Penumbral Lore; the Shadows of Science

Comment

Written by: Anonymous at 2007/12/08 - 02:23:00

Discussion

The Quantum Zeno Experiment (described in the post you responded to) isn’t some thought game to illustrate a paradox. It’s actually been carried out. It’s an illustration of δTδE< , which is another way of stating the principle, but picturing one form of complementarity doesn’t necessarily help you to picture another. I’ve found lots of experiments which relate to this conjugate pair, and to wave/particle duality, but you seem to have trouble with δVδMV< (sometimes written as δVδm< ) specifically. The fact that the two are mathematically equivalent doesn’t mean that picturing one helps to picture the other.

When I was in high school I was taught that electrons were tiny hard spheres which travel through space-time, which was a smooth and featureless substrate. The ‘pool ball’ analogy was often used, to try to get students to picture the particles travelling through space like pool balls rolling across a table. This is the situation you’re describing as how things ‘must be’. What happens if you try to picture space, time, and energy as being somehow ‘pixilated’ entitites? Does that get you closer to the concept that an object can ‘jump’ in space or energy level? Or can ‘jump’ between two positions without passing the intervening space? Or that two points which are a measurable distance from each other might be connected by quantum level wormholes?

I can picture it, actually, but I can’t relate it to the scale on which we live. That’s OK, though, because I can’t relate the results of various quantum experiments to the scale on which we live. We’re in good company, though. Einstein couldn’t grok it, either.

EPR Paradox

(paraphrased from a paper at University of California, Riverside)

One problem with demonstrating the complementarity principle with position and momentum is that the two properties are (macroscopically) on a continuum, rather than being discrete values. Spin is a much easier measurement to work with. A particle can have ’spin’ along three different axis, corresponding to X, Y, Z (three spatial directions). In any one of these axis it will be spin + or spin -. The three sets are complementary, in that the Uncertainty principle states that if you know one you do not, cannot, know the other two.

Einstein, together with Boris Podolsky and Nathan Rosen, set up a thought experiment to demonstrate that attributing this to the nature of reality rather than a limitation of measurement led to a paradox.

They imagined a physical system which created two ‘entangled’ photons - that means two particles which must have complementary spins. So, one would imagine, that measuring the X axis spin of one photon and the Z axis spin of the other photon would give you a two axis state of both photons. The uncertainty principle says it won’t, that measuring the state of one of the photons would change the state of the other, even though the photons are moving apart at the speed of light, and information travelling between them would violate the relativity principle. The only logical explanation is that both photons carry complete information about their spin state, and it’s ‘really’ there all the time.

Bell’s Theorm

(also using an article at Wikipedia)

So what happens when you take measurements along random axis of the two different photons? One would expect that they would be 100% correlated for X/X axis measurements, and 0% correlated for X/Y axis measurements.

What if the two axis of measurements were rotated by 45°? Classical physics (and logic,) which assumes that spin values along both axis exist at the time that the photons are emitted predict a different value of correlation than quantum physics does. Unfortunately, the only table I could find which represented the results is at wikipedia, so whether or not it will have changed by the time you read this is uncertain. I’m going to copy it here, attributed to the site linked above:

Quantum Mechanical models predict a correlation of 0.71.

Guess which one the experimental data supports? I’m not talking about ‘cold fusion’ here. This isn’t a single experiment, never to be repeated, it’s been done again and again and again.

Future Topics

Strangely enough, I find uncertainty a lot easier to deal with if I look at it through the lense of string (Membraner) ‘theory’ - the concept that there is a real minimum distance or time period because the ‘point’ is actually a tiny, moving squiggle of reality which doesn’t have a precise position is a little easier to deal with, but, as I said before, there is no way to design an experiment to test this… yet.

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Science for Your Amusement

Earliest chocolate drinks were alcoholic (but they already knew that), and 500 years earlier than previously thought. Archaelology Online and Dogfish Head Brewery may try to ‘reconstruct the brew. They previously did the same thing with Chateau Jiahu, which just might (probably not) be still available…

Here’s the abstract to the article “Ovulatory cycle effects on tip earnings by lap dancers: economic evidence for human estrus?” from “Evolution and Human Behavior”. A silly study, but it does have something to say about the concept of concealed ovulation.

Finally, the US Government is suggesting that, in case of a deadly influenza pandemic, Homeland Security gets preferential treatment - as though they’d be ‘on the front lines’ in response to the pandemic? Doesn’t make sense to me.

Comments

Fireweaver responded to my Big Fat Sexy Brain post by noting that all sorts of things aren’t adjusted for in the whole spate of ‘fat research’. I’d have to agree, but I want to say again that the research wasn’t about fat and intelligence in general, it was about pregnant teens, and the population was large and varied. I wouldn’t even get as far as worrying about socio-economic class and upper body vs lower body fat - I’d start off by questioning the validity of the IQ tests, and trying to figure out exactly what they measure in the first place. Until we have a good definition of intelligence, this sort of research is a curiosity at best.

Amusingly enough, I can’t help but wonder if the article about how lap dancers estrus cycles influence their tips is constructed better. At least there are two objective variables to correlate… !

Anyway, today’s post is at least peripherally related, as I’ll be using an antiobesity drug to discuss the process in which medications gain clinical approval. For readers interested in the entire obesity issue, health at any size, and anti-fat bias in the medical community, I strongly suggest checking out “First, Do No Harm” - a blog dedicated to “Real Stories of Fat Prejuidice in Health Care”.

How Drugs Are Approved (or Not)

The November 17 episode of the Lancet Podcast (12.5 Mb .mp3) discusses a meta-analysis of the drug Rimonabant (known in Europe as Accomplia, and in the US as Zimulti). This drug was approved as a weight loss aid in Europe, but not in America. I’m going to talk a little bit about the approval process, using this drug in particular as an example of how drugs get approved (or not). I’ll talk about the Lancet podcast at various points, and explain some of the terms used in the podcast discussion. This might be useful for those of you who are not familiar with clinical trials.

Pre-Clinical Research

There is a lot of research which goes into drug development before it even gets to the clinical trial stage. In this case, they probably started with the idea that a certain kind of neuro receptor (the CB1 Cannabinoid Receptor) is involved in appetite. Presumably this is one of the neuroreceptors which is involved when THC from cannabis (marijuana) is introduced in the brain. This makes a lot of sense, because one of the known effects of marijuana is appetite stimulation. Perhaps, if we block this receptor from the brains own chemicals, appetite will be repressed.

So, someone at Sanofi-Aventis came up with the idea of developing (or finding) a chemical which would block these receptors (called an agonist). They probably worked with a lot of compounds in vitro (literally in glass, but it means laboratory testing not done inside a living being) to find something which interacts with neurons in the desired way without killing cells off.

The next step is testing in living organisms, both for safety and efficacy (effectiveness). I understand that some individuals would prefer that all of this testing be done in humans, but that’s not the way it happens. For the moment, I assert that as upsetting as the idea of harming animals is, testing on humans without working in animals first is worse. I understand that there is debate on that issue - it’s far to large to address in this post. I will attempt to handle it (probably in a large series of posts) at a later time.

This process gives a single, most-favorable, substance to try in human beings. By this time there’s some confidence that the substance won’t kill people outright, nor will it damage their offspring conceived after the drug ceases to be used. Often it’s noted if the drug will damage offspring in utero. One would hope that such would be eliminated for an anti-obesity drug, but that was not the case this time around.

The fact that the drug possibly causes some sort of fetal development problems was noted during animal studies, as was the fact that it causes difficulty in breast feeding. There were other problems shown in animal studies, all at really high doses. It’s important to note that a bunch of studies were done in a number of animal systems, and that different species react differently. While this didn’t translate directly into what happens in human biochemistry, it gives researchers a list of things to look for.

Trials in humans are called “Clinical Trials”. They are controlled by the international agreements and national regulations I discussed in my previous bioethics post, they require informed consent of the volunteers, and they are (ideally) very closely monitored by organizations outside the institution or company performing the research.

Phase I Trials

Once the animal data is in, and possible risks are identified, the Phase I clinical trials start. These are the sort of studies where healthy individuals are given various doses of the medication, and observed closely. There are rare cases a substance is so toxic that it would be unethical to use healthy volunteers, and phase I participants are recruited from the patient population which might potentially benefit from the drug, but that’s a whole other set of complications.

I do mean volunteers are observed closely here. At the end of these trials the researchers know how fast the medicine gets into the system, how long it stays in the system, and whether it is eliminated or excreted - check out the Summary of Product Characteristcs from Accomplia (Rimonabant) - note that in the US the report would probably look different, but the same things are covered. If you ever hear of someone being paid for participation in a clinical trial, they probably took part in a phase I study.

Some of the volunteers for Phase I studies are medical students. Some are homeless people. In any case, they are individuals who will spend several days under very intense scrutiny. They are not without risk, and it’s worth thinking about their legal implications (pdf download) as well as their ethical ones.

Phase II Trials

After Phase I trial, the studies move to the patient population. Phase II trials are usually done with small numbers of volunteers - perhaps 100 to 300. Small studies are done first because (as you will often hear) studies are expensive. There’s no point on spending huge amounts of money to get thousands of data points until you’ve looked at those first couple of hundred.

We look to Phase II trials to see signs of clinical effect (not proof - not enough data points for that). We also take careful note of Adverse Events. Volunteers are told to keep close watch for any unusual reactions, particularly those which were evident from the animal models. Everything is noted and logged, but researchers must be especially aware of Severe Adverse Events. The National Cancer Institute has a pamphlet discussing Adverse Event Reporting (pdf download), which will give you an idea of what’s involved.

These studies are usually (but not always) double blind, placebo controlled studies. In other words, the volunteer will have either the drug or a biologically inert placebo, and neither the Investigator nor the volunteer will know which. There are all sorts of structures for this, in some studies the volunteer is given both a placebo and the investigative drug at different times, and in some the patient receives a non investigational drug in addition to the drug or placebo. The reason the placebo is involved is that there will often be some improvement in the condition of some patients simply because they are ‘under treatment’. The effect is small, but can be statistically significant. The only way to correct for this is to provide ‘treatment’ which is not biologically active, and compare that to the treatment being studied. Those of you who read my initial earlier posts will recall that placebos fit my working definition of magic.

Phase III Trials

Pretty much just like Phase II studies, but bigger. There are typically Phase III centers spread across the country, each seeing dozens or hundreds of patients, each with its own Principal Investigator who is responsible for the way the study is conducted at that site. This includes seeing that the experimental protocol is strictly followed, that Informed Consent is properly obtained from each and every Volunteer, and that scrupulous records are kept. It’s a huge effort, and it generates truckloads (literally - we’re talking several 18 wheelers) full of paperwork which the FDA is required to go over.

Phase III studies are VERY expensive, and few pharmaceuticals even get that far. So, when Acomplia/Riombanant completed the Phase III study and didn’t get FDA approval in the US it was a huge setback for Sanofi-Aventis. I don’t know the regulatory process in the EU as well, but it’s not a stretch to say that the process is similar.

Why didn’t it get the approval? Or why didn’t it get approval yet? (Different sources claim differently states) Because a lot of the study participants became severely depressed. This makes sense, actually, since the drug blocks a ‘reward center’ in the brain. The CB1 cannabanoid receptor may be primarily involved in appetite, but primarily isn’t the same as totally, and reactions to all sorts of psychoactive substances differ wildly from person to person.

So, why did the drug get approved in Europe? It looks like the same data was used, but different people were looking through it. Unless problems are excruciatingly severe (in which chase, they’d almost certainly be screened out in Phase II) there will be a judgement call involved. Sometimes even the thousands of data points obtained from a Phase III Trial can give ambiguous results.

This is why Clinical Trials no longer stop at approval, and why SAEs continue to be collected once drugs are approved. If you are in the US, you can report medication related medical problems through Medwatch at the FDA.

Phase IV Trials

After a drug is approved, any study it is in is a Phase IV trial. These can be the ‘A is proved better than B’ type trial done for advertising purposes. They can be long term (longitudinal) studies which follow individuals using a given drug for years and years. Anything at all.

In the case of Acomplia/Riombanant there were four published studies, in particular. The meta-analysis means that data from all four studies was drawn upon, and analyzed. Note that these studies excluded volunteers with a history of depression - they were ’screened out’. So when the meta-analysis showed that something like three percent of those in the study had depression related SAEs. That’s a HUGE percentage, and I can’t see how it would be worth the risk.

Further Reading and Resources

ZimultiAcompliaReport claims to be an independant source of news and information. I didn’t see any obviously biased or skewed information there, but they seem to be pretty excited about the idea of a weight loss drug which works.

FDA Guide to Buying Medicines Online - I’m putting this link here because the first three google spots on Zimulti led me to on line sellers - all three acknowledged that the FDA has not approved the drug. One mentioned suicidal thoughts (one of the scary SAEs). One claimed a 10% weight loss (the Accomplia Report site above says 5%). It’s just scary business.

Daily Med - the National Library of Medicine’s database of curent package inserts. It’s not complete, but the package inserts which are there have the most recent revisions. Learn to read them (at least how to get the highlights) - it’s a very good resource.

Comments and Notes

Yesterday I mentioned that the eldest child got recognition for her scholastic achievements. This morning I will add that the youngest did as well. Not that we’re hung up on grades in this household, but every so often I’m moved to share how well the kids are doing.

Chelseaknits, who seems to have set up a blog.com account just to write a comment on Monday’s post on QM, stopped in the middle of it. I hope her brain didn’t explode.

Lisa tagged me for the ‘Crazy Eights‘ meme. I’m flattered, really - I just met Lisa on NaBloPoMo a few days ago, she seems like a wonderful person, and I’ve added her to my blogroll. I just don’t want to get caught up in meme-madness in this journal. I barely do memes on LJ, even.

Speaking of Memes

Richard Dawkins coined the term ‘meme’ to describe a unit of cultural information, analogous to a gene as a unit of genetic memory. Yes, Dawkins is a scientist. He was writing about genes, not cultural information, so he wasn’t really trying to claim expertise outside his field. “The Selfish Gene” is on the reading list and I will get around to it eventually.

Even so, the idea was picked up and pushed by parties claiming that it was ‘really’ the way cultures operate. Hey, if memes are like genes then maybe cultures evolve like organisms, and share genetic information… Maybe there are meme viruses and infections. This fits in so many ways, it must be science. (Nope!)

The idea of a unit of information, which then changed the way other information was processed, acted recursively. Suddenly, once the idea of explaining ideas as ‘memes’ came up, it propogated through society and changed the way every idea was seen (which is what memes are described as doing). I call this magic, not science (before any students of memetic theory take offense I ask that they see “Working Definitions“. ) It’s an interesting philosophical framework, but hardly an actual ‘Theory’. The more you press the gene/meme analogy the further out on a limb you’re going. It’s certainly descriptive, but String Theory comes closer to being predictive (supposedly the LHC may make some ST verification possible, but I won’t start discussing ST until I’ve worked further through QM). Darwin had a HUGE body of carefully recorded data to work with when he came up with the idea of ‘Survival of the Fittest’, that data just isn’t there as far as ideas and cultures. We all ‘know’ this stuff, but that’s just a starting point.

The second, related but distinct, use of the term ‘meme’ applies to an internet phenemonon. I first encountered this type of meme as “the hamster dance” (but don’t click) (just trust me) then “All Your Base Are Belong To Us” (click if you must). Both of these were catchy video files which were linked to ad nauseum and then altered and propagated all over the place. People put huge amounts of time and effort into creating their own versions, and the kept going until (or well past the time when) they were no longer funny.

The third kind of meme is also an internet phenomenon. They are typially small applications, or ‘quizzes’, which are made to be filled out and propagated across blogs and social networking sites. They were made to be altered and propogated, and the amount of effort that goes into propagating them is minimal.

“Quantum physics, as a field, makes my brain go all soft and squishy. I have a very difficult time convincing myself that rules of physics in our macro-scaled world do not really apply and in fact might be contraindicated once you get really, really small. I find myself staring at some solid bit of matter trying to figure out if I am changing it by staring at it, or if the individual particles thereof are, at that “moment”, sailing through time in the opposite direction of the way I’m experiencing it. My frame of reference is that of the Neanderthal.”

The squishy brain (or ’splody brain’) feeling is half the fun of it, actually. It’s like riding a roller coaster, you go up and down and upside down and your head just about explodes then you’re suddenly standing on solid ground, and the world is still spinning all around you. I’m getting too old to enjoy riding physical roller coasters, but my appreciation of the radically shifted point of view from quantum physics just grows and grows.

So, today, a quick overview of the interpretations of quantum physics - not the science itself, but the mental gymnastics people have gone through in order to make sense of it. These rely on ‘models’ - an important part of science in general. Models are verbal and mathematical analogies which help explain what we believe is happening in a given system. So when Copernicus’ Model’ of the solar system replaced ‘Ptolemy’s Model’ what changed was our understanding of the solar system, not the solar system itself. It’s important to remember that the the current models of the universe change all the time, and that this means that scientists are doing the right thing, learning from the data.

Copenhagen Interpretation

The earliest attempt to make sense of quantum mechanics was the effort of Niels Bohr and William Heisenberg in Copenhagen in the late 1920’s. The Interpretation is a whole series of statements, the most mind-blowing part of which is that the equation which describes a particle does not come into existence until the particle is actually observed. This apparently nonsensical statement makes sense in mathematical context, but leads to a whole series of equally nonsensical conclusions.

A lot of the confusion comes from the concept of ‘observer’ - there is nothing in the mathematics which requires consciousness, but the term ‘observer’ implies a conscious mind. The most extreme interpretation of this requires that there be a conscious mind outside the universe which can ‘collapse the wave’ and make the universe real. Stephen Hawking took a stab at trying to deal with this logically, he even included a (comparatively) simple explanation of his disproof of the extra-universal observer in “A Brief History of Time”. It might be good physics, but it was a poor debating strategy, since the proof relied on imaginary numbers, which he had previously stated were not correlated with physical properties… It’s also important to note that the Copenhagen collapse can be looked at as a strictly subjective phenomenon, so it’s not necessary to take the ‘observer changes the system’ literally in order to use this interpretation.

The Copenhagen Interpretation is my least favorite quantum model.

Ensemble Interpretation

The Ensemble Interpretation is much less esoteric than Copenhagen. It simply says that the mathematical models of quantum systems describe all the possible systems, not each individual one. This sounds completely rational and reasonable, but doesn’t really explain the two-slit experiment.

In other words, nice try, but it doesn’t fit the data.

Hidden Variable Theory

When Einstein said “I am convinced God does not play dice (with the universe),” he was implying that quantum theory could predict results rather than just give probabilities if we knew all the facts about a system. In 1935 he got together with Boris Podolsky and Nathan Rosen they demonstrated that, if this were NOT the case, there could some day be an experiment which proved that information traveled at faster than light speed.

That experiment will the the topic of a future post.

Transactional Interpretation

The idea came from one John Cramer in 1986. Imagine that every particle produces two standing waves. One moves forwards in time, the other moves backwards in time - yes, I know it sounds loopy, but if Richard Feynman said it (he did) and was speaking in his field (he was) it is, by definition, brilliant whether or not it was and not at all nuts. So, the forward moving (retarded) wave from a particle meets the backwards moving (advanced) “offer” wave from another particle, the two interact and determine which of the possible interactions the two particles will have.

This is my favorite explanation.

Many Worlds Interpretation

This interpretation states that any quantum event which the equations say could take place does, in fact, take place. For example, Carbon-14 has a half life of 5730 years. For any given quantity of (radioactive) ¹⁴C, half of the atoms will decay to become (stable) ¹⁴N in 5730 years. When will any given atom decay? It has an infinitesimal chance of happening in any given Heisenberg unit of time (some ridiculous fraction of a second). In this interpretation, it happens in ALL of them, each with its own world line.

Pretty cool stuff, and a great staple for science fiction - alternate universes are amazingly useful. You’ve got to remember that it means that every possible interaction (on the quantum level) has and will happen somewhere. This doesn’t mean that every imaginable event has happened, unless you require that imagination is limited by possibility… Note that the infinite multiverse from this interpretation is not the same as the infinite multiverse implied by M-theory (which I haven’t gotten to yet).

There may be some problems involved in combining Many Worlds with string theory or quantum gravity.

There, I’m all happy and brain-splody now, and I hope you are too.   I’ll go through some of the mind bending experiments in future posts.

I’ve spent most of my blog time this morning working on the show notes for my husband’s new podcast, which will go live (hopefully) later on today or tomorrow. I’ve still got a bunch of tinkering to do on the graphics, but check out the Secret Frequency blog this afternoon - hopefully we’ll have a treat up for you.

So I’m going to put up a very short post today, on a subject everyone’s been hearing a lot about - waterboarding. There are a number questions which keep coming up about it:

1. Is it torture?
2. Is it legal?
3. Does torture work to get information?

Well, from the United Nations Convention Against Torture, torture is:

“any act by which severe pain or suffering, whether physical or mental, is intentionally inflicted on a person for such purposes as obtaining from him or a third person information or a confession, punishing him for an act he or a third person has committed or is suspected of having committed, or intimidating or coercing him or a third person, or for any reason based on discrimination of any kind, when such pain or suffering is inflicted by or at the instigation of or with the consent or acquiescence of a public official or other person acting in an official capacity. It does not include pain or suffering arising only from, inherent in or incidental to lawful sanctions.”

I don’t see any ambiguity there. It’s torture.

As far as the legality of waterboarding, and Congress having the ability to specifically make it illegal - the US is party to numerous treaties and conventions (the UN Convention cited above being only one of them) against torture. Therefore, it’s not only legal, it’s unconstitutional.

So, regardless of whether or not torture works, it’s not legal.

…but since this blog is about science, the question of ‘does it work’ is still an interesting question. As I discussed before, a reputable scientist could not just go out and torture people and see what information was good and what was not. Nor can we just rely on anecdotal evidence like “it worked for the Nazis“, because anecdotal evidence isn’t data.

No, in order to seriously research this, we need thousands of data points (which were not used in formulationg our theory, by the way), which were all obtained under controlled conditions, and which we can obtain from willing subjects. Sound impossible? Well, it’s not.

What we need, in this case, is something called a ‘retrospective study’. Look over old data, and use some serious data analysis. So, if someone has collected this kind of data in the past… who might have done that?

How about the Pentagon. Surely they’ve debriefed thousands of Americans who have survived torture, and given information under it. There must be records which can be obtained (with names and identifying information removed) of who said what, how much was true and how much was false.

Or we can just remember that, as Americans (and as citizens of the world) what makes us great is the fact that we’re the ones who don’t play those games, and we won’t let the enemies of Democracy turn us into the people who do.

’nuff said.

“It really isn’t “check the math yourself”…you have to check more than that. There’s more to the use of stats - often common sense will tell you enough to know that a number is being misused, even when the overall math is correct. Sure, 2+2=4, but what does “4″ mean in this context? And so forth…”

I’d agree with that absolutely. As I said,watch out for “liars, damn liars and statisticians”, and remember that sincere misinterpretation can be as dangerous as deliberate manipulation of the numbers.

meloukhia commented on “Why I Go Orgo” with the observation

“I hadn’t read that study on the nutritional value of organic foods; thanks for linking to that post. I had always suspected that organic foods were probably more sound nutritionally because of the healthier soil on which they are grown, which would suggest more absorption of useful minerals.

I would offer a note of caution with so-called “Free Range” foods, because that label doesn’t mean what most people think it means. Food labeling is extremely deceptive, and a “free range” chicken can still be raised inside a shed, as long as the farmer leaves a door open for a few hours every day. “Pasture raised” is probably a better bet, since it suggests that the animal actually gets to go outside and eat grass.”

This is a really good point. The truth is, while there are several agencies which will certify “Organic” produce (each of which has publicly available criteria) I don’t think there’s any agency doing the same thing for ‘free range’ or ‘pasture raised’ meat or dairy products. Your best bet is to buy locally, talk to the farmer yourself, and maybe even drive by for a visit. It’s not always possible, but it’s worth a shot.

Finally, I thought you’d enjoy this response to Bioethics, a Retrospective:

Well, that’s out there. I’m all for solving the problem of overpopulation, but I’m bothered by the doctor’s suggestion of ‘dramatic’ solutions and ‘forceful steps’. I suppose it’s possible to construct a set of ethics which allows for it, but that doesn’t mean that it could ever be moral. How about you, readers?

Oh, and a final, timely note:

Like any other language, speaking in mathematics does not guarantee speaking truthfully. False statements can be made mathematically, and any useable mathematical system will have its own internal contradictions and paradoxes. See Göedel’s Incompleteness Theorm (opens in new window) at Agronus.com for more discussion on the theorum and on proofs in general.

Mathematics is often taught in a maddening way. I remember when I was first taught subtraction, I was told “you can’t take a bigger number from a littler number”. Well, you can’t if you’re dealing with positive numbers. Later, I was told “You can’t take the square root of a negative number”. Well, you can’t if you’re dealing with real numbers. For every basic, common sense rule I was taught about how numbers behave, I later on came on an exception. The exception not being a special number, but a different way of speaking in numbers. Each of which is easier to learn if we think of numbers as a language in which information is transmitted, not the information itself. Do you have to learn to talk before you can understand poetry? Maybe. Maybe not. It really depends on how your own mind works. Many of the ‘lies’ of mathematics have to do with bad teaching methods.

Mathematics can lie in a third way. Every heard the statement “Liars, damn liars and statisticians”? Sometimes numbers are manipulated to persuade or even fool us. Even respected scientists can do this on occasion. Don’t let numbers convince you until and unless you are comfortable with how they are used, and take the time to understand what they are saying themselves… check the math yourself!

Having said all this, I wanted to tell you about the website. As you can see from the button in my sidebar, they’re the hitcount site I trust to tell me how well read this blog is.

Oh, and thank all of you for reading this. I’m really pleased that people are starting to follow along.

Reader Comments

Thank you to everyone who has left comments. Yes, I do read them and appreciate them. I’m trying to figure out the best way to respond to people. For now, I’ll save them up and periodically give on-blog responses. Here are two from the post “All Strung Out”:

Kugelblitz - Yes, I did mean Bureau of Indian Affairs when I referred to the BIA.

Anonymous - That sounds creepy to the point of being scary.