cohen, a major set theorist who invented 'forcing' in Set Theory for independence results and used by Badiou in his philosophy has this to say about the formalisation of logic in this paper about the discovery of forcing.

The attempts to formalise mathematics and make precise what the axioms are, were never thought of as attempts to explain logic, but rather to write down those rules and axioms which appeared to correspond to what contemporary mathematicians were using.

He also adds:

An unnatural tendency to investigate, for the most part, trivial minutae of the formalism has unfortunately given the subject a reputation for abstruseness that it doesn't deserve.

Essentially Cohen is saying that formalising mathematics and logic means that one can begin to think about the structure of the formalism in a way that one could not before. Its like drawing up a map and naming different areas of it so one can begin to talk about it in a structured way mathematical way.

In fact earlier he writes:

I can assure that, in my own work, one of the most difficult parts of proving independence results was to overcome the psychological fear of thinking about the existence of various models of set theory as being natural objects in mathematics about which one can use natural mathematical intuition.

Given this being said by one of the past masters of the subject, why is it taken by many people that a formalisation of logic has been fully achieved?

This shouldn't be taken as an attack on the formalism in logic or mathematics, nor as a Sokalian atack on Badious use of Set Theory.

  • I personally do not agree with "so negative" a reading of Cohen's paper. Formalisation is a tool, and not the "essence" of mathematical thinking. So certainlly, the formal approach does not take the place of the "use [of] natural mathematical intuition". The modern epistemologists call attention to the distinction between the "context of discovery" and the context of justification". Formalization is an essential tool for the study of mathematica theories and languages as mathematical objects themselves, but it hardly the source of new axioms and theorems. – Mauro ALLEGRANZA Feb 17 '14 at 13:47
  • @Allegranza: I fully agree with what you're saying. I'm not sure where you're seeing a 'so negative' reading of Cohens paper. Formalisation is indeed a tool and not the essence. I'm asking why do so many people equate the essence of mathematics with its formal apparatus. – Mozibur Ullah Feb 17 '14 at 14:31
  • I don't think that "so many people ..." is true if you think to mathematicians. It more appropriate as a (simplified) approach of many (so-called) analytical philosophers. About mathematicians, I think that the "standard" approach is like : "In the first chapter of my treatise about ... I'll show you that there are basic logical and set-theoretical notions that can be used to formalize ... Then forget them and let's go on." – Mauro ALLEGRANZA Feb 17 '14 at 14:37
  • I'm not thinking about mathematicians - they know & understand the tools of their trade. I'm thinking of the wider world outside who see physicists & mathematicians at work, have seen the result of their work - computers, the internet - and come to the understandable but simple-minded conclusion that its the formalisation & the formulae is most important. The question is really aimed and understanding why this kind of thinking is so pervasive. Perhaps its skewed by some of the comments I've seen here on SE. But my hunch is that its becoming an increasingly common phenonomen - probably – Mozibur Ullah Feb 17 '14 at 14:44
  • not helped by the gee-whiz 'ain't science wonderful' popular science programmes. This shouldn't be construed as an attack on science and mathematics which may be it looks as though the question is. – Mozibur Ullah Feb 17 '14 at 14:47

About the question :

Given this being said by one of the past masters of the subject [Paul Cohen], why is it taken by many people that a formalisation of logic has been fully achieved?

I think that is not a "good practice" in science to ask about "fully completed" ... solutions, theories, and so on.

History of science (mathematics included) shows us that the history so far has been "open ended".

The impressive amount of results achieved by mathematical logic during the past century, does not preclude that new ideas and discoveries will in the future change our understanding of the discipline.

The quote is from the second page of the paper :

They [mathematicians] may feel that the “official” exposition of set theory, i.e., all of mathematics, using formal systems and particular axiom systems, has little relevance to their work as research mathematicians. On the other hand, the existence of a whole series of surprising results has to some extent shattered the complacency of many mathematicians, and there is an unjustified aura of mystery and awe that tends to surround the subject. In particular, the existence of many possible models of mathematics is difficult to accept upon first encounter, so that a possible reaction may very well be that somehow axiomatic set theory does not correspond to an intuitive picture of the mathematical universe, and that these results are not really part of normal mathematics. In these lectures I will try to clear up some of these confusions and convince you that indeed these results are easily accessible, even to a nonspecialist. I can assure that, in my own work, one of the most difficult parts of ...

I think that these words are "consonant with my comments.

Mathematician, after the end of the struggle between "foundational schools" (Logicism, Intuitionism and Formalism) during the '30s and after the discovery of Godel's Incompleteness Theorem, developed an attitude of "disnterest" with respect to mathematical logic : it is useful for "setting the scene" right ... but the job of the "working mathematician" is different.

Also set theory was perceived as a "framework": a common languge, like latin or esperanto ...

The results of Cohen (but remeber Godel (1937) on the same subject : again Godel!) was important becuase developed new and powerful tools for "first line" research in set theory: following Cohen, set theory becomed again a living field of research (ask @Asaf Karagila on Math.Stackexchange) with a richness of methods and problems (see Imre Lakatos about problems as the driving force of mathematical "evolution").

THe brief but interesting historical sketch traced by Cohen (Cantor, Frege, Zermelo) point to an interesting issue : the role of logic (i.e. formalization) as a driving force towards new results, and this the "message" I read in his words: logic is not only mathematical "hygiene".

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  • I'm not saying that it is fully formalised. I'm asking why is this point of view so common? Perhaps my question isn't so clear. – Mozibur Ullah Feb 17 '14 at 14:33
  • @MoziburUllah - We have two possible reading: (i) a "final solution" about the foundational issues; this is clearly not yet achieved, and I think that it is quite impossible (as for every "deep" philosophical problem). (ii) the current "fundamental* language of mathematics (but is it set-lang or category/topos-lang ???) is the "definitive" one, i.e. we do not expect possible new "big" discovery: the tone of my statement shows that also in this second case I do not think it is absolutely true. – Mauro ALLEGRANZA Feb 17 '14 at 14:41
  • Yes, again I agree. But I think you're answering a question I'm not asking. I specifically said in my question that Cohen wrote 'attempts to formalise mathematics...were never thought of as explaining logic'. Doesn't this support what you're saying? – Mozibur Ullah Feb 17 '14 at 14:51

To some extent, the underlying problem here (and something that Cohen tacitly discusses in the linked paper) is one of Demarcation. Where does the mathematics of Set Theory end and the logic of mathematical truth begin?

I really liked Cohen's description of the Lowenheim-Skolem theorem on p.1085 as

the first nontrivial result of logic.

To summarize, so as to keep this answer relatively self-enclosed, the L-S theorem is a result in classical first-order logic that shows that wherever we have a transfinite model of some theory, we can show that there is a countable one, and indeed one of any arbitrary cardinality. This is a logical result, not a set-theoretic one, in the sense that it concerns a property about the abstract satisfiability of theories rather than something particularly semantic. In fact, the traditional understanding of this result seems at odds in many ways with what set theory tries to say about mathematical resources beyond the finite ordinals - a seeming called the Skolem Paradox (SEP Link).

Cohen knows about and understands this theorem incredibly well - it is importantly involved in the technique of forcing over Set theory models to get "new" models through a countable treatment adding new "generic" sets. This treatment is in some sense also independent of its set-theoretic interpretation, since what really matters for the purposes of using forcing isn't that the models are set-theoretic, but more that they follow a Boolean-valued model structure, which guarantees that there is some partial ordering on how we add new forcing conditions respecting classical negation closure and exclusion; set-theory is simply the most standard interpretation and representation of boolean valued models.

The question of whether Cohen's technique challenges the coherence of a full formalization of Logic is thus a very nuanced one. Using Forcing, Cohen shows us that given some accepted groundwork in the axioms of Set theory, it is possible to construct a diversity of models, some of which may adequately reflect hypothetical "axioms" or desirable properties that mathematicians may want to further dwell on, and others which may demonstrate properties not immediately reflected in how mathematicians go about practicing their craft. There may not necessarily be a sense of a right and wrong way to formalize our axioms, in that these various models might all be independently mathematically interesting, and all of them can be looked at and theorized about within the scope of mathematical logic as it currently stands.

But on the other hand, the logic that he uses to demonstrate this technique doesn't itself appear to be under threat by that observation, for the simple reason that the modality of Forcing only branches out in the realm of the transfinite, thanks to the Lowenheim-Skolem theorem. Nothing in forcing extensions will tell you anything unusual about whether 5+7=12 - they might intuitively semantically differ on what set, exactly, the sum of ordinal 5 and ordinal 7 is (unless we also have the axiom that we have a countable standard model), but thanks to the way forcing preserves satisfaction of the ZF axioms from model to model, truth of the relativised versions of statements about the finite ranks of the set theoretic hierarchy is preserved between forcing extensions of ZF models.

What this seems to turn on then is what kind of notion of Consequence is at stake when we look at the question of what makes something Logic versus Mathematics. An intuitive sort of "Truth across all models" conception would say that there is something seriously up with what Cohen is doing to logic here, because he's essentially creating "new" logical models that he ought, surely, to have recognised prior to using the L-S theorem as a logical result. But that doesn't seem to be what Cohen thinks he's doing - let's take a section from p1089:

Nevertheless, the feeling of elation was that I had eliminated many wrong possibilities by totally deserting the proof-theoretic approach. I was back in mathematics... Thus I assumed immediately that I had a standard model of set theory, which although 'obvious' cannot be proved in ZF.

Cohen's results although logical are properly mathematical consequences, rather than logic; Cohen via Godel had managed to get a certain amount of the way in using tactics in pure logic, but needed to step outside for additional power to get his interesting results in his classification of mathematical structures. The notion of mathematical truth seems to be a very particular species of truth once you get beyond a certain accepted kernel of logic (this would be in line with Hilbert's ideas about the role of consistency proofs of theories defined over finite arithmetic).

If this is right, then the idea that we have "settled" on first order classical predicate logic can be considered consistent with Cohen's reflections on the vast plurality to be found in set theory, because what we agree on and what logic in abstraction from strictly mathematical content can decide for us is that stable finite core that all of the possible models accept. That would be my way of lending some credence to the thesis that logic has in some sense been properly formalized.

But maybe that's not all there is to say; maybe we might want further to consider that we can add more logical structuring and higher cardinals to talk about degrees of truth according to how typical some propositions are across the various forcing extensions that we might come across. If you want to check out some of this more esoteric analysis, you could do worse than checking out some of the ideas in Woodin's Ω-logic (Bagaria et al have an advanced Primer for further reading).

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