Foundations of logic and reasoning in natural languages

Question

My intuition tells me that any theory, whether expressed using mathematics(and therefore more precise and structured) or argued for using natural languages have to involve blind faith in certain propositions (or statements). In science (except mathematics) these propositions appear as postulates, whose truth value and validity are ascertained by observation of the natural world. But this becomes tricky when you argue in mathematics, where these propositions may not have their validity based in observations of the natural world. Such statements then become the axioms of that theory. My first question is

1. Can every mathematical theory be proven to have axioms which are blindly believed in as a necessity?

Next, when we enter the realm of the internal world of thoughts and feelings to form logical theories of how to conduct ourselves in the world, which is the task of philosophy, one can't objectively prove the validity of many statements. My next question therefore is

2. Can philosophical theories of how best to live life also be shown to adopt propositions which are blindly believed to be true?

I haven't been able to find answers to these questions anywhere online. Could you please answer while also citing the sources for them so I may read further?

Answer 1

What you're saying is generally true. This is partly how you get philosophical scepticism. Of course, what you said about proving there is blind faith is kind of nonsensical if you're using it as a criticism of maths - any kind of formal proof of the necessity for axioms is going to involve blind faith or axioms itself somewhere. Although the words used have certain connotations - "blind faith" sounds a lot more negative than "certainty."

I think Mauro's comment is off in this case. Mauro said we assume mathematical axioms are true until proven false. I'd put it a different way - mathematical axioms are the rules we use in maths, so they're not true or false. Taking an axiom like "the successor of any natural number is also a natural number" is not something we hold to be true until proven false. Saying it can be proven true or false is without sense. It is a rule, and it doesn't make much sense to say a rule is true or false. A statement like "1+1=2" is more of a rule/definition than a proposition that can be true or false (saying it is true seems vacuous to me).

Answer 2

Firstly, mathematics is circular even in the notion of natural numbers. Worse still, there is not only no viable alternative but also no apparent physical model of Peano Arithmetic. Moreover, the generalized incompleteness theorems can be proven in weak meta-systems such as ACA, which imply that there is absolutely no way to pin down the natural numbers by any useful formal system whatsoever. Hence even if we assume that PA is correct about natural numbers (whatever they are), it is really a strange kind of 'blind faith' because we cannot even mathematically specify what the natural numbers are, and yet we are making assumptions about them! Also, you may be interested in this brief account of the increasing philosophical assumptions needed to express more or prove more.

So, yes, all mathematics is based on faith in the sense that there is no valid justification for PA being completely correct about some structure in the real world, and yet formal systems are based on the essentially equivalent string manipulation properties. But, no, PA was originally designed to capture what we thought was correct about what we conceived of as natural numbers, and seems to work at human scales, so is that really 'blind faith'? If it is, how can you explain why RSA decryption works?

Secondly, your question about philosophical theories is quite answered by the above considerations regarding formal systems. Let me put it explicitly. Every formal system needs the basic properties of finite strings to hold, otherwise one cannot even affirm that the very notion of logical deduction is valid. But there is no viable substitute for formal systems in rigorous logical reasoning, so any philosophical theory that has objective proof validity already relies on the same circularities as mathematics. And if a philosophical theory does not have objective proof validity, it cannot be claimed to be objective, and is arguably worse than having 'blind faith' in classical arithmetic. The reason is that any claimed proof in classical logic can be checked syntactically and its validity over the deductive system is decidable and unambiguous, so anyone who thinks their semantic interpretation of the axioms are true, then they are forced to accept the semantic truth of the proven conclusions. In contrast, any non-syntactic system is no better than just arbitrary opinions, since there is no precise delineation of valid arguments.