Doubt and Belief

Doubt is most associated with science, and with good reason. Doubt is how the process works. No matter how slick your ideas are they have to be testable. Skepticism is a healthy thing. We live in a sea of audio-visual garbage and absorb it through the pores daily. Having a strong doubting sense helps. You have to sort out the knowledge from all the information, the signal from all the noise. Advertising and politics depend on gullibility. They know Barnum was right and they peddle their un-testable stuff shamelessly and they tell you what you want to hear. TV evangelists and mega-churches are the best at it. You relax your doubting muscle and they get you in an arm bar and you have to tap out. Next thing you know it you’re chanting Hare Krishna.

So you keep that doubting muscle firmed up and “play the doubting game” (as Peter Elbow would say). But science isn’t so simple and one-sided. You can’t have doubt without belief. When a new idea comes along, especially if it bears fruit right away, scientists can jump on the believe-train just as easily as anyone else. Think about the Big Bang Theory. Not a very satisfying notion from either a philosophical or common-sense point of view. If God made the universe, then who made God? When did it all start and how? If the Big Bang made the universe, then what came before? When did it all start and how? Neither idea gets you anywhere.

But believing in the Big Bang, or at least suspending doubt and dis-belief, is enormously useful. It’s a testable notion, this idea that there was a gigantic explosion. Because such a thing, applying the known laws of physics, would presumably have observable consequences. The so-called “cosmic microwave background” is such a thing. It’s real, measurable radio astronomy and it could be the remnants of this immense ancient event. The theory has value as an organizing idea, a way to tie together observations of unusual and difficult phenomena. At some future date, I sincerely hope, schoolkids will sit around giggling about the silly concepts that their ancestors used to describe the universe back in the day. But a model, no matter how crude, that accounts for things that are happening right now is too valuable to discard until something better comes along. What’s better? One that survives more experimental tests and describes a greater swath of the known universe.

I don’t mean to make it seem that belief is just a hat to put on when you are at work. Scientists are just like everybody else, they believe in all kinds of nutty stuff. They have well-developed doubting skills, just like ballplayers have well-developed hand-eye skills, but these are not unique to them. We can all activate the doubting schemes we have when we need to. In fact it’s so much more necessary today with our 24/7 saturation culture, but I said that already.

Scholarship and other intellectual endeavors suffer from too much doubt. It’s the nature of these things to probe and ask questions, to think critically, to analyze and to de-construct. These are essential academic skills. In the oneupmanship  culture of academic publishing you can go far by how cleverly you can skewer another’s work. Being a critic is easier than being an original. But that’s only part of thinking and learning. The other part is belief. Now no one likes a zealot, especially a reformed one. And the earnestness of true believers is wearisome. Skeptics need to furrow their brows and there’s nothing wrong with that. But they have to “play the believing game” too.

So how do you do that? How do you embrace something without losing your footing? For me I read less and write more. I talk less and listen more. Communication is all about belief. You believe in the meaning you ascribe to your scribbles and you also believe that I can find the same meaning in them. I’m here creating bits to send into cyberspace because I believe a consciousness is on the other end assigning meaning to them.

I’m a skeptic by nature. I’m the doubting-est of the doubters, man. Thomas the Apostle was my favorite of The Twelve because he had to see for himself. I’m like that. I don’t trust it unless it’s the source and even then I have my doubts. But that’s not enough to live on. You have to balance doubt with belief. They aren’t really contradictory, rather they are complementary. And why worry about a few contradictions? I used to think that consistency was an important thing. Salt-water taffy has consistency. That’s why we buy it and eat it. If I’m selling taffy, I’ll worry about consistency. Real life is nothing but confusion and contradiction, it’s only in art, music, and fiction that things can make sense.

Maybe that’s it. Creating things requires belief. So when in doubt, paint. Or sing. Or dance. Or carve wood. Or cook. Or build something. When I get up at the crack of dawn so I can be fit and fed and ready to ride my mountain bike up a steep goddamn hill I gotta believe. I can’t do it otherwise. It’s way nicer in bed. But once I get going and overcome my inertia I find it to be one of the best things I do. It makes me feel great.

You play the doubting game to survive, you play the believing game to thrive. Oh, is that clever, or what? I think I might have the stuff, finally, to fulfill one of my dreams: writing a self-help best-seller! With that and eumentics I’m on my way.


My late friend Mr B was a pharmacist’s mate during WWII. Working at a stateside base he called his job “chancre mechanic.” Which means he supplied gonorrheal servicemen with penicillin. The rest of the time he was plying pilots with amphetamines. Hey, there was a war on. These guys were saving the world. What’s a few uppers in the grand scheme of things? Once we stopped being at war, amphetamines became just another street drug instead of a pharmaceutical solution to a vexing problem (long flights by worn-out flyers).

People seek out intoxication like they seek food or sex. At least they seek out altered states. Even the mystic, normally abstemious, is not content with everyday life and seeks a kind of meta-reality, that above-the-plane-of-existence sort of thing. In the 60s it became fashionable to eschew the lengthy discipline and self-abnegation of the searcher and simply eat acid. LSD is a great experience, I have to vouch for that, but I had no Saul-on-the-road-to-Damascus moments. Not that I was really looking for any. Seems like if you want to trip out on drugs, you should trip out on drugs. If you want to trip out on god, you should trip out on god. I hear the desert, prayer, and fasting work well for that.

Among other teetotalers and drug-free types are extreme athletes. They get high jumping off mountains or whatever. They tune their minds to achieve complete immersion in their activity. I get the same kind of buzz—much less intense, I suppose—when I’m skiing and conditions are perfect and I’m floating effortlessly in the snow and my only thoughts are the sensations of the turns and my breathing. I’m not really conscious of the rest of reality, just fully in the moment and enveloped in the experience. It’s a fleeting thing and I see why people chase it, it’s much like a drug high.

I think we ought to stop worrying about intoxication. In fact, I don’t like the word. What’s so toxic about it? There was a time when businessmen had three-martini lunches and then went back to work. (President Ford, a politically maladroit but otherwise intelligent man, once said “The three-martini lunch is the epitome of American efficiency. Where else can you get an earful, a bellyful and a snootful at the same time?” Pretty damn funny for a guy who often came across as clueless.) Sounds like a grand tradition that ought to be revived. Of course martinis are a hell of a lot fancier and more expensive today than they were then, what with all this small-batch micro-distilling going on. Everyone wants the coolest new spirit. No one in these high-dollar crowds worries about alcoholism. That’s strictly working-class woes. The mob can’t handle their Bud Light and Jack and they wreck their cars and beat their wives so we have to pass draconian laws to regulate the poor bastards. We retired civil servants can rest easy. Of course, I walk to my local watering hole so I don’t have to drive back. I learned that trick in Ireland where a town of Yreka’s size (7500 souls) would have 50 pubs.

Now that marijuana is on the brink of legalization in California we will see a whole new regulatory scheme. That ought to be fun. I remember the bleachers at Candlestick Park in the 70s where dope-smokers would be oohing and aahing over foul balls and getting booted for passing a joint while beer drinkers would be starting fights and the vendors would be racing over to sell them more Old Milwaukee. Hilarious! Seems like a weed section at sporting events might be a good idea. A pleasantly stoned crowd would not curse or throw things or run on the field. Hey, they might even cheer for the other team if they saw something cool. Imagine what FIFA could do with cannabis treats at some of these over-heated soccer matches where they have to police for hooliganism and keep the sections separated to avoid violence. (Sporting crowds in the States are milquetoasts compared to Europeans and South Americans.) Seems like there’s a pharmaceutical solution waiting to be tried.

Now there are many, the religiously scrupulous for example, who avoid poisoning their temples (er, bodies) because it’s seen as an affront to god. After all, we are made in the image of god. But all the intoxicants in the world were made by god, too. So that’s a weak argument by my reckoning. But to each his own, right? No doubt our pharmacopeia can be poisonous, but it’s important to remember the most important rule in medicine (teachers always double-down on ‘important’):

It’s the dose that makes the poison. (–Theophrastus von Hohenheim, aka Paracelsus)

Yup. A little of something may be grand. A lot may not be. But it’s also important to remember another important lesson from life:

Everything in moderation, including moderation. (–Oscar Wilde)

I suppose now that I’m retired the orderly and the business-like and the scheduled are giving way to the chaotic and the impractical and the spontaneous. (I just made adjectives into things. You can only lose points for grammar violations in school, in real life it is called ‘creativity.’) The relentless ticking of the 9-to-5 capitalist clock will beat us all into submission eventually so we might as well dull the pain with a few mood-enhancing substances. Like I said I don’t like these things being described as intoxicants and I think we need some new words. Nutrients? God’s gifts? Wellness-icants? Help me out here.


In northern latitudes the summer solstice is seen as the mid-point of the summer season and not the beginning. Day lengths have been increasing since the winter solstice and from this point forward day lengths will start to decrease. Around the vernal equinox the sun is out for half the day and the daylight gobbles up more and more of the night until around this time when we reach our maximum day lengths. The weather typically lags the calendar. The hottest days here are usually in August. Although the amount of time the sun spends up and out each day is less and less it doesn’t feel like it. To me the summer solstice is the start of my nocturnal season.

I don’t do well in the heat. I try to get up early and get some things done in the morning hours and then hunker down to avoid the solar onslaught to come. Then when the sun moves far enough down to cast shadows I’m ready to venture out. By evening it is civilized again and I can live like a normal human. The trick is to stay up late and enjoy the hours up to midnight and still be ready to get up at the crack of dawn. If I can manage an afternoon nap then I can make it work. Once I’ve had lunch I retreat inside my turtle shell and don’t make an appearance until the cocktail hour. Each day I praise and thank the people who invented air-conditioning.

But this nocturnal thing is not so easy. People aren’t nocturnal. They work all day. They screw up the clocks in the summer in order to get more sunlight, not less. They sleep at night. I think we need to flip it around in these hot places and stay inside during the day and sleep and go out and play (or work) at night. I live in a hot, sunny place. Lots of places have pleasant summer climates. This place doesn’t. It’s a goddamn inferno. If you spend time in the desert you don’t see many critters out during the day. They are smarter than that. They use the early morning and the evening twilight or are completely nocturnal. They avoid the sun’s harsh rays and the eyeball-shriveling heat.

A nocturnal way of life has some drawbacks. Happy Hour, for one. Bars don’t have Happy Hour at 10 pm. Usually around 4 pm, plus or minus. That’s the time I need to be in my hidey-hole. But the lure of Happy Hour forces me out and I trudge wearily through the pavement-melting incandescence so I can get my discounted pints and check in with the other regulars.

I’m whining. I know it. Heat makes me a whiner. I’m usually not much of a complainer. I try have a good attitude. But summer does this to me, man. It’s not my fault!  I know this nocturnal notion of mine is mostly wishful thinking. I do change it up in the summer and try to avoid the heat and try to enjoy the beautiful nights and early mornings. But life gets in the way. The tyranny of 9-to-5 and that silly Darkness Squandering Time reign over all. If I went the way of the scorpions and barn owls I’d probably lose most of my friends. “Sure, I’d love to come over, how about 9 pm? . . . What? 2 pm? Are you crazy? It’ll be 100 degrees!” That sort of thing doesn’t work so well in this breakfast-at-8 and dinner-at-6 world of ours.

I was on the Sonoma coast last weekend, at Jenner-by-the-sea and Portuguese Beach. The onshore breeze was stiff and cold and the temperatures were in the 60s. It was divine. The big bad ocean out there was keeping me and the tiny sliver of coastline marvelously chilled. Inland there was a heat wave across the state and in the valleys the mercury hit triple digits. I enjoy that maritime climate, but I enjoy even more living in a remote part of the state that’s free of crowds and traffic and urban blight. And I do love to ski so being close to ski parks and snowy mountains is a worthwhile trade-off for the summer torture.

Stay cool out there. And stay hydrated, too.


It’s estimated that 1 in 9 people worldwide don’t have access to a clean, safe source of water. That’s almost 800 million people or a little more than the population of Europe. I got to thinking about that because we had a mini-plumbing crisis yesterday. Our water line from the street sprouted a small leak. We were able to get it patched but we will probably have to replace the line sooner rather than later as it is old and corroded. Nonetheless I take it for granted that I will have a steady supply of fresh, clean water. We have superb municipal water, in fact many many many communities in the United States have superb municipal water. One could argue that Americans have, on the whole, the best water in the world.

It’s not just about the water. It’s about the distribution systems and the schemes for filtering and treating the water as well. Most Americans enjoy these things as part of ordinary everyday life. The recent troubles in Flint, Michigan over contaminated water bear this out: that was a national news story and a political scandal. We turn on our taps and our spigots and our hydrants and let the water flow freely. This is DRINKING water, mind you. We have so much of it that we pour drinking water on our lawns. We hose off our driveways. We wash our cars and our clothes with it. Hundreds of millions in the world have next to nothing and we splash our supplies around like they’ll last forever.

Think about bottled water for a minute. The stuff is an environmental catastrophe, of course. The plastic packaging is a nightmare, not only from the manufacturing side (the plastics are petroleum-based) but the obvious disposal problem. American consumers have so much water that they can be fussy about it and buy it by the case in the supermarket. Nestlé will sell you 12 Liters (24 0.5 L bottles) for about 12 bucks or roughly 3 cents an ounce. That’s about the cost of gasoline! ($4.00 gallon ÷ 128 oz/gal = $0.03125 or ≅ 3¢) How stupid is that? Do the world a favor and drink water from the tap! And if you don’t like that then drink draft beer, which is 95% water, and is packaged in re-usable kegs. Even bottled beer is good because glass is a much better product than polyethylene terephthalate (PET). Not that polyesters aren’t useful (I ferment my beer in PET carboys, for example), and many are recyclable, but I think we all know that glass is better.

But that’s just an aside. It really points out the absurdity of our marketing culture. Just watch TV and see those idiotic commercials about people drinking water out of plastic bottles with silly labels on them smiling and running and playing as if the stuff was magic. Water IS magic, but we don’t need Nestlé or other corporate fucks to tell us that. That’s why communities have built water supply systems for their residents. You can’t live more than three days or so without water so if you want to live someplace permanently you better take care of your water sources. Only the über-rich will be able to get Icelandic glacial meltwater or what-have-you in the near future so it behooves us to conserve our local watersheds.

Oh, water is water, just in case you were wondering. Water has stuff dissolved in it. They call water the ‘universal solvent’ in case you didn’t know. There is nothing inherently healthier about one kind of water over another. This is assuming that known poisons like lead or arsenic are not present, or micro-organisms that cause disease. But water in a plastic bottle is not superior to water from the tap, no matter what kind of propaganda the bottle-sellers throw at you. They want your money and they will tell you what you want to hear so that you will give it to them. Chemical and nutritional mis-information is their stock-in-trade.

I’m the ruminative type, not to be confused with ruminants. Those are critters (like cows) that can digest grass and turn it into protein. You can’t do that, nor can I. But we can ruminate, or think things over like cows re-chewing their cud. In psychology rumination has a negative connotation, it means to re-think things to the point of paralysis-by-analysis, or to re-visit upsetting events. But for me ruminating is normal, the mental motor hums along and I turn things over in my mind looking for new angles or insights. Sadly, those are rare.

The water line leak got me ruminating about those poor folks for whom water is the central problem of their existence. We don’t worry that we won’t get enough to drink. We worry about imaginary problems with our water. They worry they might not make it through the year. It’s a crazy world that some have so much and some don’t have much at all. And when I say much I mean basic shit like water and shelter and sanitation. How can we live in a world where there are more cell phones than toilets? Did you know that about a billion people still defecate in the open? Can you imagine that?

Life is filled with cruel things. The idea that millions of my fellow humans can’t even be sure of a proper drink of water for the day while I can stand in the shower for twenty minutes is a brutal, hard reality. And it’s mostly luck. Happenstance. The vagaries of existence. Some of us caught a break and were born into relative wealth, peace, and security. Some of us got a raw deal. I don’t mean to say that I don’t enjoy or appreciate my good fortune. Far from it. I know how lucky I am that my ancestors survived their many ordeals and that my parents did their best for me. But if I’d been born in South Sudan life would be very different.

There will always be a little sand in the vaseline for me. I don’t think that’s a bad thing. It’s just a way to stay grounded and to be reminded of how tenuous the links that hold together our civilization really are.


My phone’s battery is getting flaky. It won’t hold a charge for as long and it runs out quick in cold weather. So I’m on-line looking for a battery and then I think I ought to just junk the thing and upgrade. My phone is one of these:


Yes, it’s a dumb phone. Or in the marketing jive they use these days, a “feature” phone. What the fuck does THAT mean? Problem is, I like my phone. I like my cheap plan. I make and receive a few calls. I make and receive a few texts. That’s it. That’s all I use it for. I don’t navigate. I don’t play games. I don’t take pictures (it has a camera). I don’t access the web. I don’t app.

I don’t have a problem with any of those things. I just don’t need ’em. When I’m out at the pub quaffing pints I don’t need to look anything up or follow any events. I just like to drink my beer, bullshit with the locals and the barmaids, and enjoy the general goofiness going on around me. Lots of other folks are tapping screens, talking to Siri, and doing other smartphone stuff. After all, the new phones are really portable computers. I like computers and I even have a laptop that I can travel with. But I like to sit in a comfortable chair and not have any distractions while I do my computing. That mostly consists of blogging, checking email and Facebook, and surfing around reading and doing research on things that interest me. I don’t have to do any of those things when I’m out socializing.

Sometimes I worry that my aversion to a fancy new phone is making me a neo-Luddite. Horrors! No self-respecting science nerd ought to be accused of Luddism. I’m not a techno-phobe. After all the miracle of the world wide web is making my virtual dialog with you possible. Think of the enormous technological infrastructure required to make that happen! I’m all for it. But a couple of things keep me from advancing from dumb to smart in the phone department.

One, cost. My phone is really cheap. I like that. I don’t use it much, so why should I spend a lot of money? My cars are all paid for and I don’t drive that much. Perfect match, don’t you think? Two, I use my computers quite a bit when I’m home. I need a break from them when I go out. Three, I like to be free of stuff when I’m out and about. I’m a guy. I don’t have a purse or a murse or a bag or anything except pockets. I try to keep it simple: wallet, money clip, keys, handkerchief. It’s bad enough dealing with reading glasses or sunglasses or a comb or lip balm or a jacket & hat or what-have-you. I don’t need any more shit to lug around. My current phone is pretty small, so if I do have to carry it I can fit it in a small pocket. I like that.

I got to thinking about this notion that the gigantic international computer network that we are a part of contains “all the knowledge of humanity” and that we can “access it at any time” with our mobile devices. Well, OK. It beats carrying around the Encyclopedia Britannica. But any collection of human information, achievement, and wisdom must also surely contain human misinformation, failure, and stupidity. Those are at least as abundant as the other. Why would I want to enhance those things? All human individuals already carry around with them the idiocy of the race. To be fair, individuals also possess the virtues of the species. The computer does not sort the good from the bad. Just the opposite: everything is equal is cyberspace. The human user has to impart judgement and separate the wheat from the chaff. The sublime from the shit.

A computer can’t help me do that. At least not right now. Right now we are in the Stone Age of the bio-cybernetic revolution. I don’t have any fear of trans-humanism, cyborgs, or artificial intelligence. We’ve got a long way to go before we understand the mind sufficiently to enhance it. Note I said “enhance” and not “improve”; one does not presume the other. Like I said even the iPhone is a Stone Age implement. It’s the first glimpse of a potential future, but it’s the equivalent still of flint-knapping in the grand scheme of things. The technological interface will someday be seamless and invisible and won’t require advertising carpet-bombing to convince anyone to participate.

The last thought that sticks with me is the ubiquity of the Google search. The word is now generic. You can google anything. And if you have your smartphone you can google anytime and anywhere. But googling just gives us answers to questions that are already known. It’s Jeopardy! stuff, usually. Don’t get me wrong. I google all the time. Beats flipping through the almanac. But we still google things that already exist and we just need to find them. We are riffling through the file cabinet. The greatest filing cabinet of all time, to be sure, but still just a filing cabinet.

Steven Chu, the eminent physicist (1997 Nobel Prize, former director of the Lawrence Berkeley Lab, former Secretary of Energy, on the faculty at Stanford) was once asked if he liked puzzles. No, he said, he didn’t like spending time on things in which the solution was already known. What a wonderful attitude! He liked thinking about things that NEEDED solutions.

That’s what I want to do. Think things that haven’t been thunk. Create. Invent. Imagine. I do that just fine right now. And it’s easy enough—later—to compare what I thunk up with stuff others have done and are doing. That’s what the web is for and it’s great for that. But I can wait until I get home to do it.


In Ireland and the UK they use ‘maths’ as short for ‘mathematics’ and not ‘math’ as we do here in the States. I like ‘maths’ as it more properly conveys the richness and dimensions of mathematics. It should be a plural word as it’s an enterprise of many facets.

I was thinking about maths yesterday when my friend told me of her ambition to be a high school math teacher. The first thing I told her was the people most responsible for everyone hating math are high school math teachers. I’m very encouraging, don’t you think? But it’s not really their fault. There’s only so much time and so many school days and you have to have calculus-ready youngsters for college so you teach to that cohort of kids and let the rest fall by the wayside. There are two consequences to this weed-out process. One, you find out which kids are good at math. Two, you convince a bunch of other kids they can’t do math.

The problem with high school algebra is that it is only good for one thing: learning more algebra. You have to get to a certain facility with algebra before any kind of larger understanding or appreciation emerges. It’s just grammar before then, the rules of the road for mathematical relationships. Imagine English class as being nothing but grammar. No poetry, no speeches, no drama, no creative writing, no literature. Just grammar. That’s basic algebra. It gets dull quick. If you are good at it, it can be fun, like a puzzle. Even if you aren’t, it is pretty clear-cut if you are the analytical sort, and you can get through it without much trouble.

But if you are put off by the abstraction, or have some skill weaknesses in things like fractions, it can be rough going. I remember discovering at a very young age that I was never going to be a baseball player. I did not possess the necessary natural grace, hand-eye-coordination, and fast reaction times to succeed at the sport. Like athletic talent, math talent varies from person to person. Some people have mathematical intuition, like some people have a good ear for music.

For those of us who aren’t athletes, discovering physical things that we can do for fun and fitness is very important. I found out I can downhill ski, ride a mountain bike, and play slo-pitch softball with sufficient competence that I make time to do all of them. Lots of people do this. Music is another thing I stink at. I’ve got no sense of rhythm. I can sing in a choir because I can listen and match my voice to the notes I hear, but I can’t start a song because I don’t hear the note in my head. But I listen to music, read and learn about music, and pester my musically talented friends to show me things I’m curious about. I even took Music Appreciation in college.

What’s this all to do with maths? Well, do you hear of anyone taking Math Appreciation? Do you talk to your friends about which new math you are going to try to learn? Do you go to the math club to tone your math muscles? No, of course not. Math, for most, is something to endure. Then forget. Which is a shame, as mathematics is not only one of the great accomplishments of civilization but it is fundamental to the natural sciences, engineering, and economics. You can’t do those things without math. Math is also powerful and beautiful for it’s own sake. People sometimes make music for no other reason than the joy it gives them. Believe it or not, math can be that way.

But we don’t have the time, inclination, or infrastructure to develop a larger pool of mathophiles. I do think we could reduce the pool of mathophobes, and that would be a good thing. People who aren’t good at sports don’t necessarily hate sports. They might even like sports. People who don’t make music don’t generally hate musicians, and most enjoy hearing other people make music. But people who find out they “aren’t good at math” run away from math and never come back.

How to fix that? And why? I contend that mathematical illiteracy, or innumeracy, is a bad thing. We are asked to look at quantities and rates all the time and make personal, financial, and political decisions based on these things. If we don’t have some mathematical competence then we are crippled in our ability to think clearly and we will make poorer choices. Some basic facility with maths is just as crucial to the big ideas of the modern world as being able to read critically. It’s also important to fight innumeracy because math phobia closes doors. Kids who are willing to get through advanced math classes have more choices later on.

I think we ought to figure out what kind of maths kids can learn even if they are lousy at math. That is, if they can’t stay on the algebra-to-advanced-algebra-to-calculus treadmill that doesn’t mean they can’t learn something. Lots of people learn a workable Spanish so they can travel around Central and South America. They may never read One Thousand Years of Solitude in the original but they can function in a foreign land. That’s quite a skill. Can’t people learn some useful, operational math like that? Just because you can’t do differential equations doesn’t mean you can’t do other stuff.

So I would say to prospective math teachers and to the next generation of schoolmasters not to neglect the rest of the people. Not just in math, but everywhere. Kids who get cut from the ball team still like to play. Find a way so they can. Kids who aren’t going to be chemists can still learn a lot of interesting chemistry. There are many mathematical skills and ideas that lots of students can learn, retain, and appreciate if given half a chance. The way things are set up now we’ve got plenty of scientific and/or technical people who learned enough math so they can do their thing. But it’s the non-scientists and the non-technical types who are increasingly being asked to grasp and act on scientific and technical, that is mathematical, notions. Wouldn’t it be a good idea to bring them along?