Friday, December 2, 2016

Why I Bake Bread Using a Scale, and You Should Too

I am doing some posts about my adventures in baking bread, including some recipes, and I realized that I have become completely attached to weighing ingredients rather than using measuring cups. So the first recipe had volume measurements but if you try that and want to keep following you will need to get a scale. You can get a good one for very little money. I picked one out for you on Amazon for 9 bucks but now it costs $40, so forget that. But if you look on Amazon you will find digital kitchen scales for $10-15 that will works great.  I would stick with one that gets 4.5 stars or more.

People assume I'm advocating for a scale because it's more precise. It is true that weighing ingredients is more accurate, since flour especially varies in how much it compacts depending on how you store and measure it. I have taken a certain amount of good-natured guff in response to my admittedly-crazily-obsessive post about the inaccuracy of tablespoon measures.  My cousin Anna, a far more accomplished baker than I will ever be, wrote, "When one is not in a medical setting, that level of precision rarely matters. That's what I love about baking!" And her mother Barbara, award-winning cookbook writer/culinary historian, who first introduced me to bread-baking when I was 8, was even more pointed in a good natured way: "What is the takeaway message? My answer would be, walk away when a mathematician is in the kitchen. I am trying to think of a recipe that demands precision in tablespoon measures. I'll get back to you when something turns up."

They are right, of course.

So why weigh your ingredients, really?

  1. It's easier, quicker, and has less cleanup.  You put your mixing bowl on the scale, zero it, and throw in the ingredients one by one, zeroing the scale again after each one. No measuring spoons and cups to wash out (OK, you might still occasionally use a measuring spoon). Have you ever measured out tahini in a measuring cup? Or peanut butter? It's not pleasant to try to get air bubbles out, and it's not pleasant to wash the measuring cup afterwards.
  2. It's a snap to scale your recipes, at least if the weights are in grams (g). If you want to make a one-and-a-half-recipe of something with 1 and half cups of flour, it's annoying.  If you want to make a one-and-a-half-recipe of something with 120g of flour, use 180g of flour.
  3. This is especially true if you are experimenting with baking bread, where you actually might want to make a loaf 25% bigger, for example.
  4. Europeans cook by weight, and it's another reason they think Americans are crazy.
  5. Bakers give recipes in "baker's percentages," which means that all ingredients are measured by weight, relative to the amount of flour. So if water ("hydration") is 70% and salt is 2%, and you have 400 grams of flour, you need to use 280g of water and 8g of salt.
  6. It makes it easy to improvise with recipes. Suppose you have a yeast bread recipe and you want to use sourdough starter instead.  If you use 200g of starter, just subtract 100g from the flour and 100g from the water.  (Assuming your starter is equal parts water and flour by weight, which I strongly recommend.)
  7. It makes it easy to refine your recipe. If you make the same recipe a lot, which I mostly do not, you will remember how it's supposed to feel. But if you make it infrequently, and the dough was too sticky this time, you can make a note to cut the water by 20g next time.

The one problem I have is that I sometimes add too much of one ingredient by accident. But if you do that and you want to correct it, it's easy to scale the others up.

So give it a try!

Wednesday, November 30, 2016

A Simple, Purely Geometric Proof that the Square Root of Two is Irrational—With a Couple of Bonus Side-Trips

Lots of proofs are based on simple ideas, but get bogged down in notation or exposition that doesn't bring out the salient points, as if the soloists and all the members of the choir were singing at equal volume.

Sometimes some warm-ups can help people understand what is essential in a proof and what is extra. They can help the simplicity of an idea shine through.

So the first warm-up is a purely geometric proof that the golden ratio is irrational. A number of years ago, I saw such a proof... but the diagram that went with it was laid out on a single line, and it got bogged down in a bunch of notation, and I kind of got it but it certainly didn't excite me.

Then one day I was looking at my business card and I realized that the proof was right there.  When I was designing my business card, I tried to figure out a good logo, and I eventually settled on a golden rectangle and golden spiral:

So here's how the proof works. Saying that the golden ratio is irrational is the same as saying that line segments whose ratio is "golden" (AB and BD in the figure above) are incommensurable. Two (or more) segments are "commensurable" (meaning "can be measured together") if there is some (typically very short) ruler segment such that each of the commensurable segments can be "measured by" the ruler segment, i.e. can be constructed by laying the ruler segment end-to-end some (positive integral) number of times (or, equivalently, if it is a union of segments congruent to the ruler, which overlap only in their endpoints). Two segments are commensurable if and only if the ratio of their lengths is rational; if that is not fairly obvious to you, I encourage you to spend some time figuring it out.

So to prove our result, we are going to argue by contradiction, assuming that AB and BD are commensurable.  Measure off AB laying out m copies of the ruler segment, and measure off BD by laying out n copies of the ruler segment. It follows that the segment CD is covered by n–m copies of the ruler segment, so CD is commensurable with the two previous segments. Now look at the top segment: DF is congruent to AB, and DE is congruent to CD, so the same argument shows that EF is can be measured with an integral number of ruler segments. The same argument applies to GH. So on the one hand these segments are getting arbitrarily small, and on the other hand they can't get shorter than a single ruler segment, which is the desired contradiction. Nice, huh? What could be simpler!

[OK, for anyone who is a little squiffy about the rigor of "arbitrarily small" in this context, you can look at the sequence of how many rulers it takes to measure off each successive segment. It is an infinite strictly decreasing sequence of positive integers, and good luck finding one of those...]

One major side excursion you could take here is to understand how this picture relates to the fact that the continued fraction expansion of the golden ratio is [1 1 1 ...]. But I'm not going there now.

Something about having the proof laid out in two dimensions and using the spiral pattern of squares managed to make the argument much clearer. And if you're sitting with someone with the picture (or card) in front of you, it's even easier. You just point, and you don't need to label anything. So, after marveling over this for a while, I started to wonder about a purely geometric proof that the square root of two is irrational, or, put geometrically, that the diagonal of a square is incommensurable with its side.

Now I have to admit, this is a perverse question. The algebraic (or perhaps more accurately the number-theoretic) proof is certainly one of the crowning gems of mathematics. Utterly simple, and unimprovable.  But the truth is that my mind can be perverse. This question rolled around in my brain for a number of years and for some reason, I got serious about it the other night and figured out a proof I am happy with.

But first, so you don't get freaked out by a mildly complicated picture, let's take another warm-up. Let's look at the circle below on diameter AC (well, semicircle, but you know what I mean):

Figure 2

For any point D on the circle, ADC is a right angle. (If, like me, you don't remember the proof of that, it's fun to reconstruct it.) If you drop the altitude to the point B, you get two additional right triangles, and all three are similar.  In particular ABD is similar to DBC, and it follows that c is to b as b is to a (c:b :: b:a). That is, b is the geometric mean of a and c.  So already you can turn this into a way to construct the geometric mean of two segments. But also, note that the length of BD is less than or equal to the length of EF, with equality only if a = c. On the other hand, EF is a radius of the circle, so its length is (a+c)/2, since the diameter of the circle is a+c. So in our little warm-up we've proved that the geometric mean is less than or equal to the arithmetic mean (average), with equality only when the two numbers are equal. OK, and here is the only little bit of algebra I'm going to use in this article: if you cross-multiply c/b = b/a, you get b^2 = ac, so b = sqrt(ac), the algebraic formula for the "geometric mean."

So two more things before we get started: first, I'm not actually going to prove that sqrt(2) is irrational, I'm going to prove that sqrt(2)+1 is irrational, and let you take it the rest of the way. Second, I hope you will forgive this kind of crappy drawing, I'm sure there are great drawing programs that could produce this in a snap, but I couldn't find one and learn it in a snap.  Make believe that the two horizontal-ish lines are actually horizontal (and therefore actually parallel).

Figure 3

So CDGH is a square with side of length s and diagonal of length d, as is CHIB. The (semi)circle has center C and radius d. Now we are going to focus on the big rectangle ADGJ, whose height is s and whose base is s+d. We will prove that the height and base of this rectangle are incommensurable. When s=1, d=sqrt(2), and we will have shown that 1+sqrt(2) is irrational as promised. As in the first warm-up, we assume that the height and base are commensurable and argue to a contradiction. As in the second warm-up, the heavy-dashed line triangles ADG and GDE are similar, so our big rectangle ADJG is similar to the little rectangle GDEF. And the little rectangle GDEF is congruent to its mirror image on the left, JABI (or if you want to be pedantic, which I know many of you do, the mirror image is IBAJ). So if there is a common ruler segment that measures both AD and DG, it also measures AB, since both BC and CD are congruent to DG.  In other words: take the big rectangle ADGJ, knock two squares off of it, and you end up with a similar, smaller rectangle JABI, whose sides are also measured by our ruler segment. Because IBAJ is similar to the big rectangle, knocking two squares off of it gives yet a smaller similar rectangle, whose sides are also measured by the ruler segment. Continuing the process leads the same contradiction as before, with an arbitrarily small segment needing to be one or more of our fixed ruler segments. Done!

So I hope that was easy to follow. By the way, I didn't actually use any result that I proved in the second warm-up, I just used the picture and the idea. But hopefully that—and the heavy dashed lines—helped you focus on the essentials in what otherwise would be a moderately complicated figure. For those of you who are really disappointed to leave that warm-up behind, we have proved that s is the geometric mean of d+s and d-s.

Also, for those of you who dug into the continued fraction handwaving above, we have proved that the continued fraction expansion of sqrt(2)+1 is [2 2 2 ...]. So the continued fraction expansion of sqrt(2) is [1 2 2 ...].

Once I figured this out, I poked around a bit on the internet, since obviously I was not the first to give this kind of geometric proof. I have to say that I was not impressed by the clarity of what I found. I was aided by an offhand comment from a mathematician acquaintance who remarked that this is "anthyphairesis," which turns out to refer to this process of using a "divisor" to lop off segments, and using the remainder as a new divisor to lop segments off of the previous divisor.  In our case, we are applying anthyphairesis to the two sides of our big rectangle ADGJ.  Our "divisor" is the height DG, and we can lop off two segments of length s (the height) from the base AD, before the remainder, the segment AB, is shorter than s.  We now reverse roles, and use AB as the divisor to lop off segments from the height. But the result we just proved—that ADGJ is similar to JABI—shows that we will again lop off two segments, and the process will continue ad infinitum.  This is why the continued fraction expansion of sqrt(2)+1 is [2 2 2 ...]; and as a result, the continued fraction expansion of sqrt(2) is [1 2 2 ...].

It is not hard to see that anthyphairesis terminates if and only if the starting segments are commensurable.  Applied to segments of integer length anthyphairesis is Euclid's GCD algorithm; when applied to commensurable segments it will actually give you the "longest common ruler." Looking at it this way clarifies the relationship between Euclid's GCD algorithm and continued fractions. It is always humbling to see what the ancient Greeks accomplished more than a thousand years before the invention of the equal sign.

In general I don't like to complain in public about the work of others, especially if I haven't checked it out thoroughly.  But my sophisticated search techniques (which you will see if you click on this link) brought me to page 189 of "From China to Paris: 2000 Years Transmission of Mathematical Ideas," by Yvonne Dold-Samplonius, where I found this quote:
Fowler has shown that it is historically misleading to interpret anthyphairesis in terms of continued fraction expansion, because the ancient Greeks saw anthypairesis as a process of subtraction, whereas continued fractions are the result of a process of division [Fowler 1999: 30, 313 (n.13), 366].
Now I am fully attuned to the dangers of regarding anthyphairesis as the Greek's failed attempt to do what we do correctly as continued fractions; it needs to be considered on its own merits, separate from how it may or may not map to modern concepts. And I have not read the Fowler 1999 reference where he may say something that makes sense. That said... Hello?! What is division but repeated subtraction until you can subtract no more, leaving a remainder?!

On pp. 2-3 of a PDF on continued fractions, Paul Hewitt does fairly well, showing that if you start with a square with side s and diagonal d, then a smaller square with side s' = d-s has diagonal d' = 2s-d.  If s and d are integral multiples of a common ruler, then d' and s' will be integral multiples of the same ruler, which is the result we need. His result starts with the Pythagorean theorem s^2 = 2d^2, and proceeds with a lot of algebra to the result; a key intermediary is the fact, noted above, that s is the geometric mean of d+s and d–s. Note that I have drawn (using a dotted line KL) the new square, and labeled s' and d' above in Figure 3. I put all that stuff in parentheses because we didn't use it in the main proof.

The third reference I found is a 1979 paper by (I assume the same) Fowler; on p. 819 (p. 13 of the PDF), there is some discussion, including a figure which is poorly labeled and explained. He does, however, quote Proclus as saying:
The Pythagoreans proposed this elegant theorem about the diameters and sides, that when the diameter receives the side of which it is the diameter [that is, d–s], it becomes a side [s'=d–s], while the side, added to itself [2s] and receiving the diameter [2s–d], becomes a diameter [d'=2s–d, as in the Hewitt paper].
I have added the bracketed text. Two things stand out for me from this passage. First, we tend to take the technology of algebraic notation for granted, in much the same way that we tend to think of pre-industrial agriculture as "bucolic" rather than "technological."  But both were massive innovations. Without the bracketed notes, disentangling this paragraph is a significant cognitive load. Algebraic notation bears that load effortlessly. Second, once you appreciate the massive cognitive load the ancient Greeks were working against, their mathematical achievements are that much more awe-inspiring.

OK, let's get back to Proclus's "elegant theorem of the Pythagoreans," to close the loop.  Going back to our big rectangle ADGJ, its short side is s and its long side is d+s. Proclus is talking about making a new square of side s' = d–s, which is the short side of the far-right rectangle GDEF. What is the diagonal d' of this square? By our similarity result, the long side of GDEF is d'+s', as pictured.  It is also s. So d' = s–s' = s – (d–s) = 2s–d, as we were to have proved.  OK, remember above where I said there was only that one little bit of algebra? I lied. No doubt there is a purely geometric way to demonstrate this relationship.

If you have read this far, I salute you! And I hope that you have also gotten some (perhaps perverse) pleasure on this little ramble of ours.

PS Thanks to Dylan Thurston who got me thinking about the second warm-up by posting it on his Facebook page.

PPS When I showed Dylan a draft of this post, he immediately wondered whether it is possible to given an alternate proof using what we might call an "A-rectangle," that is, a rectangle where the ratio of long side to short side is sqrt(2). It has the property that when you bisect its long axis, you get two A-rectangles, at a 90 degree angle from your original. It turns out that A-series European paper (like A4 etc.) has this aspect ratio, so that when you do side-by-side copying of two sheets, the double sheet has the same aspect ratio—a cool fact that I had not been aware of.  (Numberphile video here, more history here; it turns out that the aspect ratio goes back to 1786 and specifics of the A-series go back to the early days of the metric system, with Lazare Carnot in 1798.)

He was absolutely right... starting with such a rectangle and subdividing continuously, the proof is immediate. But again, it does rely on a little bit of algebra. If we define an "A-rectangle" as one whose long side is equal to the diagonal of the square whose side is the short side, can you find a purely geometric proof that bisecting such a rectangle divides it into two A-rectangles? It is possible to give a short proof of that fact, and I'll let you have the fun of looking for it, if you want. But that's the thing about math... you work on a problem, come up with a solution you're happy with, and someone who is either smarter or has been exposed to different things (or both) comes up instantly with something simpler. But we get something out of having multiple approaches... in this case, some of the interesting side-notes on anthyphairesis and continued fractions.

Sunday, October 23, 2016

Making Really Good Bread is Really Easy!

OK,  I have been meaning to do this post for almost a year now. Back in November 2015, I put on a workshop with three other colleagues, and had dinner with four out-of-town colleagues afterwards. They asked if I was happy with how it went; I said that I was very happy with the workshop, but I mentioned that I was also happy because the previous week I had finally managed to make a loaf of bread that I was truly pleased with. All four were quite interested, and I thought I should put together a post about it.

That rolled around in my head for a while until in April I decided to do a demonstration using only equipment that most people have in their kitchens.  So  I made some videos, but I got distracted and then let it sit until now.  I have some subsequent posts in mind and we'll see how long they take.

The difference between this recipe and the bread I was so happy with is that this one is made with commercial yeast rather than natural leavening, also known as sourdough starter (or, in France, levain). It's not particularly sour, but it is more flavorful than this bread.  This bread is definitely good, but a bit flat in comparison. I am planning on doing some subsequent posts for folks that get you excited about baking bread, but I will warn you that the later posts will all involve weighing ingredients. If you want to follow those you will need a decent kitchen scale, but I will post some recommendations for inexpensive scales.

This recipe is from Jim Lahey of Sullivan Street Bakery in New York, who pretty much started the no-knead bread movement after his recipe was popularized by Mark Bittman in the New York Times. I had some problems with this and other recipes because the dough would often be too sticky, and instead of being able to manipulate the dough I would end up with my fingers covered in a sticky mess. The bread can come out tasting good but I was doing a lot of swearing and not having a very good time. The watchword for dough is "tacky, not sticky."

The Lahey recipe is on the Time website. I found it very helpful to watch the video there to get a sense of how to handle the dough and what it's supposed to look like, and that's part of why I wanted to make my own videos for this post. Interestingly, the Times recipe on the web page is a bit different from the one that Lahey gives in the video. The dough is slightly wetter and there is an extra step where the dough "rests" for 15 minutes. I am following the recipe in the video.

Fancy bread ovens have steam injectors, which is how you get great crusty bread. The key element of the Lahey recipe is to bake the bread in a Dutch oven or other suitable pot.  Not everyone has one, and fancy enameled ones are quite expensive, so I'm not doing it that way in this recipe. Instead, I'm using the method I learned from Jeff Hertzberg and Zoë François's Artisan Bread in 5 Minutes a Day, which involves pouring water into a pan—I use a metal roasting pan, which works nicely—in the oven to make steam.  As you will see, it works very well.  Other than that, the recipe here is largely copied from the New York times page, but I have simplified it in a few ways (e.g., using flour instead of cornmeal).

Although this bread takes only a few minutes of hands-on time, it needs some advanced planning because it rises for 18 hours (OK, 12 will do but Lahey says 18 is better) before "shaping," and then rises for another two, before baking for half an hour or more.  So if you mix up the ingredients at 6pm on Friday, you can shape it at noon on Saturday and have a great loaf of bread in the mid-afternoon.

I am a little over-obsessive about measuring and other things. But honestly, bread is very forgiving, and if you do something approximating this and get it in the oven, it will look less and less messy as it bakes, and it will probably taste great no matter how it looks.

So, without further ado, here's the recipe. When I made the recipe I also weighed the ingredients so that I could give them here, since one of the people at dinner was from the Netherlands.

  • 3 cups all-purpose flour (500g), more for dusting
  • 1/4 teaspoon instant yeast (1 ml)
  • 1 1/4 teaspoons salt (8g)
  • 1 1/2 cups Water (340g or ml)
  • another cup or so of water for the steam (250 ml)

Step 1: In a large bowl combine flour, yeast and salt. Add the water, and stir until blended; the dough will be shaggy and sticky. Cover the bowl with plastic wrap. Let the dough rest at least 12 hours, preferably about 18, at warm room temperature, about 70 degrees F (a little over 20 C). I put it on top of the refrigerator, which seems to work even in winter. When it's ready, the surface of the dough should be dotted with bubbles. If you are determined to use a stand mixer (like a Kitchen Aid), you can, but it is no problem to do this with a spoon or even with your bare hands, and there is a lot less clean-up.

Step 2: Using a rubber spatula and just enough flour to keep the dough from sticking to the work surface or to your hands, move the dough from the mixing bowl to the work surface. Flour your hands to avoid sticking (you can also use water or oil on your hands). Gently fold the dough over on itself to make a roundish shape; in the video I make four folds, one from each side. The folding makes some "seams," which are now on top; the bottom side of the dough, in contact with the work surface, is smooth. Transfer the dough to a lightly oiled bowl, placing the "seam side" down. Cover with plastic wrap and let rise, again in a warm place.

You will notice in the video that I have my dough scraper handy but I don't use it. If the dough sticks to the work surface, use a scraper (or a  knife if you don't have a scraper) to pry it up, and dust with a bit more flour. You want enough flour on the outside to get a "gluten cloak," a dull-ish surface that is not sticky, and you want to be careful not to tear the gluten cloak. (See the end of this post for some explanation of why the location of the seams makes a difference.)

Step 3: After about an hour and a half—half an hour before the dough is done rising—prepare the oven. Put a metal roasting pan on the bottom shelf; you will be putting water in it to make steam. Put a baking pan (cookie sheet) on the shelf just above it, and preheat the oven to 450 F (230-235 C, I have no idea how european oven settings work). I think I may have lightly oiled the cookie sheet, I can't remember. A non-stick cookie sheet is not a bad idea.

Alternatively, you can use a 6- to 8-quart heavy covered pot (cast iron, enamel, Pyrex or ceramic), but be a little careful of sticking. The first couple times, you may want to lightly oil the bottom of the pot, or sprinkle some cornmeal or flour in there just before you plop the bread in.

Step 4: After another half an hour (two hours of rising), the dough should have just about doubled in size.  Using a little more flour for your hands and the top of the dough, remove the dough from the oiled bowl and transfer to the baking sheet, flipping it over in the process so that the seam side is up.

Pour about a cup of water into the roasting pan, being careful not to burn yourself on the steam. Close the oven and don't open it for at least half an hour. If you are using a glass or pyrex roasting pan you might want to use very hot water.

If you're using the pot, cover it.

Step 5: Check the bread after half an hour. In the video, mine was probably done at that point, but I gave it about 5 more minutes. It gave a satisfying "thud" when I tapped on it, and the color was a lovely deep brown. Because the cookie sheet clangs, you can't really hear the hollow thud until some point in the second video.

If you are using the pot, uncover after half an hour and cook for another 15-30 minutes, "until the loaf is beautifully browned."

Transfer your loaf to a wire rack to cool. At this point the inside, or "crumb," is still cooking, so it is important to let the bread cool for a while. The experts seem to want you to let it cool completely. In my house it never makes it that far, but it is probably good to let it go for 20 minutes or so before you slice into it.

Here's a video of how mine looked at the end:

Happy baking! Let me know how it goes for you!

If you are still reading, here is the explanation of why we are so concerned about where the "seam side" goes. As the bread cooks, the volume expands faster than the surface area, creating pressure on the surface. If the seams are up, they come apart a bit to relieve the pressure. If the seams are down, the weight of the dough keeps them from opening, and the loaf may tear. Most bread recipes have you put the seam side down, because the irregular seams don't look so nice, but then you have to "slash" the top (making cuts), to allow the bread to expand. If you look at artisan-style loaves, you will see the results of slashing... it's not just for decoration. But even if your loaf tears, it may look a little funny but it will still taste great

Tuesday, April 19, 2016

OK, how big is a tablespoon, REALLY?

One of the handiest cooking factoids in our crazy US measurement system is that there are 4 tablespoons in a quarter cup.  If you are doubling a recipe for a sauce that calls for 2 tablespoons of flour, don't measure 4 tablespoons, just measure 1/4 cup and you're all set.

So I was a little surprised when I was looking at a source I consider reliable (Samuel Fromartz's In Search of the Perfect Loaf, p. 87) to see "Mix 3 tablespoons (30 grams) lukewarm water..."  Now, "everyone knows" that a tablespoon is 15 ml, and 1 ml of water weighs a gram, so 3 tablespoons of water should be 45 grams.  OK, precision is overrated in cooking, but it is more important in baking, and this is a 50% difference (45 being 50% more than 30), and that is actually significant.

1 tablespoon "=" 15 ml
Luckily, before I sent a know-it-all email to Mr. Fromartz, I decided to make sure, and when I weighed a tablespoon of water on my lovely digital kitchen scale (more on that next post) it was 10 grams. Oops. The plot was thickening. Three tablespoons got me up to 31 grams, so maybe a bit more than 10 grams. Googling "one tablespoon of water to grams," the top few hits are all say 15, you get to a precision calculator whose default is "5 digits after the decimal point" (OK, that is insane, we are not detecting gravitational waves here), but when you turn it down to a respectable "1 digit after the decimal point" it says 14.8, which is acceptable (though incorrect), for reasons below. The next hit says 15.00 which is not acceptable.  If you are going to stick in places after the decimal point, you cannot round up.

You have to get to the next-to-last entry on the first page before you find someone who actually tried it.  They say 12 grams... and interestingly, the scale they photographed says 11 grams... and the page I first saw, on the same site, seems to have the same photo, and that page says 11 grams.  But 10-11-12? This is getting into a more acceptable range of accuracy. Also, their tablespoon looks like a table spoon, not the Precision Measuring Instrument I use, pictured above.

This left me with only one question, which I will abbreviate: WTF!?

So I got out my 1/4 cup measure, stuck it on my scale, and measure out 4 tablespoons of water. Here's what I saw:

You can see 41 grams, right in line with Sam Fromartz, and if you look carefully, you can see that the measuring cup is nowhere near full.  (Also, when I first did this, I made the tablespoons as full as I could, which meant, I think, that there was a meniscus of water sticking up above the full mark.  That was 50 grams of water, or 12.5 grams per tablespoon, still in line with the web page where they actually measured it.  But in terms of everyday cooking it was still Sam Fromartz all the way.)  When I filled the 1/4 cup measure to the top, it was 56 gm, which is about what it should be, see below.

OK, so how many (of my) tablespoons are actually in 1/4 cup? To do that I needed to measure a non-liquid.  Flour is too compressible to get a reliable answer, so I used salt. The answer?

Not 4... but you knew that.  And of course I'm saying "my tablespoons" because I now have zero confidence that there is really any such thing as a standard tablespoon.

More than 5.  Really?

Less than 6.  Phew!  I scooped up the overflow, which was 2 teaspoons, or 2/3 tablespoon, so it appears that there are 5 1/3 of my tablespoons in 1/4 cup.  I say "appears" because I am not convinced at this point that there will be 3 of my teaspoons in one of my tablespoons—which is what there should be—if I measure it carefully.  But to stay sane (or, I should say, as sane as possible), I'm not going to do that.

In medical school we used metric units, but when you get to pediatrics they do talk about ounces of formula, so you need to convert.  If you ask a pediatrician how many grams in an ounce, they will say 30, because medicine is maybe a little more precise than cooking but not all that much.  Now there are 8 fluid ounces in a cup, so 1/4 cup is two fluid ounces, so if the pediatrician was right, my scale should have read 60 grams when I filled the 1/4 cup measure with water.  (A fluid ounce of water should weigh one ounce, right? So I thought, but, uh, no, though pretty close. Apparently we were thinking of Imperial fluid ounces. You knew that, right?)   But my scale said 56, what's up with that?

If you ask a pot smoker how many grams in an ounce they will say 28, because they are paying for the stuff. If you ask a drug dealer, they will probably say 28.35 because they are even more highly motivated to precision.  So it was always interesting in med school when the professor said, "How many grams in an ounce?" and the folks who usually knew all the answers were stumped, but the hand shoots up of the kid who is not considered the sharpest knife in the drawer, and he says, proudly, "28!"  "Well, 30, actually," says the professor, incorrectly. I felt bad for that guy but I felt like I knew him a little better after that.

So 1/4 cup is really 2 ounces, and an ounce of water is really pretty much 28 grams, so 2 ounces of water is 56 grams, which is what my scale said, so that part checks out.

The take-home message?  Cook with a scale, like the europeans do, and use grams.  (More on that in my next post, if I ever write it.)  All my European friends would be shaking their heads about how crazy we Americans are to cook this way, except that they are too busy shaking their heads about how crazy we are to have so many guns. Well, maybe we can't do anything about the guns, but we can do something about cooking.  I haven't locked up my measuring cups yet, though.

But if anyone can shed some light on how this state of affairs came to be, I'd love to know. I am confident that it's not because I happen to have a bad tablespoon...

Friday, January 8, 2016

Data Ain't What They Used to Be. Or Is They?

"Hmm... that makes sense," said Ben.

"Uh, you sound surprised," was my response.

"Well, usually when people start out by saying 'Here's my take on that,' it tends to not be that helpful." Well Ben's a smart guy, and if he hints that it might be helpful, that was enough to get me to write this post, which has been rattling around in my head probably about since blogs were invented.

My "take" had been prompted by another iteration of a typically tedious discussion that you may also have been involved in periodically: whether "data" is singular or plural. For those of you too young to know, or too smart to care, "data" is the Latin plural of "datum," meaning "a piece of information." So when a hapless person would say "There isn't enough data," grammar snoots would correct them and try to get them to say "There aren't enough data."  It's been a losing battle.  (David Foster Wallace fans will know that he actually called these people SNOOTS, all caps; an executive summary is here.)

So why do all of us troglodytes (we troglodytes? no, I guess it is "us troglodytes") insist on treating "data" as singular?  But wait, we don't!  "Car" is singular, but we don't say "There isn't enough car," at least not routinely.  The answer is that we are treating the word "data" as a mass noun, like "rice," "water," or "sand," as opposed to a "count noun" like "car."  We say "there isn't enough rice" without any problem. (Incidentally, I had originally thought that the word for this is "collective noun," but that refers words like "baggage" [a number of bags], "library" [a number of books], or phrases like "a pride of lions" and other terms of venery.)

The conclusion I had come to is that before the "information age," there just weren't "that many data," and you could count them one by one.  Since the arrival of the computer, we are, to paraphrase Torricelli, "swimming at the bottom of a sea of data," and it is no more practical to enumerate data one-by-one than it is to count grains of sand on a beach. So it seemed to me that this was a shift in usage prompted by a technological change, not simply by ignorance.

I was very satisfied with this explanation until, in the process of writing this post, I went to check out what I thought might be an early use of the word, recalling Sherlock Holmes saying "Not enough data" as he mulled over a difficult case.  So I fired up the OED on line (so awesome not need the magnifying glass!), and eventually found (after clicking on "full entry"), the following usage note under "data":
The use of data as a mass noun became increasingly common from the middle of the 20th cent., probably partly popularized by its use in computing contexts, in which it is now generally considered standard (compare sense 2b and the recent uses cited at datum n. 1b, some of which are ambiguous as to grammatical number). However, in general and scientific contexts it is still sometimes regarded as objectionable. Compare the plural uses cited at datum n. and the following:
1949   Nature 19 Nov. 890/1   ‘Data’ was a plural noun; for literate English writers it still is, and I contend that it always should be.
1978   P. Howard Weasel Words xiii. 63   Data stubbornly persists in trying to become an English singular.
1990   Psychologist 13 31/1   A staggeringly large number of psychologists fail to appreciate that ‘data’ should be followed by the plural form of the verb.

Snoots on parade!  And pretty much what I expected.  But when you read the entry on data, you find some shockers:
1645   T. Urquhart Trissotetras 53   The verticall Angles, according to the diversity of the three Cases being by the foresaid Datas thus obtained.
1764   Gentleman's Mag. Nov. 509,   I collected the datas chiefly from those excellent coin notes.
1807   Salmagundi 24 Nov. 366   My grandfather..took a data from his own excellent heart.
1910   Oologist 27 20/1   To make the markings on the eggs gibe with the datas is something of a chore.
2006   Cancer Causes & Control 17 1055/2   These datas were likely not missing at random.
This is "data," used as a singular count noun, with "datas" as plural.  WTF!?

Perhaps more interesting, the use of "data" as a mass noun has a long history:
1702   R. Morden Introd. Astron. i. 103   And by this Data there are twelve Problems resolved.
1826   Edinb. New Philos. Jrnl. 1 340   Inconsistent data sometimes produces a correct result. This, however, only happens..when part of the data is allowed to lie dormant.
1888   Pump Court 5 May 56/2   In the Northampton [table] the data is taken from the actual deaths of a floating population.
1902   A. S. Tompkins Hist. Rec. Rock Co., N.Y. 46   There is but little data to estimate Indian populations.

In fact, to find unambiguously plural usages of the word "data" you need to look at the OED entry on "datum." The earliest one is:
1691   Philos. Trans. (Royal Soc.) 16 498   From these data..the time of this Invasion will be determined to a day.

Merriam-Webster online has what seems to me a very sane usage note:
Data leads a life of its own quite independent of datum, of which it was originally the plural. It occurs in two constructions: as a plural noun (like earnings), taking a plural verb and plural modifiers (as these, many, a few) but not cardinal numbers, and serving as a referent for plural pronouns (as they, them); and as an abstract mass noun (like information), taking a singular verb and singular modifiers (as this, much, little), and being referred to by a singular pronoun (it). Both constructions are standard. The plural construction is more common in print, evidently because the house style of several publishers mandates it.

Of course this sent me scurrying to the Chicago Manual of Style; in 5.220 of the 16th edition ("Good usage versus common usage"), we see:
data. Though originally this word was a plural of datum, it is now commonly treated as a mass noun and coupled with a singular verb. In formal writing (and always in the sciences), use data as a plural.
Uh, so I guess "computing contexts" are not considered "science." Oh well.

The usage examples from the OED suggest to me that if you are pining for the days when an educated person would know beyond question that "data" is the plural of "datum," your nostalgia is for first-century Latin, not eighteenth-century English.  Condemning "data is" has a bit more historical basis than some other snoot shibboleths, such as the proscription on ending a sentence with a preposition, but that historical basis is not from English. In fact, many of these "rules" may arise from the same group of Latin-obsessed 17-century introverts.  There seems to be a rabbit-hole the size of the internet to slide down here, in which even cherished stories about Winston Churchill are demolished.

But back up and out the rabbit-hole: "data is"-ers, take heart! Not only does it make more sense, for the last half-century or so, to use data as a mass noun—and hence say "data is"—but you have over three centuries of English usage to back you up!

PS With apologies to Duke Ellington for the title, and to you for having to put up with an ad on that youtube video.

PPS Of course, my own inner snoot was interested to find this in the OED under datum:
The plural form data reflects the Latin plural; within English, this has given rise to a new singular and collective noun: see data n. and discussion at that entry.
Wait, don't you mean "mass noun"? I felt better about not have previously understood the difference.

PPPS I was delighted to find our old friend Cal Mooers in the OED with the first "mass noun" reference under "Computing" (2b).
1946   C. N. Mooers in Moore School Lect. (1985) 524   The data is stored in the memory in a systematic fashion with the points numbered in sequence.

PPPPS The dialog with Ben at the start was reconstructed from (my poor) memory, and might get edited if I can manage to communicate with him.

Saturday, March 12, 2011

Final comment on Errol Morris's series

I can’t resist taking one last stir at the dying embers here.  @LG #6: I join many others in thanking you for what seems to me to be a very able and cogent summary.  And as you say, it has been an interesting week, certainly very different from just reading a series in the newspaper.  @Skoorby #66: thank you for your very interesting description of the process of following this and other writings and discussion; I can relate to pretty much all of what you said, and I think it’s a valuable contribution as we adapt to this brave new world (in small letters, i.e. Shakespeare more than Huxley) of the web.

@Mike G (#78 from installment #4): dude, the question about whether the ashtray was thrown or not is certainly peripheral, and perhaps inconsequential to all but a handful of readers here.  If by “keeper of the faith” you mean that I believe we do better as a society when people engage with one another with honesty and humility, and try to understand the strengths and weaknesses of their own and their interlocutor’s positions, then I happily accept what I take as a high compliment.  “The problem” which I failed to spell out clearly enough is that the “dog fight” or entertainment model of how to deal with (much less settle) differences is continuing to eat away at our public discourse with disastrous consequences, and if we want something different we gotta start being the change we want in the world.  From that point of view, the question of whether the ashtray was thrown or not has more weight.  (Incidentally, “bloodline” is a little tricky here, because although my father might well have subscribed to that faith or something like it, there is no disputing that by its standards he could be a pretty significant sinner.)  On the other hand, if you mean “defender of Kuhnianism” (if that’s what it’s called), you don’t know me very well.  I’ll buy you a beer sometime.

@sba #67: I had not heard your remark that “those wounds which we do not somehow allow to transform us we will in some way continue to transmit.”  As a psychiatrist and psychotherapist I live with the truth of it every day and I am very happy to have these words to say it.

@Errol Morris: It has been fascinating to be part of the reading community as this unfolded, and for that you have my thanks.  I look forward to seeing more of your films.

Thursday, March 10, 2011

My comment on Errol Morris's 4th installment of "The Ashtray" ran over 5,000 characters

I finally hit my limit today, and I had some time, so I wrote a comment on Errol Morris's 4th installment of his  "Ashtray" series on the Times web site.  Comments are limited to 5,000 characters and the site cheerfully informed me that I had minus 1520 characters remaining.  So here it is in its entirety:

There is apparently yet another Thomas Kuhn here, one I don’t think he would have ever anticipated: the Thomas Kuhn who threw the ashtray.  Speaking as his son I have to say that, try as I might, I just can’t get myself to believe that he threw that ashtray.

I am not someone to take the ramparts to defend my father against every allegation.  He was a complicated guy and he did a lot of things.  Many were admirable.  Some were absolutely indefensible.

What we’re seeing here is not a rejection of his views; it’s a rejection of a caricature of his view.  He never believed in any sort of relativism that says there is no truth other than the point of view people take on it.  He believed very much in truth, but he also knew that understanding what it is to be true is much more complicated than it might first appear.

He certainly made mistakes, and I certainly heard him say things that I knew to be false but that he believed based on his own distorted point of view.  But I don’t believe I ever saw him say anything that he knew to be untrue.  He believed in truth, and he believed in truthfulness.  He had a bad temper at times.  He could be angry, he could yell, he could behave quite badly, but I never, ever saw him be violent, threaten violence, or throw anything, not even the pencil that was perpetually tucked behind his ear.  I’m prepared to believe quite a few unflattering things about him, and to say some myself (though mostly in private), but I just can’t get myself to believe that he threw that ashtray, and neither can anyone I’ve talked to who knew him well—among whom there is quite a spectrum of overall opinion about him.  (I should say here that, as a few commenters have noted, he could also be generous, helpful, understanding, encouraging, and more).

So I don’t believe he threw the ashtray.  I don’t know whether he threw it or not.  But I have a great deal of certainty that he either threw it or he didn’t.  I know he would join me in that certainty; clearly Mr. Morris would and so, I think, would the vast majority of readers of this series.

I also have a hard time believing that he would “forbid” someone to go to a lecture.  Again, there’s plenty that he could say in such a discussion that might not reflect well on him, but “forbidding” simply wasn’t his style.  On the other hand, I don’t have any difficulty imagining that he might have said something that could honestly but inaccurately be remembered that way.

But the ashtray is harder.  I can’t think of anything he might have said or that could honestly be remembered as throwing an ashtray.  Of course I have learned time and again that the strangeness of the world surpasses my imagination, but that knowledge doesn’t seem to help me here.

Which leaves me only two conceivable conclusions: that (A) my deep conviction about this man I knew intimately is simply wrong, or that (B) the “remembering” here is not honest.  (Of course it’s possible that (C) the truth of the situation is inconceivable to me, but thankfully I will be silent about that).

But why would someone fabricate something unflattering, when there are many truthful unflattering things one could say?  I had to wait to the third installment to get a clue.  Incidentally, references to the ashtray mostly started out with readers being appalled (obviously an appropriate reaction, if the story is true); by the third installment they were running more like “I can see why he threw the ashtray.”  Only one commenter did anything but take the story at face value (my sister, Sarah).  In the third installment, we hear extensively about the legend of Hippasus of Metapontum—a name I never heard my father utter, though he did talk extensively about the genesis of his various ideas, which confirms my belief that this story has nothing to do with why he used the word "incommensurable."  As many commenters have pointed out, he was in fact using the term for its mathematical meaning, which is quite apt: two line segments are incommensurable when there is no segment of which they are both integral multiples: in other words, there is no simple, single standard by which they can both be straightforwardly characterized.  (That does not mean they can’t be compared, and it is a crowning technical achievement  that Greek mathematicians found a rigorous way to do this.  Modern mathematics had to wait until the 19th century—long after Newton, Leibniz, Euler, and others—to assimilate this achievement).

But buried in this perhaps interesting but wholly irrelevant side-trip, we find the following: “One of the oddities of history is that legends often supersede facts. Historical evidence accumulates, monographs are written; but the number of popular accounts retelling the apocryphal story … proliferate. Why? Because we love to read about crisis and conflict. It’s drama. It makes a better story.”

We can say many things about my father, but he would never knowingly sacrifice the facts to “make a better story.”  In our cultural context, that job falls to fiction writers and non-documentary filmmakers.  It is disturbing to me to be drawn to the conclusion that that line is being crossed without acknowledgment.

Reading today’s installment, I have to wonder if I am witnessing an elaborate but subliminal staging of a purported “Thomas Kuhn’s nightmare” in which he screams that truth and reality exist only insofar as we say things are true or real, and the off-screen voice screams back, “In that case, I’m saying that you threw an ashtray at me, so you did!”  Even if my father did actually believe those things, it’s hard to fathom why this would be worth both the trouble and the risk involved, but since he didn’t the strangeness only deepens; he did have nightmares but that ain't one of them.  By this point things are sufficiently creepy that I find myself wishing that I could find a way to believe (A), that I am simply mistaken.

Ultimately, I don’t think it makes that much difference whether my father is remembered as an SOB or not, or perhaps as a different degree of SOB.  Ultimately, I don’t even know whether my father’s intellectual legacy, or Wittgenstein’s for that matter, makes that much difference.  What I do believe makes a difference to us, as a species, is whether we can find a way to disagree in some way other than caricaturing our antagonist’s position and then scorching the earth that the straw man stands on.   Whether you believe that “talking past each other” is a result of incommensurability or not, we are in the process of talking past each other to ecological catastrophe.  Even if the ashtray was thrown, this series seems to me to be part of the problem rather than part of the solution.