Summary:

• Chronic Fatigue and Fibromyalgia look very like Hypothyroidism.
• Thyroid Patients aren't happy with either the diagnosis or treatment of Hypothyroidism.
• It is possible that lots of FMS/CFS cases are 'something wrong with the thyroid system that doesn't show up on laboratory hormone level tests'.
• It's possible that it's not too difficult to fix these CFS/FMS cases with thyroid hormones.
• I believe that there may have been a stupendous cock-up that's hurt millions.
• Less Wrong should be interested, because it could be a real example of how bad inference can cause the sciences to come to false conclusions, as well as a good practice problem for the things we really care about.

Edit:

I found a possible answer here:
http://lesswrong.com/lw/nbm/thyroid_hormones_chronic_fatigue_and_fibromyalgia/
I do not believe it, because I do not understand it, but contemplation of it seems to be enlightening. In particular, the problem is much broader than I originally thought.

A summary of the argument in the first two posts, together with links to lots of evidence in the literature:

http://lesswrong.com/r/discussion/lw/nef/the_thyroid_madness_core_argument_evidence/

And this is pretty much proof, I think:

http://lesswrong.com/lw/nhs/the_thyroid_madness_two_apparently_contradictory/

At this point, I think I'm as confident as I can be without some sort of formal trial (so 25% maybe?)

But certainly, if you're suffering from Chronic Fatigue Syndrome/Fibromyalgia/Major Depression/Irritable Bowel Syndrome, or any of the many similar disorders which just seem to be different names for 'hypothyroidism with normal TSH', I reckon this is worth trying!

I have done, and it worked for me. For about four months now...

Original Post:

I believe that I've come across a genuine puzzle, and I wonder if you can help me solve it. This problem is complicated, and subtle, and has confounded and defeated good people for forty years. And yet there are huge and obvious clues. No-one seems to have conducted the simple experiments which the clues suggest, even though many clever people have thought hard about it, and the answer to the problem would be very valuable. And so I wonder what it is that I am missing.

I am going to tell a story which rather extravagantly privileges a hypothesis that I have concocted from many different sources, but a large part of it is from the work of the late Doctor John C Lowe, an American chiropractor who claimed that he could cure Fibromyalgia.

I myself am drowning in confirmation bias to the point where I doubt my own sanity. Every time I look for evidence to disconfirm my hypothesis, I find only new reasons to believe. But I am utterly unqualified to judge. Three months ago I didn't know what an amino acid was. And so I appeal to wiser heads for help.

Crocker's Rules on this. I suspect that I am being the most spectacular fool, but I can't see why, and I'd like to know.

Setting the Scene

Chronic Fatigue Syndrome, Myalgic Encephalitis, and Fibromyalgia are 'new diseases'. There is considerable dispute as to whether they even exist, and if so how to diagnose them. They all seem to have a large number of possible symptoms, and in any given case, these symptoms may or may not occur with varying severity.

As far as I can tell, if someone claims that they're 'Tired All The Time', then a competent doctor will first of all check that they're getting enough sleep and are not unduly stressed, then rule out all of the known diseases that cause fatigue (there are a very lot!), and finally diagnose one of the three 'by exclusion', which means that there doesn't appear to be anything wrong, except that you're ill.

If widespread pain is one of the symptoms, it's Fibromyalgia Syndrome (FMS). If there's no pain, then it's CFS or ME. These may or may not be the same thing, but Myalgic Encephalitis is preferred by patients because it's greek and so sounds like a disease. Unfortunately Myalgic Encephalitis means 'hurty muscles brain inflammation', and if one had hurty muscles, it would be Fibromyalgia, and if one had brain inflammation, it would be something else entirely.

Despite the widespread belief that these are 'somatoform' diseases (all in the mind), the severity of them ranges from relatively mild (tired all the time, can't think straight), to devastating (wheelchair bound, can't leave the house, can't open one eye because the pain is too great).

All three seem to have come spontaneously into existence in the 1970s, and yet searches for the responsible infective agent have proved fruitless. Neither have palliative measures been discovered, apart from the tried and true method of telling the sufferers that it's all in their heads.

The only treatments that have proved effective are Cognitive Behavioural Therapy / Graded Exercise. A Cochrane Review reckoned that they do around 15% over placebo in producing a measurable alleviation of symptoms. I'm not very impressed. CBT/GE sound a lot like 'sports coaching', and I'm pretty sure that if we thought of 'Not Being Very Good at Rowing' as a somatoform disorder, then I could produce an improvement over placebo in a measurable outcome in ten percent of my victims without too much trouble.

But any book on CFS will tell you that the disease was well known to the Victorians, under the name of neurasthenia. The hypothesis that God lifted the curse of neurasthenia from the people of the Earth as a reward for their courage during the wars of the early twentieth century, while well supported by the clinical evidence, has a low prior probability.

We face therefore something of a mystery, and in the traditional manner of my people, a mystery requires a Just-So Story:

How It Was In The Beginning

In the dark days of Victoria, the brilliant physician William Miller Ord noticed large numbers of mainly female patients suffering from late-onset cretinism.

These patients, exhausted, tired, stupid, sad, cold, fat and emotional, declined steeply, and invariably died.

As any man of decent curiosity would, Dr Ord cut their corpses apart, and in the midst of the carnage noticed that the thyroid, a small butterfly-shaped gland in the throat, was wasted and shrunken.

One imagines that he may have thought to himself: "What has killed them may cure them."

After a few false starts and a brilliant shot in the dark by the brave George Redmayne Murray, Dr Ord secured a supply of animal thyroid glands (cheaply available at any butcher, sautée with nutmeg and basil) and fed them to his remaining patients, who were presumably by this time too weak to resist.

They recovered miraculously, and completely.

I'm not sure why Dr Ord isn't better known, since this appears to have been the first time in recorded history that something a doctor did had a positive effect.

Dr Ord's syndrome was named Ord's Thyroiditis, and it is now known to be an autoimmune disease where the patient's own antibodies attack and destroy the thyroid gland. In Ord's thyroiditis, there is no goiter.

A similar disease, where the thyroid swells to form a disfiguring deformity of the neck (goiter), was described by Hakaru Hashimoto in 1912 (who rather charmingly published in German), and as part of the war reparations of 1946 it was decided to confuse the two diseases under the single name of Hashimoto's Thyroiditis. Apart from the goiter, both conditions share a characteristic set of symptoms, and were easily treated with animal thyroid gland, with no complications.

Many years before, in 1835, a fourth physician, Robert James Graves, had described a different syndrome, now known as Graves' Disease, which has as its characteristic symptoms irritability, muscle weakness, sleeping problems, a fast heartbeat, poor tolerance of heat, diarrhoea, and weight loss. Unfortunately Dr Graves could not think how to cure his eponymous horror, and so the disease is still named after him.

The Horror Spreads

Victorian medicine being what it was, we can assume that animal glands were sprayed over and into any wealthy person unwise enough to be remotely ill in the vicinity of a doctor. I seem to remember a number of jokes about "monkey glands" in PG Wodehouse, and indeed a man might be tempted to assume that chimpanzee parts would be a good substitute for humans. Supply issues seem to have limited monkey glands to a few millionaires worried about impotence, and it may be that the corresponding procedure inflicted on their wives has come down to us as Hormone Replacement Therapy.

Certainly anyone looking a bit cold, tired, fat, stupid, sad or emotional is going to have been eating thyroids. We can assume that in a certain number of cases, this was just the thing, and I think it may also be safe to assume that a fair number of people who had nothing wrong with them at all died as a result of treatment, although the fact that animal thyroid is still part of the human food chain suggests it can't be that dangerous.

I mean seriously, these people use high pressure hoses to recover the last scraps of meat from the floors of slaughterhouses, they're not going to carefully remove all the nasty gristly throat-bits before they make ready meals, are they?

The Armour Sausage company, owner of extensive meat-packing facilities in Chicago, Illinois, and thus in possession of a large number of pig thyroids which, if not quite surplus to requirements, at the very least faced a market sluggish to non-existent as foodstuffs, brilliantly decided to sell them in freeze-dried form as a cure for whatever ails you.

Some Sort of Sanity Emerges, in a Decade not Noted for its Sanity

Around the time of the second world war, doctors became interested in whether their treatments actually helped, and an effort was made to determine what was going on with thyroids and the constellation of sadness that I will henceforth call 'hypometabolism', which is the set of symptoms associated with Ord's thyroiditis. Jumping the gun a little, I shall also define 'hypermetabolism' as the set of symptoms associated with Graves' disease.

The thyroid gland appeared to be some sort of metabolic regulator, in some ways analogous to a thermostat. In hypometabolism, every system of the body is running slow, and so it produces a vast range of bad effects, affecting almost every organ. Different sufferers can have very different symptoms, and so diagnosis is very difficult.

Dr Broda Barnes decided that the key symptom of hypometabolism was a low core body temperature. By careful experiment he established that in patients with no symptoms of hypometabolism the average temperature of the armpit on waking was 98 degrees Fahrenheit (or 36.6 Celsius). He believed that temperature variation of +/- 0.2 degrees Fahrenheit was unusual enough to merit diagnosis. He also seems to have believed, in the manner of the proverbial man with a hammer, that all human ailments without exception were caused by hypometabolism, and to have given freeze-dried thyroid to almost everyone he came into contact with, to see if it helped. A true scientist. Doctor Barnes became convinced that fully 40% of the population of America suffered from hypometabolism, and recommended Armour's Freeze Dried Pig Thyroid to cure America's ills.

In a brilliant stroke, Freeze Dried Pig's Thyroid was renamed 'Natural Desiccated Thyroid', which almost sounds like the sort of thing you might take in sound mind. I love marketing. It's so clever.

America being infested with religious lunatics, and Chicago being infested with nasty useless gristly bits of cow's throat, led almost inevitably to a second form of 'Natural Desiccated Thyroid' on the market.

Dr Barnes' hypometabolism test never seems to have caught on. There are several ways your temperature can go outside his 'normal' range, including fever (too hot), starvation (too cold), alcohol (too hot), sleeping under too many duvets (too hot), sleeping under too few duvets (too cold). Also mercury thermometers are a complete pain in the neck, and take ten minutes to get a sensible reading, which is a long time to lie around in bed carefully doing nothing so that you don't inadvertently raise your body temperature. To make the situation even worse, while men's temperature is reasonably constant, the body temperature of healthy young women goes up and down like the Assyrian Empire.

Several other tests were proposed. One of the most interesting is the speed of the Achilles Tendon Reflex, which is apparently super-fast in hypermetabolism, and either weirdly slow or has a freaky pause in it if you're running a bit cold. Drawbacks of this test include 'It's completely subjective, give me something with numbers in it', and 'I don't seem to have one, where am I supposed to tap the hammer-thing again?'.

By this time, neurasthenia was no longer a thing. In the same way that spiritualism was no longer a thing, and the British Empire was no longer a thing.

As far as we know, Chronic Fatigue Syndrome was not a thing either, and neither was Fibromyalgia (which is just Chronic Fatigue Syndrome but it hurts), nor Myalgic Encephalitis. There was something called 'Myalgic Neurasthenia' in 1934, but it seems to have been a painful infectious disease and they thought it was polio.

Finally, Science

It turned out that the purpose of the thyroid gland is to make hormones which control the metabolism. It takes in the amino acid tyrosine, and it takes in iodine. It releases Thyroglobulin, mono-iodo-tyrosine (MIT), di-iodo-tyrosine (DIT), thyroxine (T4) and triiodothyronine (T3) into the blood. The chemistry is interesting but too complicated to explain in a just-so story.

I believe that we currently think that thyroglobulin, MIT and DIT are simply by-products of the process that makes T3 and T4.

T3 is the hormone. It seems to control the rate of metabolism in all cells. T4 has something of the same effect, but is much less active, and called a 'prohormone'. Its main purpose seems to be to be deiodinated to make more T3. This happens outside the thyroid gland, in the other parts of the body ('peripheral conversion'). I believe mainly in the liver, but to some extent in all cells.

Our forefathers knew about thyroxine (T4, or thyronine-with-four-iodines-attached), and triiodothyronine (T3, or thyronine-with-three-iodines-attached)

It seems to me that just from the names, thyroxine was the first one to be discovered. But I'm not sure about that. You try finding a history-of-endocrinology website. At any rate they seem to have known about T4 and T3 fairly early on.

The mystery of Graves', Ord's and Hashimoto's thyroid diseases was explained.

Ord's and Hashimoto's are diseases where the thryoid gland under-produces (hypothyroidism). The metabolism of all cells slows down. As might be expected, this causes a huge number of effects, which seem to manifest differently in different sufferers.

Graves' disease is caused by the thyroid gland over-producing (hyperthyroidism). The metabolism of all cells speeds up. Again, there are a lot of possible symptoms.

All three are thought to be autoimmune diseases. Some people think that they may be different manifestations of the same disease. They are all fairly common.

Desiccated thryoid cures hypothyroidism because the ground-up thyroids contain T4 and T3, as well as lots of thyroglobulin, MIT and DIT, and they are absorbed by the stomach. They get into the blood and speed up the metabolism of all cells. By titrating the dose carefully you can restore roughly the correct levels of the thyroid hormones in all tissues, and the patient gets better. (Titration is where you change something carefully until you get it right)

The theory has considerable explanatory power. It explains cretinism, which is caused either by a genetic disease, or by iodine deficiency in childhood. If you grow up in an iodine deficient area, then your growth is stunted, your brain doesn't develop properly, and your thyroid gland may become hugely enlarged. Presumably because the brain is desperately trying to get it to produce more thyroid hormones, and it responds by swelling.

Once upon a time, this swelling (goitre) was called 'Derbyshire Neck'. I grew up near Derbyshire, and I remember an old rhyme: "Derbyshire born, Derbyshire bred, strong in the arm, and weak in the head". I always thought it was just an insult. Maybe not. Cretinism was also popular in the Alps, and there is a story of an English traveller in Switzerland of whom it was remarked that he would have been quite handsome if only he had had a goitre. So it must have been very common there.

But at this point I am *extremely suspicious*. The thyroid/metabolic regulation system is ancient (universal in vertebrates, I believe), crucial to life, and it really shouldn't just go wrong. We should suspect either an infectious cause, or a recent environmental influence which we haven't had time to adjust to, an evolved defence against an infectious disease, or just possibly, a recently evolved but as yet imperfect defence against a less recent environmental change.

(Cretinism in particular is very strange. Presumably animals in iodine-deficient areas aren't cretinous, and yet they should be. Perhaps a change to a farming from a hunter-gatherer lifestyle has increased our dependency on iodine from crops, which crops have sucked what little iodine occurs naturally out of the soil?)

It's also not entirely clear to me what the thyroid system is *for*. If there's just a particular rate that cells are supposed to run at, then why do they need a control signal to tell them that? I could believe that it was a literal thermostat, designed to keep the body temperature constant at the best speed for the various biological reactions, but it's universal in *vertebrates*. There are plenty of vertebrates which don't keep a constant temperature.

The Fall of Desiccated Thyroid

There turned out to be some problems with Natural Desiccated Thyroid (NDT).

Firstly, there were many competing brands and types, and even if you stuck to one brand the quality control wasn't great, so the dose you'd be taking would have been a bit variable.

Secondly, it's fucking pig's thyroid from an abattoir. It could have all sorts of nasty things in it. Also, ick.

Thirdly, it turned out that pigs made quite a lot more T3 in their thyroids than humans do. It also seems that T3 is better absorbed by the gut than T4 is, so someone taking NDT to compensate for their own underproduction will have too much of the active hormone compared to the prohormone. That may not be good news.

With the discovery of 'peripheral conversion', and the possibility of cheap clean synthesis, it was decided that modern scientific thyroid treatment would henceforth be by synthetic T4 (thyroxine) alone. The body would make its own T3 from the T4 supply.

Alarm bells should be ringing at this point. Apart from the above points, I'm not aware of any great reason for the switch from NDT to thyroxine in the treatment of hypothyroidism, but it seems to have been pretty much universal, and it seems to have worked.

Aware of the lack of T3, doctors compensated by giving people more T4 than was in their pig-thyroid doses. And there don't seem to have been any complaints.

Over the years, NDT seems to have become a crazy fringe treatment despite there not being any evidence against it. It's still a legal prescription drug, but in America it's only prescribed by eccentrics. In England a doctor prescribing it would be, at the very least, summoned to explain himself before the GMC.

However, since it was (a) sold over the counter for so many years, and (b) part of the food chain, it is still perfectly legal to sell as a food supplement in both countries, as long as you don't make any medical claims for it. And the internet being what it is, the prescription-only synthetic hormones T3 and T4 are easily obtained without a prescription. These are extremely powerful hormones which have an effect on metabolism. If 'body-builders' and sports cheats aren't consuming all three in vast quantities, I am a Dutchman.

The Clinical Diagnosis of Hypothyroidism

We pass now to the beginning of the 1970s.

Hypothyroidism is ferociously difficult to diagnose. People complain of 'Tired All The Time' well, ... all the time, and it has literally hundreds of causes.

And it must be diagnosed correctly! If you miss a case of hypothyroidism, your patient is likely to collapse and possibly die at some point in the medium-term future. If you diagnose hypothyroidism where it isn't, you'll start giving the poor bugger powerful hormones which he doesn't need and *cause* hypermetabolism.

The last word in 'diagnosis by symptoms' was the absolutely excellent paper:

Statistical Methods Applied To The Diagnosis Of Hypothyroidism

by W. Z. Billewicz, R. S. Chapman, J. Crooks, M. E. Day, J. Gossage, Sir Edward Wayne, and J. A. Young

Connoisseurs will note the clever and careful application of 'machine learning' techniques, before there were machines to learn!

One important thing to note is that this is a way of separating hypothyroid cases from other cases of tiredness at the point where people have been referred by their GP to a specialist at a hospital on suspicion of hypothyroidism. That changes the statistics remarkably. This is *not* a way of diagnosing hypothyroidism in the general population. But if someone's been to their GP (general practitioner, the doctor that a British person likely makes first contact with) and their GP has suspected their thryoid function might be inadequate, this test should probably still work.

For instance, they consider Physical Tiredness, Mental Lethargy, Slow Cerebration, Dry Hair, and Muscle Pain, the classic symptoms of hypothyroidism, present in most cases, to be indications *against* the disease.

That's because if you didn't have these things, you likely wouldn't have got that far. So in the population they're seeing (of people whose doctor suspects they might be hypothyroid), they're not of great value either way, but their presence is likely the reason why the person's GP has referred them even though they've really got iron-deficiency anaemia or one of the other causes of fatigue.

In their population, the strongest indicators are 'Ankle Jerk' and 'Slow Movements', subtle hypothyroid symptoms which aren't likely to be present in people who are fatigued for other reasons.

But this absolutely isn't a test you should use for population screening! In the general population, the classic symptoms are strong indicators of hypothyroidism.

Probability Theory is weird, huh?

Luckily, there were lab tests for hypothyroidism too, but they were expensive, complicated, annoying and difficult to interpret. Billewicz et al used them to calibrate their test, and recommend them for the difficult cases where their test doesn't give a clear answer.

And of course, the final test is to give them thyroid treatment and see whether they get better. If you're not sure, go slow, watch very carefully and look for hyper symptoms.

Overconfidence is definitely the way to go. If you don't diagnose it and it is, that's catastrophe. If it isn't, but you diagnose it anyway, then as long as you're paying attention the hyper symptoms are easy enough to spot, and you can pull back with little harm done.

A Better Way

It should be obvious from the above that the diagnosis of hypothyroidism by symptoms is absolutely fraught with complexity, and very easy to get wrong, and if you get it wrong the bad way, it's a disaster. Doctors were absolutely screaming for a decisive way to test for hypothyroidism.

Unfortunately, testing directly for the levels of thyroid hormones is very difficult, and the tests of the 1960s weren't accurate enough to be used for diagnosis.

The answer came from an understanding of how the thyroid regulatory system works, and the development of an accurate blood test for a crucial signalling hormone.

Three structures control the level of thyroid hormones in the blood.

The thyroid gland produces the hormones and secretes them into the blood.

Its activity is controlled by the hormone thyrotropin, or Thyroid Signalling Hormone (TSH). Lots of TSH works the thyroid hard. In the absence of TSH the thyroid relaxes but doesn't switch off entirely. However the basal level of thyroid activity in the absence of TSH is far too low.

TSH is controlled by the pituitary gland, a tiny structure attached to the brain.

The pituitary itself is controlled, via Thyroid Releasing Hormone (TRH), by the hypothalamus, which is part of the brain.

This was thought to be a classic example of a feedback control system.

hypothalamus->pituitary->thyroid

It turns out that the level of thyrotropin TSH in the blood is exquisitely sensitive to the levels of thyroid hormones in the blood.

Administer thyroid hormone to a patient and their TSH level will rapidly adjust downwards by an easily detectable amount.

So:

In hypothyroidism, where the thyroid has failed, the body will be desperately trying to produce more thyroid hormones, and the TSH level will be extremely high.

In Graves' Disease, this theory says, where the thyroid has grown too large, and the metabolism is running damagingly fast, the body will be, like a central bank trying to stimulate growth in a deflationary economy by reducing interest rates, 'pushing on a piece of string'. TSH will be undetectable.

The original TSH test was developed in 1965, by the startlingly clever method of radio-immuno-assay.

[For reasons that aren't clear to me, rather than being expressed in grams/litre, or mols/litre, the TSH test is expressed in 'international units/liter'. But I don't think that that's important]

A small number of people in whom there was no suspicion of thyroid disease were assessed, and the 'normal range' of TSH was calculated.

Again, 'endocrinology history' resources are not easy to find, but the first test was not terribly sensitive, and I think originally hyperthyroidism was thought to result in a complete absence of TSH, and that the highest value considered normal was about 4 (milli-international-units/liter).

This apparently pretty much solved the problem of diagnosing thyroid disorders.

Forgetfulness

It's no longer necessary to diagnose hypo- and hyper-thyroidism by symptoms. It was error prone anyway, and the question is easily decided by a cheap and simple test.

Natural Desiccated Thyroid is one with Nineveh and Tyre.

No doctor trained since the 1980s knows much about hypothyroid symptoms.

Medical textbooks mention them only in passing, as an unweighted list of classic symptoms. You couldn't use that for diagnosis of this famously difficult disease.

If you suspect hypothyroidism, you order a TSH test. If the value of TSH is very low, that's hyperthyroidism. If the value is very high then that's hypothyroidism. Otherwise you're 'euthyroid' (greek again, good-thyroid), and your symptoms are caused by some other problem.

The treatment for hyperthyroidism is to damage the thyroid gland. There are various ways. This often results in hypothyroidism. *For reasons that are not terribly well understood*.

The treatment for hypothyroidism is to give the patient sufficient thyroxine (T4) to cause TSH levels to come back into their normal range.

The conditions hyperthyroidism and hypothyroidism are now *defined* by TSH levels.

Hypothyroidism, in particular, a fairly common disease, is considered to be such a solved problem that it's usually treated by the GP, without involving any kind of specialist.

Present Day

It was found that the traditional amount of thyroxine (T4) administered to cure hypothyroid patients, was in fact too high. The amount of T4 that had always been used to replace the hormones that had once been produced by a thyroid gland now dead, destroyed, or surgically removed appeared now to be too much. That amount causes suppression of TSH to below its normal range. The brain, theory says, is asking for the level to be reduced.

The amount of T4 administered in such cases (there are many) has been reduced by a factor of around two, to the level where it produces 'normal' TSH levels in the blood. Treatment is now titrated to produce the normal levels of TSH.

TSH tests have improved enormously since their introduction, and are on their third or fourth generation. The accuracy of measurement is very good indeed.

It's now possible to detect the tiny remaining levels of TSH in overtly hyperthyroid patients, so hyperthyroidism is also now defined by the TSH test.

In England, the normal range is 0.35 to 5.5. This is considered to be the definition of 'euthyroidism'. If your levels are normal, you're fine.

If you have hypothyroid symptoms but a normal TSH level, then your symptoms are caused by something else. Look for Anaemia, look for Lyme Disease. There are hundreds of other possible causes. Once you rule out all the other causes, then it's the mysterious CFS/FMS/ME, for which there is no cause and no treatment.

If your doctor is very good, very careful and very paranoid, he might order tests of the levels of T4 and T3 directly. But actually the direct T4 and T3 tests, although much more accurate than they were in the 1960s, are quite badly standardised, and there's considerable controversy about what they actually measure. Different assay techniques can produce quite different readings. They're expensive. It's fairly common, and on the face of it perfectly reasonable, for a lab to refuse to conduct the T3 and T4 tests if the TSH level is normal.

It's been discovered that quite small increases in TSH actually predict hypothyroidism. Minute changes in thyroid hormone levels, which don't produce symptoms, cause detectable changes in the TSH levels. Normal, but slightly high values of TSH, especially in combination with the presence of thyroid related antibodies (there are several types), indicate a slight risk of one day developing hypothyroidism.

There's quite a lot of controversy about what the normal range for TSH actually is. Many doctors consider that the optimal range is 1-2, and target that range when administering thyroxine. Many think that just getting the value in the normal range is good enough. None of this is properly understood, to understate the case rather dramatically.

There are new categories, 'sub-clinical hypothyroidism' and 'sub-clinical hyperthyroidism', which are defined by abnormal TSH tests in the absence of symptoms. There is considerable controversy over whether it is a good idea to treat these, in order to prevent subtle hormonal imbalances which may cause difficult-to-detect long term problems.

Everyone is a little concerned about accidentally over-treating people, (remember that hyperthyroidism is now defined by TSH<0.35).

Hyperthyroidism has long been associated with Atrial Fibrillation (a heart problem), and Osteoporosis, both very nasty things. A large population study in Denmark recently revealed that there is a greater incidence of Atrial Fibrillation in sub-clinical hyperthyroidism, and that hypothyroidism actually has a 'protective effect' against Atrial Fibrillation.

It's known that TSH has a circadian rhythm, higher in the early morning, lower at night. This makes the test rather noisy, as your TSH level can be doubled or halved depending on what time of day you have the blood drawn.

But the big problems of the 1960s and 1970s are completely solved. We are just tidying up the details.

Doubt

Many hypothyroid patients complain that they suffer from 'Tired All The Time', and have some of the classic hypothyroid symptoms, even though their TSH levels have been carefully adjusted to be in the normal range.

I've no idea how many, but opinions range from 'the great majority of patients are perfectly happy' to 'around half of hypothyroid sufferers have hypothyroid symptoms even though they're being treated'.

The internet is black with people complaining about it, and there are many books and alternative medicine practitioners trying to cure them, or possibly trying to extract as much money as possible from people in desperate need of relief from an unpleasant, debilitating and inexplicable malaise.

THE PLURAL OF ANECDOTE IS DATA.

Not good data, to be sure. But if ten people mention to you in passing that the sun is shining, you are a damned fool if you think you know nothing about the weather.

It's known that TSH ranges aren't 'normally distributed' (in the sense of Gauss/the bell curve distribution) in the healthy population.

If you log-transform them, they do look a bit more normal.

The American Academy of Clinical Biochemists, in 2003, decided to settle the question once and for all. They carefully screened out anyone with even the slightest sign that there might be anything wrong with their thyroid at all, and measured their TSH very accurately.

In their report, they said (this is a direct quote):

In the future, it is likely that the upper limit of the serum TSH euthyroid reference range will be reduced to 2.5 mIU/L because >95% of rigorously screened normal euthyroid volunteers have serum TSH values between 0.4 and 2.5 mIU/L.

Many other studies disagree, and propose wider ranges for normal TSH.

But if the AACB report were taken seriously, it would lead to diagnosis of hypothyroidism in vast numbers of people who are perfectly healthy! In fact the levels of noise in the test would put people whose thyroid systems are perfectly normal in danger of being diagnosed and inappropriately treated.

For fairly obvious reasons, biochemists have been extremely, and quite properly, reluctant to take the report of their own professional body seriously. And yet it is hard to see where the AACB have gone wrong in their report.

Neurasthenia is back.

A little after the time of the introduction of the TSH test, new forms of 'Tired All The Time' were discovered.

As I said, CFS and ME are just two names for the same thing. Fibromyalgia Syndrome (FMS) is much worse, since it is CFS with constant pain, for which there is no known cause and from which there is no relief. Most drugs make it worse.

But if you combine the three things (CFS/ME/FMS), then you get a single disease, which has a large number of very non-specific symptoms.

These symptoms are the classic symptoms of 'hypometabolism'. Any doctor who has a patient who has CFS/ME/FMS and hasn't tested their thyroid function is *de facto* incompetent. I think the vast majority of medical people would agree with this statement.

And yet, when you test the TSH levels in CFS/ME/FMS sufferers, they are perfectly normal.

All three/two/one are appalling, crippling, terrible syndromes which ruin people's lives. They are fairly common. You almost certainly know one or two sufferers. The suffering is made worse by the fact that most people believe that they're psychosomatic, which is a polite word for 'imaginary'.

And the people suffering are mainly middle-aged women. Middle-aged women are easy to ignore. Especially stupid middle-aged women who are worried about being overweight and obviously faking their symptoms in order to get drugs which are popularly believed to induce weight loss. It's clearly their hormones. Or they're trying to scrounge up welfare benefits. Or they're trying to claim insurance. Even though there's nothing wrong with them and you've checked so carefully for everything that it could possibly be.

But it's not all middle aged women. These diseases affect men, and the young. Sometimes they affect little children. Exhaustion, stupidity, constant pain. Endless other problems as your body rots away. Lifelong. No remission and no cure.

And I have Doubts of my Own

And I can't believe that careful, numerate Billewicz and his co-authors would have made this mistake, but I can't find where the doctors of the 1970s checked for the sensitivity of the TSH test.

Specificity, yes. They tested a lot of people who hadn't got any sign of hypothyroidism for TSH levels. If you're well, then your TSH level will be in a narrow range, which may be 0-6, or it may be 1-2. Opinions are weirdly divided on this point in a hard to explain way.

But Sensitivity? Where's the bit where they checked for the other arm of the conditional?

The bit where they show that no-one who's suffering from hypometabolism, and who gets well when you give them Desiccated Thyroid, had, on first contact, TSH levels outside the normal range.

If you're trying to prove A <=> B, you can't just prove A => B and call it a day. You couldn't get that past an A-level maths student. And certainly anyone with a science degree wouldn't make that error. Surely? I mean you shouldn't be able to get that past anyone who can reason their way out of a paper bag.

I'm going to say this a third time, because I think it's important and maybe it's not obvious to everyone.

If you're trying to prove that two things are the same thing, then proving that the first one is always the second one is not good enough.

IF YOU KNOW THAT THE KING OF FRANCE IS ALWAYS FRENCH, YOU DO *NOT* KNOW THAT ANYONE WHO IS FRENCH IS KING OF FRANCE.

It's possible, of course, that I've missed this bit. As I say, 'History of Endocrinology' is not one of those popular, fashionable subjects that you can easily find out about.

I wonder if they just assumed that the thyroid system was a thermostat. The analogy is still common today.

But it doesn't look like a thermostat to me. The thyroid system with its vast numbers of hormones and transforming enzymes is insanely, incomprehensibly complicated. And very poorly understood. And evolutionarily ancient. It looks as though originally it was the system that coordinated metamorphosis. Or maybe it signalled when resources were high enough to undergo metamorphosis. But whatever it did originally in our most ancient ancestors, it looks as though the blind watchmaker has layered hack after hack after hack on top of it on the way to us.

Only the thyroid originally, controlling major changes in body plan in tiny creatures that metamorphose.

Of course, humans metamorphose too, but it's all in the womb, and who measures thyroid levels in the unborn when they still look like tiny fish?

And of course, humans undergo very rapid growth and change after we are born. Especially in the brain. Baby horses can walk seconds after they're born. Baby humans take months to learn to crawl. I wonder if that's got anything to do with cretinism.

And I'm told that baby humans have very high hormone levels. I wonder why they need to be so hot? If it's a thermostat, I mean.

But then on top of the thyroid, the pituitary. I wonder what that adds to the system? If the thyroid's just a thermostat, or just a device for keeping T4 levels constant, why can't it just do the sensing itself?

What evolutionary process created the pituitary control over the thyroid? Is that the thermostat bit?

And then the hypothalamus, controlling the pituitary. Why? Why would the brain need to set the temperature when the ideal temperature of metabolic reactions is always 37C in every animal? That's the temperature everything's designed for. Why would you dial it up or down, to a place where the chemical reactions that you are don't work properly?

I can think of reasons why. Perhaps you're hibernating. Many of our ancestors must have hibernated. Maybe it's a good idea to slow the metabolism sometimes. Perhaps to conserve your fat supplies. Your stored food.

Perhaps it's a good idea to slow the metabolism in times of famine?

Perhaps the whole calories in/calories out thing is wrong, and people whose energy expenditure goes over their calorie intake have slow metabolisms, slowly sacrificing every bodily function including immune defence in order to avoid starvation.

I wonder at the willpower that could keep an animal sane in that state. While its body does everything it can to keep its precious fat reserves high so that it can get through the famine.

And then I remember about Anorexia Nervosa, where young women who want to lose weight starve themselves to the point where they no longer feel hungry at all. Another mysterious psychological disease that's just put down to crazy females. We really need some female doctors.

And I remember about Seth Robert's Shangri-La Diet, that I tried, to see if it worked, some years ago, just because it was so weird, where by eating strange things, like tasteless oil and raw sugar, you can make your appetite disappear, and lose weight. It seemed to work pretty well, to my surprise. Seth came up with it while thinking about rats. And apparently it works on rats too. I wonder why it hasn't caught on.

It seems, my female friends tell me, that a lot of diets work well for a bit, but then after a few weeks the effect just stops. If we think of a particular diet as a meme, this would seem to be its infectious period, where the host enthusiastically spreads the idea.

And I wonder about the role of the thyronine de-iodinating enzymes, and the whole fantastically complicated process of stripping the iodines and the amino acid bits from thyroxine in various patterns that no-one understands, and what could be going on there if the thyroid system were just a simple thermostat.

And I wonder about reports I am reading where elite athletes are finding themselves suffering from hypothyroidism in numbers far too large to be credible, if it wasn't, say, a physical response to calorie intake less than calorie output.

I've been looking ever so hard to find out why the TSH test, or any of the various available thyroid blood tests are a good way to assess the function of this fantastically complicated and very poorly understood system.

But every time I look, I just come up with more reasons to believe that they don't tell you very much at all.

The Mystery

Can anyone convince me that the converse arm has been carefully checked?

That everyone who's suffering from hypometabolism, and who gets well when you give them Desiccated Thyroid, has, before you fix them, TSH levels outside the normal range.

In other words, that we haven't just thrown, though carelessness, a long standing, perfectly safe, well tested treatment, for a horrible disabling disease that often causes excruciating pain, that the Victorians knew how to cure, and that the people of the 1950s and 60s routinely cured, away.

What is your opinion on rationality-promoting articles by Gleb Tsipursky / Intentional Insights? Here is what I think:

(Crossposted from the EA forum.)

Summary: The universe may very well be infinite, and hence contain an infinite amount of happiness and sadness. This causes several problems for altruists; for example: we can plausibly only affect a finite subset of the universe, and an infinite quantity of happiness is unchanged by the addition or subtraction of a finite amount of happiness. This would imply that all forms of altruism are equally ineffective.

Like everything in life, the canonical reference in philosophy about this problem was written by Nick Bostrom. However, I found that an area of economics known as "sustainable development" has actually made much further progress on this subject than the philosophy world. In this post I go over some of what I consider to be the most interesting results.

NB: This assumes a lot of mathematical literacy and familiarity with the subject matter, and hence isn't targeted to a general audience. Most people will probably prefer to read my other posts:

•  Ridiculous math things which ethics shouldn't depend on but does, which includes such interesting tidbits as: why standard calculus teaches children to be immoral and why the Banach-Tarski paradox implies we should means test Medicare and
• Kill the young people, whose name speaks for itself

1. Summary of the most interesting results

1. There’s no ethical system which incorporates all the things we might want.
2. Even if we have pretty minimal requirements, satisfactory ethical systems might exist but we can’t prove their existence, much less actually construct them
3. Discounted utilitarianism, whereby we value people less just because they are further away in time, is actually a pretty reasonable thing despite philosophers considering it ridiculous.
   1. (I consider this to be the first reasonable argument for locavorism I've ever heard)

2. Definitions

In general, we consider a population to consist of an infinite utility vector (u₀,u₁,…) where u_i is the aggregate utility of the generation alive at time i. Utility is a bounded real number (the fact that economists assume utility to be bounded confused me for a long time!). Our goal is to find a preference ordering over the set of all utility vectors which is in some sense “reasonable”. While philosophers have understood for a long time that finding such an ordering is difficult, I will present several theorems which show that it is in fact impossible.

Due to a lack of latex support I’m going to give English-language definitions and results instead of math-ey ones; interested people should look at the papers themselves anyway.

3. Impossibility Results

3.1 Definitions

• Strong Pareto: if you can make a generation better off, and none worse off, you should.
• Weak Pareto: if you can make every generation better off, you should.
• Intergenerational equity: utility vectors are unchanged in value by any permutation of their components.
  • There is an important distinction here between allowing a finite number of elements to be permuted and an infinite number; I will refer to the former as “finite intergenerational equity” and the latter as just “intergenerational equity”
• Ethical relation: one which obeys both weak Pareto and intergenerational equity
• Social welfare function: an order-preserving function from the set of populations (utility vectors) to the real numbers

3.2 Diamond-Basu-Mitra Impossibility Result¹

1. There is no social welfare function which obeys Strong Pareto and finite intergenerational equity. This means that any sort of utilitarianism won’t work, unless we look outside the real numbers.

3.3 Zame's impossibility result²

1. If an ordering obeys intergenerational equity over [0,1]^N, then almost always we can’t tell which of two populations is better 
   1. (i.e. the set of populations {X,Y: neither X<Y nor X>Y} has outer measure one)
2. The existence of an ethical preference relation on [0,1]^N is independent of ZF plus the axiom of choice

4. Possibility Results

We’ve just shown that it’s impossible to construct or even prove the existence of any useful ethical system. But not all hope is lost!

The important idea here is that of a “subrelation”: < is a subrelation to <’ if x<y implies x<’y.

Our arguments will work like this:

Suppose we could extend utilitarianism to the infinite case. (We don't, of course, know that we can extend utilitarianism to the infinite case. But suppose we could.) Then A, B and C must follow.

Technically: suppose utilitarianism is a subrelation of <. Then < must have properties A, B and C.

Everything in this section comes from (3), which is a great review of the literature.

4.1 Definition

• Utilitarianism: we extend the standard total utilitarianism ordering to infinite populations in the following way: suppose there is some time T after which every generation in X is at least as well off as every generation in Y, and that the total utility in X before T is at least as good as the total utility in Y before T. Then X is at least as good as Y.
  • Note that this is not a complete ordering! In fact, as per Zame’s result above, the set of populations it can meaningfully speak about has measure zero.
• Partial translation scale invariance: suppose after some time T, X and Y become the same. Then we can add any arbitrary utility vector A to both X and Y without changing the ordering. (I.e. X > Y ó X+A > Y+A)

4.2 Theorem

1. Utilitarianism is a subrelation of > if and only if > satisfies strong Pareto, finite intergenerational equity and partial translation scale invariance.
   1. This means that if we want to extend utilitarianism to the infinite case, we can’t use a social welfare function, as per the above Basu-Mitra result

4.3 Definition

• Overtaking utilitarianism: suppose there is some point T after which the total utility of the first N generations in X is always greater than the total utility of the first N generations in Y (given N > T). Then X is better than Y.
  • Note that utilitarianism is a subrelation of overtaking utilitarianism
• Weak limiting preference: suppose that for any time T, X truncated at time T is better than Y truncated at time T. Then X is better than Y.

4.4 Theorem

1. Overtaking utilitarianism is a subrelation of < if and only if < satisfies strong Pareto, finite intergenerational equity, partial translation scale invariance, and weak limiting preference

4.5 Definition

• Discounted utilitarianism: the utility of a population is the sum of its components, discounted by how far away in time they are
• Separability:
  • Separable present: if you can improve the first T generations without affecting the rest, you should
  • Separable future: if you can improve everything after the first T generations without affecting the rest, you should
• Stationarity: preferences are time invariant
• Weak sensitivity: for any utility vector, we can modify its first generation somehow to make it better

4.6 Theorem

1. The only continuous, monotonic relation which obeys weak sensitivity, stationary, and separability is discounted utilitarianism

4.7 Definition

• Dictatorship of the present: there’s some time T after which changing the utility of generations doesn’t matter

4.8 Theorem

1. Discounted utilitarianism results in a dictatorship of the present. (Remember that each generation’s utility is assumed to be bounded!)

• Sustainable preference: a continuous ordering which doesn’t have a dictatorship of the present but follows strong Pareto and separability.

4.10 Theorem

1. The only ordering which is sustainable is to take discounted utilitarianism and add an “asymptotic” part which ensures that infinitely long changes in utility matter. (Of course, finite changes in utility still won't matter.)

5. Conclusion

I hope I've convinced you that there's a "there" there: infinite ethics is something that people can make progress on, and it seems that most of the progress is being made in the field of sustainable development.

Fun fact: the author of the last theorem (the one which defined "sustainable") was one of the lead economists on the Kyoto protocol. Who says infinite ethics is impractical?

6. References

1. Basu, Kaushik, and Tapan Mitra. "Aggregating infinite utility streams with intergenerational equity: the impossibility of being Paretian." Econometrica 71.5 (2003): 1557-1563. http://folk.uio.no/gasheim/zB%26M2003.pdf
2. Zame, William R. "Can intergenerational equity be operationalized?." (2007).  https://tspace.library.utoronto.ca/bitstream/1807/9745/1/1204.pdf
3. Asheim, Geir B. "Intergenerational equity." Annu. Rev. Econ. 2.1 (2010): 197-222.http://folk.uio.no/gasheim/A-ARE10.pdf

Following in the path of So8res and others, I’ve decided to work my way through the textbooks on the MIRI Research Guide. I’ve been working my way through the guide since last October, but this is my first review. I plan on following up this review with reviews of Enderton’s A Mathematical Introduction to Logic and Sipser’s Introduction to the Theory of Computation. Hopefully these reviews will be of some use to you.

Discrete Mathematics and Its Applications

Discrete Mathematics and Its Applications is wonderful, gentle introduction to the math needed to understand most of the other books on the MIRI course list. It successfully pulls off a colloquial tone of voice. It spends a lot of time motivating concepts; it also contains a lot of interesting trivia and short biographies of famous mathematicians and computer scientists (which the textbook calls “links”). Additionally, the book provides a lot of examples for each of its theorems and topics. It also fleshes out the key subjects (counting, proofs, graphs, etc.) while also providing a high level overview of their applications. These combine to make it an excellent first textbook for learning discrete mathematics.

However, for much the same reasons, I would not recommend it nearly as much if you’ve taken a discrete math course. People who’ve participated in math competitions at the high school level probably won’t get much out of the textbooks either. Even though I went in with only the discrete math I did in high school, I still got quite frustrated at times because of how long the book would take to get to the point. Discrete Mathematics is intended to be quite introductory and it succeeds in this goal, but it probably won’t be very suitable as anything other than review for readers beyond the introductory level. The sole exception is the last chapter (on models of computation), but I recommend picking up a more comprehensive overview from Sipser’s Theory of Computation instead.

I still highly recommend it for those not familiar with the topics covered in the book. I’ve summarized the contents of the textbook below:

Contents:

1.     The Foundations: Logic and Proofs

2.   Basic Structures: Sets, Functions, Sequences, Sums, and Matrices

3.     Algorithms

4.     Number Theory and Cryptography

5.     Induction and Recursion

6.     Counting

7.     Discrete Probability

8.     Advanced Counting Techniques

9.     Relations

10.  Graphs

11.  Trees

12.  Boolean Algebra

13.  Modeling Computation

The Foundations: Logic and Proofs

This chapter introduces propositional (sentential) logic, predicate logic, and proof theory at a very introductory level. It starts by introducing the propositions of propositional logic (!), then goes on to introduce applications of propositional logic, such as logic puzzles and logic circuits. It then goes on to introduce the idea of logical equivalence between sentences of propositional logic, before introducing quantifiers and predicate logic and its rules of inference. It then ends by talking about the different kinds of proofs one is likely to encounter – direct proofs via repeated modus ponens, proofs by contradiction, proof by cases, and constructive and non-constructive existence proofs.

This chapter illustrates exactly why this book is excellent as an introductory text. It doesn’t just introduce the terms, theorems, and definitions; it motivates them by giving applications. For example, it explains the need for predicate logic by pointing out that there are inferences that can’t be drawn using only propositional logic. Additionally, it also explains the common pitfalls for the different proof methods that it introduces.

Basic Structures: Sets, Functions, Sequences, Sums, and Matrices

This chapter introduces the different objects one is likely to encounter in discrete mathematics. Most of it seemed pretty standard, with the following exceptions: functions are introduced without reference to relations; the “cardinality of sets” section provides a high level overview of a lot of set theory; and the matrices section introduces zero-one matrices, which are used in the chapters on relations and graphs.

Algorithms

This chapter presents … surprise, surprise… algorithms! It starts by introducing the notion of algorithms, and gives a few examples of simple algorithms. It then spends a page introducing the halting problem and showing its undecidability. (!) Afterwards, it introduces big-o, big-omega, and big-theta notation and then gives a (very informal) treatment of a portion of computation complexity theory. It's quite unusual to see algorithms being dealt with so early into a discrete math course, but it's quite important because the author starts providing examples of algorithms in almost every chapter after this one.

Number Theory and Cryptography

This section goes from simple modular arithmetic (3 divides 12!) to RSA, which I found extremely impressive. (Admittedly, I’ve only ever read one other discrete math textbook.) After introducing the notion of divisibility, the textbook takes the reader on a rapid tour through base-n notation, the fundamental theorem of arithmetic, the infinitude of primes, the Euclidean GCD algorithm, Bezout’s theorem, the Chinese remainder theorem, Fermat’s little theorem, and other key results of number theory. It then gives several applications of number theory: hash functions, pseudorandom numbers, check digits, and cryptography. The last of these gets its own section, and the book spends a large amount of it introducing RSA and its applications.

Induction and Recursion

This chapter introduces mathematical induction and recursion, two extremely important concepts in computer science. Proofs by mathematical induction, basically, are proofs that show that a property is true of the first natural number (positive integer in this book), and if it is true of an integer k it is true of k+1. With these two results, we can conclude that the property is true of all natural numbers (positive integers). The book then goes on to introduce strong induction and recursively defined functions and sets. From this, the book then goes on to introduce the concept of structural induction, which is a generalization of induction to work on recursively-defined sets. Then, the book introduces recursive algorithms, most notably the merge sort, before giving a high level overview of program verification techniques.

Counting

The book now changes subjects to talk about basic counting techniques, such as the product rule and the sum rule, before (interestingly) moving on to the pigeonhole principle. It then moves on to permutations and combinations, while introducing the notion of combinatorial proof, which is when we show that two sides of the identity count the same things but in different ways, or that there exists a bijection between the sets being counted on either side. The textbook then introduces binomial coefficients, Pascal’s triangle, and permutations/combinations with repetition. Finally, it gives algorithms that generate all the permutations and combinations of a set of n objects.

Compared to other sections, I feel that a higher proportion of readers would be familiar with the results of this chapter and the one on discrete probability that follows it. Other than the last section, which I found quite interesting but not particularly useful, I felt like I barely got anything from the chapter.

Discrete Probability

In this section the book covers probability, a topic that most of LessWrong should be quite familiar with. Like most introductory textbooks, it begins by introducing the notion of sample spaces and events as sets, before defining probability of an event E as the ratio of the cardinality of E to the cardinality of S. We are then introduced to other key concepts in probability theory: conditional probabilities, independence, and random variables, for example. The textbook takes care to flesh out this section with a discussion about the Birthday Problem and Monte Carlo algorithms. Afterwards, we are treated to a section on Bayes theorem, with the canonical example of disease testing for rare diseases and a less-canonical-but-still-used-quite-a-lot example of Naïve Bayes spam filters. The chapter concludes by introducing the expected value and variances of random variables, as well as a lot of key results (linearity of expectations and Chebyshev’s Inequality, to list two). Again, aside from the applications, most of this stuff is quite basic.

Advanced Counting Techniques

This chapter, though titled “advanced counting techniques”, is really just about recurrences and the principle of inclusion-exclusion. As you can tell by the length of this section, I found this chapter quite helpful nevertheless.  

We begin by giving three applications of recurrences: Fibonacci’s “rabbit problem”, the Tower of Hanoi, and dynamic programming. We’re then shown how to solve linear homogenous relations, which are relations of the form

a_n = c₁ a_n-1 + c₂ a_n-2 + … + c_k a_n-k+ F(n)

Where c1, c2, …, ck are constants, ck =/= 0, and F(n) is a function of n. The solutions are quite beautiful, and if you’re not familiar with them I recommend looking them up. Afterwards, we’re introduced to divide-and-conquer algorithms, which are recursive algorithms that solve smaller and smaller instances of the problem, as well as the master method for solving the recurrences associated with them, which tend to be of the form

f(n) = a f(n/b) + cn^d

After these algorithms, we’re introduced to generating functions, which are yet another way of solving recurrences.

Finally, after a long trip through various recurrence-solving methods, the textbook introduces the principle of inclusion-exclusion, which lets us figure out how many elements are in the union of a finite number of finite sets.

Relations

Finally, 7 chapters after the textbook talks about functions, it finally gets to relations. Relations are defined as sets of n-tuples, but the book also gives alternative ways of representing relations, such as matrices and directed graphs for binary relations. We’re then introduced to transitive closures and Warshall’s algorithm for computing the transitive closure of a relation. We conclude with two special types of relations: equivalence relations, which are reflexive, symmetric, and transitive; and partial orderings, which are reflexive, anti-symmetric, and transitive.

Graphs

After being first introduced to directed graphs as a way of representing relations in the previous chapter, we’re given a much more fleshed out treatment in this chapter. A graph is defined as a set of vertices and a set of edges connecting them. Edges can be directed or undirected, and graphs can be simple graphs (with no two edges connecting the same pair of vertices) or multigraphs, which contain multiple edges connecting the same pair of vertices. We’re then given a ton of terminology related to graphs, and a lot of theorems related to these terms. The treatment of graphs is quite advanced for an introductory textbook – it covers Dijkstra’s algorithm for shortest paths, for example, and ends with four coloring. I found this chapter to be a useful review of a lot of graph theory.

Trees

After dealing with graphs, we move on to trees, or connected graphs that don’t have cycles. The textbook gives a lot of examples of applications of trees, such as binary search trees, decision trees, and Huffman coding. We’re then presented with the three ways of traversing a tree – in-order, pre-order, and post-order. Afterwards, we get to the topic of spanning trees of graphs, which are trees that contain every vertex in the graph. Two algorithms are presented for finding spanning trees – depth first search and breadth first search. The chapter ends with a section on minimum spanning trees, which are spanning trees with the least weight. Once again we’re presented with two algorithms for finding minimum spanning trees: Prim’s Algorithm and Kruskal’s algorithm. Having never seen either of these algorithms before, I found this section to be quite interesting, though they are given a more comprehensive treatment in most introductory algorithms textbooks.  

Boolean Algebra

This section introduces Boolean algebra, which is basically a set of rules for manipulating elements of the set {0,1}. Why is this useful? Because, as it turns out, Boolean algebra is directly related to circuit design! The textbook first introduces the terminology and rules of Boolean algebra, and then moves on to circuits of logic gates and their relationship with Boolean functions. We conclude with two ways to minimize the complexity of Boolean functions (and thus circuits) – Karnaugh Maps and the Quine-McCluskey Method, which are both quite interesting. 

Modeling Computation

Who should read this?

If you’re not familiar with discrete mathematics, this is a great book that will get you up to speed on the key concepts, at least to the level where you’ll be able to understand the other textbooks on MIRI’s course list. Of the three textbooks I’m familiar with that cover discrete mathematics, I think that Rosen is hands down the best. I also think it’s quite a “fun” textbook to skim through, even if you’re familiar with some of the topics already.

However, I think that people familiar with the topics probably should look for other books, especially if they are looking for textbooks that are more concise. It might also not be suitable if you’re already really motivated to learn the subject, and just want to jump right in. There are a few topics not normally covered in other discrete math textbooks, but I feel that it’s better to pick up those topics in other textbooks.

What should I read?

In general, the rule for the textbook is: read the sections you’re not familiar with, and skim the sections you are familiar with, just to keep an eye out for cool examples or theorems.

In terms of chapter-by-chapter, chapters 1 and 2 seem like they’ll help if you’re new to mathematics or proofs, but probably can be skipped otherwise. Chapter 3 is pretty good to know in general, though I suspect most people here would find it too easy. Chapters 4 through 12 are what most courses on discrete mathematics seem to cover, and form the bulk of the book – I would recommend skimming them once just to make sure you know them, as they’re also quite important for understanding any serious CS textbook. Chapter 13, on the other hand, seems kind of tacked on, and probably should be picked up in other textbooks.

Final Notes

Of all the books on the MIRI research guide, this is probably the most accessible, but it is by no means a bad book. I’d highly recommend it to anyone who hasn’t had any exposure to discrete mathematics, and I think it’s an important prerequisite for the rest of the books on the MIRI research guide.

I'm pleased to announce a new paper from MIRI about The Value Learning Problem.

Abstract:

A superintelligent machine would not automatically act as intended: it will act as programmed, but the fit between human intentions and formal specification could be poor. We discuss methods by which a system could be constructed to learn what to value. We highlight open problems specific to inductive value learning (from labeled training data), and raise a number of questions about the construction of systems which model the preferences of their operators and act accordingly.

This is the sixth of six papers supporting the MIRI technical agenda. It motivates the need for value learning, a bit, and gives some early thoughts on how the problem could be approached (while pointing to some early open problems in the field).

I'm pretty excited to have the technical agenda and all its supporting papers published. Next week I'll be posting an annotated bibliography that gives more reading for each subject. The introduction to the value learning paper has been reproduced below.

I'm pleased to announce the release of Aligning Superintelligence with Human Interests: A Technical Research Agenda written by Benja and I (with help and input from many, many others). This document summarizes and motivates MIRI's current technical research agenda.

I'm happy to answer questions about this document, but expect slow response times, as I'm travelling for the holidays. The introduction of the paper is included below. (See the paper for references.)

Discussion article for the meetup : Warsaw, next week!

Discussion article for the meetup : Warsaw, next week!

Discussion article for the meetup : Warsaw Meetup - This Friday

Discussion article for the meetup : Warsaw Meetup - This Friday