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Explanatory Vs non-exp. proofs for a given theorem. Any examples?

  1. Jan 18, 2014 #1
    Can you think of any theorems that admit of both explanatory and non-explanatory proofs?
    Roughly, a proof is "explanatory" if the proof illuminates why the theorem is true.
     
  2. jcsd
  3. Jan 18, 2014 #2

    SteamKing

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    I'm afraid I don't understand the distinction. What's the point of a proof if it doesn't follow that the theorem to be proved is true? Or are you trying to illustrate how a proof shows that one particular theorem is true, while another proof for a different theorem shows that theorem to be false?
     
  4. Jan 18, 2014 #3
    The unexplanatory proof may well entail that the theorem is true without explaining why it is true. Perhaps some examples will help...

    Reductio proofs are arguably unexplanatory proofs e.g. Euclid's proof that there are infinitely many primes. This proof establishes that there must be infinitely many primes because otherwise we would get a contradiction. Arguably, this does not help us see why there are infinitely many primes. In general, many reductio proofs just crank out a contradiction any way possible - that's not explanatory. Also, brute-force methods such as proof by exhaustion) often seem unexplanatory (e.g. the relevant proof of Rolle's theorem lacks unity, which detracts from explanatory capacity). Finally, it has been argued that mathematical induction is unexplanatory.

    Do these examples help make clear the distinction?
     
  5. Jan 18, 2014 #4
    jgens what happened to your post did you delete it? I'm very keen on advanced / complex examples!

    Simple examples can be useful but their simplicity can sometimes affect how explanatory we judge them to be.
     
  6. Jan 18, 2014 #5

    jgens

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    So other readers readers know the example in question:

    There are a couple reasons I deleted this. First because it is a rather advanced answer to a simple question. Second because while the adjoint functor argument probably satisfies your definition of non-explanatory, in some sense it is morally the more explanatory proof.
     
  7. Jan 18, 2014 #6

    D H

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    There exist irrational numbers a, b such that ab is rational.

    A nonconstructive way to prove this conjecture is by looking at ##\surd 2##, which is irrational. If ##\surd 2^{\surd 2}## is rational, then case closed. If ##\surd 2^{\surd 2}## isn't rational, then since ##(\surd 2^{\surd 2})^{\surd 2}## is rational (it's 2), case closed. Either ##\surd 2^{\surd 2}## is rational or it isn't, but either way, we have an example that proves the conjecture.

    A constructive way to prove this conjecture is to note that ##\surd 2^{\surd 2}## is provably irrational by the Gelfond-Schneider theorem. Since ##(\surd 2^{\surd 2})^{\surd 2}## is rational, there does indeed exist such an (a,b) by construction.
     
  8. Jan 18, 2014 #7
    Thanks jgens. Advanced examples are good, though I will have to study the concepts involved in your example. I'm curious about your comment though - are you thinking that the constructive truth is explanatory because it shows how a free group can be constructed, while the functor proof does something more general?
     
  9. Jan 18, 2014 #8

    DH, thanks that's a nice example! I was wondering if you had a view as to which of these proofs is more explanatory and why?
     
  10. Jan 19, 2014 #9
    There exists two non-zero elements, a and b (not necessarily distinct) of [itex]\mathbb{Z}_4[/itex] that yield a zero product, i.e. a * b = 0.

    A demonstrative proof would be a proof by exhaustion. Just build a multiplication table of the elements {0, 1, 2, 3}. The witness is [itex]2 \cdot 2 \equiv 0 \mod{4}[/itex].

    A proof by contradiction could be used. In this case you would suppose that EVERY non-zero element of [itex]\mathbb{Z}_4[/itex] yields a nonzero product. Then, every nonzero element is relatively prime to 4. But if every element less than 4 is relatively prime to 4, we must conclude that 4 is a prime number. Here is the contradiction.
     
    Last edited: Jan 19, 2014
  11. Jan 19, 2014 #10

    jgens

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    The constructive proof gives you a detailed picture of the free group. Intuitively this guy is just a group whose generators have no relations between them, so the construction provides a concrete description of its elements and how you multiply them. In this sense the argument is quite explanatory since it lays out the internal structure of the group explicitly.

    On the other hand, understanding free groups by looking at their elements is not the most natural thing. In practice we care very little about what a free group actually is and are more concerned about its mapping properties. In particular, one of these (the universal property of free groups) determines the group uniquely up to isomorphism. The adjoint functor proof works essentially by saying this is the property we care about and then uses some abstract nonsense arguments to magic these objects up.
     
  12. Jan 19, 2014 #11
    I don't think reductio proofs are necessarily not explanatory. In fact they help me understand better than some direct proofs.

    Like Euclid's proof. I think it illustrates why there's infinitely many primes pretty clearly. You can always "build" a new prime.
     
  13. Jan 20, 2014 #12
    I think induction is usually "non-explanatory" (for me at least). The classic example would be proving the formula for 1+2+...+n. The inductive proof doesn't do much for me, but Gauss' argument or some variation is much more satisfying. Likewise, there are various visual proofs or proofs by rearrangement of the sums of squares or cubes.

    A related discussion below is about the Binomial Theorem and related proofs. I think the combinatorial (combinatoric?) proof is more illuminating than the inductive one.

    There are some inductive proofs that are pretty illuminating, though. The number of subsets of a set with n elements for example, or the fact that the rows in Pascal's Triangle add to powers of two (each row contributes twice to the row below it).
     
  14. Jan 22, 2014 #13
    Yes the paper linked to in my second post argues that the arbitrariness of the base case renders mathematical inductions unexplanatory. In particular one typically appeals to P(1), and shows that 1 has the relevant property P. One then shows that P(n) for any number n>1. But we could take any such n>1 as our base case and thereby prove that P(1). The idea is that in real explanations you can't just arbitrarily swap explanans with explanandum.

    Gauss' proof of the n(n+1)/2 theorem is a nice contrast. His explanans does not seem interchangeable with the explanandum and so avoids the criticism of induction. His explanans just includes the sum S written from 1 to n and then that same sum S written from n to 1; which makes clear that 2S=n(n+1). Thanks for the example.

    Sorry, what discussion are you referring to here? I would be interested to look these examples up.
    Why do you think these are illuminating? After all, if the objection to the inductive proof of the n(n+1)/2 theorem works, then the objection will work just as well against the proofs that you mention, don't you think?
     
  15. Jan 22, 2014 #14

    jgens

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    I am sure you already know this, but as it is one of the most common mistakes people make about the infinitude of primes, it warrants a note: The procedure for building a new prime in Euclid's proof only works under the assumption that there are finitely many primes. It does not, in general, produce a prime at all!

    Not unless you add another step to the proof showing your argument works backwards from the base case as well. Or alternatively establish each of these cases on its own.
     
  16. Jan 22, 2014 #15
    The worry is that the proof works simply by finding a contradiction any way possible. I don't see why such a procedure explains why it is that there are infinitely many. It just explains that there must be. The real explanation would state what it is in virtue of which there are infinitely many.

    I agree that some reductio proofs will seem explanatory, but I suspect that is only because they resemble a direct proof, and suggest how we can construct a direct proof. Consider a reductio proof that 2 is the only even prime: assume there is some prime p > 2; since p is even p is divisible by 2 so p=2q for some q; but then p is composite contradicting the assumption. Here we really do get an explanation and it's because the reductio exploits explanatorily relevant concepts, unlike Euclid's.
     
  17. Jan 22, 2014 #16
    Correct. Rather than using "the standard" rule of inference:

    For any property P and natural numbers n,k:
    if P(1) &
    if P(k) then P(k+1)
    then P(n)

    ...we could instead use "Upwards and downwards from 5":

    For any property P and natural numbers n,k:
    if P(5) &
    if P(k) then P(k+1) &
    if P(k) then P(k - 1) [k>1]
    then P(n).

    This interchangeability of explanans and explanandum suggests that all mathematical inductions are unexplanatory proofs.

    p.s. I'm still working through your free group example.
     
  18. Jan 22, 2014 #17

    jgens

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    Out of curiosity, why the fuss about (non)explanatory proofs? It does not sound like you doubt the logical status of arguments like induction and contradiction, just the explanatory value. So I am curious where the motivation comes from.
     
  19. Jan 24, 2014 #18
    Yeah I could have worded that better--I mean the first thread under "Related Discussions" at the bottom of the page. I didn't even look at that thread, but it reminded me of the Binomial Theorem so I mentioned it. I find the proof where you count the ways to get the term x^ky^(n-k) in the expansion of (x+y)^n more enlightening than the inductive proof that uses Pascal's rule for the binomial coefficients, although to be fair I don't think it's too hard to see the intuition behind the inductive proof if you approach it correctly.

    I checked out that link after I posted but to be honest I only skimmed it. I didn't really get what the author was getting at by saying that P(1) can be used to prove P(5) and vice versa and it wasn't the criterion I had in mind for deciding whether something is explanatory or not. I could be overlooking something more profound, but it's just my personal preference: I guess I can just visualize what happens to all the subsets of a set when a new element is added, or how adding the rows according to Pascal's rule (which is easily visualized) leads to the doubling result that cascades down. The induction is a key part of this visualization, whereas the more visual proofs of, say, the sum of squares, aren't really inductive. That's the best I can do to try to explain it.
     
  20. Jan 26, 2014 #19
    I think explanatory proofs have pedagogical importance. They help in understanding what's going on and gaining an intuition.
     
  21. Feb 1, 2014 #20
    I think it's an interesting question as to whether pure mathematics provides explanations at all, rather than mere guarantees of truth (for theorems). Fields medallist Michael Neilson said in a 2009 publication: "For mathematicians, proofs are more than guarantees of truth: they are valued for their explanatory power, and a new proof of a theorem can provide crucial insights." If he's right then we should be able to characterise what makes a proof (that is, a guarantee of the truth of a theorem) explanatory versus non-explanatory. Either way we learn something about the role of mathematics in our overall scientific understanding of reality.

    I have a question for you. I haven't been able to find a good discussion of the adjoint functor proof of free groups. Can you please point me in the right direction? I think the proof I found (of "theorem 11.1") is somewhat different, and perhaps there are many different kinds of proofs of free groups?
     
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