Are there clear counter-examples to this definition of mathematics?

Question

Here I'll re-present the question about a definition of mathematics as being about deduction, that I've given in a prior posting, but here I'll further clarify that this might not be what is usually known about that term, I mean "deduction". Here before I'll go to that I'll present what I believe it to be the two main sources of factual verification attainable to us humans, that is the :empirical and analytic. A factual empirical proof is an observation or otherwise a consistent result of experimentation under controlled conditions. This is factual evidence as present to the senses. The other source of factual evidence, i.e. of truths, is the analytic source which begs no empirical validation at all, i.e. it is empirical free, and philosophical free, i.e. not subject to philosophical debate, it merely depends on controlled string manipulation substitution rules (I'll call those inference rules), example: All bald men are bald. No one needs to go see all bald men and check whether they are bald or not in order to verify such a statement. Another example of such a string manipulation is reciprocity, i.e. if at rule 1 x is replaced by y, and at rule 2 y is replaced by x, then if we apply rules 1 then 2 to object x we get x. A simple rule following symbolic substitution game. Of course we can begin with any strings of symbols, define any amount of manipulation rules, and then define a symbolic system as the closure of all strings of symbols under the specified inference rules, the resulting system is a set of all strings of symbols so obtained. I'll call such a system generally as a "symbolic deductive system". Now a "deductive proof" is a minimal subset of a deductive system in which every string in it is derived by an inference rule from a string in it. Of course axioms are strings derived by 'stipulation rules', [i.e. stipulated directly to be in the system], a privilege not offered to any string in the system. Now deductive provability in this general sense is what is meant by "Analytic fact", in other worlds the fact is the statement that if such and such strings belongs to a specified deductive system then such and such strings also belong to it. Not only that it can also include a specific assignment of "meaning" in an exact manner to the symbols of that system, and so the result would also be analytic for that meaning, i.e. the fact would be if such and such sentences in the system are about meaning such and such, then it follows that sentences such and such would be in the system and it would be about meaning such and such. Of course the specification of a deductive system is by specifying its inference rules and the primary string of symbols, i.e. those privileged to be derivable by stipulation rules, and sometimes the specific meanings attached to its symbols especially the axioms. The meaning assigned to a deductive system, i.e. it's semantics, is considered as "part" of the deductive system.

I'm personally not aware of a standard of factual knowledge other than being either empirically validated or analytically validated. What I mean here by "factual" is having high degree of certainty about it that the endeavor of opposing it is a negligible quest. Of course there are other kinds of knowledge some might be based on extensively trained philosophical insights like in metaphysics, ethics, epistemology, etc..., I'm not sure if such matters bear a kind of certainty about some of its material similar to that in empirical and analytic spheres, I think there should be some!? supposing that some do, then those are pretty much complex and definitely non-trivial extra-analytic pieces of knowledge. Also proofs by appeal to "intuition" about extra-analytic matters, are extra-analytic, while argumentation based on intuition about deductive systems that are not based on non-trivial extra-analytic provability, are considered an intuition about analytic matters, although in itself is not an analytic fact, yet it is permitted to be the basis of axioms for pure mathematical systems (see below).

Now the main question that I want to answer is what is the definition of mathematics? in other words what is subject matter of mathematics? i.e. mathematics is the study of ...? what we are to put in the blank? Botany is the study of "plants", Astronomy is the study of "Celestial objects", Medicine is the study of "management of diseases", pretty much specific subject matters. The question here is more about what "Pure Mathematics" is really about.

My answer to this is that Pure Mathematics is the study of what takes part in non trivial deductive systems whose axioms, inference rules and definitions are all not the results of provability by non-trivial extra-analytic methods.

Applied mathematics is defined by exactly the same definition above but with removing "not" from it.

Now Mathematics in the most general sense is the study of what takes part in non trivial deductive systems.

It's important to realize that the definition given here to "deduction" is not the traditional one, the traditional one is about sentences when the inferential rules are logical. While here this method applies to any exact way of specifying a string of symbols, including individual symbols, terms, diagram sketching, model representations,..any piece of syntax, so an inference rule can be a function on any piece of syntax, not necessarily being functions on "sentences" as it is usually held. Actually we can even have inference rules being non-functions, i.e. having multiple outputs, so they are definable relations on pieces of syntax, whatever those pieces were, but they must be exactly specifiable. It is also important to realize that the sentential inference rules (and therefore sentential deduction) can be extra-logical also, provided that they are exact! I think this would be the widest possible definition of "deduction", that might be even undesirable in some sense.

Also to be understood is that once we've said "study" we mean treatment of all aspects involved in the definition, so the informal reasoning leading to axioms, definitions, inferences, etc.. are all involved in this study of course, as long as the result at the end abide by the definitions given above, as well is the technical manipulations involved with the study of having good terminology, clear processing, etc...

My question here: are there clear counter-examples to this definition?

Answer 1

Pure Mathematics is about what can be proved deductively in a non trivial manner in deductive systems whose axioms, inference rules and definitions are all not the results of provability by non-trivial extra-analytic methods.

I don't mind that your definition falls prey to objections as "what is non-trivial?", "your definition of deductive is too broad" which of course it does.

Here are some other, in my opinion far more serious objections:

• You are assuming that mathematics is definable, and even worse, that it needs a definition

This is an implicit assumption which I have serious trouble finding arguments for. By the way, while it is easy to agree that botanists study plants, it is not so clear whether mathematicians study mathematics or whether whatever mathematicians study is mathematics.

• You don't define mathematics, you define (if anything at all) a part of mathematics that logic can make sense of

No, mathematicians don't talk all day about proofs. They like ideas, nifty tricks, pathological counterexamples, clear exposition, good notation, useful applications and so on. This is part of what mathematicians think about/what they do, which means it is part of mathematics but you don't even mention these things.

• You have a misunderstanding about what applied mathematics is

In applied math, you seem to think, the axioms come somehow from outside mathematics. But there are plenty of cases in pure math, where central definitions are motivated from outside mathematics (for example symplectic geometry), and lots of cases that were considered pure, but after having found applications have partly turned into applied math (for example in graph theory, heck, even type theory).

• Nobody cares what can be proven

There are several aspects of this point. Most mathematicians haven't seen a formal definition of proof. And they don't care, since they do not want to know what you can prove with the axioms of an abelian group, they want to know what is true in abelian groups.

But even this step from syntax to semantics won't help you everywhere. No group theorist would say he wants to know what properties in all groups hold. He has long recognised, that such a property is almost certainly trivial. He has set himself quite a different challenge, maybe how to meaningfully classify groups, or maybe just try to understand groups. Understand is something entirely different from knowing that it is true/being able to prove it.

And lastly, if something is not provable but should be so, it is simply going to be made provable. What does that tell you about the object of study?

_{These are only some rough points, I will try to elaborate a little bit when I find some more time.}