Here I formulate a few critical points in various areas of mathematics
Also I need tools and instruction to write mathematical notation from the comp.
As totally unrelated issue, I need piecewise explanations of small pieces of text in English and Spanish.
The math issues.
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No particular points hard to pass. Just give me a motivation to learn it. The best known use of topology is as "rubber geometry". But all books I saw are extremely bad in providing it. When they try it at all, they introduse geometric shape, appealing to geometric intuition, and link them to the topologic concepts learned, in the way those concepts don't appear any useful. Given the shapes popular in studying topology, how are they defined, and their properties derived, in purely topological axiomatic terms?
Of course, there are other uses of topology, but I am too unfamiliar to their contexts. So I'll need help on those of the contexts that intersect with my interests.
Geometry, or, better said, geometries.
Not to learn the subject, but: is there definition of "geometry" like that of topology, that is not as an area of studies (what all sources say in an indecisive, common-sense style) but as a mathematical structure, or at least a theory, with clearly defined properties?
I'll need directions to fundamental axiomatics of different geometries, especially minimal axiomatics, "periodic table" of geometries, sketchy description of how axiom systems are developed into entire geometries, in particular - how is a Euclidean geometry of whatever dimension defined isotropically, without use of coordinates (and using some minimal structure instead of field of reals).
(currently nearly reserved to a person;
non-investing acquaintance is welcome, anything more must be aligned with that person;
situation to be updated)
I need explanations of different kinds of logic: their purposes, use cases, rationales, what-ifs. All it in small pieces, in most cases I'll just abandon them when learn they are not what I need.
Systematics of logic systems, notation in the context of this systematics (books I've read until now rarely use this notation and never explain it).
As well as for topology, help me to decide what and why I really need of logics. If and when I decide to proceed with some book (to whatever distance, ranging from acquaintance to learning throughout), I'll need tutoring in form of supported reading (explication of separate obstructing points and consultation about what I can safely skip in a book, given my goals).
Mostly probable, those books will be on intuitionistic theory, lambda calculus and like, not calssical logic.
Also, a few particular items.
It's a very ugly idea to separate between implication and inference. I understand many different modalities of inference (as well as of other kinds of formulas): merely a syntactic structure, various claims about it: idea (to be treated further) of it being "true about an object", claim of it being, again, "true about an object", claim of it being "true in an interpretation", claim of it being valid. But why inference as a concept separate from implication? I found many reasons for it but few justifications (while logic is intended for use as logic, not as one more purely mathematical structure to challenge).
Why not do the following:
define expresiions as graphs rather than strings (use strings only as metalanguage means to describe the graphs). Define conjunction/disjunction (defined independently or one from the other, it's of choice) as governing arbitrarily big (or, alternatively, any finite) set of expression appearances (it will be more difficult to express in a text, but more adequate conceptually). Have axiom shemas like multivalent version of
(no matter whether implication is primary or derived),
have no inference rules, instead of syntactic proofs develop an axiom of form:
where a is again an axiom, b->c is another axiom, b is conjunction of the premises, c is what we wanted to derive from b.
Of course, one doesn't need to write down entire axiom even once, there must be shortcut form (similar to existing syntactic proofs, but intended to be metalanguage means) from which the entire axiom can be recovered in a tedious but straightforward way.
Apparently, the classical logic cannot be translated to that form because it allows infinite set of premises but not infinite con/disjunction (finite but nonbinary and unordered one is't allowed either but can be easily emulated). But why at all have infinity at one end and not in-between? Even why have it at all? As much as I have seen (but I could miss, I skip much) in the books, infinite set of premises in mentioned only to demonstrate use of the compactness theorem.
I have idea of accretional logic, working following way:
beginning from nearly empty language and theory, add step-by-step more predicates and axioms (referring to those already present), as well as functions and constants (but those are not essential as they can be expressed through predicates and axioms).
Extra predicates are always allowed (together with axioms to build from them functions and constants), but sentences to be tested by or added to the system can be of some limited form (a plausible form is: an inference from a conjunction of a few atomic sentences to single atomic sentence, negation doesn't exist, but T and F are bacic predicates, T is in the theory, F is not; existence quantifier doesn't exist, entire sentence under consideration is governed by implicit universal quantifiers for all variables involved).
Those sentences are subject to test through some algorithm, with following possible results:
either the sentence is inconsistent with the theory in its curent state (F is derived from it);
or it'not the case, and we can (but not have to) "accrete" the theory by adding the sentence to it;
the last case divides into two:
either the sentence under question is derivable from the current theory (in which case adding it is futile),
or it is proved to be underivable from the system, so adding it will "proper accrete" the system.
With accretion, results of the tests mentioned can change as follows:
what was incompatible with old system, will remain incompatible with new one;
what was compatible, can become incompatible;
what was derivable, will remain derivable;
what was underivable, can become derivable.
Universe and, respectively, model is not needed at all.
Such systems can form a set partially ordered by derivability of one system from another by accretion.
Do you know something of such idea? Sources, keywords?
A function, a relation of whatever arity (with operands generally from different sets) can be presented as one more operand under "upper evaluator": f(a,b) can be presented as F(f,a,b), where F is an "applicator", then make further step: f is not anymore a function, it's merely an element of some set, while F is a function that, having arguments properly positioned, emulates the situation that "f is a function". Also we can move in the other direction: "a" can be considered a function q: (f,b)|->f(a,b) .
So we get kind of unification (that still does't cancel the very concept of function/relation, only pushes it one step away).
Now apply it to first order logic. Replace n-ary predicates with new constants and introduce an n+1-ary predicate to emulate previous semantics:
we had predicate P and Q terms a and b, and formulas P(a,b) and Q(a,b), and now we have a "superpredicate" S while P and Q became constants, and instead of P(a,b) and Q(a,b) we have S(P,a,b) and S(Q,a,b).
Such system is very imperfect, arity must not be of special importance, instead everything must be typed - but let it all aside. Concentrate on some fixed arity to avoid distraction to catch the idea.
Now, what if we plug variables instead of constants, former predicates? Interpretation needs to "grow" for the cases when those variables are not equal to any of constants mentioned above. What was a predicate, now can be quantified. But we didn't get a second order logic: here "predicate" variables are conceptually equal to usual ones, they are quantified over usual universes - the common one in one-sorted case, and a separate but not special one in multisorted case, never over a set of tuples.
Again, do you know something of such idea? Sources, keywords?
Conditions, depending where you have arrived from:
Primary responce here in comments.
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