mathematical bullshit: a confession

[gusl]

Confession time: when proving stuff, I feel the need to justify things that I consider to be intuitively obvious. In the course of this, I sometimes look up theorems in a textbook that I couldn't possibly prove, and cite them. I feel slightly dishonest when I do this.

Why? Because the proof in my head didn't need the theorem in the textbook. So why should the paper expression of my thoughts use that theorem?

I am *sure* that the statement is true but I'm unable to justify this intuitive knowledge in a formal language, without the help of the theorem in the textbook. All this talk about eigenvalues feels artificial; but it's the only way I found to connect my intuitive idea with the "objective" mathematics that "everyone" must accept.

Maybe my mind is wrong in being so secure about intuitions whose foundations it has trouble fleshing out (but I doubt it). Instead, I think that my mathematical language skills are deficient, i.e. a kind of "aphasia". If you are a visual thinker, the "informal yet rigorous" language of sentential proofs is a foreign language.

I'd say that communication is one of the hardest and most important problems faced by humanity.
See Jukka Korpela on "How all human communication fails, except by accident".

Comments

[roseandsigil.livejournal.com]

What level of mathematics do you consider "objective"? Are real numbers? Integers? Positive integers? Inductive definitions? Sets?

[gustavolacerda.livejournal.com]

I'm not going there.

[gustavolacerda.livejournal.com]

A piece of mathematics is "objective" iff a reasonable person with mathematical education must be compelled to believe it.

[roseandsigil.livejournal.com]

Gaaah!

[gustavolacerda.livejournal.com]

I am describing the reality of how humans communicate about math, and the norms they follow.

[gustavolacerda.livejournal.com]

<< the proof in my head didn't need the theorem in the textbook. >>

I should be skeptical of this. It seems likely that I am using a version of the theorem that I looked up, and I just can't recognize it as such.

[jcreed.livejournal.com]

I don't believe it's dishonest to use theorems whose proof you don't fully understand. Math is a big enterprise; delegating parts of the work to other (trustworthy) provers is fine. Anyhow while the knowledge that A implies B isn't as good as A together with A implies B, still knowing A implies B is better than nothing. (Here A = "some dead dude didn't screw up the proof of this hairy theorem" and B = "my desired corollary actually holds") The modularity of modus ponens means you can always put off really understanding the proof of A until later.

That being said, I don't find "intuitively obvious" to count for much of anything. I'm tempted to say that the entire point of mathematics is to replace one person's unanalyzable intuitive certainty that a conclusion follows from some assumptions with an argument that it does that can be examined and analyzed by anyone.

[gustavolacerda.livejournal.com]

<< I don't believe it's dishonest to use theorems whose proof you don't fully understand. Math is a big enterprise; delegating parts of the work to other (trustworthy) provers is fine. >>

I completely agree, of course.

But do you ever find yourself unsatisfied because your "informal yet rigorous"^* proof is a poor expression of your intuition, of your original reasons for believing the theorem?

^{* - by this I mean the standard of formality required by math journals, etc.}

[gustavolacerda.livejournal.com]

To borrow philosophy jargon, one could say that I'd like to unify the context of discovery and the context of justification.

[ikeepaleopard]

Is this any different than comparing pseudocode or a vague implementation strategy to actual code, where you have to deal with library details you didn't care about in order to make the code go through?

[gustavolacerda.livejournal.com]

I like the analogy. It might be correct.

[ikeepaleopard]

Proofs are programs.

[jcreed.livejournal.com]

Ah! That clarifies what you meant my "informal yet rigorous", thanks.

But to answer your question, if my original hunch as to why a theorem is true differs from the structure of the proof, I'd say my hunch was wrong - unless perhaps there's a different, still undiscovered proof that matches it more closely.

Why should I find my unrigorous, unjustified belief that something is true more weighty than a proof?

[jcreed.livejournal.com]

I guess I'm a bit of a pessimist on this issue, or rather I think that it's useful to have many tools for discovery and (in a certain sense) few tools for justification. "Having few tools for justification" is a terrible way of marketing it :) but I hope you know what I mean: the fewer axioms and more highly constrained rules I used to achieve a theorem, the stronger the theorem is.

[gustavolacerda.livejournal.com]

What about proofs that use excluded middle? What about proofs you'd find in journals other than JFM?

[jcreed.livejournal.com]

I think I'm with akiva in believing that it's a foregone conclusion that the analogy is correct, almost an identity.

Hoping that informal sketches of proofs should be persuasive somehow is like hoping psuedocode can be run without turning it into real code.

[jcreed.livejournal.com]

Excluded middle is definitely just a library that some programs call and some don't.

I think in the long term things that are publishable in JFM (or some equivalent) will be called mathematics, and things that are not will not be.

[roseandsigil.livejournal.com]

To say something similar to what jcreed said below, intuition is not a reason to believe something is true. Intuition is a useful pointer to a proof that something is (mathematically) true.

(Of course, what "mathematically true" means is a different subject.)

[ikeepaleopard]

It is exactly the same as, say running a piece of code that isn't rigorously proven. It might be right, but we don't know for sure. It depends on how much we care about trusting our code works. I'm more likely to trust it if I can follow it myself, or I know it was written or checked by good engineers. Same with math. The difference is that code is so much more complicated relative to your average proof and even slightly wrong code is often useful, whereas what use is a mathematical theorem that isn't true?

[gustavolacerda.livejournal.com]

Not more weighty...

But by throwing out the informal bits, you may be sacrificing your readers' understanding.

Also, I'd like readers to see how I came to this hypothesis. In cases where I came to know the theorem was true before I could prove it, the "intuitive proof" precedes the hypothesis. The formalization comes after that.

[roseandsigil.livejournal.com]

Proofs with excluded middle can be encoded as programs with callcc.

The snarky PL answer to your second question is that those aren't "proofs". Those are "proof sketches". I'm not quite sure what are more legitimate answer is. It might be that we can argue that mathematicians writing less formal proofs generally have an idea that a more formal proof exists, and they just don't want to go through with the effort of encoding it (and the community does not expect them to). In this case, we are somewhat justified in calling them "proof sketches", because that's what they are. I suspect most mathematicians think this from time to time, but I am not sure.

[gustavolacerda.livejournal.com]

<< intuition is not a reason to believe something is true >>

I don't think you mean this.

[roseandsigil.livejournal.com]

Err? No, I mean exactly that. Intuition is not a reason to believe something is true. Intuition is way to find a proof. A proof is a reason to believe that something is true.

My understanding of a proof may be intuitive, but, it is my proof, that just means I don't actually believe it yet. In my experience, the bits of the proof that one doesn't examine because they are "obvious" often turn out to actually be hard or wrong.

If it is someone else's proof, I might believe it because they say so, but that's a sort of accepted mathematical weakness, not a real reason to believe it.

[gustavolacerda.livejournal.com]

By your standard, we don't have reason to believe any theorem that hasn't been published by JFM and the like.

[ikeepaleopard]

To clarify, putting together a bunch of axioms and lemmas that say some thing, is essentially the same act as programming, it doesn't matter what those axioms say.