What does it mean to truly prove something?

Let me point you to the following recent blog post from Prof Keith Devlin, entitled “What is a proof, really?”

After a lifetime in professional mathematics, during which I have read a lot of proofs, created some of my own, assisted others in creating theirs, and reviewed a fair number for research journals, the one thing I am sure of is that the definition of proof you will find in a book on mathematical logic or see on the board in a college level introductory pure mathematics class doesn’t come close to the reality.

For sure, I have never in my life seen a proof that truly fits the standard definition. Nor has anyone else.

The usual maneuver by which mathematicians leverage that formal notion to capture the arguments they, and all their colleagues, regard as proofs is to say a proof is a finite sequence of assertions that could be filled in to become one of those formal structures.

It’s not a bad approach if the goal is to give someone a general idea of what a proof is. The trouble is, no one has ever carried out that filling-in process. It’s purely hypothetical. How then can anyone know that the purported proof in front of them really is a proof?


Click the link to read the rest of the article. Also read the comments below the article to see what conversation has already been generated.

I won’t be shy in saying that I disagree with Keith Devlin. Maybe I misunderstand the subtle nuance of his argument. Maybe I haven’t done enough advanced mathematics. Please help me understand.

Devlin says that proofs created by the mathematical community (on the blackboard, and in journals) are informal and non-rigorous. I think we all agree with him on this point.

But the main point of his article seems to be that these proofs are non-rigorous and can never be made rigorous. That is, he’s suggesting that there could be holes in the logic of even the most vetted & time-tested proofs. He says that these proofs need to be filled in at a granular level, from first principles. Devlin writes, “no one has ever carried out that filling-in process.”

The trouble is, there is a whole mathematical community devoted to this filling-in process. Many high-level results have been rigorously proven going all the way back to first principles. That’s the entire goal of the metamath project. If you haven’t ever stumbled on this site, it will blow your mind. Click on the previous link, but don’t get too lost. Come back and read the rest of my post!

I’ve reread his blog post multiple times, and the articles he linked to. And I just can’t figure out what he could possibly mean by this. It sounds like Devlin thoroughly understands what the metamath project is all about, and he’s very familiar with proof-checking and mathematical logic. So he definitely isn’t writing his post out of ignorance–he’s a smart guy! Again, I ask, can anyone help me understand?

I know that a statement is only proven true relative to the axioms of the formal system. If you change your axioms, different results arise (like changing Euclid’s Fifth Postulate or removing the Axiom of Choice). And I’ve read enough about Gödel to understand the limits of formal systems. As mathematicians, we choose to make our formal systems consistent at the expense of completeness.

Is Devlin referring to one of these things?

I don’t usually make posts that are so confrontational. My apologies! I didn’t really want to post this to my blog. I would have much rather had this conversation in the comments section of Devlin’s blog. I posted two comments but neither one was approved. I gather that many other comments were censored as well.

Here’s the comment I left on his blog, which still hasn’t shown up. (I also left one small comment saying something similar.)

Prof. Devlin,

You said you got a number of comments like Steven’s. Can you approve those comments for public viewing? (one of those comments was mine!)

I think Steven’s comment has less to do with computer *generated* proofs as it does with computer *checked* proofs, like those produced by the http://us.metamath.org/ community.

There’s a big difference between the proof of the Four Color Theorem, which doesn’t really pass our “elegance” test, and the proof of e^{i\pi}=-1 which can be found here: http://us.metamath.org/mpegif/efipi.html

A proof like the one I just linked to is done by humans, but is so rigorous that it can be *checked* by a computer. For me, it satisfies both my hunger for truth AND my hunger to understand *why* the statement is true.

I don’t understand how the metamath project doesn’t meet your criteria for the filling in process. I’ll quote you again, “The trouble is, no one has ever carried out that filling-in process. It’s purely hypothetical. How then can anyone know that the purported proof in front of them really is a proof?”

What is the metamath project, if not the “filling in” process?


If anyone wants to continue this conversation here at my blog, uncensored, please feel free to contribute below :-). Maybe Keith Devlin will even stop by!

When will she pass me for the first time? [solution]

Recently, my dad posed the following question here:

My wife and I walk on a circular track, starting at the same point.  She does m laps in the time that it takes me to do n laps.  She walks faster than I do, so m > n.  After how many laps will she catch up with me again?

If you haven’t solved it yet, give it a crack. It’s a fun problem that has surprising depth.

Here’s my solution (in it, I refer to “mom” rather than “my wife” for obvious reasons!):

Since mom’s lap rate is m laps per unit time, and dad’s lap rate is n laps per unit time, in time t, mom goes mt laps and dad goes nt laps.

They meet whenever their distance (measured in laps) is separated by an integer number of laps k. That is, mom and dad meet when

mt=nt+k, k\in\mathbb{Z}.

This happens at time


Mom will have gone


laps and dad will have gone


laps when they meet for the kth time.

And that’s it! That’s the general solution. This means that:

  • At time t=0, dad and mom “meet” because they haven’t even started walking at all (they are k=0 laps apart).
  • At time t=\frac{1}{m-n}, dad and mom meet for their first time after having started walking (they are k=1 lap apart). This is the answer to the problem as it was originally stated. Mom will have gone mt=\frac{m}{m-n} laps and dad will have gone nt=\frac{n}{m-n} laps when they meet for the first time.
  • At time t=\frac{2}{m-n}, dad and mom meet for their second time (now k=2 laps apart).
  • At time t=\frac{k}{m-n}, dad and mom meet for their kth time.

Here are two examples:

  • If mom walks 15 laps in the time it takes dad to walk 10 laps, when they meet up for the first time, mom will have gone \frac{m}{m-n}=3 laps and dad will have gone \frac{n}{m-n}=2 laps.
  • If mom walks 12 laps in the time it takes dad to walk 5 laps, when they meet up for the first time, mom will have gone \frac{m}{m-n}=1\frac{5}{7} laps and dad will have gone \frac{n}{m-n}=\frac{5}{7} laps.

Boom! Problem solved! :-)

Catch yourself up on the world of origami

Have you been doing other things and failed to notice the origami world evolve without you? Have you fallen asleep and been left behind? If you want to get caught up on what you’ve been missing in the world of origami, I suggest you visit Hannah’s origami blog, A Soul Made of Paper. She’ll have you caught up in no time.

I especially like giving her blog a shout-out because Hannah is a student of mine. She often comes by my classroom to show me her latest paper creations. I like origami, and I’ve dabbled in it–stuck my toe in the stream, if you will–but Hannah is like a scuba diver in the origami world. She’s loves modular origami, but she’s also great at the artsy curved creations (like origami roses), textures, and tessellations.

If you haven’t clicked over to her blog yet, here are a few more pictures to whet your appetite.

Go get lost at A Soul Made of Paper. Maybe you can join her other 3000 followers on tumblr :-).

*All of the above are Hannah’s pieces and Hannah’s photos.

Looking back on 299 random walks

This is my 300th post and I’m feeling all nostalgic. Here are some of the popular threads that have appeared on my blog over the last few years. If you’ve missed them, now’s your chance to check them out:

Thanks for randomly walking with me over these last few years (though, some say it’s a “drunken walk” :-) ). Either way, I’ll raise a glass to another 300 posts!

Challenge Problems

Want to enrich your Precalculus course with difficult problems? Look no further!

Very-Difficult-Mazes-Coloring-Page-1I teach a high-octane version of Precalculus to students in our magnet program. Our course, like most Precalculus courses, covers a very wide variety of topics. As often as possible, I like to give them more difficult problems that enrich the material from the book. These documents are a work in progress, but feel free to steal them (just email me a copy if you improve them!):

If you want solutions for any of these, shoot me an email.

These aren’t 100% polished by any means, but I’m sharing them anyway! Long live the spirit of sharing :-).

By the way, many of these problems are collected from other sources but I’m too far removed from those sources to properly attribute the problem-creator. My sincere apologies!

When will she pass me for the first time?

[Guest post by Dr. Chase]

My wife and I walk on a circular track, starting at the same point.  She does m laps in the time that it takes me to do n laps.  She walks faster than I do, so m > n.  After how many laps will she catch up with me again?

Example:  For m = 4, and n = 3, she will catch up when I have finished 3 laps.  Reason:  When I have finished 1 lap, she finished 1 1/3 laps, so she is 1/3 of the track ahead of me.  (But hasn’t passed me yet.)  When I have finished 2 laps, she has finished 2 2/3 laps around the track, still ahead of me.  When I finish 3 laps, she has finished 3 3/3 laps, which is to say 4 laps.  So we are together for the first time since starting.  If m = 2, n = 1, she will catch up in just 1 lap.  If m = 7, n = 6, she will catch up in 6 laps.  Will she always catch up in n laps?  In how many laps will she catch up for arbitrary m and n?

What does a point on the normal distribution represent?

Here’s another Quora answer I’m reposting here. This is the question, followed by my answer.

What does the value of a point on the normal distribution actually represent, if anything?


It’s important to note the difference between discrete and continuous random variables as we answer this question. Though naming conventions vary, I think most mathematicians would agree that a discrete random variable has a Probability Mass Function (PMF) and a continuous random variable has a Probability Density Function (PDF).

The words mass and density go a long way in helping to capture the difference between discrete and continuous random variables. For a discrete random variable, the PMF evaluated at a certain x gives the probability of x. For a continuous random variable, the PDF at a certain x does not give the probability at all, it gives the density. (As advertised!)

So what is the probability that a continuous random variable takes on a certain value? For example, assume a certain type of fish has length X that is normally distributed with mean 22 cm and standard deviation 1.6 cm. What is the probability of selecting a fish exactly 26 cm long? That is, what is P(X=26)?

The answer, for any continuous random variable, is zero. More formally, if X is a continuous random variable with support \mathcal{S}, then P(X=x)=0 for all x\in\mathcal{S}.

For the fish problem, this actually does make sense. Think about it. You pull a fish out of the water which you claim is 26 cm long. But is it really 26 cm long? Exactly 26 cm long? Like 26.00000… cm long? With what precision did you make that measurement? This should explain why the probability is zero.

If instead you want to ask about the probability of getting a fish between 25.995 and 26.005 cm long, that’s perfectly fine, and you’ll get a positive answer for the probability (it’s a small answer :-).

Let’s return to the words mass and density for a second. Think about what those words mean in a physics context. Imagine having a point mass–this is in an ideal case–then the mass of that point is defined by a discrete function. In reality, though, we have density functions that assign a density to each point in an object.

Think about a 1-dimmensional rod with density function \rho(x)=x, x\in (0,10). What is the mass of this rod at x=5? Of course, the answer is zero! This should make intuitive sense. Of course, we can get meaningful answers to questions like: What is the mass of the rod between x=5 and x=6? The answer is \int_5^6 xdx=5.5.

Does the physical understanding of mass vs density clear things up for you?