Math class has two goals:

**Verify that**a statement is true**Understand why**a statement is true

There's a tendency to put the goals in opposition, assuming concepts are either "easily understood but wrong" or "difficult to understand yet correct".

It's like a restaurant that believes in having taste *or* nutrition, but not both. Why choose?

Our goal is a deep intuition for correct things. And it's ok to start with an understood "sorta-true" concept and refine it to an understood "very-true" version:

Correctness isn't binary. Our standards for a valid proof have evolved over the years, and in a century our notion of what's true may seem embarrassingly primitive. That's ok: let's get a decent understanding and work to make it better.

We have to balance two roles: the safety inspector who makes sure the food is safe, and the customer who wants to enjoy the meal. In my head I think about "inspection mode" and "tasting mode": the secret is inspecting things that already taste good.

## Example: The Pythagorean Theorem

The Pythagorean theorem is usually introduced as a statement about *triangles*. A common proof is a visual rearrangement, like this:

This is nutritious and correct, but not tasty to me. It seems like a special case, an optical illusion: with *just* the right shape, things can be re-arranged.

A tastier proof is that the Pythagorean Theorem is really about the nature of 2d area. A big shape, when split, yields two smaller shapes. The total area must be the same:

The split-apart area can come from a triangle, circle, or cardboard cutout of Thomas Jefferson. It doesn't matter: two pieces, when cut from a larger one, must have the same total area.

Aha!

This intuition can then be refined into a more formal statement.

## Example: Euler's Formula

Here's a trickier example: Euler's Formula.

It's a baffling statement, and here's the common justification:

It's crisp and concise, but unsatisfying to even other math fans:

I agree: it's a bunch of symbols that happen to line up. Here's a tastier version:

- e
^{x}represents continuous growth (interest earning interest, which earns interest…) - sin(x) and cos(x) represent vertical and horizontal directions
- i represents rotation

If we create "continuous rotation" (e^{ix}) then we move in a circle, which can be separated into horizontal and vertical components (cos(x) and i sin(x)).

Again, this intuition can be sharpened further:

- e
^{x}can be seen as an infinite series, starting with an initial value (1), the interest it earns (x), the interest that earns (frac(x^{2})(2!)), and so on. - Sine (and cosine) are infinite series based on an initial impulse, which creates a restoring force, which creates a restoring force, and so on. This is like interest that earns interest in the opposite direction (and why sine oscillates without going to infinity: its motion opposes itself.)
- Plugging i into e
^{x}means we earn "imaginary interest" (i), which earns "imaginary imaginary interest" (-1) which earns "imaginary imaginary imaginary interest" (-i), and so on. Some of the interest opposes the previous terms, and we can collect them into patterns matching sine and cosine.

Rather than staring at a dry proof and trying to understand it directly, get a rough intuition (ADEPT method) and then see if the proof makes sense. It's a bit of math inception, where we try to *understand* the verification step, not simply *verify* the verification step.

Happy math.

## Appendix: On Proof and Progress in Mathematics

William Thurston (Fields Medal Winner) wrote a great essay, *On Proof and Progress in Mathematics*. It's full of ideas I found interesting:

- The question for mathematicians is: "How do mathematicians advance human understanding of mathematics?"

For instance, when Appel and Haken completed a proof of the 4-color map theorem using a massive automatic computation, it evoked much controversy. I interpret the controversy as having little to do with doubt people had as to the veracity of the theorem or the correctness of the proof. Rather, it reflected a continuing desire for human understanding of a proof, in addition to knowledge that the theorem is true...They discover by this kind of experience that what they really want is usually not some collection of “answers”—what they want is understanding.

- We're never done explaining a concept:

We may think we know all there is to say about a certain subject, but new insights are around the corner. Furthermore, one person’s clear mental image is another person’s intimidation.

- On the role of intuition:

Personally, I put a lot of effort into “listening” to my intuitions and associations, and building them into metaphors and connections. This involves a kind of simultaneous quieting and focusing of my mind. Words, logic, and detailed pictures rattling around can inhibit intuitions and associations.

- The "emperor's clothes" problem in math happens even for professionals:

Nonetheless, most of the audience at an average colloquium talk gets little of value from it. Perhaps they are lost within the first 5 minutes, yet sit silently through the remaining 55 minutes. Or perhaps they quickly lose interest because the speaker plunges into technical details without presenting any reason to investigate them. At the end of the talk, the few mathematicians who are close to the field of the speaker ask a question or two to avoid embarrassment.

A further issue is that people sometimes need or want an accepted and validated result not in order to learn it, but so that they can quote it and rely on it.

- On the difference between everyday explanations and and technical ones:

Why is there such a big expansion from the informal discussion to the talk to the paper? One-on-one, people use wide channels of communication that go far beyond formal mathematical language. They use gestures, they draw pictures and diagrams, they make sound effects and use body language...In papers, people are still more formal. Writers translate their ideas into symbols and logic, and readers try to translate back.

It’s like a new toaster that comes with a 16-page manual. If you already understand toasters and if the toaster looks like previous toasters you’ve encountered, you might just plug it in and see if it works, rather than first reading all the details in the manual.

- On what motivates us to do math:

What motivates people to do mathematics? There is a real joy in doing mathematics, in learning ways of thinking that explain and organize and simplify. One can feel this joy discovering new mathematics, rediscovering old mathematics, learning a way of thinking from a person or text, or finding a new way to explain or to view an old mathematical structure.

I love the "aha!" moments when a concept click. People willing seek out mysteries and puzzles (movies where we don't know the ending, games like Tetris). Math is an experience with similar emotional payoffs when approached correctly.

## Leave a Reply

20 Comments on "Math Proofs vs. Explanations (aka Nutrition vs. Taste)"

Inspiring!

This post – not just an Aha!, not just Aha! vs. Huh? but also meta-Aha! vs. meta-Huh?

So in a way a much deeper discussion than the “regular” brilliant explanations we are used to see here every week.

IMHO this could be a basis for a mathematical paper because it sheds (new?) light on whether there is a *mathematical* need to understand – and take math forward. It’s known that old “proofs” from 300 yrs ago do not meet today’s proofs’ standards – maybe it’s time to set higher standards for mathematical proofs, that go beyond pure logic.

Thanks Boaz!

Interesting question on the nature of understanding. I’d be curious about a “standard of understanding” (similar to the standard of proof that we have, which keeps getting stricter) where there’s some notion of how easily understood a concept is. A purely logical conclusion isn’t quite satisfying enough.

Great as usual, Kalid!

We have learnt a lot of equations and formulae but have understood very little.

You are airlifting all of us from marsh fields and dropping on the highway of understanding.

Thanks a million.

Thanks Nandeesh! :)

Great, great as can be. I enjoyed that Eulers Formula explanation. Suddenly, the whole thing makes sence to me.

Don’t know if its appropriate to present here but i just want to know why we often write integrals as, say: (dy/dx)dx, as in the incremental angle (dQ/dx)dx due to an increamental element dx. In otherwords, how is Q + (dQ/dx)dx different from Q + (dQ/dx)?

I hope that makes sense, and sorry if i have presented the question in the wrong place. I will be happy for a link on any work you have done on calculus

Hi Mustafa,

Glad it helped! Good question, it would depend on the context. Check out https://betterexplained.com/guides/calculus, it walks through the intuition for what dy, dx, etc. can mean. (In my head they are tiny slices/changes of a larger pattern.)

I haven’t often seen the integral of (dy/dx)dx, but the idea is that we are taking a pattern (y), slicing it up by dx (giving dy/dx), then re-assembling the pattern (integral (dy/dx) dx). You’d expect to get the original pattern, y, back out.

you are my math God

In your Cumulative Imaginary Interest you mention that growth follows a circle. Can you explain further and is this a general statement for all growth?

Hi Robert, regular exponential growth keeps compounding: interest earns interest, which earns interest, and so on.

With imaginary interest, the interest we earn goes “sideways” so doesn’t push us in the same direction. We end up spinning instead of getting faster. (I like to imagine a rocket mounted sideways… you start spinning around, instead of moving along faster and faster in your direction.)

Check out https://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/ for more.

This is great keep it coming

Thanks Ray!

A big shape, split up into two smaller shapes, must have the same area. Got that part. What I didn’t get, with the triangles diagram, is why the areas of those shapes are proportional to the square of those sides. I think it must have to do with similar triangles having the same ratio of sides to area, but that connection was missing. What I didn’t get with the circles diagram is how those circles add up. You could’ve drawn any size circle and made the same claim, no?

Great question. The missing step (I should have mentioned it explicitly, it’s in the article https://betterexplained.com/articles/surprising-uses-of-the-pythagorean-theorem/) is that Area is proportional to (any line segment)^2.

In a square, we usually write Area = side^2. But we could also say because perimeter=4s, then

Area = 1/16 p^2

Or, we could measure the diagonal (s * sqrt(2)) and get

Area = 1/2 diagonal^2

The constant factor F (1, 1/16, 1/2, etc.) just depends on the shape and the line segment we pick. That gives us:

Area(big) = Area(part1) + Area(part2)

Fc^2 = Fa^2 + Fb^2 [assume c, a and b are the same line segment in each shape]

c^2 = a^2 + b^2 [divide out scaling factor: the line segments in each shape are related using this formula]

Yep, in the circle diagram, I could have picked circles with circumference 3, 4, 5 instead of radius 3, 4, 5.

Makes me wonder why more people presenting the sorts of talks you describe aren’t changing the thrust of them to make them more vital and accessible to a wider swath of professional mathematical colleagues. I wouldn’t expect people at this level to try to make their work accessible to ME, but if the phenomenon you describe is accurate (and it’s hardly the first time I’ve heard these talks described in similar ways), it seems crazy not to refocus them and change the style. I refused to do talks at mathematics education conferences that were like the ones I heard that were basically unfathomable as presented unless you already were intimately familiar with the research (and which were sure-fire sleep inducers for me). I made my presentational style friendly, informal, familiar, colloquial, humorous, and, I hope, fun for more than just me. Seems perfectly feasible in mathematics, too.

Hi Michael, thanks for the note. I wonder that too — I think there may be an Emperor’s Clothes problem where a concept presented too simply may seem unimpressive. (That’s it? The magical formula we suffered centuries to unravel is just about XYZ?)

Thurston’s essay is excellent and says how the informal explanations that take minutes in person (with handwaving, diagrams, analogies, etc.) take an hour in a conference presentation, with much less retention.

You make a very good distinction between a rigorous, technical presentation of information vs a purposeful, focused presentation supported by rigorous, technical details. As you say, the details aren’t the “tasty” part; for non-experts, the way the details connect together to form a cohesive “ambient experience” defines one’s enjoyment much more than the minute, distinct details.

As a math teacher, I can speak from experience – almost none of my students care very much about the technical details, and because they don’t care, they rarely put the effort in to understanding it. I would estimate (from my own experience!) that most adults aren’t much different in this regard. I would love to see more of this dialogue in math classrooms at every level.

Thanks Josh. That’s exactly it, we have to recognize that there’s role for both flavor/experience and detailed analysis, and people seldom suffer through the latter without a hint of the former. Keeping a sense of empathy for what works for the audience is key.

Nice article. I also agree about 4-color map theorem. I hope someday I can see the “understandable” proof of it.

– Hitoshi

beautifully explained very soothing to read

Hey Kalid, your points inspired me thinking about how i want to learn, how we should learn, and why i was struggling before. What i found most difficult is the inability to phrase a proper questions that can help myself. Your great article get me into thinking where things go wrong.

As i have been reading more and more science textbooks, suddenly i realized that most authors tend to present math formulas rather than talking about ideas. The issue is, quite often those math formulas are just correlations rather than causality. I was troubled before with lots of topics such as physics, and now i see why: i got confused by math correlations and causality. Why mathematical correlations give us the ability to predict one thing based on others, they offer no intuition as to why. And without understanding the causality, i constantly found myself unable to connect things, explain things. Unfortunately nowadays people confuse the two as well.