# What to do if there is a disagreement on fundamentals, e.g. axioms or inference rules?

Sometimes it happens that a person doesn't want to accept your argument, because he claims not all the inferences are valid. There's a famous example of Lewis Carroll, namely What the Tortoise Said to Achilles, where nonacceptance of a basic inference rule leads to infinite regression.

Sometimes, it's the other way around, where a student doesn't want to acknowledge your proof. It might be more subtle, like with the non-constructive proof that an irrational raised to the power of another irrational can be rational, or it could be less subtle, e.g. when pupil doesn't concur that $$(\forall \varepsilon > 0\;. \ |x| < \varepsilon) \implies x = 0$$ for $x \in \mathbb{R}$ (I remember one guy that agreed on $x \neq 0 \implies |x| \not<\frac12|x| > 0$, but wouldn't accept the statement above).

To give a different kind of example, consider the following. For propositions $P(x)$ and $Q(x)$ we could infer that $\exists x. P(x) \land Q(x)$ from $\exists x. P(x)$ and $\exists x.\ Q(x)$. This is not true in general, but there are cases where it works, e.g. if $P$ and $Q$ are independent (this could be formalized as, for any considered model $\mathcal{M}$ and element $x \in \mathcal{M}$ we have $\mathcal{M},P(x) \not\models Q(x)$ and $\mathcal{M},P(x) \not\models \neg Q(x)$, etc.). While a scrutinizing approach would be to work out all the details, one might want the students to apply a more intuitive, higher-level perspective; in my opinion, such exercises are helpful to develop a deeper understanding. Yet, some don't even want to start participating.

This happens also in non-math fields, like psychology or biology where papers on non-widely-accepted theories (like group selection) are ridiculed, for example, because the audience doesn't handle well the concept of "assumption", or because the author insists on something bizarre without considering it might be false.

Question: How can we resolve conflicts that stem from disagreements on the fundamentals?

• @senshin Thank you. May 31 '14 at 1:00
• Disagreements about inference rules are opportunities for education. There are two qualitatively different types of opportunities. (1) The student's position is poorly thought out, inconsistent, or betrays a misconception. (2) The student finds some framework like non-Aristotelian logic or finitism more intuitively appealing than what you'd been assuming. #2 probably happens far less often than #1.
– user507
May 31 '14 at 3:58
• @BenCrowell Could you expand on how would you handle each of these cases and make it an answer? May 31 '14 at 11:22

A small partial-answer: relativize it, as you can easily anticipate. Be up-front and explicit about logical principles and such. Someone might still (reasonably!) object, but if you conscientiously acknowledge your assumptions, the issue changes from "is this right?" to the relative one.

I did once try to give a seminar for advanced undergrads about ideas of modern mathematics, and the philosophy students gave me endless trouble, ... :) ... in the sense that I felt (at the time, much younger...) that we couldn't get off the ground for all the quibbling. :)

In particular, I think it is a bad thing to squelch intellectually honest queries or skepticism. Heaven forbid a calculus student gets access to Bishop Berkeley's essay? :)

"Making the context (more) explicit" is often the answer to such questions.

• "Intellectual honesty" wasn't always Berkeley's strongpoint, to say the least. Jun 4 '14 at 16:37

You could ask the student to come up with his own rules and axioms from which he can prove whatever theorems he thinks are valid. Usually when they really get down to doing that systematically and formally, they will understand the rules and axioms better anyway, and then they can precisely pinpoint what is it that they don't agree with. Otherwise, often they won't be able to give justification for their disagreement, and arguments would just go in (periodic) circles.

• Such a discussion might take quite a bit of time. It would be fine in logic course, but not really in other classes. Yet delaying it until the end (or break if there is one) might result in students being anxious over it and interfering with their learning. May 31 '14 at 9:34
• @dtldarek: I fully agree that it would take substantial time, but I sometimes feel that there is insufficient clarity in the logical underpinnings in the standard textbooks and course materials, ranging from hand-waving to failure to mention the use of some axiom like the axiom of dependent choice in the standard proof of the Bolzano-Weierstrass theorem. All this is very unhelpful to the student. So personally I would rather spend time to clearly show the foundations on which the material is built, such that any disagreement would have to be with the foundations, which can be dealt with later. May 31 '14 at 9:50

If there is a disagreement, then have an argument!

"Argument" in the sense of "reasoned discussion" - it is not an accident that the word is used commonly in mathematics. Even if you are sure that you are objectively "right", you may learn something by taking the student's perspective more seriously. They are clearly wrong in your sense, but perhaps they are correct in their sense, and are just having trouble communicating their ideas.

In the end, mathematics is about proof. If you cannot prove something to someone's satisfaction, then it probably means that you should re-evaluate your understanding of that concept. These issues are often more subtle than you would expect.

• I can't agree that "mathematics is about proof". This is too limited. And, also, the legitimate means of "proof" are open to discussion... Perhaps this "answer" is in-the-end too naive. May 30 '14 at 23:21
• @paulgarrett - are you confusing proof with proofs? The distinction to me is that proofs are that two column thing you do for geometry (mostly) but proof is simply being able to solve any problem one step to the next. Proof here, meaning that one step logically flows to the next. May 31 '14 at 18:37
• @JoeTaxpayer, no, I don't think I'm confused about such alleged distinctions, although I would dispute the possibility of meaningful distinctions of this sort. The notion of "one step logically flowing to the next" is very subjective, etc. May 31 '14 at 18:41
• @paulgarrett "'mathematics is about proof' [is] too limited." Sure, I agree. What does that have to do with my answer though? The question revolves around how to react to disagreement, not how to explicate the nature of mathematics. Jun 5 '14 at 1:39

At least when trying to explain mathematical ideas to my friends, one way that I try to give them an intuitive feel for why something is true is to ask them to provide counterexamples. If the "counterexamples" are invalid I try to explain why. This does not constitute a valid proof, but at the very least it change their "gut feeling" for it.

Example: One person refused to believe that $0.\overline{9}=1$. I asked him the following: "In between any two distinct real numbers, there is a third distinct real number, eg. via taking the average. Find me a number between $0.\overline{9}$ and $1$." After some reflection, he acquiesced.

This is rather similar, I believe, to your example of $\forall \varepsilon, |x|<\varepsilon\implies x=0$. "Find me $x$ such that this statement is false."

• Finding counterexamples is a valid strategy for gaining intuition. However, this is in essence shifting the burden of proof, which, if overused, I dislike very much. May 31 '14 at 1:05
• There's a problem with your example. When someone disagrees with the equality of the two decimal expansions of $1$, it is because they haven't been properly taught the meaning of the infinite decimal expansion, which is why they do not realize that its value is defined as the limit to which the expansions approach, and is not necessarily equal to any of the truncations. May 31 '14 at 9:06
• I don't think it's a bad example; it's actually exactly the type of example where things like this happens. A student will insist, because they were wrongly first introduced to the real numbers as "infinite decimals" rather than with a proper definition, that there's inherently such a thing as "point nine repeated" rather than it being notation for a limit. May 31 '14 at 18:21

Don't claim that the axioms are true, nor that the rules of inference are valid.

Just claim that if the axioms qualify as true, as well as rules of inference qualify as valid, then the rest follows.

Consequently, it follows that everything can get proved from modus ponendo ponens (which ends up residing in the meta-meta-language here). If the student wants to try and deny modus ponendo ponens, then tell him or her to get to work on developing propositional calculi without them (it has gotten done before, and though I haven't seen it myself, it seems to involve a lot of work).

For example, say I write the following "push-back" proof in (a subsystem of) Lukasiewicz infinite-valued logic (in Polish notation):

axiom 1 CCpqCCqrCpr

axiom 2 CpCqp

axiom 3 CCNpNqCqp

  1 p/Np, q/CNqNp, r/Cpq


4 CCNpCNqNpCCCNqNpCpqCNpCpq

  2 p/Np, q/Nq


5 CNpCNqNp

  3 p/q, q/p


6 CCNqNpCpq

  detachment 4, 5


7 CCCNqNpCpqCNpCpq

  detachment 7, 6


8 CNpCpq

Then I say that theorem 8 CNpCpq implies that from the negation of a proposition p, and the proposition p, by detachment we can infer any proposition at all in infinite-valued Lukasiewicz logic. But, the student objects, because as is well-known, because the principle of contradiction fails in infinite-valued Lukasiewicz logic. Consequently, I would respond by saying that if those are permissible axioms, if the rule of inference of substitution comes as permissible, if the rule of formal detachment: {C$\alpha$$\beta$, $\alpha$} $\vdash$ $\beta$ comes as permissible, and if soundness holds, then the above suffices for my argument (and then I'd probably explain that it's CKpNpq that fails in Lukasiewicz infinite-valued logic, but that's another issue).

• Lukasiewicz logic a bit high-brow to illustrate the general idea... Also: "Just claim that if the axioms qualify as true, as well as rules of inference qualify as valid, then the rest follows." Of course, your "claim" here is of the form "if... then." Jun 4 '14 at 17:17
• @BenjaminDickman Well I like the example I use here, because it can easily get seen that the student has a reasonable objection to the conclusion here. Jun 4 '14 at 17:27