Advanced elegance

I think it's a natural human tendency to be awed by what we don't understand.

I know when I see some abstruse concept that is far beyond my grasp, I'm impressed not only by how complex the universe can be, but that there are people who can comprehend it.  I first ran into this in a big way when I was in college, and took a class called Classical Mechanics.  The topic was the mathematics of why and how objects move, how that motion affects other objects, and so on.

It was the first time in my life I had ever collided with something that regardless of my effort, I couldn't get.  The professor, Dr. Spross, was a very patient man, but his patience was up against a classical-mechanics-proof brain.  On the first exam, I scored a 19.

Percent.

And I'm convinced that he had dredged up the 19 points from somewhere so I wouldn't end up with a single-digit score.  I ended that class with a C-, which I think Dr. Spross gave me simply because he didn't want me back again the following semester, spending another four months ramming my poor physics-deficient head up against a metaphorical brick wall.

There's one memory that stands out from that experience, over forty years ago, besides the overwhelming frustration.  It was when Dr. Spross introduced the concept of the "Hamiltonian function," a mathematical framework for analyzing motion.  He seemed so excited about it.  It was, he said, an incredibly elegant way to consider velocity, acceleration, force, momentum, and so on.  So I thought, "Cool!  That sounds pretty interesting."

Following that cheerful thought was an hour and a half of thinking, "I have no fucking idea what any of this means."  It was completely opaque.  The worst part was that a number of my classmates were nodding their heads, writing stuff down, and seemed to get it with no problem.

So either I was the only complete dunderhead in the class, or they were just better at hiding their dismay than I was.

Anyhow, I think that was the moment I realized a career in research physics was not in the cards for me.

To this day, the "Hamiltonian function" remains something that in my mind symbolizes the Unknowable.  I have deep and abiding admiration for people for whom that concept makes sense (first and foremost, William Rowan Hamilton, who developed it).  And I'm sure it is elegant, just as Dr. Spross said.  But experiencing that elegance was (and probably still is) entirely beyond me.

It's this tendency to find what we can't understand awe-inspiring that has led to the idea of the God of the gaps -- in which gaps in our scientific knowledge are attributed to the incomprehensible hand of the divine.  Theologian Dietrich Bonhoeffer realized what the problem with this was, at least for people who are religious:

How wrong it is to use God as a stop-gap for the incompleteness of our knowledge.  If in fact the frontiers of knowledge are being pushed further and further back (and that is bound to be the case), then God is being pushed back with them, and is therefore continually in retreat.  We are to find God in what we know, not in what we don't know.

Anyhow, that was a long-winded preamble as an explanation of why all of this comes up in today's post.  I immediately thought of the awe-inspiring nature of what we don't understand when I read an article yesterday about two researchers at the University of Rochester, Tamar Friedmann and Carl Hagen, who found that a method for calculating the energy levels of a hydrogen atom generates the well-known number pi.

[Image is in the Public Domain]

It turns out to have something to do with a mathematical function called the Wallis product, which says that you can generate π/2 by a simple series of multiplications:

π/2 = (2/1) x (2/3) x (4/3) x (4/5) x (6/5) x (6/7) x (8/7) x (8/9)....

The pattern is that the numerators of the fractions are 2, 2, 4, 4, 6, 6, 8, 8... and the denominators 1, 3, 3, 5, 5, 7, 7, 9, 9...  And the cool thing is, the more terms you add, the closer you get to π/2.

Now, as for why this is so... well, I tried reading the explanation, and my eyes started spinning.  And I've taken lots of math courses, including calculus and differential equations, and like I said earlier, I majored in physics (as much of a mistake as that turned out to be).  But when I took a look at the paper about the energy levels of hydrogen and the Wallis product and gamma functions, I almost could hear Dr. Spross's voice, explaining it in a tone implying it would be immediately clear to a small child, or even an unusually intelligent dog.

And all of those feelings from Classical Mechanics came bubbling up to the surface.

So I'm left with being a little in awe about it all.  Somehow, even though I have no real understanding of why, the same number that I learned about in geometry class as the ratio between a circle's circumference and its diameter shows up in the energy levels of hydrogen atoms.  Predictably, I'm not inclined to attribute such correspondences to the hand of the divine, but they do seem to be (in Dr. Spross's words) "elegant."  And even if I never get much beyond that, I can still appreciate the minds of the people who can.

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A slice of pi

Note to my loyal readers: This will be my last post before a two-week break so I can go to my publisher's annual writers' retreat in the Ozarks.  Please keep sending links & ideas -- I'll be right back in the saddle when I return.  Look for the next Skeptophilia post on Monday, August 6!

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Sometimes you have to admire the woo-woos' dogged determination to fashion the universe into their own bizarre version of reality.

Most of us, I'd like to think, just see what we want to see and believe what we want to believe, and don't make such a big deal out of it.  If we want to believe in a Higher Power That Guides Everything, we do, and don't spend endless hours crafting abstruse proofs of the conjecture.  We're content to have a beer, watch a hockey game, and let god have some much-needed quiet time.

There are a few people, however, who just aren't content if they're not actively beating the matter into submission.  Such a person is Marty Leeds, Wisconsin-born writer, mystic, philosopher, and the origin of dozens of highly entertaining YouTube videos.

Just yesterday, I was sent a link to one of Leeds' creations, entitled, "Number Magic: Gematria."  The link was accompanied by a message stating, and I quote: "Words cannot describe the level of derp in this video."  So of course I had to watch it.  And I wasn't disappointed.

Turns out he has an obsession with the number pi.  Okay, it's a pretty cool number,  being transcendental and all, but he thinks there's... more than that.  Way more, as it turns out.

What you get when you divide the circumference of an apple by its diameter.  [Image licensed under the Creative Commons Matman from Lublin, Apple pie Pi Day 2011, CC BY-SA 3.0]

If you're unwilling to sacrifice an hour and a half of your precious time (and to be honest, I made it through about twenty minutes, and just skimmed the rest), and countless innocent cells in your prefrontal cortex that will die in agony, allow me to present to you the main points of Leeds' argument.

1. There's this thing called gematria that was made up a while back by some Hebrew mystics who had overactive imaginations and way too much free time.   The idea behind gematria is that each letter in the alphabet (whether Hebrew, English, or other) is assigned a number, and when you add up the numbers for a word or name, you get a number that "means something."
2. You get to decide what the numbers mean.
3. If two words add up to the same thing, they are mystically linked.  Leeds uses a form of gematria which takes the English alphabet, splits it into two lists of thirteen letters each (A-M, and N-Z), and numbers each list from 1 through 7 and then back down to 1.  So my first name, Gordon, would be 7+2+5+4+2+1 = 21.  "Sharp" is 6+6+1+5+3, which also adds up to 21.  So you can see that thus far, we have a pretty persuasive theory here.
4. Leeds then does a gematria addition for four words or phrases.  We have "man" = 3, "woman" = 9, "Christian" = 39, and "The Holy Spirit" = 61.  Note that he had to add a "the" to the last one to make it work out the way he wanted.
5. So, let's look at the first thirteen digits of pi.  He picked thirteen because we had split the alphabet into two groups of thirteen letters each, which seems like impeccable logic to me, given the obvious connection between pi and the English alphabet.  Anyhow, we have 3.141592653589. It starts with 3 and ends with 9 -- giving you "39."  So right away, we can see that there's something wonderfully Christian about pi, not to mention having a man on one end and a woman on the other.  Also, 3+9 = 12, and 3x9 = 27, and 12+27 = 39. So you get your 3 and 9 back, so "man + woman" + "man x woman" = "Christian."  Or something like that.
6. Take the middle number in the sequence (2) and the two on either side (9 and 6).  Why?   Because tridents, that's why.  Stop asking questions.
7. If you multiply 9x2x6, you get 108, which is a very holy and important number.  Myself, I just thought it was the most convenient way of getting from 107 to 109, but what do I know?  But the Hindus liked the number 108, and plus, it's the number of stitches on a baseball, so there you are.
8. Now, take the remaining digits of pi, and basically draw a menorah under them.  You then put them together in pairs, flip 'em around, and add 'em together.  I really don't want to go into all of how he does that, because my cortical neurons are already whimpering for mercy, so you'll just have to either watch the video or else just accept on faith that somehow all of the numbers and flipped numbers and all add up to 352.  Then, you add that to the 2 and 6 from the trident bit, and you get 360, which is the number of degrees in a circle.  Get it?  Circle?  Pi?  Are you blown away?  (Okay, he left out the 9. But still.)
9. If you multiply the first through eighth digits of pi, you get 6,480.  If you multiply the eighth through the thirteenth digits, you get 32,400. Subtract them, and you get 25,920, which he says is the number of years for the precession of the Earth's axis to complete one rotation.  Except that according to the Cornell University Astronomy Department's webpage on the precession of the Earth's axis, the length of the precession of the Earth is said to be "about 26,000 years" -- the imprecision being because a motion that slow is almost impossible to measure accurately.   But I think we call all agree that since we're using gematria as our jumping-off point, being off by eighty years or so is plenty accurate enough.
10. Of course, like any good performer, he saves his most amazing bit for the end, wherein we find out that the first thirteen digits of pi add up to 61, which you will recall is the number of "The Holy Spirit."  So pi "encodes" (his word) The Holy Spirit and the precession of the equinoxes.
11. Therefore god.  Q.E.D.

Well, I hope you've enjoyed our little ramble through woo-woo arithmetic. Me, I'm planning on watching some of Leeds' other videos when I have the time.  (Two especially fascinating-sounding ones are "Pizzagate, Symbolism, and Secret Societies" and "Flat Earth Implications, NASA Lies, and Gematria."  Of course he's a Flat Earther.  You're surprised by this?)  However, I think next time I won't launch into this without something to insulate my poor brain against further damage.  I'm thinking that a double scotch might do the trick.

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This week's Skeptophilia book recommendation is a must-read for anyone concerned about the current state of the world's environment.  The Sixth Extinction, by Elizabeth Kolbert, is a retrospective of the five great extinction events the Earth has experienced -- the largest of which, the Permian-Triassic extinction of 252 million years ago, wiped out 95% of the species on Earth.  Kolbert makes a persuasive, if devastating, argument; that we are currently in the middle of a sixth mass extinction -- this one caused exclusively by the activities of humans.  It's a fascinating, alarming, and absolutely essential read.  [If you purchase the book from Amazon using the image/link below, part of the proceeds goes to supporting Skeptophilia!]

Advanced elegance

I think it's a natural human tendency to be awed by what we don't understand.

I know when I see some abstruse concept that is far beyond my grasp, I'm impressed not only by how complex the universe can be, but that there are people who can comprehend it.  I first ran into this in a big way when I was in college, and took a class called Classical Mechanics.  The topic was why and how objects move, how that motion affects other objects, and so on.

It was the first time in my life I had ever collided with something that regardless of my effort, I couldn't get.  The professor, Dr. Spross, was a very patient man, but his patience was up against a classical-mechanics-proof brain.  On the first exam, I scored a 19.

Percent.

And I'm convinced that he had dredged up the 19 points from somewhere so I wouldn't end up with a single-digit score. I ended that class with a C-, which I think Dr. Spross gave me simply because he didn't want me back again the following semester, spending another four months ramming my poor physics-deficient head up against a metaphorical brick wall.

There's one memory that stands out from that experience, nearly forty years ago, besides the overwhelming frustration.  It was when Dr. Spross introduced the concept of the "Hamiltonian function," a mathematical framework for analyzing motion.  He seemed so excited about it.  It was, he said, an incredibly elegant way to consider velocity, acceleration, force, momentum, and so on.  So I thought, "Cool!  That sounds pretty interesting."

Following that cheerful thought was an hour and a half of thinking, "I have no fucking idea what any of this means."  It was completely opaque.  The worst part was that a number of my classmates were nodding their heads, writing stuff down, and seemed to get it with no problem.

So I was either the only dumb one in the class, or they were just better at hiding their dismay than I was.

Anyhow, I think that was the moment I realized a career in research physics was not in the cards for me.

To this day, the "Hamiltonian function" remains something that in my mind symbolizes the Unknowable.  I have deep and abiding admiration for people for whom that concept makes sense (first and foremost, William Rowan Hamilton, who developed it).  And I'm sure it is elegant, just as Dr. Spross said.  But experiencing that elegance was (and probably still is) entirely beyond me.

It's this tendency to find what we can't understand awe-inspiring that has led to the idea of the god of the gaps -- in which gaps in our scientific knowledge are attributed to the incomprehensible hand of the divine.  Theologian Dietrich Bonhoeffer realized what the problem with this was, at least for people who are religious:

How wrong it is to use God as a stop-gap for the incompleteness of our knowledge.  If in fact the frontiers of knowledge are being pushed further and further back (and that is bound to be the case), then God is being pushed back with them, and is therefore continually in retreat.  We are to find God in what we know, not in what we don't know.

Anyhow, that was a long-winded preamble as an explanation of why all of this comes up in today's post.  I immediately thought of the awe-inspiring nature of what we don't understand when I read an article yesterday about two researchers at the University of Rochester, Tamar Friedmann and Carl Hagen, who found that a method for calculating the energy levels of a hydrogen atom generates the well-known number pi.

[image courtesy of the Wikimedia Commons]

It turns out to have something to do with a mathematical function called the Wallis product, which says that you can generate π/2 by a simple series of multiplications:

π/2 = (2/1) x (2/3) x (4/3) x (4/5) x (6/5) x (6/7) x (8/7) x (8/9)....

The pattern is that the numerators of the fractions are 2, 2, 4, 4, 6, 6, 8, 8... and the denominators 1, 3, 3, 5, 5, 7, 7, 9, 9...  And the cool thing is, the more terms you add, the closer you get to π/2.

Now, as for why this is so... well, I tried reading the explanation, and my eyes started spinning.  And I've taken lots of math courses, including calculus and differential equations, and like I said earlier, I majored in physics (as much of a mistake as that turned out to be).  But when I took a look at the paper about the energy levels of hydrogen and the Wallis product and gamma functions, I almost could hear Dr. Spross's voice, explaining it in a tone that implies that it would be immediately clear to a small child, or even an unusually intelligent dog.

And all of those feelings from Classical Mechanics came bubbling up to the surface.

So I'm left with being a little in awe about it all.  Somehow, even though I have no real understanding of why, the same number that I learned about in geometry class as the ratio between a circle's circumference and its diameter shows up in the energy levels of hydrogen atoms.  Predictably, I'm not inclined to attribute such correspondences to the hand of the divine, but I do think they're (in Dr. Spross's words) "elegant."  And even if I never get much beyond that, I can still appreciate the minds of the people who can.

Pi à la mode

Well, it's Pi Day -- 3/14 -- at least if you live in the United States, where we put the month first.  Otherwise, it's 14/3, which is much less momentous.  But we can do what we want, rest of the world be damned, because, you know, we're 'Murica and all.

But if you accept that the month comes first, this is actually a really special Pi Day, because at a little before 9:30 AM, it will be 3/14/15 9:26:53, which is pretty cool.  And pi, you have to admit, is an awesome number.  It's the first number most of us run into in school that's irrational -- that cannot be expressed as the ratio of two integers, and whose decimal expansion never terminates or repeats.

Pi is formally defined as the ratio between the circumference of a circle and its diameter.  Its approximate value was known to Chinese, Indian, and Greek mathematicians as far back as 2,500 years ago; a more precise method for determining the value of pi was discovered by the extraordinary Indian mathematician Madhava of Sangamagrama in the 14th century, and rediscovered by Gottfried Leibniz of Germany three hundred years later.  Both of them had uncovered the peculiar relationship now called the Madhava-Leibniz formula:

This was just the first of many appearances of pi in unusual places -- it's a number that crops up over and over in physics, for example.

All of which has made it important that we know pi to a great degree of accuracy.  Computers have now been used to calculate pi to 13.3 trillion digits, which is pretty damned impressive, and which makes what a wingnut who calls himself "Jain108" is claiming even crazier.

Jain108 says we've got pi wrong. Pi, he says, is actually 3.144.  Give or take.  Which, I think we can all agree, is far less than accurate to within 13.3 trillion digits. But this doesn't dissuade Jain108 at all.   He says if we just accept his value, we'll find our pipes fitting together better:

(A)n ex-Engineer from NASA, "Smokey” admitted (via email) that when he was making metal cylinders for this same Mooncraft, finished parts just did not fit perfectly, so an adjusted value for Pi was also implemented.  At the time, he thought nothing about it, but after reading an internet article called The True Value of Pi, by Jain 108, he made contact.

Because that couldn't have anything to do with inaccuracies of construction, or anything.

But that's not his only argument.  We should accept his higher value for pi because, I dunno, mysticism or something:

For reasons that will be soon explained, traditional Pi is deficient because historically it has awkwardly used logical straight lines to measure illogical curvature.  Thus, by using the highest level of mathematics known as Intuitive Maths, the True Value of Pi must be a bit more than anticipated to compensate for the mysterious "Area Under The Curve”.  When this is done, the value, currently known as JainPi, = 3.144… can be derived, by knowing the precise Height of the Cheops Pyramid which is based on the Divine Phi Proportion (1.618…).  Instead of setting our diameter at 1 unit or 1 square, something magical happens when we set the diameter at the diagonal length of a Double Square = 2.236… which is the Square Root of 5 (meaning 2.236… x 2.236… = 5). This is the critical part of the formula that derives Phi (1+√5)÷2, and was used by ancient vedic seers as their starting point to construct their most important diagram or ‘Yantra’ or power-art called the Sri Yantra.  With a Root 5 diameter, the translation of the Phi’s formula into a geometric construct derives the royal Maltese Cross symbol, concluding that Phi is Pi, that Phi generates Pi, and that Pi must be derived with a knowledge of the Harmonics of Phi.  When this is understood and utilized, we will collectively enter into a veritable Space Age.

Oh! Okay! I mean, my only question would be, "What?"  This is about as lucid as when people like Diane Tessman start to babble about using quantum physics to explain astrology.

But I thought I'd soldier on, so I kept reading.  And then I ran into a section where Jain108 claims that his ideas are proven because Queen Elizabeth and "Pope Benedictine XVI" both have worn robes with Maltese crosses, and something about the Ark of the Covenant.  So despite my earlier objections, I think we can all agree that what we have here is an "air-tight case."

Other fun stuff abounds in this website, such as when he calls 3.144 "the PIN number of the universe."  But my favorite part of what Jain108 calls "3.144ology" is when he starts talking about a guy named Billy Meier, who is "one of the most controversial members of the UFO community:"

Billy Meier is a Swiss farmer in contact with Plearjen people (humans like you and me) since the age of 6.  In Contact 251 it is stated :
"In the process they will discover that the base for pi was miscalculated.  By eliminating the error in pi, and correcting future computations based on pi, scientists and their amazing, highly developed technology will have the capability to make unimaginable energies accessible to the people of Earth”.

So if the "Plearjen people" are "humans like you and me," what makes them "Plearjen people?"  And how could changing the value of pi make us access "unimaginable energies?"

The only thing in the way is the pride and arrogance of mathematicians who cannot conceive of such a notion that Pi could be anything else than what their books have instructed them to believe.  It is their lack of understanding of Fractal Harmonics (based on the cascading proportions of the Fibonacci Sequence) which hinders them to comprehend the elegance of the geometric solution known as the Fairywand Method identical to Saint Germain’s Maltese Cross.  Fractal Harmonics allows us to zoom forever into the Area Under The Curve and detect more infinitesimal Areas Under The Curve, concluding that old Pi is only an approximation, a limit of millions or billions of straight lines, that it is deficient, that the True Value of Pi must be a fraction more than we estimated, and that it must be based on Phi, The Golden Mean Harmonics.  Thus 4 divided by the Square Root of Phi (1.272…) gives the correct frequency of 3.144… its really very simple. 

True Pi is like a crystalline seal or a Recoding ready to unlock, to override the old genetic imprint of degeneration, of lack and sickness.  It will, in effect, spark a Process of Purification.

The "Fairywand Method" allows you to "zoom forever into the Area Under The Curve."  So there you are, then.

In any case, I'm sorry if you thought you were going to be able to celebrate the extra-special Pi Day today.  Since Jain108's version of pi is, more exactly, 3.14460551..., you'll have to wait until March 14, 2046, at five minutes and 51 seconds past midnight, to be able to celebrate your Pi Moment.

I hope the thousands of pies people have purchased to celebrate the occasion will keep until then.

A slice of pi

Sometimes you have to admire the woo-woos' dogged determination to fashion the universe into their own bizarre version of reality.

Most of us, I'd like to think, just see what we want to see and believe what we want to believe, and don't make such a big deal out of it.  If we want to believe in a Higher Power That Guides Everything, we do, and don't spend endless hours crafting abstruse proofs of the conjecture.  We're content to have a beer, watch a hockey game, and let god have some much-needed quiet time.

There are a few people, however, who just aren't content if they're not actively beating the matter into submission.  Such a person is Marty Leeds, Wisconsin-born writer, mystic, philosopher, and the origin of dozens of highly entertaining YouTube videos.

Just yesterday, I was sent a link to one of Leeds' creations, entitled, "The Holy Spirit, Pi, and the English Alphabet."  The link was accompanied by a message stating, and I quote: "Words cannot describe the level of derp in this video."  So of course I had to watch it.  And I wasn't disappointed.

If you're unwilling to sacrifice ten minutes of your precious time, and countless innocent cells in your prefrontal cortex that will die in agony, allow me to present to you the main points of Leeds' argument.

1)  There's this thing called gematria that was made up a while back by some Hebrew mystics who had overactive imaginations and too much free time.  The idea behind gematria is that each letter in the alphabet (whether Hebrew, English, or other) is assigned a number, and when you add up the numbers for a word or name, you get a number that "means something."

2)  You get to decide what the numbers mean.

3)  If two words add up to the same thing, they are mystically linked.  Leeds uses a form of gematria which takes the English alphabet, splits it into two lists of thirteen letters each (A-M, and N-Z), and numbers each list from 1 through 7 and then back down to 1.  So my first name, Gordon, would be 7+2+5+4+2+1 = 21.  "Sharp" is 6+6+1+5+3, which also adds up to 21.  So you can see that thus far, we have a pretty persuasive theory here.

4)  Leeds then does a gematria addition for four words or phrases.  We have "man" = 3, "woman" = 9, "Christian" = 39, and "The Holy Spirit" = 61.  Note that he had to add a "the" to the last one to make it work out the way he wanted.

5)  So, let's look at the first thirteen digits of pi.  He picked thirteen because we had split the alphabet into two groups of thirteen letters each, which seems like impeccable logic to me, given the obvious connection between pi and the English alphabet.  So, we have 3.141592653589.  It starts with 3 and ends with 9 -- giving you "39."  So right away, we can see that there's something wonderfully Christian about pi, not to mention having a man on one end and a woman on the other.  Also, 3+9 = 12, and 3x9 = 27, and 12+27 = 39.  So you get your 3 and 9 back, so "man + woman" + "man x woman" = "Christian."  Or something like that.

6)  Take the middle number in the sequence (2) and the two on either side (9 and 6).  Why?  Because tridents, that's why.  Stop asking questions.

7)  If you multiply 9x2x6, you get 108, which is a very holy and important number.  Myself, I just thought it was the most convenient way of getting from 107 to 109, but what do I know?  But the Hindus liked the number 108, and plus, it's the number of stitches on a baseball, so there you are.

8)  Now, take the remaining digits of pi, and basically draw a menorah under them.  You then put them together in pairs, flip 'em around, and add 'em together.  I really don't want to go into all of how he does that, because my cortical neurons are already whimpering for mercy, so you'll just have to either watch the video or else just accept on faith that somehow all of the numbers and flipped numbers and all add up to 352.  Then, you add that to the 2 and 6 from the trident bit, and you get 360, which is the number of degrees in a circle.  Get it?  Circle?  Pi?  Are you blown away?  (Okay, he left out the 9.  But still.)

9)  If you multiply the first through eighth digits of pi, you get 6,480.  If you multiply the eighth through the thirteenth digits, you get 32,400.  Subtract them, and you get 25,920, which he says is the number of years for the precession of the Earth's axis to complete one rotation.  Except that according to the Cornell University Astronomy Department's webpage on the precession of the Earth's axis, the length of the precession of the Earth is said to be "about 26,000 years" -- the imprecision being because a motion that slow is almost impossible to measure accurately.  But I think we call all agree that since we're using gematria as our jumping-off point, being off by eighty years or so is plenty accurate enough.

10)  Of course, like any good performer, he saves his most amazing bit for the end, wherein we find out that the first thirteen digits of pi add up to 61, which you will recall is the number of "The Holy Spirit."  So pi "encodes" (his word) The Holy Spirit and the precession of the equinoxes.

11)  Therefore god.  Q.E.D.


Well, I  hope you've enjoyed our little ramble through woo-woo arithmetic.  Me, I'm planning on watching some of Leeds' other videos when I have the time (two especially fascinating-sounding ones are "The Isis and Osiris Myth" and "The Holy 108").  However, I think next time I won't launch into this without something to insulate my poor brain against further damage.  I'm thinking that a double scotch might do the trick.