Intuitionist Mathematics and Physics

Posted on September 5, 2008
Filed Under Peirce, Philosophy, Science | 9 Comments

There’s a great post at Mathematics and Computation on Intuitionist Mathematics and Physics. (HT: Philosophy, Science, and Method) I always found Intuitionism a bit vague in some key places but there’s a lot about it I like. The big issue is in what counts as evidence. I tend to favor the Peirce view of falsification. (Something is true unless shown false) The Inuitionists, as I understand it, are more positive. Something must be positively shown true. They end up being pretty similar outside of that though in many key aspects.

The Peircean view, of course, only makes sense if the determination of truth is something off in the indefinite future when all possible falsifications have been processed and the community of inquirers has reached consensus. It’s debatable whether this conception of truth was something “real” or just a normative notion explicating what we mean by truth. (I favor the former but a lot of Peirceans favor the latter)

What this means is that Peircean truth while appearing negative (like Popper) ends up being tied to positive experience. It just acknowledges that meaning is always caught up in a context. (Roughly similiar to what Duhem and Quine argued) That’s why I see a lot of similarity.

Of course there are problems between Intuitionist logic and Peirce. Consider that Peirce’s law, ((p → q) → p) → p, can’t be proven. Of course for Peirce that’s not necessarily a problem since he adopted what today we’d call quasi-empirical methods in logic and mathematics. That is it wasn’t just a matter of absolute proof but what an ideal community of inquirers would arrive at given that there were real universals in the universe. Since these universals “act” they affect the universe and are knowable. While he’s not quite a Platonist and definitely not a Platonist of the traditional sort he does think that there are real structures in the universe that are somewhat independent of any particular thing. They are objective in that they are knowable and don’t depend upon any particular finite mind.

One of the big parallels between Peirce and the Intuitionists was the Law of the Excluded Middle. Peirce felt it didn’t apply to Generals. (His term for universals) And of course it’s rejected in Intuitionist logic. One has to be careful since depending upon one was reasoning about particulars, generals, or vague descriptions different logical principles applied.

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9 Responses to “Intuitionist Mathematics and Physics”

To me this all feels very much like the sighted man watching the six blind men of Hindoostan.

Given Lukasiewicz’ axioms of three valued logic and suitable (though slightly nonstandard) definitions and interpretations of the symbols, Peirce’s axiom is true and I can prove it. So are Heyting’s axioms for intuitionism. So are Lewis’s axioms of Modal logic.

What I can’t do (yet) is convince anyone that my proofs are worth serious attention.

Confutus, I do not know the details of your proofs, but from your comment it sounds like your method is that of validating certain laws of logic by interpeting them in a particular three-valued model, or interpreting them inside Lukasiewicz’s logic. This is ok, but it is insufficient to show that the law of logic is also valid in other situations and under other interpretations. If everybody only cared about Lukasiewicz, then we would care about your proofs. But we do not. We are interested in many other aspects of logic, its connection with computation, geometry, etc.

I was wrong in claiming that Peirce’s axiom is true. I rechecked my work, and it fails.

I expected that simpler and more general methods of logic that are capable of reproducing, reconciling, and explaining the results of various seemingly incompatible systems of logic would be of some interest to someone. I was naively overoptimistic. What I’ve found instead is resounding indifference.

It seems to me that once you have a sound theory, the applications to such things as computation, geometry, and so forth all fall much more easily into place, although I’ll admit that I haven’t found those sufficiently interesting to compare the different approaches. I’m primarily interested in the comparison of different systems of logic.

The problem I’ve seen with intuitionism is that there is a subtle flaw in its foundations which, although it doesn’t invalidate it, makes it much more difficult and cumbersome to apply and interpret than it needs to be.

I’d second what Andrej said. (And indicated something along those lines in the final sentence in my post) I think what logic one uses depends really about what you are reasoning about. How one reasons about possibilities and how one reasons about existents are just different. I believe that was one of the big things Peirce brought forth to logic even if many of his discoveries weren’t acknowledged at the time. (And in one case “rediscovered” by Russell who probably should have known Peirce had discovered it earlier)

The idea that there is one set of rules for everything seems difficult to accept.

One set of rules for everything? I wouldn’t go so far. I can think of several things the system I use doesn’t cover.

I don’t blame you for being skeptical. Back when I was just trying to figure out why three valued logic was a mostly useless curiosity and not a *real* logic, I never dreamed that the results would be so far-reaching and extensive.
I would express the concepts you have given as
Peirce (not P) = ~[]P (not necessarily P)
Intuitionism, (not P) = ~P (not possible P) and from this draw some quick conclusiona about how these negations ought to behave. For instance, Double negation is not equivalent to a given proposition in either case, but there is an interesting symmetry.

Nevertheless, (to go all Mormon about logic) none but those who arouse some particle of interest sufficient to experiment with it will ever be enlightened by it. If no one is interested here and now, I’ll go away and try someone else, somewhere else, some other time.

Note that as a Peircean of course I value three valued logic. (See the SEP for a brief discussion of Peirce and three valued logic) That was one of Peirce’s main innovations. And it is key for realistic semiotics as a general logic. I just think one has to consider the nature of what one is reasoning about. (Existents vs. Possibles being the obvious place this matters)

So I’m definitely sympathetic to what you say limited primarily by not nearly as much knowledge of Lukasiewicz as I should. (And honestly I haven’t spent much significant time on formal logic since the 90’s although I hope to remedy that one of these days)

I suppose I’m equally limited by the fact that I’ve never studied Peirce. Although I do note that the some of the ideas Peirce pursued as indicated in the SEP article are significantly simiilar bar operator is the same as Lukaseiwicz’s negation, and his Z operator is the same as Lukasiewicz’s ‘and’ The most important weakness I see in Peirce’s 3 valued logic is that he didn’t have a good conditional for purposes of implication, inference, and deduction.

Lukasiewicz used an unfamiliar prefix notation, one that is good for some technical reasons, but not easily read, written, or understood. He did trouble to invent a conditonal, which *almost* works, but doesn’t quite.

I prefer the notation and some of the concepts of C. I Lewis, who attempted to make a comparison with his modal logic and that of Lukasiewicz, but wound up rejecting it, partly because of the inferior conditional, and partly because he, like the intuitionists, held tightly to the principle of the excluded middle. This similarly creates a subtle flaw in his modal logic, so that his theory is forced to use comparatively difficult and cumbersome methods of proof and semantics.

Lukasiewicz apparently overlooked the possibility, which is implicit in his definitions and axioms, that he could define a strict or exact conditional. In practical terms, that’s a huge oversight. I can hardly blame his logic for having a poor reputation, because without a strict conditional, it’s crippled to the point of paraplegia. With it, any superlative sufficient to describe the difference looks like hyporbole.

I think Peirce ends up with those things. As I said I’m way out of practice on formal logic – so excuse egregious errors on my part. And I don’t have time to look it up. But I seem to recall him dealing with those (which would have been key given his trichotomy of categories).

But I’m just not familiar enough with the papers to say much without getting back up to speed sadly.

Here’s a perspective on Peirce’s Law as it appears in a version of Peirce’s own logical graphs:

http://planetmath.org/encyclopedia/PeircesLaw.html

Jon Awbrey

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