This question that you are baffled by has been one of the most prurient topics discussed in Philosophy and has been confronted and tackled by some of the most notorious Intellectuals throughout History. There are many different Philosophical theories of which could offer enlightenment to your predicament, all of which has its strengths and weaknesses. I believe though that One should really clarify what we are trying to answer. To some, Belief and Understanding are synonymous with one another, both requiring experience for both to be pertinent. If I were to attempt to clarify with more precision by assigning these terms to another designation, it may help both board readers and yourself to become acquainted with a suitable philosophical suffice. If I denoted Belief to Notion, and left Understanding resolute, it maybe easy for the question to be more intelligible and comprehensible. I proposed this change as both Belief and Understanding requires clarification, whilst other phenomenons of which are debated within Philosophy are present without such a perception.
If one takes a belief within the Philosophical concept of Rationalism, One would come to the belief that knowledge precedes Experience to a large degree. For the ability to understand, It would seem to most that the requirement is experience of the Objective World firstly, but the views held by rationalists differ. Rationalism is the philosophical view that regards reason as the chief source and test of knowledge which asserts that reality itself has an inherently logical structure, and a class of truths exists that the intellect can grasp directly. There are, according to the Rationalist concept, certain rational principles-especially in logic, Mathematics and even in ethics and Metaphysics-that are so fundamental that to deny them is to fall into contradiction. This being so, if One were to take a Rationalist approach to the issue it would seem that within the Options of Notion or Understanding being the priori, the former would be the answer.
If One asserted to comply with the Empiricist approach to the issue, they would be agreeable to a totally opposite approach to the notion held by rationalists. In an Empiricist theory, knowledge is a posteriori to Experience of which requires a verification like any other Science. Empiricists argue that the statement "2=2=4" is only a truth after verification and thus is not a priori notion in itself. As long as the statement can be perceived and verified by deductible truths, it will be accepted. One example in particular was John Stuart Mill, who asserted that all knowledge comes to us from observation through the senses. This applied not only to matters of fact, but also to "relations of ideas" in themselves, as Hume called them: the structures of logic which organize, interpret and abstract observations. The logical theory which seems to be somewhat intuitive, is only indubitable after observation.
The next argument one could be lead to after perplexing over the Mind's tangibility of Mathematics, is to whether Mathematics exists independently of the human mind as that of which is asserted by realism, or whether it is a part of the Mind's rationalism (whether posteriori or priori of experience) as held by Logicism.
There is no palpable answer to the question, but many propositions exist. I personally am inclined to the theory that the Mind has the ability to form many concrete manifestations of rational truth of which experience is required for verification. I also am swayed to the believe that for which we are presented to and become acquainted with; is potentially minimal. And our limited physical apparatus which subjects the Body to stimuli to form Understanding and sculpt the logical axioms has the capacity and faculty to form other concrete concepts of a different denomination. If alternative realities existed which incorporated different forms of laws and stimuli, would the Mind have the ability to perceive and conceive and thus form completely contrary logical constituents? I am inclined to agree. I believe the mind has the potential to receive many alternative forms of objective stimuli to construe logic and understanding-but we are currently only able to compile a particular amount of sensual stimuli to subsequently form ideas of which maybe only partial to the completeness of reality
Thanks for this detailed as well as exquisite answer. I agree with your opinion that many propositions exist; and it might be irrational of us to lean to any of them without sufficient reasons.
But, as I have also pessimistically observed that, in this world, most people, if not all, have the pyschological tendencies to find a definite answer to what they will to learn, no matter whether the final answer found is correct by nature or not. In order to justify what they believe, they claim with solemnity: truth is out there; truth is revealed because it exists prior to our revelation of it. Well, in many cases, I totally consent to this notion; but in some others, especially for those arithmetical problems, I find the "truth-is-out-there" theory not so appealing compared to its validity in the proof of such as, the existence of elephants; for the existence of a species seems to be independent of our knowledge.
It appears to me that our recognition of the validity of 1+1=2 is based on a circular journey. By circular journey I mean, when we were young, we were tought that 1+1=2 as something axiomatically right. As the firmness of the validity of this formula rooted deeply in our minds could we tackle more complex arithmetical problems. Then we know 1+2=3, 2+2=4..etc. By encountering with these more complex formulas could we say that, "ah! 1+1 indeed equals 2," for if 1+1 does not equal 2, then 1+2 cannot equal 3.
The education in morals simulates very much the way we were taught in mathematics, for we were taught that honesty was a virtue before we knew the actual reasons why honesty was appreciated for both its own sakes and some utilitarian merits it could bring upon mankind. Only after we finally step into the real society, could we taste the joy and bitterness brought by honesty and dishonest respectivelly.
Axiom, as something evidently true, has a two-sided impact upon the growth of our knowledge. In a positive sense, it makes scientifc discovery more convenient, for we can further our experiments on what has been already proved as axiomatically trustworthy; the negative side of it is that, it prevents many people from re-inspecting the validity and reliability of an "indubitable truth". This is exactly the same reason why so many metaphysicians have been cold-shouldered for their astoundingly absurd questions such as whether we exist or not, given the too obvious fact that we exist. But, as I have been unswervingly believed that, there must be some valuable knowledge behind absurdity performed by those metaphysicians, given the excellent logic staged in their philosophical propositions such as "I think, therefore I am."
Of course, I also understand that it is impossible for everyone to know how a rocket is designed, since this is only the expertise belonging to certain scientists who may have been spending a dozen years studying in this field. Nonetheless, this confession still does not refrain me from generalizing in a sad manner that, we human beings are indeed imperfect, for most of us only have the audacity to learn what is already learned, proved, examined and acertained without putting our own intellectual endeavor into the request of truth, given that we either do not have enough time to do so (after all, one's vigor and stamina are finite), or we simply do not care compared to the glamore of making money and enjoying luxury, or our timidity towards authority.
There are two things that I fear and refuse when philosophizing: first, being too extreme and cynical; second, over-simplifying and being irresponsible. I hope, so far, I haven't committed to either category with you, my dear friend, as the judge.
---------- Post added 09-27-2009 at 12:56 PM ----------
I would break it down as such:
1) The teacher is teaching a new habit: 1+1=2. Habits are necessary to get along in the culture one lives in.
2) The understanding comes the first time you put the habit to practical use: e.g. when your parents ask you how much is 1+1 and you answer 2.
3) The reward to continue learning this habit (for some anyway) is the big smile you get from your parents, and maybe a new game or something.
4) The belief is that if you keep learning the habits that they teacher is teaching you then you will get more rewards from your parents. Sometimes it works and sometimes not.
5) The belief is shattered the first time you learn the teacher is teaching habits that can be challenged! Now, what should one do with the darn habit?
Ha! I like the way you show your arguments! It simply reminds me of Spinoza's work in his "ethics." Simplicity is always yearned and preferred! Thank you for contributing!