Ockham's Razor & Mathematical Proofs

Occam's Razor (also written as Ockham's razor from William of Ockham (c. 1287 – 1347), and in Latin lex parsimoniae) is a principle of parsimony, economy, or succinctness used in problem-solving. It states that among competing hypotheses, the one with the fewest assumptions should be selected. Other, more complicated solutions may ultimately prove correct, but - in the absence of certainty - the fewer assumptions that are made, the better.

"Quoted from Wikipedia"

In mathematics there are theorems which have several proofs, some short and some long. Based on the limitation of time I cannot explain more than one proof for each multi-proof theorem in my courses but there is a problem here.

Question. Which proof should I choose?

Based on the Occam's razer I should choose the shorter one because it is simpler and less confusing but it seems some of the longer proofs for a particular theorem have more intuitive content. They explain the nature of the phenomenon better. Also they show the complexity and importance of the theorem clearly. Maybe students can gain a deeper intuition using some of these longer proofs. In other words my question is about possible (dis)advantages of using Occam's Razor in mathematics. Precisely, is there any exception for Occam's Razor principal in mathematics? Is there a longer proof of a particular theorem which has a significant educational advantage to all other shorter proofs?

Example. In my "Introduction to Large Cardinal Theory" graduate course, there are several approaches to Hanf-Tarski theorem which says The least inaccessible cardinal is not measurable. The current standard proof in textbooks (e.g. Drake, An Introduction to Large Cardinal Theory) is based on a very short simplified proof but Tarski and Hanf's original sophisticated argument (using introducing the notion of a weakly compact cardinal and analyzing the expression power of infinitary language $\mathcal{L}_{\kappa,\kappa}$ for weakly compact cardinal $\kappa$) is really useful for understanding the complexity of this theorem which was one of the most important open problems of set theory for many years.

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This could be an interesting question with some examples of which proofs to choose. As it stands, I don't understand what situation is prompting or motivating this. I am voting to close. –  Matt F. Mar 30 '14 at 14:55
@MattF. I added an example. However I think the question is clear even without examples. You can find many similar examples amongst famous theorems of mathematics in different levels (research, undergraduate & school levels) e.g. Fundamental theorem of algebra, Pythagoras theorem, etc. –  user230 Mar 30 '14 at 15:24
Great. Now there is a real question. I retracted my vote to close. –  Matt F. Mar 30 '14 at 15:29
I don't think this is a valid invocation of Occam's Razor. Occam's Razor applies more to inductive arguments (not mathematical induction). It says that the explanation with the fewest unproven assumptions is the most likely to be correct. For a deductive argument like a mathematical proof, correct is correct regardless of complexity. Of course, the standards for which correct proof is "elegant" or "better" or most informative to students is an interesting question (hence no downvote from me), but not really related to Occam's Razor. –  Jsor Mar 31 '14 at 9:08
I think the question tied to your example -- which proof do I present, given one is much longer but more indicative of the nature/history of the theorem? -- is quite interesting (and specific), but I don't see how this an invocation of Occam's Razor. This just sounds like a pedagogy decision: Do I show the short and unenlightening proof to save time, or not? Indeed, sometimes short proofs can be more confusing to students because they require an intimidating level of insight to construct (which is daunting) and more careful reading on the student's part (also daunting). –  brendansullivan07 Apr 1 '14 at 2:51

This is a fairly complicated question, it has several aspects, each contributing to the final answer. Consider

• Is the intuition from the longer proof better? It might explain some phenomena of why the question was hard, but it could also limit other, perhaps even better explanations.
• Is it possible to supplement the shorter proof with appropriate intuition? Perhaps you could elaborate on each part so the student can learn more, but keep the final formal derivation short.
• Which proof is easier to understand? Sometimes the shorter proof is nicer, cleaner and have a more intuitive feel, but the longer one could be still more accessible, e.g. it doesn't need any complex machinery, just the most basic and mundane tools are enough.
• Which proof would let the students learn more? It might happen, that student's are yet unable to appreciate that truly brilliant idea in the shorter proof, while the boring, longer proof has a number of mid-results that would still be useful to students at this point.
• Do you have enough time? It is possible that the students could benefit more from something else that wouldn't fit if the longer proof was chosen.

That's not all, but let's finish here. It's easy to guess, there's no general solution, but perhaps this list might clarify some issues. If you have serious doubts which one to choose, you could also try to present both proofs.

I hope this helps $\ddot\smile$

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The issue is not length itself, but intelligibility, I think.

On one hand, over time, proofs are polished and "perfected", in part exactly to make them as short and efficient as possible. Obviously this is good in some ways, but it does sometimes have the bad side effect of making the details become unmotivated. Indeed, since the details of the perfected proof became clear only in hindsight, that is, only upon study of the not-perfected arguments, a student reading the perfected argument without experience with the imperfect one may reasonably be baffled by motivations and choices.

At the same time, sometimes the improvements in the argument are very great, and represent conceptual improvements in their own right, and part of our task is to improve our intuition to match.

But, in general, I find it desirable to try a "natural" line of argument, clumsy or inefficient though it may be, rather than wait for or present some seemingly-magical, ineffable argument whose motivations are obscure except after having tried a more natural approach.

Also, methodologically, admitting to students that it's reasonable to "try the obvious thing", even if revision is required or advantageous, is helpful and honest.

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(+1) Thank you for your useful answer and exact explanation. –  user230 Mar 30 '14 at 16:35
It is common to state too restrictive axioms for e.g. group theory (two-sided identity, two-sided inverses) in order to make the theory easier to follow (and to be able to cut to the chase, i.e., get to the meat of the matter, sooner). –  vonbrand Mar 31 '14 at 3:20

I'd choose the proof which helps to understand the theorem the most.

If the long one is too technical, the message might get lost. If the short one is too mysterious, the time saved with the more elegant version might need to be spent to digest the proof.

I know one book on calculus & linear algebra for physicists (in German, R. Wüst: Mathematik für Physiker und Mathematiker, Bd. 2) where the implicit function theorem is first stated and proven in a two-dimensional version, and then dealt with in the general case. As R. Wüst mentions, an approach which is completely unnecessary from a mathematical point of view. However, I must add, it is really helpful from a pedagogical point of view.

In lectures, elegance is still a factor, but one should not sacrifice legibility in favor of elegance.

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I would be in favor of giving a proof of a very-simple case first, and then the "general case". Or, often, only doing the simplest (representative) case, leaving the general case to supplemental reading. –  paul garrett Mar 30 '14 at 18:02
In undergraduate integral calculus, our professor first "introduced" us to the one dimensional Riemann integral (took about 90min). Then he introduced us to the two dimensional Riemann integral (took about 90min). Afterwards he introduced us to the three dimensional Riemann integral (took about 90min). Then he said "Now, this is taking rather long." and introduced us to the $n$-dimensional Riemann integral (took about 90min). After playing with it for again 90min, he went on to measure theory. –  Toscho Apr 1 '14 at 15:42

Firstly, Occam's razor is completely unsuitable to Mathematics and Natural sciences. Propositions and Theorems have different qualities:

• few assumptions
• many results
• intelligibility
• practical applications
• scientific usefulness

Occam's razor reduces all that to only one quality.

Secondly, proofs are no hypotheses and they are not based on assumptions. They are based on technique and knowledge, and this is something, that you should regard:

Do the students know and understand all the techniques used in the proof? (Techniques, which you delibaretly want to introduce by the proof, don't count.)

And lastly, although shortness may be a quality of proofs it is no didactic one. Choose the proof, that illustrates better, what you want the students to learn.

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I think it may be too strong to say "completely unsuitable", unless the philosophical outlook is that mathematics is independent of the physical world (whatever that might mean). My own 40-word philosophy here is that (for example) mathematics allows us to make both mundane and surprising predictions, both "within" mathematics and about the obviously-physical world. Ok, then, I become interested about the causality, or perhaps explanation. Simplicity of explanation/causality is what is addressed by Occam. –  paul garrett Mar 30 '14 at 22:44
Either way: Why should Occam’s razor be completely unsuitable to natural sciences? Also Occam’s razor does not “care” about practical applications and scientific usefulness and does at most care indirectly about intelligibility – so what makes you say that it reduces these aspects to one quality? –  Wrzlprmft Mar 31 '14 at 8:18
@Wrzlprmft That's exactly what I said: Occam's razor doesn't care about anything but the "fewness" of assumptions (which might relate to intelligibility in some cases). But natural sciences have to provide answers with different qualities. In a way, you could say, that giving "God" as an explanation is always the possibility with the least assumptions and therefore prefferable by Occam's razor. –  Toscho Apr 1 '14 at 12:05
@Toscho: 1) That’s not what you said. You said that “Occam’s razor reduces all that [few assumptions, many results, intelligibility, practical applications, scientific usefulness] to only one quality.” 2) Apart from the “counting” formulation of the razor being somewhat simplified anyway (as there is no clear way to count assumptions), you have to add a lot of assumptions to ∃ God to actually obtain a hypothesis which one could regard as actually explaining something and which thus would be a competing hypothesis to the hypothesis of the natural sciences. –  Wrzlprmft Apr 1 '14 at 14:10
@Wrzlprmft 1) I don't see your difference. 2) The metric on assumptions is indeed the problem here. Verbally, $\exists$ God is exactly one assumption. Conformingly theologically, $\exists$ God is no assumption at all but certainty. Heretically, $\exists$God is a hell lot of assumptions. –  Toscho Apr 1 '14 at 15:45

I don't particularly know the Hanf-Tarski theorem or its proofs, but based on what you say:

Teach the shorter proof. A complicated proof shows that it used to be a complicated theorem, but apparently it isn't today. Whatever your goals in teaching set theory, complexity should not be a goal in itself.

The Pythagoarean theorem has so many short proofs that it can make sense to teach more than one.

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My reading of Occam's Razor is that it is a heuristic principle about (essentially) nature or the world: it is sometimes quoted as "Do not multiply hypotheses"; in other words if a simple assumption will explain something then there is no need for more complicated assumptions. I don't think it really applies to mathematical proofs. For proofs, certainly one driving force is to cut down the number of assumptions to a minimum so as to find the widest domain of applicability of a theorem, but this may result in a more complicated proof (for example, constructive proofs that avoid unrestricted application of the excluded middle). As other posters have said, different proofs may serve different purposes, and having more than one proof may bring a range of insights about the theorem under consideration.

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