# Logic and math as a study of possibilities and not so much about human reasoning

Most of what I've come across about the "hierarchy of disciplines" seem to say that logic/math is more fundamental than physics, physics more fundamental than chemistry ... biology more fundamental than psychology, etc. This seems all well and good. However, some of them also say that psychology loops back into logic/math, as if logic/math is also about thinking and language, and manipulation of abstract symbols. Does this not confuse the activity of studying with the subject matter? Sure we use language in logic and math, just as we do for the other disciplines, but that doesn't necessarily mean that logic and math IS ABOUT symbols and their manipulations. I believe that it makes more sense to say that logic and math studies what's possible and what's impossible which secures its place as more fundamental than physics.

I'm interested in what others think about this, and if there's a school of philosophy that would cohere to my line of thinking so I can study it further to see where it takes me.

• Modal neo-logicism comes to mind, but why not let the tree grow towards its roots? May 28 at 16:43
• What do you mean by "logic and math studies what's possible and what's impossible"? In math, when we manipulate symbols to find a solution, we say "It is impossible to divide by zero." In statistics we say that the impossible event or outcome has probability zero written as P(X) = 0 where X is the impossible possibility. Some keyword search turns up questions of proof of impossibility using math, but such proof relies on the manipulation of symbols according to rules, and I think there are always Axioms in any math system that cannot be proven true or false within the math system. May 28 at 17:46
• "Fundamental" in what sense? May 28 at 18:09
• @SystemTheory basically saying them being deductive, leads me to think the subject matters are what must be true (logically true), satisfiable (possible), contradictory (impossible). Proofs are showing what must be true given axioms, lack of proof still shows satisfiablity in a sense, I believe. The symbols seem more like tools, as experiments are to sciences.
– csp
May 28 at 19:05
• There is a divide between the traditional view of logic as "laws of thought" and Frege's rejection of this as mere psychologism. Most of modern logic since Frege has followed his view that logic is about relationships of consequence between sentences, not about how human beings think. So many, maybe most, logicians would agree with you that logic is about possibility or consistency. There are also some who maintain we can combine the two approaches. May 28 at 20:57

Looking into the epistemology of modality would be a good starting point. More specifically in relation to mathematics, Hamkins and Linnebo  go over a "potentialist" account of mathematical (or at least set-theoretic) ontology, which roughly hearkens back some to Zermelo's own "unfinished totality = potential infinity of actual infinities" viewpoint. You could also try out Zalta's modal neo-logicism:

Zalta (1999) proposes an interestingly different, because modal-logical, route to the natural numbers. ... The first-order Barcan formula [as used by Zalta] forces one to interpret quantifiers as ranging over all possible individuals, whatever world one is ‘in’—no ‘expansion’ or ‘contraction’ of the domain can be involved as one traverses the accessibility relation from possible world to possible world.

The logic is free, and descriptive terms (the description operator ι is primitive) are interpreted rigidly—that is, the denotation of a descriptive term in the actual world, if it has one there, is its denotation in any other possible world.

Now, so far as (deductive) logic is often characterized in terms of the necessity of conclusions from their premises and by the inference rules, there is a "modalization" of logic-talk from the get-go. Induction then might be handled by interpolating possibility-talk and probability-talk; in Łukasiewicz logics, for example:

This revolutionary development came in the context of discussing modality, in particular possibility. ... For if values other than “0” and “1” are interpreted as “the possible”, only two cases can reasonably be distinguished: either one assumes that there are no variations in degrees of the possible and consequently arrives at the three-valued system; or one assumes the opposite, in which case it would be most natural to suppose, as in the theory of probabilities, that there are infinitely many degrees of possibility, which leads to the infinite-valued propositional calculus.

Will circularity really be avoidable, though? Consider impossible-worlds talk:

... another definition has it that impossible worlds are worlds where the laws of logic fail (where these may be theorems of the target logic, or its logical truths, or valid consequences, etc.). This depends on what we take the laws of logic to be. Given some logic L, an impossible world with respect to the L-laws is one in which some of those laws fail to hold (see e.g. Priest 2001, Chapter 9).

I've even seen people express doubts over whether modal logic itself is "really" logic. But so now rather than think of all domains of discourse as layered in a hierarchy, one can imagine them as sections of a sphere, and we might want to identify both the union of all the domains, as the entire sphere, and then also the intersection of the domains. Does logic occur as the intersection, here? Or does language? If both logic and language in different ways occupy the intersection, perhaps we could hold to both "logic is about objective possibilities" and "logic is about subjective linguistics" (c.f. Kant's critique of metaphysics, which is much invested in judging modalized knowledge claims).

Have you encountered..? Munchausen's Trilemma

Hofstadter's solution is that Strange Loops, the kind of self-model feedback loops intrinsic to having intentions, relate to 'tangled hierarchies'. The clearest example of which I think is, The Fabric of Reality: The Four Strands, by David Deutsch.

This is as an extended version of XKCD 435: Fields Arranged by Purity: Recursion, feedback, higher-order logic. That is the domain of life, of intelligence, of creativity. Don't let the simplicity that enables effective models, convince you that therefore reality must be simple. It is not; it is turbulent, fractal, self-referencing, and stranger than we can yet imagine.