Are there counter-examples to this broad characterization of mathematics?

Question

Mathematics can be broadly characterized as the study of non-trivial apriori symbolically displayed axiomatic systems. Or more elaborately the study of non trivial apriori implicit or explicit axiomatic systems that are usually written in a symbolic inferential language.

If I want to put the above in an exact terms, then I'd say that mathematics is the study of non-trivial apriori justified sets of sentences that are interpretable in an axiomatic system.

In that sense what I've previously said to be an axiomatic system can be taken here to mean just a part of that system. An axiomatic system is composed to two parts the semantics and the syntactical side, it has axioms which are sentences that are considered as true by stipulation, and has inference rules by which we derive sentences from prior ones, the deductive closure of all these sentences constitute the axiomatic theory. The formal system is the syntactical part of an axiom system. (see Shoenfield) (1.1, 1.2)

However here what I mean by an axiomatic system is more appropriately understood as a "part of an axiomatic system". Now arithmetic is written in sentences with inference rules guiding formation of sentences from prior ones and with primary sentences that are considered as rules of arithmetic that are not derived by arithmetic inference rule from other sentences, although one doesn't see the logical background governing many of the processing in mathematics, and one doesn't see statements explicitly called "axioms", nor statements called "inference rules", however in action definitely they are all there, one can easily interpret arithmetic studied in our schools in an axiomatic system like Peano arithmetic, now the arithmetic we study is apriori justified, and it is non-trivial, so it satisfies all the conditions in this definition. Geometry I mean Euclid's is clearly an axiomatic system, and even Geometry preceding it, since ever, all is interpretable in an axiomatic system, and is non-trivial and apriori justified.

I personally don't know of a discipline belonging to mathematics that evades that definition.

One word needs to be said is that this definition is not a formalist stand point, nor belong to deductivism or the alike. To the formalist mathematics is just manipulation of string of symbols, there is no meaning (except of some basic entities like the naturals and logic) to the symbols, and virtually has no semantics in the true sense of the word, so they don't advocate something like "apriori" justification (which is related to semantics of an axiomatic system) or the alike (abstract, necessary, etc...). So definitely this definition incorporates a semantic side, which are semantics the truth of which is apriori justified, in other words to a formalist this account translates to saying that mathematics is applied pieces of formal systems to semantics with apriori justification, so he'll sum it up by saying that what is said here is that mathematics is non-trivial apriori applied formalism, and to the formalist this is not mathematics as he conceives, since to him he doesn't define mathematics to be about a certain kind of concepts or semnatics, he just think it is the formal side, i.e. the syntactical symbolic stream. The account here greatly differs, of course here when I said apriori then it can be apriori analytic which is logic in my understanding, or apriori synthetic which is the extra-logical parts of mathematics. That mathematics was not largely axiomatized until later in history is irrelevant to this definition, just because it wasn't axiomatized, it doesn't mean it is not using actually an axiomatic framework in implicit manner, and it doesn't mean that it cannot be interpretable in an axiomatic system, all mathematics before the 20th century are actually interpretable in an axiomatic system. Almost all of mathematical presentations can be easily axiomatized since they are written using string of symbols with clear mathematical operators that are deductively closed under mathematical inference rules, all of those can be added as primitives, and added on top of a logical frame, and the mathematical inference rules can be added as inference rules of the system, and as long as the system is deductively non-trivial and is apriori justified, then it is clearly mathematical, so the criterion of interpretability in an axiom system is straightforward for almost all of mathematically presented material. There might be some informal concepts the stance of which from axiomtizability is unknown, and that are considered as mathematical. I'd like to know of those really, since they'd constitute counter-examples to this broad characterization.

So in nutshell what I'm saying here is broadly speaking:

Mathematics= Interpretability in an axiomatic system + apriori justified semantics + Non-trivialism.

Are there counter-examples to this broad characterization?

Answer 1

Your characterization sounds like a version of formalism or deductivism in the philosophy of mathematics. Some important mathematicians including Hilbert and Turing adopted some version of this idea. It assumes that we can completely expel semantics from mathematics and treat it as a symbol manipulation game. This is a contentious claim. Mathematicians do not typically work just by constantly manipulating symbols; they usually have their own mental grasp or understanding of the subject matter. It's been said of mathematicians that they are platonists from Monday to Friday and formalists at weekends. Even if they profess formalism, they tend to work as if they are engaging with 'real' albeit abstract things of which we can make true or false statements.

Some considerations that might weigh against the formalist notion are:

1. Some theorems in geometry are such that we can just look at a diagram and grasp the necessity of some proposition without needing to transform it into a symbolic representation and construct a proof in the conventional way.

2. The fact that a theorem is provable is not the only thing about it that is important to a mathematician. We want to understand 'why' it is provable. There is surely a difference between someone who feeds a proposition into a computer theorem prover and receives the result that it is a theorem, and someone who works through the proof step by step and understands it.

3. Complete separation of syntax from semantics lies at the root of arguments about the strong AI thesis. Depending on your views about the possibilities and limitations of computation and AI, you may find it implausible to suppose that everything is reducible to symbol manipulation.

4. When you speak of mathematics, you are describing only pure mathematics and the attempt to find some foundation for it. Propositions in applied mathematics clearly have a semantics. Formalists would have to say that these are just intended interpretations of a mathematical theory, but again, it is contentious to claim this holds in all cases.

5. Your reference to the a priori is also potentially a moot point. Although mathematical propositions are not usually considered to be straightforwardly empirical, some philosophers maintain that there is no absolute a priori and that even mathematics is justifiable only insofar as it contributes to our scientific knowledge of the universe.

Answer 2

This is my second answer, since my previous one was to an earlier version of the question.

I think there are two main issues with your suggestion. One is that it is contentious to claim that mathematical sentences are justified a priori. The other is that the concept of implicit axiom makes no sense except as an ex post facto rationalisation.

Let's start with the a priori claim. No doubt many mathematicians historically thought of their work as intuiting a priori truths, but this is potentially misleading. The development of non-Euclidean geometries in the mid-19th century was the result of mathematicians being brave enough to say, Let's discard this fifth postulate of Euclid and try some different ones. If Euclid's axioms are supposed to be a priori, how would that make sense? Indeed, Gauss actually developed non-Euclidean geometry in the early 19th century and refrained from publishing his work because he was afraid it would ruin his career. Other mathematicians at the time thought Euclid's geometry was a priori true and that erroneous belief held up the development of geometry for nearly half a century.

Is there really an a priori justification for all the axioms within axiom systems? What is the a priori justification for the axiom of choice? After all, some mathematicians use it and some don't. Are some mathematicians just wrong and incapable of recognising a priori truths when they see them? The axiom of choice has some highly counterintuitive consequences: it entails the Banach-Tarski theorem, for example. Is this a reason to think it is false? Or just a limitation on its interpretability?

The same consideration applies to the logic in use. There are, after all, many different logics, but if you want to say that a theory is a priori then you would need an a priori justification of the accompanying logic as well. Classical logic is the most common choice, but it too has counterintuitive consequences. It entails the principle of explosion. It entails that statements of the form "All A's are B's" lack existential import. It entails the law of excluded middle. Are these supposed to be a priori true? They seem rather to me to be just choices. I can use a logic without explosion; I can use a logic with existential import. Classical logic is a tool that I choose to use: it has lots of nice features and the advantages outweigh the disadvantages, so I am happy to work around the counterintuitive elements. I use other logics too.

A more general criticism of the a priori comes from naturalism. Two of the main advocates of this are Quine and Putnam. They maintain that mathematics derives its justification from the contribution it makes to our scientific knowledge. A rather oversimplified gloss might be that we need physics in order to understand the universe and we need mathematics in order to do physics, so mathematics pays its epistemological debts indirectly but naturalistically. To object that we don't test mathematical theories directly by experimentation and observation is missing the point. It is the systematic contribution that mathematics as a whole makes to our scientific knowledge that ultimately gives us reason to accept it. If our best systematisation of science includes the principles and rules of mathematics then that is reason enough: to ask for more is pointless.

Incidentally, you say: "axioms are sentences that are considered true by stipulation" but then later say that mathematical sentences are a priori justified. I don't see how you can have it both ways. If axioms are stipulated to be true then why would they need justification? Formalists can axiomatise things too. The axiom set of some theory may be treated just as uninterpreted formulae and they may be true under some interpretations and false under others.

On the second issue of implicit axioms, you say "one doesn't see statements explicitly called 'axioms', nor statements called 'inference rules', however in action definitely they are all there." Where are they, exactly? Axioms are sentences: they are written down or they are not. If they are not written down they are not anywhere. For 2000 or more years mathematicians were mostly not bothered with trying to axiomatise their work. Mathematicians did their stuff and in most cases the axiomatisations came much later. The only way I can understand your claim is to interpret it as meaning that from our modern perspective and with access to modern tools of logic, we can now symbolise all the mathematical work of bygone centuries and represent it in axiomatic form. This may be true, but it is an ex post facto rationalisation of what mathematics is about. I can hardly imagine that Pythagoras used to think to himself, I hope some clever person manages to axiomatise my work some day, otherwise it won't qualify as mathematics. And it may not even be true. One way of understanding Gödel's first incompleteness theorem is to say that no theory can be (i) consistent, (ii) complete, (iii) recursively axiomatisable, and (iv) sufficiently strong to represent arithmetic. One can perm any three from these four, but not have all four. This leaves open the possibility for someone to say, I have an arithmetic that is consistent and complete, so by Gödel's theorem it is not axiomatisable. In other words, we can understand Gödel's theorem as being itself a statement about the limitations of axiomatisability.