This is a response to the following question posted by fiftyducklings as a comment to Fraction Satisfaction:
In geometry, we learned things like “segment addition postulate” and “area addition postulate,” which were pretty intuitive statements. But then I looked at Euclid’s five postulates, and although he included several intuitive statements, he never bothered to list the “postulates” listed above. Would those statements be considered too intuitive to bother to list? Or is there actually a way to prove them (not demonstrate them)? Because I used to think they were too obvious to bother to attempt to prove, and then I found about the whole Banach-Tarski sphere dissection…
Your questions seem to be about what should or should not be explicitly stated as a postulate and whether or not it is necessary to provide proofs for “obvious” statements. Your questions raise a number of complex issues and their answers depend on context. Here are some comments.
Regarding Euclid’s Elements, one point to keep in mind is that despite having tremendous historical significance, it is not a suitable foundation for modern geometry. Since the time of the ancient Greeks, human understanding of logic has advanced considerably and gone through major upheaval. By modern standards, Euclid’s Elements is not a very good model of logical development; there are unstated assumptions commensurate with ones that are stated. (And, by the way, the postulates commonly called the “five postulates of Euclid” are not the only axioms in the Elements. There are statements of “common knowledge” which are also axioms, such as: “Things which are equal to the same thing are also equal to one another.” Probably people will generally regard these as “intuitive” or “obvious,” but that didn’t keep Euclid from stating them.) But knowing why Euclid chose to state certain postulates and not others is likely a question that could only be answered if we could go back in time and discuss the matter with him even though many scholars have suggested explanations.
Your questions raise a larger issue concerning the relationship between proof, axioms, obviousness, and intuitiveness. So, another point to keep in mind is that the criterion for what constitutes an acceptable proof varies depending on the context.
I’m wondering if you’ve read some proofs where the author makes a statement without proof saying only that it is “obvious,” and, if so, whether that suggested to you that obvious statements do not require proof? I’m reminded of my first graduate level course in abstract algebra. The professor had an annoying habit. He’d write a theorem on the board followed by “Proof:”. The class would hunker down eagerly awaiting some elucidation as to why the theorem was so. He’d pause in contemplation, then turn to us and say in this deep, soporific voice, “It’s obvious,” and would write “Obvious,” draw the little square that symbolizes the completion of a proof, and move on to the next theorem. At one point during the course, he even said during one of these pauses, “It’s funny, I remember when I was a grad student first learning this material… everything seemed so hard. But now, everything seems so obvious I don’t see that anything needs to be said.”
Next comment: Sometimes, mathematicians become keenly interested in the logical interplay of various theorems and axioms and they try to build an appealing set of axioms from which an entire theory follows, and, here, a “proof” might constitute an explicit deduction of a theorem from the axioms. Other times, when a mathematician asks for the proof of a statement, what is desired is a sufficient amount of information that enables that mathematician to satisfy herself of a statement’s validity. Because different mathematicians have different knowledge, what a mathematician needs to be self-convinced varies.
In the first type of proof, where the goal is to demonstrate that a theorem does indeed rest upon a certain set of axioms, the need to be explicit about an axiom does not depend on how obvious or not the axiom may or may not seem to this or that mathematician. Here, the aim is to gain an understanding of the logical dependencies of the subject.
I’m reminded of another story from when I was a TA for an undergraduate abstract algebra class. The professor, Andrew Gleason, assigned a homework assignment that asked students to prove various properties of functions whose domain and range were both finite sets. For example, “Prove that an injective function between finite sets of the same size is also surjective.” There was one student whose homework answer was, “Nothing needs proving…All these statements are obvious.” What this person failed to understand (or, perhaps I should say, what we failed to convey!) is that obviousness or non-obviousness was not relevant. The problems were about explicitly deducing the statements from the basic axioms and definitions that were provided in the class. Doing so enables one to understand that the statements need not be regarded as isolated facts, whether intuitively clear or not, but rather, as logical implications of certain axioms and definitions.
So, what I would say is that when one is considering mathematics as a logical structure, the question of whether or not to explicitly state a postulate is answered solely by the question of that postulate’s logical relationship to the other postulates and the result you are establishing. In this context, there is no statement that is “too intuitive” to not require proof.
But if you’re being asked to prove something in a paper or in a talk, you have to make a judgement call that accounts for your purpose, your audience, and even the amount of space or time that you have when you decide how much detail to include. There is subjectivity and there are disagreements. You might legitimately conclude that providing the details of a proof would muddle things and that you’d rather convey your non-rigorous intuition while referring interested parties who want a detailed proof to some other resource where a detailed proof is provided.
By the way, in theory, a mathematician would be able to produce a proof resting upon some standard set of axioms of any statement that the mathematician claims to be true regardless of whether the statement is “obvious” or not. In practice, this may not actually be the case for a variety of reasons.
I also suggest reading Terrence Tao’s blog post There’s more to mathematics than rigour and proof and the references within. Your question also implicitly asks about the relationship between intuition and proof which Prof. Tao’s blog addresses. Also, check out Part II of the marvelous Princeton Companion to Mathematics. Though not directly related to your questions, you might also be interested in reading about “formal proofs.” There was a special issue of the Notices of the American Mathematical Society devoted to formal proof which is a good place to start.
I hope this answers your questions, if not, let me know.
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